Mesh Generation in CFD

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CFD Open Series Revision 1.85.9

Mesh Generation in CFD Ideen Sadrehaghighi, Ph.D.

Cyliner Head (Polyhedral cells)

Typical Turbo-Machine Mesh (Hexahedral cells)

ANNAPOLIS, MD

Mixer (SAMM cells)

Wing-Body-Pylon-Nacelle (Tetrahedral cells)

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Contents 1

Introduction ................................................................................................................................ 19

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Computer-Aided Design (CAD) ............................................................................................. 24

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Structured Mesh Generation ................................................................................................. 32

The Black Box Dilemma .................................................................................................................................... 19 Trust the Mesh Generated by the Software, or Take a Proactive Approach? .................. 19 Not All the Meshes Created Equal ..................................................................................... 19 The Mesh Types ................................................................................................................. 20 Regional Meshing .............................................................................................................. 21 Simulation Cost .................................................................................................................. 21 Physics vs. Mesh ................................................................................................................ 22 Meshing Generalities ......................................................................................................... 22

Software and Technology ................................................................................................................................ 24 Commercially Available CAD Systems: .............................................................................. 25 Freeware and Open Source ............................................................................................... 26 Solid (Geometry) Modeling ............................................................................................................................. 26 Principal Characteristics of a Solid Modeling Software ..................................................... 26 Feature-Based Modeling ................................................................................................... 26 Constraint-Based Modeling ............................................................................................... 26 Parametric Modeling ......................................................................................................... 27 History-Based Modeling .................................................................................................... 27 Associative Modeling ......................................................................................................... 27 Constructive Solid Geometry (CSG) Representation of Solids.......................................................... 27 Basic Primitives .................................................................................................................. 27 Regularized Boolean Operators ......................................................................................... 28 The CSG Tree ......................................................................................................................................................... 28 Geometry Related Issues For Mesh Generation ..................................................................................... 28 Understanding the Analysis Requirements ....................................................................... 29 Disfeaturing ....................................................................................................................... 29 “Dirty” Geometry ............................................................................................................... 30

Classification of Mesh Generation Techniques ....................................................................................... 33 Domain Decomposition and Multi-Block Strategy ................................................................................ 34 Field (Domain) Discretization Process (Mesh Generation) ............................................................... 35 Conformal Mapping (The Sponge Analogy) ............................................................................................. 36 Domain Decomposition with Multi-Blocking (Multi-Sponge Analogy) ........................................ 36 Structured Grid Generation ............................................................................................................................ 37 Complex Variables ............................................................................................................. 38 Algebraic Methods -Transfinite Interpolation (TFI) ........................................................... 38 PDE Smoother .................................................................................................................... 39 3.6.3.1 Elliptic Schemes ............................................................................................................ 40 3.6.3.1.1 Case Study – Orthogonal Elliptic Mesh Smoother............................................. 41 3.6.3.1.2 Orthogonality Adjustment Algorithm ................................................................ 41 3.6.3.1.3 Stretching Functions ........................................................................................... 42 3.6.3.1.4 Extension to 3D .................................................................................................. 42

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1.8

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3.6.3.1.5 Mesh Quality Analysis ........................................................................................ 43 3.6.3.2 Hyperbolic Schemes ..................................................................................................... 43 3.6.3.3 Parabolic Schemes ........................................................................................................ 44 Variational Method............................................................................................................ 44 Structured Adaptive Grid.................................................................................................................................. 45 Case Study – 2D Euler Flow Over an NACA Airfoil ............................................................. 47

Un-Structured Mesh Generation .......................................................................................... 50

Advancing Front Method ................................................................................................................................. 50 Advancing Front Triangular Mesh Generator .................................................................... 51 Advancing Front Quadrilateral Meshing Using Triangle Transformations ................................ 53 Outline of Quad-Morphing Algorithm ............................................................................... 53 4.2.1.1 Initial Triangle Mesh ..................................................................................................... 53 4.2.1.2 Front Definition ............................................................................................................ 53 4.2.1.3 Front Edge Classification .............................................................................................. 53 4.2.1.4 Front Edge Processing .................................................................................................. 53 4.2.1.5 Topological Clean-up and Final Smoothing Process ..................................................... 55 4.2.1.6 Example Problems ........................................................................................................ 55 4.2.1.7 Conclusion .................................................................................................................... 58 Delaney Triangulation Method ...................................................................................................................... 58 Properties of Delaunay Triangulation................................................................................ 58 4.3.1.1 Delaunay Lemma .......................................................................................................... 59 4.3.1.2 Compactness ................................................................................................................ 59 Algorithms ......................................................................................................................... 59 Advantages ........................................................................................................................ 61 Delaunay Adaptive Refinement ......................................................................................... 62 Voronoi Diagrams .............................................................................................................. 63 Restricted Delaunay Triangulation .................................................................................... 63 Anisotropic Mesh Generation ........................................................................................................................ 64 Case Study - Anisotropic Mesh Generation via Discretized Riemannian Delaunay Triangulations ................................................................................................................................... 65 4.4.1.1 Anisotropic Delaunay Triangulations............................................................................ 67 4.4.1.1.1 Locally Uniform Anisotropic Meshes.................................................................. 67 4.4.1.1.2 Metric Tensor ..................................................................................................... 67 4.4.1.1.3 Distortion ............................................................................................................ 68 4.4.1.1.4 Locally Uniform Anisotropic Meshes.................................................................. 69 4.4.1.1.5 The Star Set ........................................................................................................ 70 4.4.1.1.6 Stars and Inconsistencies ................................................................................... 70 4.4.1.2 Refinement Algorithm .................................................................................................. 71 4.4.1.3 Discussion on the Parameters ...................................................................................... 71 4.4.1.3.1 Parameter φ0 ..................................................................................................... 71 4.4.1.3.2 Parameters r0 and ρ0 .......................................................................................... 72 4.4.1.3.3 Parameters β and δ ............................................................................................ 73 4.4.1.3.4 Parameters σ0 ..................................................................................................... 74 4.4.1.4 Results and Limitations................................................................................................. 74 4.4.1.4.1 Uniform Metric Fields......................................................................................... 74 4.4.1.4.2 Shock-Based Metric Fields on Planar Domains .................................................. 74 4.4.1.4.3 Starred ................................................................................................................ 74

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4.4.1.4.4 Hyperbolic .......................................................................................................... 75 4.4.1.4.5 Swirl .................................................................................................................... 76 4.4.1.4.6 Curvature-Based Metrics Fields on Surfaces ...................................................... 76 4.4.1.4.7 Optimization ....................................................................................................... 76 4.4.1.5 Discrete Riemannian Voronoi Diagrams....................................................................... 77 4.4.1.5.1 Advantages Over Isotropic Canvasses ................................................................ 78 4.4.1.5.2 Straight Riemannian Delaunay Triangulation..................................................... 78 4.4.1.5.3 Curved Riemannian Delaunay Triangulation ...................................................... 79 4.4.1.6 Conclusion .................................................................................................................... 81 Octree Decomposition ....................................................................................................................................... 81 Unstructured Hexahedral Meshes................................................................................................................ 83 Conversion of Triangular to Quadrilateral Meshes (2D) ................................................... 84 Overset Grids ........................................................................................................................................................ 85 Cartesian Grids ..................................................................................................................................................... 87 Background and Cartesian Grid Origins............................................................................. 87 Cartesian Grids Schemes ................................................................................................... 88 4.8.2.1 Adaptive Mesh Refinement .......................................................................................... 89 4.8.2.2 Immersed Boundary Methods...................................................................................... 91 4.8.2.3 Volume of Fluid Methods ............................................................................................. 92 4.8.2.4 Reconstruction Schemes .............................................................................................. 92 4.8.2.5 Cut Cell Based Methods................................................................................................ 92 4.8.2.6 Chimera Grid Schemes ................................................................................................. 94 4.8.2.7 Hybrid Grid Schemes .................................................................................................... 94 4.8.2.7.1 Composite Grid Approach .................................................................................. 95 Discussion .......................................................................................................................... 97 Trimmed (SAMM) Cells .................................................................................................................................... 98 Polyhedral Cells ............................................................................................................................................. 98 Cell Decomposition ............................................................................................................ 98 Mesh Duality ...................................................................................................................... 99 Methodology ................................................................................................................... 100 Treatment of Boundary Layer .............................................................................................................. 100 Domain Mesh Stretching in Unstructured Environment ........................................................... 101 Spatial (Field) Discretization ................................................................................................................ 104 Considerations for the Navier-Stokes Equation ............................................................................ 105 Unstructured Quadrilateral Mesh Generation ............................................................................... 106 Geometry Representation ............................................................................................... 106 Local Mesh Generation Algorithm ................................................................................... 107 Connectivity Information and Data Structure................................................................................ 108

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Hybrid Meshes ........................................................................................................................ 110

Accuracy Consideration................................................................................................................................. 110 Comparing Mesh Type for Viscous Accuracy................................................................... 111 Effect of Prismatic Extrusion Sub-Layer in Viscous Layer ................................................ 111 Meshing Tools in CD-Adapco® .................................................................................................................... 112 A Novel Methodology for Extrusion Layer Meshing ........................................................ 114 Mesh Refinement.............................................................................................................................................. 114 R-refinement.................................................................................................................... 116 H-refinement ................................................................................................................... 116

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5.3.2.1 Isotropic vs. Anisotropic Meshing .............................................................................. 117 P-refinement.................................................................................................................... 117 Mesh Modification Operators...................................................................................................................... 118 Coarsening Triangulation Regions ................................................................................... 118 5.4.1.1 Case Study - Numerical Testing for Engine Nacelle .................................................... 119 5.4.1.2 Coarsening With/Without Local Re-Triangulation .................................................... 120 Refinement of Triangulation Region ............................................................................... 120 5.4.2.1 Local Re-Triangulation ................................................................................................ 121 Refinement of Hexahedral Region (Near Wall) ............................................................... 122 5.4.3.1 Improvement to Near-Field Grid Generation Procedure (Hexahedral) ..................... 123 Discretization Improvement Through Chimera Technique for Sharp Corners ................ 125 Case Study 1 - Hybrid Unstructured Meshes for Common Research Model (CRM & JSM) via ANSA®................................................................................................................................................................................ 126 Geometry and Mesh Generation Background................................................................. 127 Geometry Handling.......................................................................................................... 127 5.5.2.1 The CRM Model ......................................................................................................... 127 5.5.2.2 The JSM Model ......................................................................................................... 128 Surface Meshing .............................................................................................................. 130 Volume Meshing .............................................................................................................. 133 5.5.4.1 Extrusion Layers Generation ...................................................................................... 133 5.5.4.2 Tetra Meshing............................................................................................................. 134 Sample CFD Results ......................................................................................................... 135 5.5.5.1 CRM ............................................................................................................................ 135 5.5.5.2 JSM ............................................................................................................................. 136 Case Study 2 - A 3D Hybrid Grid Generation Technique and a Multigrid/Parallel Algorithm Based on Anisotropic Agglomeration Approach ............................................................................................. 137 Statement of the Problem ............................................................................................... 137 Introduction, Background and Contributors ................................................................... 137 Hybrid Grid Generation Technique based on Anisotropic Agglomeration Approach ..... 139 5.6.3.1 Prism Grid Generation Method Based on Anisotropic Agglomeration Approach ..... 140 5.6.3.1.1 Volume Agglomeration .................................................................................... 140 5.6.3.1.2 Interface Agglomeration .................................................................................. 140 Multigrid/Parallel Algorithm............................................................................................ 141 Multi-Level Coarser Grid Generation Based on Anisotropic Agglomeration Approach .. 141 Applications and Discussions ........................................................................................... 143 5.6.6.1 Subsonic Turbulence Flow over 2D 30P30N Airfoil .................................................... 143 5.6.6.2 Transonic Turbulence Flow over ONERA M6 Wing .................................................... 144 5.6.6.3 Transonic Turbulence Flow over DLR-F6 Wing-Body Configuration .......................... 144 Concluding Remarks ........................................................................................................ 149 Recent Advances in Hybrid Mesh Generation and Literature Survey ........................................ 149 Parallel Consideration...................................................................................................... 149 Local Remeshing .............................................................................................................. 149 Background Mesh ............................................................................................................ 150 Boundary Viscous Meshes & Sharp Corners ................................................................... 151 Procedures for Mesh Generation .................................................................................... 151 Dynamic Mesh ................................................................................................................. 152 Adaptation ....................................................................................................................... 153 Special Issues ................................................................................................................... 155

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5.7.8.1 Centaur © .................................................................................................................... 155 5.7.8.2 Pointwise© .................................................................................................................. 156 Listing of Available Meshing Software .................................................................................................... 156

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Adaptive Mesh (Unstructured) ......................................................................................... 158

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Dynamic Meshing ................................................................................................................... 188

Adaptive Meshing by Subdivision ............................................................................................................. 158 Adaptive Mesh Refinement (AMR) ........................................................................................................... 159 Generalities...................................................................................................................... 160 Cell Division for a Geometry ............................................................................................ 161 6.2.2.1 Division Criteria .......................................................................................................... 162 Uniform AMR ................................................................................................................... 162 6.2.3.1 Transient Inviscid Flow ............................................................................................... 163 Case Study 1 - An Adaptive Hybrid Mesh Generation Method for Complex Geometries 163 6.2.4.1 Mesh Stitching ............................................................................................................ 164 6.2.4.1.1 Removal of Background Mesh Elements .......................................................... 164 6.2.4.2 Triangulation............................................................................................................... 164 6.2.4.3 Test Cases ................................................................................................................... 164 6.2.4.3.1 30P30N Multi-Element Airfoil .......................................................................... 164 6.2.4.3.2 2D Fuel Cell Slice............................................................................................... 165 Case Study 2 – Unstructured Mesh Adaptation for 2D Airfoil......................................... 165 6.2.5.1 Adaption Control Mechanism .................................................................................... 167 Case Study 3 – Parallel Implementation of Unstructured Mesh Refinement of Duct Flow 167 Case Study 4 – Generic Transonic Store Release............................................................. 168 Case Study 5 - Adaptive Hybrid Mesh Refinement for Multiphysics Applications .......... 170 6.2.8.1 Adaptive Hybrid Mesh Optimization .......................................................................... 170 6.2.8.2 Hybrid Adaptive Meshing ........................................................................................... 172 6.2.8.3 Meshing and Load Balancing. ..................................................................................... 174 6.2.8.4 Conclusions. ................................................................................................................ 175 Strategies for Driving Mesh Adaptation in CFD ................................................................................... 175 6.3.1.1 Feature-Based Adaption ............................................................................................. 175 6.3.1.2 Discretization Error and Recovery-Based Adaption ................................................... 177 6.3.1.3 Adjoint-Based Adaption.............................................................................................. 177 6.3.1.4 Truncation Error-Based Adaption............................................................................... 177 Current Approach for Performing Mesh Adaptation ...................................................... 177 Case Study - Mesh Adaption Results for 1D Burgers Equation (Re = 32) ........................ 179 A Solution-Based Adaptive Redistribution Method for Unstructured Meshes....................... 180 Introduction & Literature Survey .................................................................................... 180 Feature Detection ............................................................................................................ 181 Extraction of Solution Feature Surfaces .......................................................................... 182 6.4.3.1 Case Study 1 - NACA0012 Wing-Section..................................................................... 183 6.4.3.2 Case Study 2 - Capsule Model .................................................................................... 185

Type of Mesh Motion ...................................................................................................................................... 188 Mesh Deformation ........................................................................................................................................... 189

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Finite Volume in Dynamic Mesh ................................................................................................................ 189 Dynamic Mesh Techniques .......................................................................................................................... 190 Laplacian Mesh Morphing ............................................................................................... 190 Pseudo-Solid Equation ..................................................................................................... 191 7.4.2.1 Case Study – Motion of a Cylinder ............................................................................. 191 Biharmonic Equation ....................................................................................................... 192 Radial Basis Function ....................................................................................................... 193 Generalized Grid Interface .............................................................................................. 194 Overset Methods ............................................................................................................. 196 Delaunay Method ............................................................................................................ 197 7.4.7.1 Case Study - Airfoil Rotation ....................................................................................... 197 Spring Analogy ................................................................................................................. 198 Six Degrees of Ferndom (6 DOF)...................................................................................... 198 7.4.9.1 Transitional Deformation ........................................................................................... 198 7.4.9.2 Rotational Deformation.............................................................................................. 198 Dynamically Adaptive Mesh Refinement (DAMR) ............................................................................. 199 Case Study - Dynamically Adaptive Mesh Refinement FDTD: A Stable and Efficient Technique for Time-Domain Simulations ....................................................................................... 200 7.5.1.1 Numerical Results ....................................................................................................... 201

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Assessment of Mesh Types ................................................................................................. 202

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Case Studies Involving Comparisons of Structured vs. Unstructured Meshes . 213

Structured vs. Unstructured ........................................................................................................................ 203 Time and Memory ........................................................................................................... 203 Resolution ........................................................................................................................ 203 Alignment ........................................................................................................................ 203 Definable Normal............................................................................................................. 203 Effect of Cell Topology in Truncation Error ..................................................................... 204 Polyhedral vs. Tetrahedral ............................................................................................... 204 8.1.5.1 Boundary Prismatic Cells ............................................................................................ 205 Accuracy Assessment of Gradient Calculation Methods ................................................................. 207 Geometric Properties ...................................................................................................... 207 Literature Survey ............................................................................................................. 207 Gradient Calculation ........................................................................................................ 208 8.2.3.1 Green-Gauss Gradient Method .................................................................................. 208 8.2.3.2 GG-Simple Face Averaging ......................................................................................... 208 8.2.3.3 GG-Inverse Distance Weighted (IDW) Face Interpolation.......................................... 209 Visual Inspection .............................................................................................................. 210 Results Based on L2 Norm ................................................................................................ 211 Concluding Remarks ........................................................................................................ 212

Case Study 1 – Flow through Pipe with 90 degree Bend ................................................................. 213 Case Study 2 - Comparison of Hexa-Structured and Hybrid-Unstructured Meshing Approaches for Numerical Prediction of the Flow Around Marine Propellers .................................. 214 Introduction & Contributions .......................................................................................... 215 Propeller Models ............................................................................................................. 216 Numerical Method........................................................................................................... 216

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Meshing ........................................................................................................................... 217 Results ............................................................................................................................. 218 9.2.5.1 Propeller A .................................................................................................................. 219 9.2.5.2 Propeller P5168 .......................................................................................................... 220 Conclusions ...................................................................................................................... 221 Case Study 3 – Structure & Unstructured Hybrid Meshing and its effect on Quality of Solution on Turbine Blade ........................................................................................................................................ 222 Applications ..................................................................................................................... 222 Results ............................................................................................................................. 222 Case Study 4 - Evaluation of Structured vs. Unstructured Meshes for Simulating Respiratory Aerosol Dynamics ............................................................................................................................... 223 Bifurcation Model, Boundary Conditions, and Contributions ......................................... 223 Mesh Types ...................................................................................................................... 224 9.4.2.1 Structured ................................................................................................................... 225 9.4.2.2 Unstructured .............................................................................................................. 225 Governing Equations ....................................................................................................... 226 Numeric Method ............................................................................................................. 227 Results ............................................................................................................................. 229 9.4.5.1 Validation Studies ....................................................................................................... 229 9.4.5.2 Grid Convergence ....................................................................................................... 229 9.4.5.3 Velocity Fields ............................................................................................................. 231 9.4.5.4 Particle Deposition ..................................................................................................... 231 Discussion ........................................................................................................................ 234 9.4.6.1 Advantages of Hexahedral Structured Mesh ............................................................. 235 Conclusion ....................................................................................................................... 236 Case Study 5 - Comparison Between Structured Hexahedral and Hybrid Tetrahedral Meshes Generated by Commercial Software for CFD Hydraulic Turbine Analysis ........................... 236 Problem Description ........................................................................................................ 237 Geometry ......................................................................................................................... 238 Mesh Description............................................................................................................. 238 9.5.3.1 Structured Hexahedral Meshes .................................................................................. 239 9.5.3.2 Hybrid Tetrahedral Mesh ........................................................................................... 240 CFD Solution Strategy and Boundary Conditions ............................................................ 240 Results ............................................................................................................................. 241 Conclusion ....................................................................................................................... 243

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Mesh Sensitivity and Mesh Independence Study ........................................................ 245

Different Types of Mesh Sensitivity.................................................................................................... 245 Symbolic Differentiation .................................................................................................. 245 Automatic Differentiation ............................................................................................... 245 10.1.2.1 Symbolic vs Automatic Differentiation.................................................................. 245 Finite Differencing ........................................................................................................... 245 Mesh Sensitivity via Direct Differentiation (DD) .......................................................................... 246 Surface Modeling Using NURBS....................................................................................... 246 10.2.1.1 Case Study - 2D Study of Airfoil Grid Sensitivity via Direct Differentiation (DD) .. 248 Adjoint Variable Sensitivity Analysis (AV) ...................................................................................... 249 Mesh Independence Study ..................................................................................................................... 251

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Mesh Quality ............................................................................................................................ 253

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Appendix A ............................................................................................................................... 266

Background .................................................................................................................................................. 253 Mesh Quality Metric .................................................................................................................................. 253 Mesh Quality from User’s Perspective ............................................................................ 255 Mesh Quality from Researcher’s Perspective ................................................................. 255 Mesh Quality from Solver’s Perspective.......................................................................... 255 11.2.3.1 CFD++..................................................................................................................... 256 11.2.3.2 Fluent and CFX ....................................................................................................... 256 11.2.3.3 Kestrel .................................................................................................................... 256 11.2.3.4 STAR-CCM+ ............................................................................................................ 257 11.2.3.5 Deducing Results ................................................................................................... 258 Some Geometric Properties ............................................................................................ 258 11.2.4.1 Aspect ratio ........................................................................................................... 258 11.2.4.2 Orthogonality ........................................................................................................ 258 11.2.4.3 Skewness ............................................................................................................... 259 11.2.4.4 Warpage ................................................................................................................ 259 11.2.4.5 Jacobian ................................................................................................................. 259 11.2.4.6 Tetrahedral Volume............................................................................................... 259 11.2.4.7 Polygonal Face Area and Centroid ........................................................................ 260 11.2.4.8 Polyhedral Volume and Centroid .......................................................................... 261 Best Practice for Mesh Generation ..................................................................................................... 261 Geometry Modeling and Geometry Cleanup .................................................................. 262 Computational Domain ................................................................................................... 262 Choice of Grid .................................................................................................................. 262 Surface Meshing .............................................................................................................. 263 Volume Meshing .............................................................................................................. 263 Boundary Layer Meshing ................................................................................................. 263 Guidelines for Aerodynamics in General ......................................................................... 264 Guidelines for Auto Aerodynamics .................................................................................. 264 Improvement of Grid Quality .......................................................................................... 265

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Computer Code for a Transfinite Interpolation ................................................................................... 266

List of Tables Table 4.1 Nomenclature ........................................................................................................................................... 66 Table 4.2 Comparison of the number of vertices and quality of the mesh for different values of δ - (Courtesy of [Labbe]) .................................................................................................................................................. 73 Table 5.1 Near-Field Grid Details....................................................................................................................... 124 Table 5.2 Abbreviations......................................................................................................................................... 126 Table 5.3 Currently Available Grid Generation Software ........................................................................ 156 Table 9.1 Dimensions of Domains – (Courtesy of Morgut & Nobile) .................................................. 217 Table 9.2 Grids for Propeller A– (Courtesy of Morgut & Nobile) ......................................................... 217 Table 9.3 Grids for Propeller P5168 – (Courtesy of Morgut & Nobile) .............................................. 217 Table 9.4 Results of Propeller A– (Courtesy of Morgut & Nobile) ....................................................... 219

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Table 9.5 Experimental setup of Propeller P5168 ..................................................................................... 220 Table 9.6 Relative Percentage Differences of Computed Values Between Finer and Coarser Mesh for propeller P5168 – (Courtesy of Morgut & Nobile) ...................................................................... 220 Table 9.7 Grid Convergence – (Courtesy of Samir Vinchurkar & Worth Longest) ........................ 230 Table 9.8 Mesh Densities for Structured Hexahedral and Hybrid Un-Structural Tetrahedral – (Courtesy of Rousseau et al.) ................................................................................................................................... 239 Table 10.1 Pros & Cons of Different Grid Sensitivity Method (NDV = Number of Design Variable) ........................................................................................................................................................................... 250

List of Figures Figure 1.1 Meshes Created using ANSYS Mosaic-Enabled Poly-Hex Core Meshing - Courtesy of Sheffield Hallam University ......................................................................................................................................... 20 Figure 1.2 Methodology of General Grid Generation .................................................................................... 22 Figure 2.1 Anatomy of commercial CAD Systems .......................................................................................... 25 Figure 2.2 Fighter Airplane F-16 calculation ................................................................................................... 25 Figure 2.3 Example of a CSG Tree ......................................................................................................................... 28 Figure 2.4 Different Analysis Require Different Geometric Representations .................................... 29 Figure 2.5 Small Feature (Left) vs Removed (Right) .................................................................................... 30 Figure 3.1 Classification of Grid Generation Algorithms (Courtesy of Steven Owen) .................... 32 Figure 3.2 Schwarz concept of iterating between domains ....................................................................... 34 Figure 3.3 Domain Decomposition for M6 wing using TIL scripts (Courtesy of GridPro) ............ 34 Figure 3.4 Example of Unstructured Tetrahedral Grids.............................................................................. 35 Figure 3.5 Examples of Structured grids for Turbine Blade ...................................................................... 35 Figure 3.6 Sponge Analogy ...................................................................................................................................... 36 Figure 3.7 Multi Block representation for C-H mesh around a wing ..................................................... 37 Figure 3.8 Topology and Grid on a Multi-Block Wings using GridPro® ................................................ 37 Figure 3.9 Multi-block gridding over Turbine blade - (Courtesy of GridPro) .................................... 38 Figure 3.10 Dual Block Grid Topology for a Generic Wing-Fuselage Configuration ....................... 39 Figure 3.11 Grid for dual-block generic airplane geometry ...................................................................... 40 Figure 3.12 Typical Elliptic Grid for an Airfoil with Orthogonality Enforced on the Boundary . 41 Figure 3.13 Orthogonality Adjustments – (Courtesy of Chaitanya Varier) ......................................... 42 Figure 3.14 Euler Solution on a HSCT Wing-Fuselage ................................................................................. 44 Figure 3.15 Folded Grid by Transfinite Interpolation - Smooth Grid by Winslow Functional.... 45 Figure 3.16 1D Weight Function for High Gradient and Curvature........................................................ 46 Figure 3.17 Mesh and Mach Contours for Transonic Flow ........................................................................ 47 Figure 3.18 Grid Adaption and Mack Contours for Supersonic Airfoil ................................................. 48 Figure 4.1 Closing stage of a Moving Front Method ...................................................................................... 50 Figure 4.2 Mesh parameters ................................................................................................................................... 51 Figure 4.3 Surface Mesh of SGI Logo ................................................................................................................... 52 Figure 4.4 States of a front edge – (Courtesy of Owen et al.) .................................................................... 53 Figure 4.5 Steps demonstrating process of generating a quadrilateral from Front NA-NB (Courtesy of Owen et al.) .............................................................................................................................................. 54 Figure 4.6 Progression of Q-Morph- (Courtesy of Owen et al.) ................................................................ 55 Figure 4.7 Comparison of Q-Morph with Lee’s Algorithm Illustrating Element Boundary.......... 56 Figure 4.8 Results of Q-Morph Compared with Lee’s (1994) Advancing Front Indirect............... 56 Figure 4.9 Large Transition Mesh for CFD Application - (Courtesy of Owen et al.) ........................ 57 Figure 4.10 Success and failure of the in sphere test of abcd with e. .................................................... 59 Figure 4.11 Relationship Between Delaunay Triangles and the Voronoi Diagram ......................... 60 Figure 4.12 Two-Three Tetrahedral swap ........................................................................................................ 60

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Figure 4.13 Robust and Fast way to Detect if point D lies in the Circumcircle of A, B, C ............... 61 Figure 4.14 Delaunay Triangulation (white) and Voronoi Diagram (blue) – Courtesy of [Labbe]) ............................................................................................................................................................................... 61 Figure 4.15 2D Delaunay triangulation of a set of vertices (black) restricted to a curve (blue) 64 Figure 4.16 Representation of a 3D Metric with Eigenvalues λ1, λ2 and λ3 as an Ellipsoid – (Courtesy of [Labbe]) ..................................................................................................................................................... 68 Figure 4.17 An anisotropic uniform Delaunay triangulation (orange) and the corresponding stretched ............................................................................................................................................................................. 70 Figure 4.18 Two stars Sp and Sq forming an inconsistent configuration - (Courtesy of [Labbe]) ................................................................................................................................................................................................. 71 Figure 4.19 Influence of the Parameter ψ0 in a 2D (shown on the left) and 3D Domain (shown on the right) - (Courtesy of [Labbe]) ....................................................................................................................... 72 Figure 4.20 A square of side 10 and centered on the origin, endowed with the “Starred” metric field ........................................................................................................................................................................................ 75 Figure 4.21 Anisotropic Triangulation of a Rectangle Endowed with the Hyperbolic Shock Metric Field - (Courtesy of [Labbe]) ........................................................................................................................ 75 Figure 4.22 A square of side 6 and centered on the origin, endowed with the “Swirl” metric field - (Courtesy of Labbé et al.) .............................................................................................................................. 76 Figure 4.23 The optimized SRDT of 4000 seeds in a planar domain endowed with a hyperbolic shock induced metric field (left). On the right, a zoom on a rotational region of the metric field shows the difference between pre- (above) and post- (bottom) optimization – (Courtesy of Labbé et al.) ....................................................................................................................................................................... 77 Figure 4.24 Isotropic and Anisotropic Canvas Sampling - (Courtesy of [Labbe]) ............................ 78 Figure 4.25 Unit Sphere Endowed with the Hyperbolic Metric field - (Courtesy of [Labbe]) ..... 79 Figure 4.26 On the left, the discrete Riemannian Voronoi diagram of 1020 seeds on the “Chair” surface, with a curvature induced metric field; the edges of the curved Riemannian Delaunay triangulation are traced in black - (Courtesy of [Labbe]) ............................................................................... 79 Figure 4.27 Discrete Riemannian Voronoi Diagram (top) and Curved Riemannian Delaunay Triangulation (bottom) endowed with the Hyperbolic Shock Metric Field - (Courtesy of [Labbe]) ................................................................................................................................................................................................. 80 Figure 4.28 Converging of an Octree Decomposition Around an Airfoil .............................................. 81 Figure 4.29 A close-up view of nasty cheese a well-known test-case featuring 30◦ dihedral angles – (Courtesy’s of [Mar´echal]) ........................................................................................................................ 82 Figure 4.30 Hierarchy of Meshing Methodologies......................................................................................... 83 Figure 4.31 Quadrilateral Mesh Generation..................................................................................................... 85 Figure 4.32 Overset Mesh Combination............................................................................................................. 86 Figure 4.33 Two Counter-Rotating Objects Embedded in Two Overset Regions with Background Mesh – (Courtesy of Siemens) .......................................................................................................... 86 Figure 4.34 Example of Cartesian Grid Near Curved Surface – (Courtesy of NASA Ames) .......... 87 Figure 4.35 Solid Surface Over-Layer Cartesian Cell and Resulting Cut and Split Cell – (Courtesy of NASA Ames) .................................................................................................................................................................. 87 Figure 4.36 Example of Merge Cell Creation – (Courtesy of NASA Ames) ........................................... 88 Figure 4.37 Example Adaptive Grid for Supersonic Wedge Flow – (Courtesy of NASA Ames) .. 89 Figure 4.38 Schematic image of Adaptive Mesh Refinement – (Courtesy of Hiroshi Abe) ........... 90 Figure 4.39 Pressure Contours in 2D Backward Step .................................................................................. 90 Figure 4.40 Flow Around a F16XL Fighter Jet using Cut Cells and AMR – (Courtesy of M. J. Aftosmis, M. J. Berger, J. E. Melton) .......................................................................................................................... 91 Figure 4.41 Example Chimera Grid Near Curved Surface (Courtesy of NASA Ames) .................... 93 Figure 4.42 Example Hybrid Grid Near Curved Surface – (Courtesy of NASA Ames)..................... 95 Figure 4.43 Basic Superposition Example – (Courtesy of Kalinin, Mazo and Isaev) ....................... 96

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Figure 4.44 Example of Cartesian Grid on a Generic Airplane – (Source: Richard Smith 1996) 97 Figure 4.45 Meshing Types in SAMM .................................................................................................................. 98 Figure 4.46 Typical Polyhedral Cell and their Decomposition ................................................................. 99 Figure 4.47 Polyhedral meshing using Delaunay triangulation............................................................... 99 Figure 4.48 Dual surface Triangulation resulting in Polyhedron ......................................................... 101 Figure 4.49 Boundary Layer Prisms Generated on a Cascade of a 2D Triangulation and Dual Polyhedron ...................................................................................................................................................................... 102 Figure 4.50 Concept of cascading for boundary layer in 3D................................................................... 103 Figure 4.51 Dual Mesh for Mixed Triangular-Quadrilateral Unstructured Mesh ................................ 104 Figure 4.52 Conventional configuration geometry (a), final structural mesh (Courtesy of Hwang & Martins) ........................................................................................................................................................ 106 Figure 4.53 The Six Steps of the Unstructured Quad Meshing Algorithm ........................................ 107 Figure 5.1 Hybrid Grid and Steady State Solution ...................................................................................... 110 Figure 5.2 Comparison of different mesh types for RANS Computations......................................... 111 Figure 5.3 Constructions of Hybrid mesh ...................................................................................................... 112 Figure 5.4 Predominantly polyhedral meshing ........................................................................................... 112 Figure 5.5 Combined Volume and Extrusion Layer Meshes ................................................................... 113 Figure 5.6 Meshing tools in CD-adapco ........................................................................................................... 113 Figure 5.7 Meshes Generated by a) Proposed Algorithm and b) Leading Commercial Vendor .............................................................................................................................................................................................. 114 Figure 5.8 Adaptive Mesh Refinement Types ............................................................................................... 115 Figure 5.9 An H-refinement mesh about a Shuttle-like body (left) and Computed CP (right).. 116 Figure 5.10 Isotropic vs. Anisotropic Meshing............................................................................................. 117 Figure 5.11 Coarsening by Edge Collapsing – Courtesy of [Cavallo]................................................... 118 Figure 5.12 Hierarchy of Successively Coarser Meshes Obtained by Uniform ............................... 119 Figure 5.13 Coarsening ratio for coarsening with and without local Retriangulation. ............... 120 Figure 5.14 3 to 2 and 2 to 3 Swap ................................................................................................................... 121 Figure 5.15 Comparison of Coarse, Medium and Fine Grids: lateral view on fore-body with Symmetry......................................................................................................................................................................... 125 Figure 5.16 local dissipation error of drag coefficient on field cut-plane at x=1400 inch; isometric/downstream view ................................................................................................................................... 125 Figure 5.17 Comparison of SolarChimera5 and Solar Grid at x =1454 inch plane; Viscous Wall Surface in Dark .............................................................................................................................................................. 126 Figure 5.18 JSM Model with Engine Nacelle.................................................................................................. 128 Figure 5.19 Computational Domain of the HL-CRM Gapped Flaps Model ........................................ 128 Figure 5.20 Three Locations of Problematic Areas of the JSM Geometry for the Generation of Boundary Layers ........................................................................................................................................................... 129 Figure 5.21 Computational Domain and Separation of Zones of the JSM Model with Engine Nacelle ............................................................................................................................................................................... 129 Figure 5.22 Batch Mesh setup for the JSM Model with Size Boxes for Local Mesh Control....... 130 Figure 5.23 Resulting Layers for Isotropic Surface Mesh (Top) and Anisotropic (Bottom) ..... 131 Figure 5.24 Close ups of Coarse CRM Gapped Flap Model with Comparison of Tridiagonal Dominant (Top) vs. Quad Dominant (Bottom) Surface Mesh .................................................................... 132 Figure 5.25 Volume Mesh of the JSM................................................................................................................ 134 Figure 5.26 CL and CD for CRM Geometry at 8 degree AoA using OpenFOAM and STAR-CCM+ .............................................................................................................................................................................................. 135 Figure 5.27 Lift and Drag Coefficients for the JSM Geometry using OpenFOAM and STAR-CCM+ .............................................................................................................................................................................................. 136 Figure 5.28 Interface Agglomeration Procedure Wing – Courtesy of [Laiping et al.] ................. 141 Figure 5.29 Initial Hybrid Grids and Coarsen Grids Wing – Courtesy of [Laiping et al.] ........... 142

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Figure 5.30 CP Distribution on Solid Wall Wing Courtesy of [Laiping et al.].................................. 143 Figure 5.31 Initial Hybrid Grids and Coarsening Grids over 30P30N Airfoil Wing – Courtesy of [Laiping et al.] ................................................................................................................................................................ 143 Figure 5.32 Initial Hybrid Grids and Coarsening Grids over ONERA M6 Wing – Courtesy of [Laiping et al.] ................................................................................................................................................................ 144 Figure 5.33 Close-up Views of Hybrid Grids After Agglomeration Wing – Courtesy of [Laiping et al.]................................................................................................................................................................................... 145 Figure 5.34 Aerodynamic Force Coefficients for Different Angles of Attack (M∞ = 0.75) Wing – Courtesy of [Laiping et al.] ....................................................................................................................................... 146 Figure 5.35 Hybrid Grids over DLR-F6-WBNP Configuration Wing – Courtesy of [Laiping et al.] .............................................................................................................................................................................................. 147 Figure 5.36 CP Distributions at Three Typical Sections (M = 0.75, α = 1.0 deg) Wing – Courtesy of [Laiping et al.]........................................................................................................................................................... 148 Figure 5.37 Local Remeshing .............................................................................................................................. 150 Figure 5.38 Hybrid Mesh on a Wing-Body-Pylon-Nacelle Configuration – Courtesy of Centrum® .............................................................................................................................................................................................. 152 Figure 5.39 Meshing Aircraft Landing & Takeoff – Courtesy of Centaur© ........................................ 155 Figure 6.1 Example Adaptive Grid for Supersonic Wedge Flow ........................................................... 160 Figure 6.2 Schematic image of Adaptive Mesh Refinement .................................................................... 160 Figure 6.3 Octree Data Structure of Adaptive Cartesian Grid Method ............................................... 161 Figure 6.4 Schematic 2D view of angular variation of normal .............................................................. 162 Figure 6.5 Pressure Contours in 2D Backward Step .................................................................................. 162 Figure 6.6 Selected Initial Meshes for the Transient Adaptive Procedure (Meshes 3, 20, 27 and 29) ....................................................................................................................................................................................... 163 Figure 6.7 30P30N Multi-Element Airfoil & close up of slat................................................................... 165 Figure 6.8 2D Fuel Cell Slice & Zoomed........................................................................................................... 165 Figure 6.9 Grid Adaption using Supersonic Flow for an Airfoil (bow shock).................................. 166 Figure 6.10 NACA 0012 Transonic test case: M∞ = 0.8, α=1.25 ............................................................. 166 Figure 6.11 Two-Pass Approach for Parallel Coarsening and Refinement. ........................................... 168 Figure 6.12 Store position, orientation, and surface pressures at selected points in trajectory ........ 168 Figure 6.13 Adapted Mesh Partitioning During Store Dispense ................................................................ 169 Figure 6.14 Inter-Processor Partitioning Based on Laplace Coefficients ......................................... 170 Figure 6.15 Hybrid Icosahedra Surface Mesh (left) and Multi-Material Hybrid Volume Mesh (right) – (Courtesy of Khamayseh and Almeida)............................................................................................. 171 Figure 6.16 HTTR Multi-Material Geometry, Initial Coarse Mesh (left), Refined Mesh (right) ) – (Courtesy of Khamayseh and Almeida) ............................................................................................................... 172 Figure 6.17 Orography field (left), r-adaptivity (center) and h-adaptivity (right) for climate modeling ) – (Courtesy of Khamayseh and Almeida) .................................................................................... 172 Figure 6.18 Coupled orography field transfer with h-adaptivity. Planar orography field (top), .............................................................................................................................................................................................. 173 Figure 6.19 Meshing and Partitioning of Centrifugal Contactor ) – (Courtesy of Khamayseh and Almeida) ........................................................................................................................................................................... 174 Figure 6.20 Discretization Error in the Drag Coefficient for Transonic Flow over an Airfoil (Reproduced from Dwight) ...................................................................................................................................... 176 Figure 6.21 Steady-State Burgers Equation for Reynolds Number 32............................................... 179 Figure 6.22 Adaption Schemes Applied to Burgers Equation Left) numerical solutions and right) local nodal spacing Δx. ................................................................................................................................... 180 Figure 6.23 NACA0012 wing-Section Adaption........................................................................................... 184 Figure 6.24 Extraction of a Flow Feature & Redistributed Volume Meshes ...................................... 185

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Figure 6.25 Hybrid Meshes for the NACA0012 Wing-Section and Cp Distribution (-1.0 to 1.0) .............................................................................................................................................................................................. 186 Figure 6.26 Adaptive Remeshing of Capsule................................................................................................. 186 Figure 7.1 Mesh Deformation Problem ........................................................................................................... 189 Figure 7.2 Cylinder Motion in 2D....................................................................................................................... 192 Figure 7.3 Mesh Deformation via Bi-Harmonic Equations ..................................................................... 193 Figure 7.4 Mesh Deformation via Laplace & RBF Methods ..................................................................... 194 Figure 7.5 GGI interface ......................................................................................................................................... 195 Figure 7.6 Overset Method ................................................................................................................................... 196 Figure 7.7 Delaunay Method of Dynamic mesh ........................................................................................... 197 Figure 7.8 Mesh before and after the translational deformations ....................................................... 198 Figure 7.9 Mesh Before and After the x-axis Rotational Deformation ............................................... 199 Figure 7.10 Spiral Inductor Geometry where P1 and P2 denote port 1 and Part 2...................... 200 Figure 7.11 Vertical field evolution and associated mesh refinement in the microstrip spiral inductor, simulated by a two-level dynamic AMR-FDTD ............................................................................. 201 Figure 8.1 Backward facing step in a duct using Polyhedral, Hexahedral and Tetrahedral cells .............................................................................................................................................................................................. 203 Figure 8.2 Effect of truncation error on Hex and Tet cells ...................................................................... 204 Figure 8.3 Average Bees Being Smarter than CFD Engineer? (Courtesy of Stephen Ferguson) .............................................................................................................................................................................................. 204 Figure 8.4 Polyhedral cells vs Tetrahedral cells .......................................................................................... 205 Figure 8.5 Boundary prims cells for tetrahedral (left) and polyhedral (right) cells – (Courtesy of CD-Adapco) ................................................................................................................................................................ 206 Figure 8.6 GG simple face averaging ................................................................................................................ 208 Figure 8.7 GG Inverse Distance Weighted (IDW) Face Interpolation ................................................. 209 Figure 8.8 Methodologies for various Gradient Order of Accuracy..................................................... 210 Figure 8.9 Global Error Norms for x-Direction Gradient for Various Gradient Methods ........... 211 Figure 9.1 Comparison of Hex (16 K Cells) and Tet (440 K Cells) for a Pipe with 90 Degree Bend ................................................................................................................................................................................... 213 Figure 9.2 Results of Hex vs Tet Meshes as well as Hybrid Mesh in a Pipe with 90 Degree Bend .............................................................................................................................................................................................. 214 Figure 9.3 Design of Propellers, (left) Propeller P5168, .......................................................................... 216 Figure 9.4 Computational Domain– (Courtesy of Morgut & Nobile) .................................................. 216 Figure 9.5 Meshing for Propeller P5168– (Courtesy of Morgut & Nobile)....................................... 218 Figure 9.6 KT , KQ and η Curves of Propeller A – (Courtesy of Morgut & Nobile) ........................... 219 Figure 9.7 KT and KQ curves of Propeller P5168 – (Courtesy of Morgut & Nobile)....................... 220 Figure 9.8 Flow Around Turbine Blade – (Courtsy of Sasaki et al.) .................................................... 222 Figure 9.9 Geometric Blocking Used (a) Structured Hexahedral (178 Blocks) and (b) Unstructured Hexahedral (80 Blocks) – (Courtesy of Samir Vinchurkar & Worth Longest)........ 224 Figure 9.10 Four Meshing Styles of the PRB Model (a) Structured Hexahedral, (b) Unstructured Hexahedral, (c) Prismatic, and (d) Hybrid – (Courtesy of Samir Vinchurkar & Worth Longest) 225 Figure 9.11 Velocity Vectors (a) Structured Hexahedral Mesh with 214 K C.V. (b) Unstructured Hexahedral Mesh with 318 K, C. V. (c) Prismatic Mesh with 510K C. V, (d) Hybrid Mesh with 608 K C. V. – (Courtesy of Samir Vinchurkar & Worth Longest) ........................................................................ 232 Figure 9.12 Deposition Locations for 10 lm Particles in the Planar Geometry for the (a) Structured Hexahedral Mesh, (b) Unstructured Hexahedral Mesh, (c) Prismatic Mesh, and (d) Hybrid Mesh – (Courtesy of Samir Vinchurkar & Worth Longest) .......................................................... 233 Figure 9.13 Boundary Layer Transition Between Prismatic and Volume Elements – (Courtesy of Rousseau et al.)......................................................................................................................................................... 237 Figure 9.14 Example of a hydraulic turbine spiral case (half domain) .............................................. 238

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Figure 9.15 Geometry of the Stay Vanes and Wicket Gates, Left: Geometry A, Right: Geometry B – (Courtesy of Rousseau et al.) ................................................................................................................................ 238 Figure 9.16 Structured Hexahedral Mesh of the Geometry A on the Symmetrical Surface and Close Up – (Courtesy of Rousseau et al.) ............................................................................................................. 239 Figure 9.17 Hybrid Tetrahedral Medium Mesh on the Symmetric Surface of the Geometry A (left) & Mesh in the wake of a Hydraulic Profile (wicket gates trailing edge)(right) – (Courtesy of Rousseau et al.).............................................................................................................................................................. 240 Figure 9.18 Relative Total Head Loss on the Meridian Plane for the Geometry A with fine mesh, left: Structured Hexahedral, right: Hybrid Tetrahedral – (Courtesy of Rousseau et al.) ................ 241 Figure 9.19 Meridian Velocity Near a Stay Vane with fine mesh for Geometry A, left: Structured Hexahedral, right: Hybrid Tetrahedral – (Courtesy of Rousseau et al.) ................................................ 242 Figure 9.20 Meridian Velocity on the Meridian Plane for the Geometry B – (Courtesy of Rousseau et al.).............................................................................................................................................................. 243 Figure 10.1 B-Spline Approximation of NACA0012 (left) and RAE2822 (right) Airfoils ........... 247 Figure 10.2 Six Control Point Representation of a Generic Airfoil ...................................................... 247 Figure 10.3 Free Form Deformation (FFD) for Volume Grid with Control Points (Courtesy of Kenway et al.) ................................................................................................................................................................. 248 Figure 10.4 Sample Grid and Grid Sensitivity............................................................................................... 249 Figure 10.5 Effects of Mesh Density on Solution Domain ........................................................................ 251 Figure 10.6 Mesh Independence ........................................................................................................................ 252 Figure 11.1 Predicted Mesh Quality (Volume, Aspect Ratio, and Stretch) ....................................... 254 Figure 11.2 A simple Demonstration of How a Poor Mesh from a Cell Geometry Perspective 256 Figure 11.3 Using Kestrel one can Show a Correlation Between Mesh and Solution Quality .. 257 Figure 11.4 Concept of Orthogonality in Cells .............................................................................................. 258 Figure 11.5 Skewness and Warpage................................................................................................................. 259 Figure 11.6 Tetrahedral Volume ........................................................................................................................ 259 Figure 11.7 Triangulation of a polygon ........................................................................................................... 260 Figure 11.8 Tetrahedralization of a Polyhedral (showing a single face) .......................................... 261 Figure 11.9 General estimation of surface mesh element size .............................................................. 263 Figure 12.1 Symmetry plane (XY) ..................................................................................................................... 267

Contributors ➢ Roy Koomullil, Bharat Soni, Rajkeshar Singh ,”A comprehensive generalized mesh system for CFD applications”, Mathematics and Computers in Simulation 78 (2008). ➢ Narayan, K. Lalit. Computer Aided Design and Manufacturing. New Delhi, 2008. ➢ Duggal, Vijay. Cadd Primer: A General Guide to Computer Aided Design and Drafting-Cadd, Mailmax Pub. ISBN 978-0962916595, 2000. ➢ Christophe Geuzaine, Emilie Marchandise , and Jean-Francois Remacle, “An introduction to Geometrical Modelling and Mesh Generation”, The Gmsh Companion. ➢ Butlin, G., Stops C., “CAD Data Repair”, Proc. 5th Int. Meshing Roundtable, pp. 7-12, 1996. ➢ Mezentsev, A.A. and Woehler, T., “Methods and algorithms of automated CAD repair for incremental surface meshing”, Proc. 8th Int. Meshing Roundtable, Sandia report SAND 992288, pp. 299-309, 1999. ➢ Ribo, R., Bugeda, G. and Onate, E., “Some algorithms to correct a geometry in order to create a finite element mesh”, Computers and Structures, 80:1399-1408, 2002.

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➢ Richardson LF. Weather prediction by numerical process. Cambridge: Cambridge University Press; 1921. ➢ Edelsbrunner H. “Geometry and topology for mesh generation”, Cambridge: Cambridge university, 2001. ➢ Baker, T., “Mesh generation: Art or science?” MAE Department, Princeton University, Princeton, NJ. ➢ Steven J. Owen, “A Survey of Unstructured Mesh Generation Technology”, Carnegie Mellon University, PA. ➢ Steven Owen: Introduction to unstructured mesh generation, 2005. 1 Baker, T.,J., “Mesh generation: Art or science?”, MAE Department, Princeton University, Princeton, NJ. ➢ Bauer F, Garabedian P, Korn D. Supercritical wing sections I, Lecture Notes in Economics and Mathematical Systems, vol. 66. Berlin: Springer; 1972. ➢ Moretti G.”Grid generation using classical techniques”. Proceedings of the NASA Langley workshop on numerical grid generation techniques, Langley, VA, October, 1980. ➢ Caughey DA, “A systematic procedure for generating useful conformal mappings”, Int J Num Meth Eng 1978. ➢ Eriksson LE,”Generation of boundary-conforming grids around wing-body configurations using transfinite interpolation”, AIAA J 1982; 20:1313–20. ➢ An overview of Grid Pro/az3000 for automated grid generation. ➢ Churchill, R., V., “Introduction to Complex Variables”, McGraw-Hill, New York. ➢ Joe F. Thompson, Z. U. A. Warsi, C. Wayne Mastin, “Numerical Grid Generation -Foundations and Applications”, North Holland, 1985. ➢ Peter Eiseman and Robert E. Smith, “Applications of Algebraic Grid Generation”, April 1990. ➢ Sadrehaghighi, I., Smith, R.E., Tiwari, S., N., “Grid Sensitivity and Aerodynamic Optimization Of Generic Airfoils”, Journal of Aircraft, Volume 32, No. 6, Pages 1234-1239. ➢ M. Farrashkhalvat and J.P. Miles, “Basic Structured Grid Generation”, An imprint of Elsevier Science Linacre House, Jordan Hill, Oxford OX2 8DP, 200 Wheeler Rd, Burlington MA 01803, First published 2003. ➢ Feng Liu, Shanhong Ji, and Guojun Liao,” An Adaptive Grid Method and Its Application to Steady Euler Flow Calculations”, Siam J. Sci. Comput. C° 1998 Society For Industrial And Applied Mathematics Vol. 20, No. 3, Pp. 811{825. ➢ Mael Rouxel-Labbe, “Anisotropic mesh generation”, Université Côte d’Azur, 2016. ➢ Vangelis Skaperdas, and Neil Ashton, “Development of high-quality hybrid unstructured meshes for the GMGW-1 workshop using ANSA”, AIAA, January 2018. ➢ David A. Venditti and David L. Darmofal, “Grid Adaptation for Functional Outputs: Application to Two Dimensional Inviscid Flows", Journal of Computational Physics 176, 40– 69 (2002). ➢ Cavallo, P.A., Sinha, N., and Feldman, G.M.,” Parallel Unstructured Mesh Adaptation For Transient Moving Body And Aero propulsive Applications”, Combustion Research and Flow Technology, Inc. (CRAFT Tech), PA. ➢ Cavallo, P.A., Sinha, N., and Feldman, G.M.,”Parallel Unstructured Mesh Adaptation For Transient Moving Body And Aeropropulsive Applications”, Combustion Research and Flow Technology, Inc. (CRAFT Tech), PA 18947. ➢ Hrvoje Jasak, ˇZeljko Tukovi´c, “Automatic Mesh Motion for the Unstructured Finite Volume Method”, ISSN 1333–1124, UDK 532.5:519.6. ➢ JIA Huana, SUN Qin b, “A Comparison of Two Dynamic Mesh Methods in Fluid –Structure interaction”, School of Aeronautics, Northwestern Polytechnic University, Xi‘an china. 2nd International Conference on Electronic & Mechanical Engineering and Information Technology (EMEIT-2012). ➢ Fluent, “Meshing and CFD Accuracy”, CFD Summit, June 2005.

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➢ Mitja Morgut, Enrico Nobile, “Comparison of Hexa-Structured and Hybrid-Unstructured Meshing Approaches for Numerical Prediction of the Flow Around Marine Propellers”, First International Symposium on Marine Propulsions smp’09, Trondheim, Norway, June 2009. ➢ Daisuke Sasaki, Caleb Dhanasekaran, Bill Dawes, Shahrokh Shahpar, “Efficient Unstructured Hybrid Meshing and its Quality Improvement for Design Optimization of Turbomachinery”, European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006. ➢ Samir Vinchurkar, P. Worth Longest, “Evaluation of hexahedral, prismatic and hybrid mesh styles for simulating respiratory aerosol dynamics”, Computers & Fluids, 2008. ➢ Sadrehaghighi, I., Smith, R.E., Tiwari, S., N., “Grid Sensitivity and Aerodynamic Optimization Of Generic Airfoils”, Journal of Aircraft, Volume 32, No. 6, Pages 1234-1239. ➢ Philippe Martineau Rousseau, Azzeddine Soulaïmani and Michel Sabourin, “Comparison between structured hexahedral and hybrid tetrahedral meshes generated by commercial software for CFD hydraulic turbine analysis”, Conferece Paper, May 2013. ➢ Ahmed Khamayseh and Valmor de Almeida, “Adaptive Hybrid Mesh Refinement for Multiphysics Applications”, Journal of Physics: Conference Series 78 (2007) 012039. ➢ Kenneth Wong is Digital Engineering’s resident blogger and senior editor. ➢ Biswas, R., and Strawn, R.C., "Tetrahedral and Hexahedral Mesh Adaptation for CFD Problems", NAS Technical Report NAS-97-007, 1997. ➢ H.L. De Cougny and MS. Shephard, 'Local modification tools for adaptive mesh enrichment and their parallelization', Scientific Computation Research Center, RPI, NY. ➢ Zhang Laiping, Zhao Zhong W. Huang and R. D. Russell, Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J. Sci. Comput. 20(3), 998 (1999). ➢ Hector D. Ceniceros and Thomas Y. Hou, “An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions”, Journal of Computational Physics 172, 609–639 (2001). ➢ Yaxun Liu and Costas D. Sarris, “Dynamically Adaptive Mesh Refinement FDTD: A Stable And Efficient Technique For Time-Domain Simulations”, Department of Electrical and Computer Engineering University of Toronto, Toronto, ON, M5S 3G4, Canada. ➢ Chang Xinghua, He Xin, “A 3D hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration approach”, Chinese J of Aeronautics, 2013. ➢ Lars Tysell, “Hybrid grid generation for viscous flow computations around complex geometries”, Technical Reports from Royal Institute of Technology, Department of Mechanics, Stockholm, Sweden, 2010. ➢ L¨ohner, R., A parallel Advancing Front Grid Generation Scheme. Paper AIAA 2000. ➢ Larwood, B., Weatherill, N. P., Hassan, O. & Morgan, K. 2003, Domain Decomposition Approach for Parallel Unstructured Mesh Generation. International Journal for Numerical Methods in Engineering, 2003. ➢ Weatherill, N. P., Hassan, O., Morgan, K., Jones, J. & Larwood, B., “Towards Fully Parallel Aerospace Simulations on Unstructured Meshes”. International Journal for Numerical Methods in Engineering, 2001. ➢ Ito, Y., Murayama, M., Yamamoto, K., Shih, A. & Soni, B., Development of a Grid Generator to Support 3-D Multizone Navier-Stokes Analysis. Paper AIAA-2008-7180. ➢ Eliasson, P., Nordstr¨om, J., Peng, S-H. & Tysell, L.,”Effect of Edge-based Discretization Schemes in Computations of the DLR F6 Wing-Body Configuration”. Paper AIAA-2008-4153. ➢ Berglind, T., Numerical Simulation of Store Separation for Quasi-Steady Flow. FOI-R-2761-SE, FOI, Swedish Defense Research Agency, 2009. ➢ Berglind, T., Peng, S-H. & Tysell, L., FoT25: Studies of Embedded Weapons Bays - Summary Report. FOI-R-2775-SE, FOI, Swedish Defense Research Agency, 2009.

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1 Introduction The Black Box Dilemma1 Trust the Mesh Generated by the Software, or Take a Proactive Approach? Are you the type who likes to take a peek inside the black box to see how it works? Or are you one who’s willing to put your faith in the black box? Argued [Kenneth Wong] of Digital Engineering’s. The answer to that may offer clues to the type of meshing applications that appeal to you. But that’s not the only factor. Your own finite element analysis (FEA) skills also play a role. Most simulation programs aimed at design engineers offer fully or almost fully automated meshing. In other words, the software makes most or all of the mesh-related decisions required. Your part may be limited to selecting the desired resolution or the level of details fine meshing (high resolution, takes more time, but more accurate) or coarse meshing (low resolution, takes less time, but more approximations involved). There are good reasons to keep the meshing process hidden inside the black box, as it were. It takes a lot of experience and expertise (perhaps even a Ph.D.) to understand the difference between, say, a hexahedral mesh and a tetrahedral mesh; or tri elements and quad elements. It takes considerable simulation runs to know what type of meshing methods work well for a particular set of solid geometry. It requires yet another level of wisdom to know how to manually readjust the softwaregenerated meshes to more accurately account for the problematic curvatures, corners and joints in your geometry. These are beyond the scope of what most design engineers do. Therefore, many argue presenting a design engineer with a menu of these choices is counterproductive. On the other hand, expert users with a lot of analysis experience know the correlations between mesh types and accuracy, so they may want to get more involved in the meshing process. For this reason, high-end analysis software usually offers much more knobs and dials in the meshing process. Depriving expert users of these choices would force them to accept what they know to be unacceptable approximations. To navigate between the two different approaches, you need at least some understanding of how meshing works, automated or manual. Not All the Meshes Created Equal According to [Abdullah Karimi], CFD analyst for Southland Industries, uses fluid dynamics programs to examine airflow and heat distribution to develop the best residential heating solutions for his company’s clients. Via an online blog by Southland Industries, Karimi penned an article titled “How Not to Mesh Up: Six Mistakes to Avoid When Generating CFD Grids”. His first tip: Never use the first iteration of automatically generated mesh. “I’ve realized even some people with Ph.D.’s don’t have a good grasp on meshing,” he says. “People say, garbage in, garbage out. I say, good mesh equals good results. But the vast majority of the times I’ve seen the [software’s] automatically generated initial mesh is too coarse. The mesh may not even work, and if it does, the result may not be accurate.” If the automatically generated mesh significantly distorts the original geometry’s prominent characteristics—such as rounded corners, sharp angles and smooth curves it may be a sign that the mesh needs manual intervention in those specific regions. “You should at least take a look at the mesh. You can check to see if there are sudden size transitions, aspect ratio for skewness and triangular distortions. Just by visually inspecting the mesh, you can get a good idea if this may or may not work for your problem,” says Karimi. In his article, Karimi advises, “Don’t hit ‘Run’ without a mesh quality inspection. Depending on the robustness of the solution scheme, this could cause serious issues like straightaway divergence of the solution ... There are several quality metrics that need attention depending on mesh type and flow problem. Some of these metrics include skewness, aspect ratio, orthogonality [and] negative volume.” 1

Kenneth Wong is Digital Engineering’s resident blogger and senior editor.

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The Mesh Types With its designer friendly Altair Inspire (previously solidThinking Inspire) and expert-centric Altair HyperMesh software, Altair offers different approaches to meshing. “In Inspire, meshing is mostly hidden from the user,” explains Paul Eder, senior director of HyperWorks shell meshing, CAD and geometry at Altair. “The users choose to solve either in the first order [which prioritizes speed] or second order [which prioritizes accuracy].” By contrast, in HyperMesh, “We expose a lot more knobs and dials, because it’s for advanced users who understand the type of meshes they want to generate,” he adds. A similar strategy is seen in ANSYS software offerings. “Two of our products, ANSYS Forte and Discovery Live, provide a fully automated meshing experience,” says Bill Kulp, lead product marketing manager for Fluids at ANSYS. “ANSYS Discovery Live provides instantaneous 3D simulation so there is no time to make a mesh. On the other hand, our [general-purpose CFD package] ANSYS Fluent users need to solve a wide variety of fluid flow problems that can be most accurately approached by optimizing the mesh for the task at hand.” “Push-button automated meshing is our goal because we want to take this time-consuming job away from the engineers so they can concentrate on the innovation and optimization of their products,” adds [Andy Wade], lead application engineer at ANSYS. “Automated meshing will enable AI and digital twins to run simulations in the future and so this area is becoming the focus.” In theory, design engineers and simulation analysts could use different products, but in reality, some design engineers have sufficient expertise to make critical meshing decisions; and some analysis experts prefer the efficiency of automated or semi-automated meshing. So even with different products, satisfying both crowds is a difficult balancing act for vendors. Though the meshing process is mostly kept in the background in Altair Inspire, “If you’re an advanced user and want to see the meshes, you have the option to,” says Eder. “At the same time, we also offer automation in HyperMesh, because even some expert users want the same ease of use seen in Inspire.” “Tools such as ANSYS Discovery Live takes the meshing away completely from the user, whereas Discovery AIM features automatic physicsaware meshing, so the user can allow the product to do the hard work but if they want to see the

Figure 1.1

Meshes Created using ANSYS Mosaic-Enabled Poly-Hex Core Meshing - Courtesy of Sheffield Hallam University

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mesh and tweak it they can take control,” says Wade. Figure 1.1 shows meshes were created using ANSYS Mosaic-enabled Poly-Hex core meshing that automatically combines disparate meshes with polyhedral elements for fast, accurate flow resolution. ANSYS Fluent provides Mosaic-enabled meshing as part of a single-window, task-based workflow. Image courtesy of Sheffield Hallam University, Centre for Sports Engineering Research; and ANSYS. Regional Meshing The relatively new startup OnScale recently began offering on-demand multi-physics simulation from the browser. Some firms like Rescale offer high-performance computing (HPC) resources needed to run simulation, but not the software. By contrast, OnScale offers both the hardware and the multi-physics solver required to process jobs. “We offer automatic meshing as well as userdefined meshing. Users can define the level of fidelity desired,” explains Gerald Harvey, OnScale’s founder and VP of engineering. “OnScale gives you the ability to refine the grid and apply finer meshes in specific regions.” Not every corner, section or region in your geometry needs fine meshing. With simple geometry, a coarse mesh with fewer elements may suffice. But in certain regions where curvature, contact and joints create complex stress concentrations or flow patterns, a finer mesh (simply put, a higher number of meshes to cover the area) is warranted. Advanced simulation programs usually offer tools to specify how to treat these regions. Even in programs that target design engineers, some tools may be available to treat these regions differently.“ In Altair FEA products like SimLab, you can perform automatic local mesh refinement,” says Eder. “So you can run an analysis, review the results, then automatically refine the mesh in areas of high strain energy error density for subsequent runs. In [expert-targeted] HyperMesh, you also have many more manual mesh refinement options.” Simulation Cost OnScale’s Harvey suggests running a mesh study to understand the correlation between the stress effects and the mesh types and mesh density chosen. This can offer clues on how meshing affects the FEA results. “Every engineer should conduct a mesh convergence study test the meshes with some key performance indicators (KPIs) to find a happy medium,” says Harvey. “Suppose you’re looking at the design of a bracket. Then look at how the different meshes affect the bend angle of the bracket, for example.” Calculating simulation cost is complex, in part due to the mix of licensing policies in the market. But fundamentally, two parameters are involved: the time it takes and the hardware it uses. The need to find simplified meshes (as simple as possible without infringing on the accuracy of results) largely stems from the desire to keep these two parameters as low as possible. “If you have a simple solid part and you put 3D meshes on it, it takes more times than necessary to run,” notes Eder. In such a case, running simulation in a 2D cross-section of the geometry may be much more efficient. “And think of how many iterations you plan to run, because you’ll be paying that penalty for every single run,” he adds. ANSYS’ Wade points out that most solvers prefer hexahedral elements or quad surface mesh because “they fill the space very efficiently and using such elements when transient or explicit analysis is required can give massive gains in solve times (minimizing CPU effort for calculations). Hex elements can follow the flow direction better as well, which has some accuracy benefits. Tetrahedra, polys and other unstructured methods are very popular because they don’t require the decomposition (chopping up) of the space like a hex mesh; as a result, they are excellent for automation and really minimize manual effort.” Another tip from Karimi’s article: “Don’t fill the domain with a ridiculous number of tetrahedrons. So many times, I see meshing engineers filling up their CFD [computational fluid dynamics] domains [the target region for fluid analysis] with a large number of tetrahedrons and then struggling to get simulation results on time.” Certain programs are equipped to make the mesh selection easier. “With OnScale, you can conduct a study on a sweep of design, with mesh being

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one of the variables,” Harvey points out. “In OnScale, the run wouldn’t cost you significantly, because it would be a one-off cost. And the payback is well worth it.” Physics vs. Mesh Choosing the right kind of mesh, applying the right density to critical regions and selecting the right kind of coarseness or looseness affect the accuracy and speed of the simulation job. That is an exercise in tradeoffs, so there’s no black-and-white answer. “Meshing is always an exercise in tradeoffs in the quality of the mesh versus the speed of the solution,” says Karimi. “If you just want to see if a part will stand up to stresses and daily beating over time, and you’re not looking at the lowest level of details but at a high level of generality, then getting your physics correct is more important than the mesh,” says Eder. That means, at the general concept design level, the loads and boundary conditions—such as temperature, forces and direction of the forces may be more important than the type of meshes selected. Meshing Generalities A pre-processing step for the computational field simulation is the discretization of the domain of interest and is called mesh generation. The process of mesh generation can be broadly classified into two categories based on the topology of the elements that fill the domain. These two basic categories are known as structured and unstructured meshes. The different types of meshes have their advantages and disadvantages in terms of both solution accuracy and the complexity of the mesh generation process. A structured mesh is defined as a set of hexahedral elements with an implicit connectivity of the points in the mesh. The structured mesh generation for complex geometries is a time-consuming task due to the possible need of breaking the domain manually into several blocks depending on the nature of the geometry. An unstructured mesh is defined as a set of elements, commonly tetrahedrons, with an explicitly defined connectivity. The unstructured mesh generation process involves two basic steps: point creation and definition of connectivity between these points. Flexibility and automation make the unstructured mesh a favorable choice although solution accuracy may be relatively unfavorable compared to the structured mesh due to the presence of skewed elements in sensitive regions like boundary layers. In an attempt to combine the advantages of both structured and unstructured meshes, another approach in practice is hybrid mesh generation. In a hybrid mesh, the viscous region is filled with prismatic or hexahedral cells while the rest of the domain is filled with tetrahedral cells. It has been observed that a hybrid mesh in viscous regions creates a lesser number of elements than a completely unstructured mesh with a similar resolution. This type of mesh has no restrictions on the number of edges or faces on a cell, which makes it extremely flexible for topological adaptation. It is given that unstructured mesh has an advantage over the structured mesh in handling complex geometries, mesh adaptation using local refinement and de-refinements, moving mesh capability by locally repairing the bad quality elements, and load balancing using appropriate graph partitioning algorithms. In the case of a non-matched block-to-block boundary, interpolation issues have to be handled properly to satisfy the conservation principles. However, the structured mesh has a better accuracy for viscous calculations due to the fact that it can handle cells with very high aspect ratio

CAD Data

Surface Grid

Figure 1.2

Volume Grid

Optimization of Grid

Methodology of General Grid Generation

CFD

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cells in the boundary layer2. Precipitate of most grid generation procedure can be summarized as Figure 1.2 provided that everything goes according to plan.

Roy Koomullil, Bharat Soni, Rajkeshar Singh ,”A comprehensive generalized mesh system for CFD applications”, Mathematics and Computers in Simulation 78 (2008) 605–617. 2

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2 Computer-Aided Design (CAD) Computer-Aided Design (CAD) is the use of computer systems (or workstations) to aid in the creation, modification, analysis, or optimization of a design3. CAD software is used to increase the productivity of the designer, improve the quality of design, improve communications through documentation, and to create a database for manufacturing. CAD output is often in the form of electronic files for print, machining, or other manufacturing operations. The term CADD (for Computer Aided Design and Drafting) is also used4. CAD may be used to design curves and figures in two-dimensional (2D) space; or curves, surfaces, and solids in three-dimensional (3D) space. CAD is an important industrial art extensively used in many applications, including automotive, shipbuilding, and aerospace industries, industrial and architectural design, prosthetics, and many more. CAD is also widely used to produce computer animation for special effects in movies, advertising and technical manuals, often called DCC digital content creation. The modern ubiquity and power of computers means that even perfume bottles and shampoo dispensers are designed using techniques unheard of by engineers of the 1960s. Because of its enormous economic importance, CAD has been a major driving force for research in computational geometry, computer graphics (both hardware and software), and discrete differential geometry5. CAD is an important industrial art extensively used in many applications, including automotive, shipbuilding, and aerospace industries, industrial and architectural design, prosthetics, and many more. CAD is also widely used to produce computer animation for special effects in movies, advertising and technical manuals, often called DCC digital content creation. The modern ubiquity and power of computers means that even perfume bottles and shampoo dispensers are designed using techniques unheard of by engineers of the 1960s. Because of its enormous economic importance, CAD has been a major driving force for research in computational geometry, computer graphics (both hardware and software), and discrete differential geometry6.

Software and Technology Originally software for Computer-Aided Design systems was developed with computer languages such as Fortran, ALGOL but with the advancement of object-oriented programming methods this has radically changed. Typical modern parametric feature based modeler and freeform surface systems are built around a number of key C modules with their own APIs. A CAD system can be seen as built up from the interaction of a graphical user interface (GUI) with NURBS geometry or boundary representation (B-rep) data via a geometric modeling kernel. A geometry constraint engine may also be employed to manage the associative relationships between geometry, such as wireframe geometry in a sketch or components in an assembly. Unexpected capabilities of these associative relationships have led to a new form of prototyping called digital prototyping. In contrast to physical prototypes, which entail manufacturing time in the design. That said, CAD models can be generated by a computer after the physical prototype has been scanned using an industrial CT scanning machine. Depending on the nature of the business, digital or physical prototypes can be initially chosen according to specific needs. Today, CAD systems exist for all the major platforms (Windows, Linux, UNIX and Mac OS X); some packages support multiple platforms. CAD software enables engineers and architects to design, inspect and manage engineering projects within an integrated graphical user interface (GUI) on a Narayan, K. Lalit (2008). Computer Aided Design and Manufacturing. New Delhi: Prentice Hall of India. p. 3. Duggal, Vijay (2000). Cadd Primer: A General Guide to Computer Aided Design and Drafting-Cadd, Mailmax Pub. ISBN 978-0962916595. 5 Wikipedia. 6 Same Source. 3 4

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personal computer system. Most applications support solid modeling with boundary representation (B-Rep) and NURBS geometry, and enable the same to be published in a variety of formats. A geometric modeling kernel is a software component that provides solid modeling and surface modeling features to CAD management applications. Based on market statistics, commercial software from Autodesk, Dassault Systems, Siemens PLM Software and PTC dominate the CAD industry. Presently, most of commercially available CAD systems, such as SolidWorks, Autodesk or Siemens NX, calming to be able to do faster design loops, are also including a CFD analysis tool (some with limited capabilities), and Grid Generation kernel, in their product. (see Figure 2.1). For example, using SolidWorks, to solve the symmetric algebraic problem for pressure-correction, an original double preconditioned iterative procedure is used7. It is based on a specially-developed multigrid method from [Hackbusch (1985)]. This is an external flow around a F-16 fighter (Mach Number equals 0.6 and 0.85). The geometry is a native CAD model of the airplane with external tanks and armaments. Flow into the intake and exhaust from the engine’s nozzle are both taking into account. Calculations were performed with relatively coarse grid of approximately 200,000 cells. (see Figure 2.2) Calculation results are compared with the test data from [Nguyen, Luat T. et al.]. Commercially Available CAD Systems: The following is a list of major CAD applications. ➢ Alibre Design ➢ Autodesk AutoCAD ➢ Autodesk Inventor 7

CAD Management

Grid Generation

CFD

Figure 2.1

Anatomy of commercial CAD Systems

Figure 2.2

Fighter Airplane F-16 calculation

➢ Bentley Systems Micro Station ➢ Bricsys BricsCAD ➢ Dassault Systemes CATIA

Solidworks, “Numerical Basis of CAD-Embedded CFD”, White Paper.

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➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢

Dassault Systemes SolidWorks Kubotek KeyCreator Siemens NX Siemens Solid Edge PTC PTC Creo (formerly known as Pro/ENGINEER) Trimble SketchUp AgiliCity Modelur TurboCAD IRONCAD MEDUSA ProgeCAD

➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢

SpaceClaim PunchCAD Rhinoceros 3D VariCAD Vectorworks Cobalt Gravotech Type3 RoutCad SketchUp Onshape ActCAD Remo 3D

Freeware and Open Source ➢ ➢ ➢ ➢ ➢ ➢ ➢

123D LibreCAD FreeCAD BRL-CAD OpenSCAD QCad SolveSpace

Solid (Geometry) Modeling

A solid model is a computer model of a 3D solid. It is a virtual representation of the shape of a solid8. Solid models can be simple parts, or complex assemblies of multiple parts. We aim here at explaining how such solids can be described on a computer. We will principally focus on the ability of such solid models to serve as input to numerical simulations. Principal Characteristics of a Solid Modeling Software A solid modeling software may have some specific characteristics that enables to enhance both its efficiency and the productivity of the solid modeling process: Feature-Based Modeling Features are defined to be parametric shapes associated with attributes such as intrinsic geometric parameters (length, width, depth etc.), position and orientation, geometric tolerances, material properties, and references to other features. Feature-based modelers allow operations such as creating holes, fillets, chamfers, bosses, and pockets to be associated with specific edges and faces. When the edges or faces move because of a regeneration, the feature operation moves along with it, keeping the original relationships. Constraint-Based Modeling There are two types of constraints. Dimensional constraints are used to specify distances between items. Geometric constraints define positional relationships between entities in the model in terms of the geometry. Examples of geometric constraints include tangency, parallelism, symmetry, concentricity. Constraint-based modeling allows the engineer or designer to incorporate intelligence Christophe Geuzaine, Emilie Marchandise , and Jean-Francois Remacle, “An introduction to Geometrical Modelling and Mesh Generation”, The Gmsh Companion. 8

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into the design. The initial sketch of a two-dimensional profile in constraint-based solid modeling does not need to be created with a great deal of accuracy. It just needs to represent the basic geometry of the cross section. The exact size and shape of the profile is defined through assigning enough parameters to fully constrain it. Parametric Modeling Parametric modeling means that parameters of the model may be modified to change the geometry of the model. A dimension is a simple example of a parameter. When a dimension is changed, the geometry of the part is updated. Thus, the parameter drives the geometry. An additional feature of parametric modeling is that parameters can reference other parameters through relations or equations. The power of this approach is that when one dimension is modified, all linked dimensions are updated according to specified mathematical relations, instead of having to update all related dimensions individually. History-Based Modeling The last aspect of solid modeling is that the order in which parts are created is critical. This is known as history-based modeling. For example, a hole cannot be created before a solid volume of material in which the hole occurs has been modeled. If the solid volume is deleted, then the hole is deleted with it. This is known as a parent-child relation. The child (hole) cannot exist without the parent (solid volume) existing first. Parent-child relations are critical to maintaining design intent in a part. Most solid modeling software recognizes that if you delete a feature with a hole in it, you do not want the hole to remain floating around without being attached to the feature. Consequently, careful thought and planning of the base feature and initial additional features can have a significant effect on the ease of adding subsequent features and making modifications. Associative Modeling The associative character of solid modeling software causes modifications in one object to \ripple though" all associated objects. For instance, suppose that you change the diameter of a hole on the engineering drawing that was created based on your original solid model. The diameter of the hole will be automatically changed in the solid model of the part, too. In addition, the diameter of the hole will be updated on any assembly that includes that part. Similarly, changing the dimension in the part model will automatically result in updated values of that dimension in the drawing or assembly incorporating the part. This aspect of solid model software makes the modification of parts much easier and less prone to error. As a result of being feature based, constraint based, parametric, history based, and associative, modern solid modeling software captures \design intent", not just the design. This comes about because the solid modeling software incorporates engineering knowledge into the solid model with features, constraints, and relationships that preserve the intended geometric relationships in the model.

Constructive Solid Geometry (CSG) Representation of Solids We discuss here briefly the Constructive Solid Geometry (CSG) representation of solids. CSG allow to construct complex solid through primitives, Boolean operators and rigid motions. Basic Primitives The standard CSG basic primitives are the sphere, the torus, the parallelepiped (block), the cylinder and the cone. All those primitives defined bounded closed orientable domains. All basic primitives are defined in the world system of coordinates. Rigid motions (rotations, translations) and scaling can be applied to re-position the primitives.

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Regularized Boolean Operators Each primitive divides the 3D space into two parts: the one that is inside the primitive and the one that is outside. The closure of a primitive is the surface that separates its interior with its exterior. It is easy to think a primitive as a set where standard Boolean operations like union, intersection and difference can be defined. Basic primitives can be combined using Boolean operations. Three Boolean operators are defined. Consider two primitives A and B. ➢ The Union A ⋃ B operation returns of all the points x ∈ R3 that are either inside Figure 2.3 Example of a CSG Tree A or inside B. ➢ The Intersection A ⋂ B operation returns of all the points x ∈ R3 that are both inside A and inside B. ➢ The Difference A n B operation returns of all the points x ∈ R3 that are inside A and outside B. Regularized Boolean operators differ from the set-theoretic ones in that dangling lower dimensional structures are eliminated, all remaining faces, edges and vertices belonging to the closure of the resulting volume.

The CSG Tree A CSG object can be easily represented in a tree structure where the leaves of the tree are simple primitives, nodes of the tree are solids, edges of the tree are Boolean operations and where the root of the tree is a solid that is the final CSG object. Figure 2.3 shows an example of a simple CSG tree. Most of the current commercial solid modelers enable to use CSG trees. Designing robust algorithms for computing both the geometry and the topology of surface intersections is a complex problem. A few number of software enable to perform CSG computations efficiently and, to our best knowledge, only one is open source. In Gmsh, we have interfaced Open cascade primitives and operators to build the solid of Figure 2.3.

Geometry Related Issues For Mesh Generation One of the major issues of mesh generation is access to CAD geometry in an accurate and efficient manner, as addresses by [Beall et al.]9. Here, we will provide an overview the process of accessing 9

Mark W. Beall1, Joe Walsh2, Mark S. Shephard, “Accessing CAD Geometry For Mesh Generation”.

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CAD geometry for mesh generation and will review several of the issues associated with accessing CAD geometry for mesh generation. The techniques for CAD geometry access to be reviewed include: Translation & Healing, Discrete Representations, Direct Geometry Access, and Unified Topology Accessing Geometry Directly. The intent of this paper is to provide an overview to the alternative approaches and how they address the specific issues related to accessing CAD geometry for mesh generation. It is not the intent of this paper to provide detailed algorithms related to accessing or repairing CAD data. There are several issues associated with effective and efficient access of CAD geometry for mesh generation. This section will provide a quick overview of several of the major issues and the ramifications that this issues have on mesh generation Understanding the Analysis Requirements The first major issue with CAD geometry access for mesh generation is the need to understand the analysis requirements. The appropriate mesh and geometry to be used for meshing is a function of the analysis to be performed and the desired accuracy. There does not exist an optimal mesh independent of the analysis to be performed. A-prior element shape quality test have often been used as a misleading indicator of a good mesh independent of the analysis to be performed or the accuracy desired. The appropriate mesh is one that produces the desired accuracy for the problem to be solved. In practice this is only achievable through adaptively. Different types of analyses require different instances of the geometry to capture the physics. For example, we can perform a dynamic structural response analysis and a Computational Fluid Dynamics (CFD) analysis on the same part. The dynamic structural response analysis requires the solid geometry of the part while the CFD analysis requires the geometry of the cavities through which the fluid will flow. This simple illustration of different use of geometry representations is illustrated in Figure 2.4. Dynamic structural response analysis requires solid geometry of the part. While CFD analysis requires geometry of the flow cavities. Different types of analysis also require different resolutions of mesh to achieve the desired accuracy on a particular design.

Figure 2.4

Different Analysis Require Different Geometric Representations

Disfeaturing Disfeaturing is one of the most complex issues associated with CAD geometry access for mesh generation. Indeed one of the major issues that the CAD and CAE software industries have encountered is developing a consistent definition of a feature. For the purposes of this paper we will

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classify features into two main groups. The first group of features will be called “intended features”. Intended features are features that were explicitly defined as features in the model that drive the resulting geometry. In this case a feature-based modeling system was used to create a model which contains intended features. Intended features can only be created by feature-based modeling systems and can be suppressed by the original modeling system. The second group of features will be called “artifact features”. Artifact features are features that are created indirectly by the modeling process. One example of artifact features is the creation of engineering features such as holes by a modeling system that is not feature-based. The second example of artifact features is the creation of recognizable patterns of geometry / topology data that create a valid design model but also create difficulties associated with mesh generation. Artifact features can be created from any modeling system and cannot be suppressed in the original modeling system. Figure 2.5 illustrates small features removed from geometry. Part of the complexity associated with CAD geometry access for mesh generation is due to the fact that historically analyses are performed too late in the design process and the design model contains more details than are appropriate for analysis. Moving the analysis earlier in the design process will help to reduce, but will not remove, the need for defeating. Since multiple analysis types may be required for any design state there remains a need for defeating to various levels to support the range of analysis to be performed.

Figure 2.5

Small Feature (Left) vs Removed (Right)

“Dirty” Geometry Dirty geometry has been one of the most nagging issues related to geometry access. Dirty geometry consists of gaps, overlaps and other incompatibilities in the model preventing the model from being valid. These incompatibilities do not exist in the native CAD system and are introduced from translating the native CAD geometry to another format. Differences in representations, methods and tolerances between modeling engines create dirty geometry. Translators must then heal or repair

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the geometry to represent it as a valid model in the non-native system10-11-12. Note that without knowledge of the modeling system tolerances and methods, there is no a priori means to ensure a healing process will successfully recover the correct model representation.

Butlin, G., Stops C., “CAD Data Repair”, Proc. 5th Int. Meshing Roundtable, pp. 7-12, 1996. Mezentsev, A.A. and Woehler, T., “Methods and algorithms of automated CAD repair for incremental surface meshing”, Proc. 8th Int. Meshing Roundtable, Sandia report SAND 99-2288, pp. 299-309, 1999. 12 Ribo, R., Bugeda, G. and Onate, E., “Some algorithms to correct a geometry in order to create a finite element mesh”, Computers and Structures, 80:1399-1408, 2002. 10 11

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3 Structured Mesh Generation Mesh generation or Domain Discretization has evolved to the point where highly complicated domains can be covered by a variety of mesh types including hexahedral, tetrahedral and overset meshes. It is an important and very tedious aspect of computational geometry and accounts for almost 70% of CFD works. The concept of a mesh as a field or domain discretization of space has been associated with computational methods since the first attempts to obtain numerical solutions of partial differential equations13. Establishing a suitable mesh was long considered to be a rather tedious exercise and a minor part of the computational effort involved in solving partial differential equations by either a finite difference or finite element method. But mesh generation has steadily evolved into a discipline in its own right drawing on ideas from other fields, in particular mathematics and computer science, and gradually developing a distinct identity of its own. Two series of international conferences are now devoted entirely to mesh generation and adaptation, and almost all conferences on computational methods have sessions that feature this topic. In addition, it is important to recognize the growing interest of the computer science community in mesh related problems. In addition, it is important to recognize the growing interest of the computer science

Figure 3.1

13

Classification of Grid Generation Algorithms (Courtesy of Steven Owen)

Richardson LF. Weather prediction by numerical process. Cambridge: Cambridge University Press; 1921.

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community in mesh related problems14. Not only has this synergy brought new ideas and ways of viewing mesh related questions, it has also opened up whole new areas of application including medical imaging and segmentation, computer graphics and animation, and data interpolation and compression.15

Classification of Mesh Generation Techniques

As discussed before, the mesh generation techniques can be divided to two major categories of structured and un-structured mesh. Strictly speaking, a structured mesh can be recognized by all interior nodes of the mesh having an equal number of adjacent elements. For our purposes, the mesh generated by a structured grid generator is typically all quad or hexahedral. Algorithms employed generally involve complex iterative smoothing techniques that attempt to align elements with boundaries or physical domains. Where non-trivial boundaries are required, block structured techniques can be employed which allow the user to break the domain up into topological blocks16. Structured grid generators are most commonly used within the CFD field, where strict alignment of elements can be required by the analysis code or necessary to capture physical phenomenon17. Unstructured mesh generation, on the other hand, relaxes the node valence requirement, allowing any number of elements to meet at a single node. Triangle and Tetrahedral meshes are most commonly thought of when referring to unstructured meshing, although quadrilateral and hexahedral meshes can also be unstructured. While there is certainly some overlap between structured and unstructured mesh generation technologies, the main feature which distinguish the two fields are the unique iterative smoothing algorithms employed by structured grid generation. The semi-complete picture of grid generation algorithm is updated by [S. Owens ] and presented here as reference18 (see Figure 3.1). In general, on the structure side, some mapping techniques such as Transfinite Interpolation (TFI), or Elliptic operator are used extensively and proven to be sufficient for majority of applications. On unstructured side, the same could be said about Advancing Front or Delaunay triangulation. The above table is too broad and extensive for our purpose. Our concentration, as red circles indicate, would be on •

Structured Grid • Complex Variables (Restricted to 2D) • Algebraic Techniques (TFI) • PDE Methods (PDE)

•

Unstructured Grid • Delany Triangulation • Advancing Front • Octree Method • Hybrid Meshes • Overset Meshes • Cartesian Meshes Adaptive Grids • Structured • Unstructured

•

Edelsbrunner H. “Geometry and topology for mesh generation”, Cambridge: Cambridge university, 2001. Baker, T., “Mesh generation: Art or science?” MAE Department, Princeton University, Princeton, NJ. 16 Steven J. Owen, “A Survey of Unstructured Mesh Generation Technology”, Carnegie Mellon University, PA. 17 Introduction: An Initial Guide to CFD and to this Volume; page 1, 2007. 18 Steven Owen: Introduction to unstructured mesh generation, 2005. 14 15

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Domain Decomposition and Multi-Block Strategy One of the essential topics in grid generation is the topic of Domain Decomposition (DD). It is in essence a “divide and conquer” technique for arriving at the solution of problem defined over a domain from the solution of related problems posed on subdomains. The main reason is that the solution Iterate of the subdomain is qualitatively or quantitatively “easier” than Figure 3.2 Schwarz concept of iterating between domains the original one. Other factors are memory concern as well as that the subdomain can be solved with the aid of parallel programing. The issue of domain decomposition is vast and it involves a lot of math such as Schwarz concept19. He purposed that simply: ➢ Solve the PDE in the circle with boundaries taken from interior of square. ➢ Solve the PDE in the square with Boundaries taken from interior of circle. And then iterate as depicted in Figure 3.2. These days, with aid of strong work stations with visual aids, this is running on the background. The user does not know, or cared, what algorithm in running. Some of the vendors are opted for automatic DD schemes, or at least to begin with. User has options to change the topology later. But there is no free launch! There is usually a script which should be run prior to DD. An example would be GridPro® which is runes a TIL (Topology Input Language) script, written in C. The DD obtained using a TIL for an M6 wing is shown in Figure 3.3. Other venders have their own scripts or input data depending. Another example is Poitwise® which uses Glyph or newer Figure 3.3 Domain Decomposition for M6 wing using TIL Glyph2 as a scripting for the geometry. scripts (Courtesy of GridPro) There are generally two methods for generating the grid; Top to Bottom (TTB) and inversely Bottom to Top (BTT). While most of unstructured mesh engines use the TTB approaches, majority of structured ones are adapted to BTT. Some might think that multi-blocking approach is too tedious which of course is true. But the reward is in complete control of grid and its quality, something which is usually lacking in automated unstructured grid generates.

David E. Keyes, “Domain Decomposition Methods for Partial Differential Equations”, Department of Applied Physics & Applied Mathematics Columbia University. 19

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Field (Domain) Discretization Process (Mesh Generation) Once a mathematical model is selected, we can start with the major process of a simulation, namely the domain discretization process. Since the computer recognizes only numbers, we have to translate our geometrical and mathematical models into numbers which of course called discretization. The first action is to discretize the space, including the geometries and solid bodies present in the flow field or enclosing the flow domain. This set of points, which replaces the continuity of the real space by a finite number of isolated points in space, is called a grid or a mesh. The process of grid generation is in general extremely complex and requires dedicated software tools to help in defining grids that follow the solid surfaces (this is called ‘body-fitted’ grids) and have a minimum level of regularity. We wish already here to draw your attention to the fact that, when dealing with complex geometries, the grid generation process can be very delicate and time consuming. Grid generation is a major step in setting up a CFD analysis, since, as we will see the Figure 3.4 Example of Unstructured outcome of a CFD simulation and its accuracy can Tetrahedral Grids be extremely dependent on the grid properties and quality. Please notice here that the whole object of the simulation is for the computer to provide the numerical values of all the relevant flow variables, such as velocity, pressure, temperature, etc., at the positions of the mesh points. Hence, this first step of grid generation is essential and cannot be omitted. Without a grid, there is no possibility to start a CFD simulation. Figure 3.5 shows examples of 2D and 3D structured grids, while Figure 3.4 displays an example of mainly tetrahedral unstructured grids.

Figure 3.5

Examples of Structured grids for Turbine Blade

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Conformal Mapping (The Sponge Analogy) It is perhaps not surprising that conformal mapping was among the first and most effective techniques to carry out this task. The best way to the correspondence of a curvilinear grid in physical domain, with logically rectangle grid in computational domain, is through sponge analogy. Consider a rectangular sponge within which an equally spaced Cartesian grid has been drowned. Now wrapped the sponge around a circular cylinder and connect the two end of sponge together. Clearly the original Cartesian grid now becomes a curvilinear grid fitted the cylinder. But the rectangle logical form of grid lattice is still 20 preserved . Figure 3.6 spectacles a simply connected (as oppose to Figure 3.6 Sponge Analogy multiple connected) region which obviously results in O type grid. Since the difference formulae were applied in mapped space it was necessary to transform the partial differential equations to the coordinate system associated with the mapping. Conformal maps lead to a new set of fairly straightforward equations without messy cross-derivative terms. In addition, the orthogonality and smoothness properties of review of conformal mapping meshes obtained in this manner produce a high quality mesh in physical space. Perhaps the first published application of conformal mapping to Computational Fluid Dynamics (CFD) is circle plane mapping that transforms the space exterior to an airfoil onto the interior of the unit circle. This particular conformal mapping technique extends back a long way but its use for creating suitable meshes was a novel application. The same mapping was later used by [Bauer et al.]21 when they developed the first transonic flow code for solving the full potential equation. Other conformal mappings were developed to handle axisymmetric inlets and airfoil/slat combinations. A comprehensive techniques for mesh generation has been given by Moretti22. Another useful reference is the paper by23.

Domain Decomposition with Multi-Blocking (Multi-Sponge Analogy)

This problem was largely solved by the second significant development, the multi block strategy, or Domain Decomposition. The basic idea, first formulated is to break up the domain into several smaller blocks (essentially an ultra-coarse mesh) and then generate separate meshes in each individual block. Figure 3.7 illustrates this idea by showing a schematic of a three block decomposition for the region around a wing. In this example, one would use an H–H-mesh combination in blocks 1 and 3 and a C–H-mesh combination in block 2. A block corresponds to a sub domain that is geometrically much simpler than the full configuration and which can therefore be easily meshed either by solving a partial differential equation or, alternatively, by an algebraic Baker, T.,J., “Mesh generation: Art or science?”, MAE Department, Princeton University, Princeton, NJ. Bauer F, Garabedian P, Korn D. Supercritical wing sections I, Lecture Notes in Economics and Mathematical Systems, vol. 66. Berlin: Springer; 1972. 22 Moretti G.”Grid generation using classical techniques”. Proceedings of the NASA Langley workshop on numerical grid generation techniques, Langley, VA, October, 1980. 23 CaugheyDA, “A systematic procedure for generating useful conformal mappings”, Int J Num Meth Eng 1978.1. 20 21

37

method24. It is, in fact, common practice nowadays to create the mesh in any particular block by an algebraic method such as transfinite interpolation and then smooth the mesh by some iterations of an elliptic solver. A slightly more complicated topology of a dual Block for generic airplane configuration shown on Figure 3.10. An example showing a multi block conformal mapping for a M6 wing is illustrated in Figure 3.8 (a). Another example of multi-block structure gridding for a Turbine Blade is giving by Figure 3.9. GridPro© has developed a Topology Input Language (TIL) which can be used for similar geometries with minimal effort25. As an example, the topology and grid is putted for M6 wing, can be used for Reference H wing, and results are displayed in Figure 3.8 (b).

(a) M6 Wing

Figure 3.8

Figure 3.7

Multi Block representation for CH mesh around a wing

(b) Reference H

Topology and Grid on a Multi-Block Wings using GridPro®

Structured Grid Generation In general, decomposition of the physical domain produces several blocks. Each block is usually defined by six sides, and each side can be defined by either a surface, plane, line, or a point. If one side of a block collapses to a line or a point, then there would be a singularity in the block. In some instances, a block may have been defined by less than six surfaces. Once the surfaces are defined, the Eriksson LE,”Generation of boundary-conforming grids around wing-body configurations using transfinite interpolation”, AIAA J 1982; 20:1313–20. 25 An overview of Grid Pro/az3000 for automated grid generation. 24

38

interior grid can be computed by any standard grid generation technique. The cell information stored in a 3D array, in random fashion and could be easily access. Complex Variables Complex variables techniques have the advantage that the transformation used are analytic as opposed to those methods that are entirely numerical. Unfortunately, complex variable method are restricted to two dimension. For this reason, the technique has limited applicability and will not be covered here. For details readers should refer to Churchill26, Moretti, and Davis. Algebraic Methods -Transfinite Interpolation (TFI)

Figure 3.9 Multi-block gridding over Turbine blade - (Courtesy of GridPro)

Transfinite Interpolation has been used to generate the interior grid points from the boundary surfaces. In 2D (I, J), we may inscribe a linear Lagrange interpolation function as: N M ξ  η r( , ) =   n   r(ξ n , η) +  ψ m   r( , m ) − I J n =1 m =1 N M ξ η  n  ψ m   r(ξ n , ηm )  I J n =1 m =1

Eq. 3.1

Where now the "blending" functions, φn and ψm, are any functions which satisfy the cardinality conditions:

ξ  n  L  = δ nL n, L = 1,2,..., N  I 

η  and ψ m  L  = δ mL m, L = 1,2,...., M  J 

Eq. 3.2 The interpolation function defined by Eq. 3.2 can be thought of two unidirectional interpolation the corner points which has been duplicated. With N=M=2, using the Lagrange interpolation polynomials as the blending functions, is termed the transfinite bilinear interpolant. With N=M=3, this form is the transfinite bi-cubic-interpolation. Other candidates for the blending functions are the Exponential, Hermit Interpolation Polynomials and Splines. For example, for n, L = 2, Eq. 3.3 shows a typical Exponential blending function as K

1 ( ) =

e

ξ 2 −ξ ξ 2 − ξ1

−1 e −1 K

K

,

 2 ( ) =

e

ξ − ξ1 ξ 2 − ξ1

−1 e −1 K

Eq. 3.3 Where K is a negative constant greater than one. The greater the K, the less discontinuity will propagate. Similarly, a blending function could be constructed for η direction. The spline-blended 26

Churchill, R., V., “Introduction to Complex Variables”, McGraw-Hill, New York.

39

form gives the smoothest grid with continuous second derivatives27. A sample coding in FORTRAN is given in Appendix A and the resultant grid and topology for a dual-block generic airplane geometry is display in Figure 3.11. A pioneering work in control point form of Algebraic Grid Generation using a univariate interpolations can be attributed to [Eiseman and Smith]28.

Figure 3.10

Dual Block Grid Topology for a Generic Wing-Fuselage Configuration

PDE Smoother Like algebraic methods, differential equation methods are also used to generate grids. Grid construction can be done using all three classes of partial differential equations. The generation of field values of a function from boundary values can be done in various ways, e.g., by interpolation between the boundaries, etc., as is discussed previously. The solution of such a boundary-value problem, however, is a classic problem of partial differential equations, so that it is logical to take the coordinates to be solutions of a system of partial differential equations. If the coordinate points (and/or slopes) are specified on the entire closed boundary of the physical region, the equations must be elliptic, while if the specification is on only a portion of the boundary the equations would be

27 Joe

F. Thompson, Z. U. A. Warsi, C. Wayne Mastin, “Numerical Grid Generation -Foundations and Applications”, North Holland, 1985. 28 Peter Eiseman and Robert E. Smith, “Applications of Algebraic Grid Generation”, April 1990.

40

parabolic or hyperbolic. This latter case would occur, for instance, when an inner boundary of a physical region is specified, but a surrounding outer boundary is arbitrary. The present chapter, however, treats the general case of a completely specified boundary, which requires an elliptic partial differential system. 3.6.3.1 Elliptic Schemes At this stage grid is smooth enough to satisfy majority of applications, but if needed, further smoothing is obtained with solution of elliptic partial differential equations (PDE). For 2D formulation, forcing function terms are used to construct stretched layers of the cells close to the domain boundaries.

ξ xx + ξ yy = P( , η)

,

Figure 3.11 Grid for dual-block generic airplane geometry

ηxx + ηyy = Q( , η)

Eq. 3.4

Where (ξ, η) are the coordinates in the computational domain. Control functions are computed using the boundary point spacing, r, and then interpolated to the inner points29. The forcing terms (P, Q) are computed as:

P=−

r  2 r ξ ξ 2 r ξ

2

, Q=

r  2 r η η2 r η

Eq. 3.5

2

Once P and Q are obtained at each boundary the values for the inner points are obtained using a linear interpolating along lines of constant ξ and η:

P( , ) = (1 − η) P1 ( ) + ηP2 ( )

0  ξ 1

Q( , ) = (1 − ξ) Q1 ( ) + ξQ 2 ( )

Eq. 3.6

0  η 1

Grid control of orthogonality at boundaries is introduced adding a second term in P and Q as:

P=−

r  2 r ξ ξ 2 r ξ

2

−λ

r  2 r ξ η2 r η

2

,

Q=

r  2 r 2 η η r η

2

−λ

r  2 r η ξ 2 r ξ

2

Eq. 3.7

Thomas P., and Middlecoff J., ”Direct control of the Grid Point Distribution in Meshes Generated by Elliptic Equations”, AIAA Journal Vol. 18, No. 6., 1980. 29

41

Where 0 < λ < 1 is a factor that relaxes the orthogonality at the boundaries. It has been observed that the range λ∈ [0.4-0.7] produces optimal results for our configurations. The elliptic PDEs are solved using a multi-grid method and the smoother is based on a point-wise Newton solver. When the forcing terms are used the convergence of the algorithm deteriorates slightly30. An important property in regard to coordinate system generation is the inherent smoothness that prevails in the solutions of elliptic systems. Furthermore, boundary slope discontinuities are not propagated into the field. Finally, the smoothing Figure 3.12 Typical Elliptic Grid for an Airfoil with tendencies of elliptic operators, and the Orthogonality Enforced on the Boundary extremum principles, allow grids to be generated for any configurations without overlap of grid lines (see Figure 3.12). There are thus a number of advantages to using a system of elliptic partial differential equations as a means of coordinate system generation. A disadvantage, of course, is that a system of partial differential equations must be solved to generate the coordinate system. 3.6.3.1.1 Case Study – Orthogonal Elliptic Mesh Smoother This is a 2D orthogonal elliptic mesh generator which works by solving the Winslow PDE 31-32. It is capable of modifying the meshes with stretching functions and an orthogonality adjustment algorithm. This algorithm works by calculating curve slopes using a tilted parabola tangent line fitter (original discovery). A distinct feature of the elliptic mesh solver is that it corrects overlapping and misplaced grid line very well. Firstly to construct an initial mesh, the Transfinite Interpolation algorithm is applied to the given domain constrained by the specified boundary conditions. This algorithm is implemented by mapping each point within the domain (regardless of the boundaries) to a new domain existing within the boundaries. This algorithm works by iteratively solving the parametric vector equation. At the heart of the solver is the mesh smoothing algorithm, which at a high level, works by solving the pair of Laplace equations. Coordinates of every point in the target domain, mapped to a transformed, computational space using the change of variables method. This renders the calculations simpler and faster to compute. However, we wish to solve the inverse problem, where we transition from the computational space to the curvilinear solution space. Using tensor mathematics, it can be shown that this problem entails solving the equations. 3.6.3.1.2 Orthogonality Adjustment Algorithm In several computational fluid dynamics applications, an orthogonal mesh is necessary in certain regions to ensure a high enough accuracy when performing calculations. However, it is not always possible to achieve a fully orthogonal solution, and thus the problem becomes finding a nearlyorthogonal solution to an arbitrarily defined domain. The implemented solution uses an iterative approach to find the angles of intersection and adjust the position of the nodes until their respective Sorenson R. L. and Steger J. L. Numerical Generation of Two dimensional Grids by the Use of Poisson Equations with Grid Control, in Numerical Grid Generation Techniques, R. E Smith, ed.. NASA CP 2166, NASA Langley Research Center, Hampton, VA, USA, 1980. 31 Chaitanya Varier 2017. 32 M. Farrashkhalvat and J.P. Miles, “Basic Structured Grid Generation”, An imprint of Elsevier Science Linacre House, Jordan Hill, Oxford OX2 8DP, 200 Wheeler Rd, Burlington MA 01803, First published 2003. 30

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angles of intersection converge to a reasonable threshold value from 90 degrees. The exact method makes use of the linear approximation of the grid lines intersecting at each node within the mesh. (see Figure 3.13).

(a) Before

(b) After

Figure 3.13

Orthogonality Adjustments – (Courtesy of Chaitanya Varier)

3.6.3.1.3 Stretching Functions In order to further improve the quality of the mesh, one can introduce univariate stretching functions to either compress or expand grid lines in order to correct non-uniformity where grid lines are more or less dense. These functions are arbitrarily chosen and only reflect the distribution of grid lines. We can derive a new set of equations by combining our previously established differential model for grid generation and a set of univariate stretching functions of our choice. In order to do so in a straightforward manner, we can transform our Cartesian coordinates to a new set of coordinates which exists in a different space, called the parameter space. Then, we define our stretching functions as onto and one-to-one univariate functions of ξ and η respectively. For additional info, please consult the work by 33-34-35. 3.6.3.1.4 Extension to 3D If we wished to extend the elliptic solver to 3D, we would need to develop equations for transitioning Chaitanya Varier 2017. M. Farrashkhalvat and J.P. Miles, “Basic Structured Grid Generation”, An imprint of Elsevier Science Linacre House, Jordan Hill, Oxford OX2 8DP, 200 Wheeler Rd, Burlington MA 01803, First published 2003. 35 Joe F. Thompson, Z. U. A. Warsi, C. Wayne Mastin, “Numerical Grid Generation – Foundations and Application”, Mississippi State, Mississippi, January 1985. 33 34

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from a curvilinear coordinate system (ξ, η, ς) to a 3D Cartesian coordinate system (x, y, z). Using the same elliptic model as before, we get the 3D version of the Winslow equations where each of the metric tensor coefficients is determined by taking the cofactors of the contravariant tensor matrix. The contravariant tensor matrix is used to obtain the coefficients for the Winslow equations, which are the inverse of the Laplace equations as stated before. In general, if we wish to extend our elliptic mesh solver to n dimensions, then we will have n sets of equations each with n!/(2(n - 2)!) + n terms. This renders the problem gradually more and more difficult to solve for higher dimensions with the existing elliptic scheme, implying that a different type of PDE might be needed in these cases. Another complication that arises in higher dimensions is adjusting grid lines to enforce orthogonality. Using the aforementioned algorithm for adjusting grid lines to achieve either complete or partial orthogonality on the boundary, we would need to iteratively solve three sets of two linear equations for each node in the mesh, as well as solve three trigonometric equations per iteration to compute the tangents. 3.6.3.1.5 Mesh Quality Analysis In order to determine the quality of the resulting mesh, it was necessary to construct an objective means of quality measurement. Therefore, several statistical procedures were implemented in the program to produce a meaningful mesh quality analysis report. The metrics which are presented are divided into the following categories: ➢ Orthogonality Metrics • Standard deviation of angles • Mean angle • Maximum deviation from 90 degrees • Percentage of angles within x degrees from 90 degrees (x can be set as a constant in the code) ➢ Cell Quality Metrics • Average aspect ratio of all cells • Standard deviation of all aspect ratios 3.6.3.2 Hyperbolic Schemes This grid generation scheme is generally applicable to problems with open domains consistent with the type of PDE describing the physical problem. The advantage associated with Hyperbolic PDEs is that the governing equations need to be solved only once for generating grid. The initial point distribution along with the approximate boundary conditions forms the required input and the solution is the then marched outward. Steger and Sorenson36 proposed a volume orthogonality method that uses Hyperbolic PDEs for mesh generation. For a 2D problem, considering computational space to be given by Δξ = Δη =1, the inverse of the Jacobian is given by,

x ξ y η − x η yξ = I

Eq. 3.8

Where I represents the area in physical space for a given area in computational space. The second equation links the orthogonality of grid lines at the boundary in physical space which can be written as Steger, J.L; Sorenson, R.L (1980). "Use of hyperbolic partial differential equation to generate body fitted coordinates, Numerical Grid Generation Techniques". NASA conference publication 2166: 463–478. 36

44

dξ = 0 = ξ x dx + ξ y dy

Eq. 3.9

For ξ and η surfaces to be perpendicular the equation becomes:

x ξ y η + yξ y η = 0

Eq. 3.10

The problem associated with such system of equations is the specification of I. Poor selection of I may lead to shock and discontinuous propagation of this information throughout the mesh. While mesh being orthogonal is generated very rapidly which comes out as an advantage with this method. Figure 3.14 displays a C-O type hyperbolic grid around an HSCT wing-fuselage configuration, with Pressure contours mapped using an Euler solution and M∞ = 2.4. 3.6.3.3 Parabolic Schemes The solving technique is similar to that of hyperbolic PDEs by advancing the solution away from the initial data surface satisfying the boundary conditions at the end. Nakamura (1982) and Edwards (1985) developed the basic ideas for parabolic grid generation. The idea uses either of Laplace or the Poisson's equation and especially treating the parts which controls elliptic behavior. The initial values are given as the coordinates of the point along the surface η = 0 and the advancing the solutions to the outer surface of the object satisfying the boundary conditions along ξ edges. The control of the grid spacing has not been suggested till now. Nakamura and Edwards, grid control was accomplished using non uniform spacing. The parabolic grid generation shows an advantage over the hyperbolic grid generation that, no shocks or discontinuities occur and the grid is relatively smooth. The specifications of initial values and Figure 3.14 Euler Solution on a HSCT Wing-Fuselage selection of step size to control the grid points is however time consuming, but these techniques can be effective when familiarity and experience is gained. Variational Method These methods have evolved from elliptic grid generation. To solve an elliptic PDE is often equivalent to minimizing a functional. Variational methods has been used for improving quality of a given grid37. In the vibrational methods, a grid functional is defined. Grid functional is an algebraic expression of the position vectors of the internal nodes of a mesh. Optimization of the grid functional may result in a grid with desired properties such as orthogonal grid lines, equal cell areas, linear or parallelogram cells and untangled mesh. There are many algebraic functional for grid generation and optimization. For example, algebraic grid generation methods such as Transfinite Interpolations though, one of simplest method of grid generation, but can produce folded grids for curved domains as seen in the 37

J.F. Thompson, B.K. Soni and N.P. Weatherill. Handbook of Grid Generation. CRC Press, 1998.

45

Figure 3.15. One other disadvantage of algebraic grid generation is that boundary discontinuity can prorogate inside the domain. As indicated in Figure 3.15, Winslow functional smooth the grid, and removes the folded grid lines. There are far too many algebraic functional for grid generation and optimization as reader should check with38.

Figure 3.15

1.8

Folded Grid by Transfinite Interpolation - Smooth Grid by Winslow Functional

Structured Adaptive Grid

In an adaptive grid, the physics of the problem at hand must ultimately direct the grid points to distribute themselves so that a functional relationship on these points can represent the physical solution with sufficient accuracy39. The idea is to have the grid point’s move as the physical solution develops, concentrating in regions of large variation in the solution as they emerge. The mathematics controls the points by sensing the gradients in the evolving physical solution, evaluating the accuracy of the discrete representation of the solution, communicating the needs of the physics to the points, and finally by providing mutual communication among the points as they respond to the physics. The basic techniques involved then are as follows: • • • •

A means of distributing points over the field in an orderly fashion, so that neighbors may be easily identified and data can be stored and handled efficiently. A means of communication between points so that a smooth distribution is maintained as points shift their position. A means of representing continuous functions by discrete values on a collection of points with sufficient accuracy, and a means for evaluation of the error in this representation. A means for communicating the need for a redistribution of points in the light of the error evaluation, and a means of controlling this redistribution.

Several considerations are involved here, some of which are conflicting. The points must concentrate, and yet no region can be allowed to become devoid of points. The distribution also must retain a sufficient degree of smoothness, and the grid must not become too skewed, else the truncation error will be increased as noted. This means that points must not move independently, but rather each point must somehow be coupled at least to its neighbors. Also, the grid points must not move too far Sanjay Kumar Khattri, “Grid Generation and Adaptation by Functionals”, Department of Mathematics, University of Bergen, Norway. 39 Joe F. Thompson, J., F., Warsi, Z., U., A., Mastin, .W. “Numerical Grid Generation; Foundations and Applications”, North-Holland Book, 1995. 38

46

or too fast, else oscillations may occur. Finally the solution error, or other driving measure, must be sensed, and there must be a mechanism for translating this into motion of the grid. The need for a mutual influence among the points calls to mind either some elliptic system, thinking continuously, of some sort of attraction (repulsion) between points, thinking discretely. Both approaches have been taken with some success, and both are discussed below. It should be noted that the use of an adaptive grid may not necessarily increase the computer time, even though more computations are necessary, since convergence properties of the solution may be improved, and certainly fewer points will be required. With the time derivatives at fixed values of the physical coordinates transformed to time derivatives taken at fixed values of the curvilinear coordinates, no interpolation is required when the adaptive grid moves. Thus the first derivative transformation given the chain rule is given by

(

∂A ∂A ∂x =( ) + ∇A. ( ) ) ∂t ξ,η,ζ ∂t x,y,z ∂t ξ,η,ζ 3

∂A ∂A ∂A =( ) +∑ ( ) ∂t ξ,η,ζ ∂t x,y,z ∂xi i=1

or

∂xi ( ) ⏟∂t mesh movement

Eq. 3.11 The computation thus can be done on a fixed grid in the transformed space, without need of interpolation, even though the grid points are in motion in physical space. The influence of the motion of the grid points registered through the grid speeds, (xi) t, appearing in the transformed time derivative. This is the appropriate approach when the grid evolves with the solution at each Time step. Some methods, however, change the grid only at selected time steps, and here interpolation must be used to transfer the values from the old grid to the new since the grid movement is not continuous. A combination of the weight functions given by Eq. 3.11 provides the desired tendency toward concentration both in regions of high gradient and near extrema. The effect of the inclusion of the curvature illustrated below:

  A  w = (1 + β 2 K )1 + α 2     x  

2

1/2

   

where

K=

2A x 2   A  2  1 +     x    

Eq. 3.12

Figure 3.16

1D Weight Function for High Gradient and Curvature

3/2

47

Where α and β are parameters to be specified. Clearly, concentration near high gradients is emphasized by large values of α , while concentration near extrema (or other regions of large curvature) is emphasized by large β. Case Study – 2D Euler Flow Over an NACA Airfoil In a grid adaptation method for structured grids without adding or removing grid points, adaptation is achieved through moving the grid points toward the desired locations. Changes in the mesh point locations can be controlled by two methods.40 In the first method, the arc elements forming the ides of a control volume are directly related to specified functions. For a three-dimensional problem, this implies that three arc elements need to be given. In the second method, the cell volume may be altered by specifying that the volume of each element change according to a specific rule. To control the cell size, only one relationship must be specified that relates the volume to the quantity responsible for changes in the mesh. The specification of only one control function is an advantage in simplicity but may be less flexible than independently controlling arc lengths. The cell volume control method is applied successfully to calculating transonic Euler flows with shock waves. The method is applied to computing the flow field over an airfoil. Figure 3.17 shows the initial C-mesh of an NACA 0012 airfoil and the adaptive one on the right, with their respective Mach contours. Two flow cases

Figure 3.17

Mesh and Mach Contours for Transonic Flow

Feng Liu, Shanhong Ji, and Guojun Liao,” An Adaptive Grid Method and Its Application to Steady Euler Flow Calculations”, Siam J. Sci. Comput. C° 1998 Society For Industrial And Applied Mathematics Vol. 20, No. 3, Pp. 811{825 40

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were calculated over the initial grid. The first is a transonic case with free stream Mach number, M1 = 0.85 and an angle of attack, α = 1 where Left is without mesh adaption and Right is with. There is one strong shock wave on the upper side of the airfoil and a weaker one on the lower side. It can be seen from Figure 3.17 that the computed shock waves are rather thick. Shock waves zero thickness in the inviscid limit. To get better computational results, particularly to capture the shock waves more accurately, one would like to concentrate grid points around the shock waves. The deformation method is applied to get a new grid with prescribed distribution of cell sizes based on gradients of the flow field. The adaptive criterion here is to detect the shock waves. This suggests choosing the monitor function f of the form:

1 = C1 (1 + C2P) f

Eq. 3.13

Where P is the pressure and C1and C2 are constants. It can be seen that grid points are clustered closely in the areas where the two shock waves occur, although grid lines are somewhat skewed in the clustered regions because the deformation method does not guarantee orthogonality. However, since our flow solver is based on a finite volume scheme which does not require the use of an orthogonal grid, we are content with the locally reduced cell sizes. Another test case is the supersonic flow over the same airfoil with a free stream Mach number M∞ = 1.5, and α= 0. As can be seen, a strong bow shock wave appears in front of the airfoil leading edge. In addition, there are two weak shocks emanating from the trailing edge of the airfoil the Mach number distribution computed on the adapted grid. It can be seen that a sharper front of the bow shock is captured compared with that on the initial un-adapted grid. The resolution of the two trailing edge shocks is also slightly increased. The computational time needed for the grid adaptation and the flow solver for the supersonic case is the same as that for the transonic case. (See Figure 3.18).

Figure 3.18

Grid Adaption and Mack Contours for Supersonic Airfoil

49

50

4 Un-Structured Mesh Generation The 1980s witnessed the rapid development of alternative meshing techniques. The most prominent of these are the unstructured methods based on triangles in 2D and tetrahedral in 3D. Some of the earliest ideas for generating triangular meshes over planar regions can be found in a review by Thacker41. It is interesting to note that until the mid-1980s little effort had been applied to the problem of computing aerodynamic flow fields on meshes of triangles or tetrahedral. The first attempts to solve the flow around a complete aircraft by means of a finite element solution on a tetrahedral mesh are described in the papers of [Bristeau et al.] and [ Jameson et al.]. There are three essentially different approaches to generating triangular or tetrahedral meshes, the Moving (Advancing) front technique42, Delaunay base methods43,44,45 and the Octree approach46. Moving front and Delaunay based methods create a triangulation that matches a prescribed boundary (a specified set of points and edges in 2D, a specified set of points and triangular faces in 3D). The Octree method, however, determines the boundary discretization as part of the domain meshing procedure and is thus closely related to the Cartesian approach. Each method is discussed below.

Advancing Front Method This technique has the virtue of starting from a prescribed boundary definition (set of edges in 2D, set of triangular faces in 3D) which remains intact throughout the mesh generation process. The boundary triangulation is regarded as a front on which a new layer of elements is built. The original front triangles become interior faces of the mesh and a new set of front faces is created, a process that continues until the entire domain has been filled. A particular difficulty of this method occurs in the closing stages when the front is collapsing on itself and the last vestiges of empty space are replaced by new elements. This is Figure 4.1 Closing stage of a Moving Front Method illustrated in Figure 4.1 which demonstrations an almost completed triangulation of the region around an airfoil with the edges on the current front marked in bold. In Thacker WC. A brief review of techniques for generating irregular computational grids. Intentional Journal Numerical Meth Eng. 1980; 15:1335–41. 42 Lo SH. A new mesh generation scheme for arbitrary planar domains. Int J Numer Meth Eng 1985;21:1403–2 43 Baker TJ. Three dimensional mesh generation by triangulation of arbitrary point sets, AIAA 8th CFD conference. Honolulu, HI. AIAA paper 87-1124, 1987. 44 Weatherill NP. , “A method for generating irregular computational grids in multiply connected planar domains”, International Journal Number Meth Fluids 1988; 8:181–97. 45 George PL, Hecht F, Saltel E. Constraint of the boundary and automatic mesh generation. Proceedings of the second international conference on num grid gen comp fluid dyn, Miami, FL, 1988. p. 589–97. 46 YerryMA, Shephard MS., “Automatic three-dimensional mesh generation by the modified Octree technique”, International J Numerical Meth Engineering 1984; 20:1965–90. 41

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practice, there is rarely any difficulty in completing the process for a planar triangulation. In three dimensions, however, the remaining region of space can have an extremely complicated shape which may not yield to an acceptable covering by tetrahedral elements, thus preventing the volume triangulation from filling the entire region to be meshed. The basic methodology is based on action performed for a certain boundary image as described by geometric rules or tests. These rules (2D and 3D) are to optimize the shape of the new element in the advancing front method. Each methodology depends on these rule, its complexity, and how they been applied. Therefore, the algorithm has to check the rules stored in data structures. The code complexity is independent of the number of rules. The algorithm is complicated, but well defined and can be, at least theoretically, implemented failsafe. Especially in 3D, the choice of the concrete rules is based on heuristics, which is put into an easily maintainable rule description data-base47. Advancing Front Triangular Mesh Generator The original advancing front algorithm has been developed over time into a family of programs which are very reliable and flexible for an easy incorporation of mesh adaptation48. The advancing front mesh generator can be described as . ➢ ➢ ➢ ➢ ➢ ➢

Input of geometric data (using control points); Input of mesh control parameters (through a background mesh); Geometric modeling (using cubic splines); Boundary discretization (placing new points on the boundary); Domain discretization (simultaneously generating points and triangles); Mesh quality enhancement (through topological and geometrical strategies).

The computational domain is modeled through the use of cubic splines which are defined by some control points. Close to singularities extra care must be taken in the definition of these points in order to avoid failure (Thompson et al., 1999). As a “pre-processing” stage, before the mesh generation begins, we must first build an initial and very coarse triangular background mesh that covers the whole domain. This coarser mesh is used only to provide a piecewise linear spatial distribution of the nodal parameters over the mesh to be constructed. Typically, elements of the generated mesh will have a projected length of δ2 in the direction parallel to α2 a and a projected length of St δ2 in the direction normal to α2 a (see Figure 4.2), with St being the stretching factor. During the generation process, the local values of these parameters will be obtained by a linear interpolation over the triangles of the background mesh. Figure 4.2 Mesh parameters Joachim Sch¨ober, “NETGEN An advancing front 2D/3D-mesh generator based on abstract rules”, Computing and Visualization in Science, 1:41–52 (1997). 48 Paulo Roberto M. Lyra, Darlan Karlo E. de Carvalho, “A Computational Methodology for Automatic TwoDimensional Anisotropic Mesh Generation and Adaptation”, Methodology for Automatic Two-Dimensional Anisotropic Mesh Generation and Adaptation. 47

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The boundary of the domain is represented by the union of boundary segments forming closed loops. External boundaries are defined in an anti-clockwise fashion while inner boundaries are set in a clockwise manner. As described previously, the generation of a triangular mesh by the advancing front technique begins by the discretization of the boundary of the domain. New points are created according to the mesh parameters which are interpolated from those of the background mesh. At the beginning of the process, the generation front is made by a set of linear segments connecting the boundary nodes. With the initial front defined, one segment is chosen and, in general, a triangle is created through the insertion of an internal node or by simply connecting existing nodes. New triangles are built following the same procedure. During the process any segment available to build a new triangle is set as “active” and the others which are set as “non-active” are removed from the generation front. Therefore the boundary segments are not modified during the mesh generation. The procedure continues until the whole domain is discretized. When solving problems which develop some essentially one dimensional features at certain regions (e.g. boundary layer, shocks, etc.) it is not very efficient to use uniform isotropic meshes. In these cases, it is important to have the possibility to define a direction and a stretching factor for the elements close to such regions. At least for linear triangular elements, the use of anisotropic meshes can be extremely important in terms of computational effort and accuracy. To generate an anisotropic triangulation of the desired domain, it is used a transformation T which is a function of the mesh parameters, i.e. αi , i = 1, 2. This transformation49, is given by, N

1 T(α i , δi ) =  (α i  α i ) i =1 δ i

Eq. 4.1

where X denotes the tensor product of two vectors and N is the number of dimensions, here, N = 2. The effect of this transformation is to map the physical domain into a normalized domain, where a mesh is generated in which the elements are approximately equilateral with unit average size. Applying the inverse of this transformation T-1, we end up with a directional stretched mesh dictated by the mesh parameters, which are defined either by the analyst or by the mesh adaptive procedure. This mesh generator provides an accurate geometric modeling and high quality meshes, where the high level of control of the distribution of local mesh parameters eases the incorporation of mesh adaptation strategies. The quality of the meshes is strongly influenced by the mesh optimization stage. A specific mesh improvement strategy for highly anisotropic meshes and the definition of an adequate sequence of mesh enhancement procedures are incorporated into the code. Several other modifications have been introduced in the original code in order to incorporate the flexibility to deal with predefined multi-domains Figure 4.3 Surface Mesh of SGI Logo and automatically defined sub-regions, to build boundary layer meshes, to make possible Peiró, J., Peraire, J. and Morgan, K., 1994, “{FELISA SYSTEM}: Reference Manual Part1 - Basic Theory”, University of Wales Swansea Report CR/821/94. 49

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generating quadrilateral and mixed meshes and the automatic definition of which domains or subregions should be filled up by triangular or by quadrilateral elements. These features will be fully described in the correspondent sections. Figure 4.3 shows an triangulation of SGI logo by [Rypl and Bittnar].

Advancing Front Quadrilateral Meshing Using Triangle Transformations Quad-morphing is a new technique used for generating quadrilaterals from an existing triangle mesh. Beginning with an initial triangulation, triangles are systematically transformed and combined. An advancing front method is used to determine the order of transformations. An all-quadrilateral mesh containing elements aligned with the area boundaries with few irregular internal nodes can be generated. [see Owen et al.]50. Outline of Quad-Morphing Algorithm Quad-morphing is briefly outlined in the following steps: 4.2.1.1 Initial Triangle Mesh The surface is first triangulated. This may be done using any surface triangulation method. Any sizing (Owen,1997) or adaptively information should be built into the initial triangulation. The local sizing for the final quadrilateral mesh will roughly follow that of the triangle mesh. 4.2.1.2 Front Definition The initial front is defined from the initial triangle mesh. Any edge in the triangulation that is adjacent to only one triangle becomes part of the initial front. 4.2.1.3 Front Edge Classification Each edge in the front is initially sorted according to its state. The state of a front edge defines how the edge will eventually be used in forming a quadrilateral. Angles between adjacent front edges determine the state of an individual front. Front edges will be updated and reshuffled as the algorithm proceeds. Figure 4.4 shows the four possible states of a front, where the front edge is indicated by the bold line.

Figure 4.4

States of a front edge – (Courtesy of Owen et al.)

4.2.1.4 Front Edge Processing Each front edge is individually processed to create a new quadrilateral from the triangles in the initial mesh. Figure 4.5 (a) shows front NA-NB in the triangulation ready to be processed. Front edges are handled differently according to their current state classification. As quadrilaterals are formed, the front is redefined and adjacent front edge states are updated. The current front always defines the Steven J. Owen, Matthew. Staten, Scott A. Canann and Sunil Saigal, “Advancing Front Quadrilateral Meshing Using Triangle Transformations”, Conference Paper · January 1998. 50

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interface between quadrilateral elements in the final mesh and triangle elements in the initial triangle mesh. This process can be further subdivided into the following sub-steps: ➢ Check for Special Cases. Before proceeding to construct a quadrilateral from the current front, several special case scenarios are checked. These include situations where large transitions or small angles exist local to the front. In these cases a seam, or transition seam operation is performed. ➢ Side Edge Definition. Using the front edge as the initial base edge of the quadrilateral, side edges are defined. Side edges may be defined by using an existing edge in the initial triangle mesh, by swapping the diagonal of adjacent triangles, or by splitting triangles to create a new edge. In Figure 4.5 (b), side edge NB-NC shows the use of an existing edge, while the side edge NA-ND was formed from a local swap operation. ➢ Top Edge Recovery. The final edge on the quadrilateral is created by an edge recovery process. During this process, the local triangulation is modified by using local edge swaps to enforce an edge between the two nodes at the ends of the two side edges. Edge NC-ND in Figure 4.5 (c) was formed from a single swap operation. Any number of swaps may be required to form the top edge. ➢ Quadrilateral Formation. Merging any triangles bounded by the front edge and the newly created side edges and top edge as shown in Figure 4.5 (d) forms the final quadrilateral. ➢ Local Smoothing. The mesh is smoothed locally to improve both quadrilateral and triangle element quality as shown in Figure 4.5 (e). ➢ Local Front Reclassification. The front is advanced by removing edges from the front that have two quadrilateral adjacencies and adding edges to the front that have one triangle and one quadrilateral adjacency. New front edges are classified by state. Existing fronts that may have been adjusted in the smoothing process are reclassified.

Figure 4.5

Steps demonstrating process of generating a quadrilateral from Front NA-NB - (Courtesy of Owen et al.)

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Front edge processing continues until all edges on the front have been depleted, in which case an all quadrilateral mesh will remain, assuming an even number of initial front edges. When an odd number of boundary intervals is provided, a single triangle must be generated, usually towards the interior of the mesh. 4.2.1.5 Topological Clean-up and Final Smoothing Process Element quality is improved by performing local quadrilateral transformations in an attempt to improve the individual edge valences at the nodes of the mesh. A final smoothing pass is performed further improving the element qualities. The final smoothing step involves a limited number of iterations of a constrained Laplacian smoothing algorithm. Each node is moved to the centroid of its neighbors only if an improvement in element shape metric (Lee,1994) would result. In situations where Laplacian smoothing produces poor results, an optimization based smoothing (Canann,1998) operation may be performed. 4.2.1.6 Example Problems The first example, shown in Figure 4.6, demonstrates the progression of the Q-Morph algorithm on a simple planar domain with two holes. Figure 4.6 (a) shows the initial triangle mesh before QMorph begins. In this case an advancing front triangle meshed (Canann,1997) was used to create the triangles. The method used for triangulation is unimportant, inasmuch as the appropriate nodal density is provided. Figure 4.6 (b)-(g) show the progression of the algorithm as each successive layer of elements is completed. Figure 4.6 (c) shows an additional layer of small elements meshed on the internal circle loop before meshing the larger elements of the outer loop. To improve element

Figure 4.6

Progression of Q-Morph- (Courtesy of Owen et al.)

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Figure 4.8

Figure 4.7

Results of Q-Morph Compared with Lee’s (1994) Advancing Front Indirect Method on Toroidal Surface- (Courtesy of Owen et al.)

Comparison of Q-Morph with Lee’s Algorithm Illustrating Element Boundary Alignment - (Courtesy of Owen et al.)

transitions, provision is made in Q-Morph to mesh loops with smaller elements before those with

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larger elements. The mesh is completed in Figure 4.6 (h) after a final pass of cleanup and smoothing. Figure 4.8 and compares Q-Morph against Lee’s (1994) quad meshing algorithm, which uses an indirect method, coupled with an advancing front scheme to combine triangles into quadrilaterals. The toroidal surface of Figure 4.8 is composed of four surface patches represented as rational BSplines. Q Morph utilizes projection and geometric evaluation routines as part of the local and final smoothing procedures to maintain nodal locations on the three-dimensional surface. Both Figure 4.8 (a) and (b) were generated using the same initial triangle mesh as well as the same cleanup and smoothing procedures. Despite using an advancing front scheme, Lee’s algorithm shown in Figure 4.8 (b), has difficulty maintaining well-aligned rows of elements introducing many irregular internal nodes.

Figure 4.9

Large Transition Mesh for CFD Application - (Courtesy of Owen et al.)

Figure 4.7 further illustrates the ability of the Q-Morph algorithm to generate well-aligned rows of elements parallel to a complex domain boundary, while still maintaining the required element size transitions. Figure 4.9 demonstrates the use of Q-Morph with a planar surface requiring a high degree of transition. Figure 4.9 (a) shows the partially completed quad mesh with two layers of quads placed. Figure 4.9 (b) shows the same area after final cleanup and smoothing. In order to maintain a specified nodal density near the top of the area, a sizing function (Owen,1997) was used during the triangle meshing process. The algorithm’s ability to maintain the desired mesh density while still enforcing well-aligned rows of elements transitioning quickly to larger size elements is demonstrated in this example. For further and complete analysis, please consult the work by [Owen

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et al.]51. 4.2.1.7 Conclusion The Q-Morph algorithm is an indirect quadrilateral meshing algorithm that utilizes an advancing front approach to transform triangles into quadrilaterals. It generates an all-quadrilateral mesh, provided the number of intervals on the boundary is even. The resulting mesh has few irregular internal nodes and produces elements whose contours, in general, follow the boundary of the domain. Overall element quality is excellent. The Q-Morph algorithm borrows many of its techniques from the paving method (Blacker,1991; Cass,1996) but adapts them for use as an indirect method, operating on an existing set of triangles. In so doing, it is able to improve upon the paving technique by resolving some of its inherent difficulties. The intersection problem, common to most direct methods of advancing front meshing, is eliminated by relying on the topology of the initial triangle mesh to close opposing fronts. Improvements also include facility for handling individual element placement through the use of states for classifying front edges. Facility for handling transition in element sizes has also been addressed through the use of sizing information provided by the initial triangle mesh and the definition of specific transformations that enable improved mesh transitions. Additionally, the initial triangle mesh provides information that reduces the cost of direct evaluations on three dimensional surface geometry.

Delaney Triangulation Method Delaunay triangulation for a given set of discrete points in a plane is a triangulation such that no point in is inside the circumcircle of any triangle in. Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after [Boris Delaunay]52 for his work on this topic. For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors. By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean distance. However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique53. The Delaunay triangulation of a discrete point set points in general position corresponds to the dual graph of the Voronoi diagram for the same points, as revealed in Figure 4.11. Special cases include the existence of three points on a line and four points on circle. Properties of Delaunay Triangulation54 Definition 1. The Delaunay triangulation (DT) of a finite set of points S in R3, denoted as D(S), is a triangulation with a special property that no point of S lies in the interior of the circumsphere of any tetrahedron of D(S). The special property of the Delaunay triangulation is called empty circle property in R2 and empty sphere property in R3. This definition of Delaunay triangulation can be generalized to any higher dimension. Definition 2. A simplex s of the Delaunay triangulation D(S) is said to be Delaunay if there exists an empty circumsphere of s. Steven J. Owen, Matthew. Staten, Scott A. Canann and Sunil Saigal, “Advancing Front Quadrilateral Meshing Using Triangle Transformations”, Conference Paper · January 1998. 52 Delaunay, Boris (1934). "Sur la sphère vide". Bulletin de l'Académie des Sciences de l'URSS, Classe des sciences mathématiques et naturelles. 6: 793–800. 53 From Wikipedia, the free encyclopedia. 54 Ashwin Nanjappa, “Delaunay Triangulation In R3 on The Gpu”, A Thesis Submitted For The Degree of Doctor of Philosophy, Department of Computer Science, National University of Singapore, 2012. 51

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From the definition of circumsphere of a triangulation, it follows that every k-simplex of D(S) has an empty circumsphere. If k = d, then the circumsphere of s is unique, else s has infinitely many circumspheres. 4.3.1.1 Delaunay Lemma There is an alternate local property to the empty sphere property that is related to the Delaunay triangulation. Definition 3. A facet abc ∈ T(S) is said to be locally Delaunay if • •

It belongs to only one tetrahedron and therefore belongs to the boundary of the convex hull, or It belongs to two tetrahedral abcd and abce, and e lies on the exterior of the circumsphere of abcd.

The second test is called the in sphere test and its result is the same no matter if abcd is tested with e or if abce is tested with d. (See Figure 4.10). Lemma 1. (Delaunay Lemma) If every facet of a triangulation T is locally Delaunay, then T is the Delaunay triangulation of S. A face that is locally Delaunay is no guarantee Figure 4.10 Success and failure of the in sphere test of abcd with e. that it belongs to the Delaunay triangulation. However, if a triangulation T consists of only locally Delaunay faces then T = D. 4.3.1.2 Compactness In R2, the Delaunay triangulation maximizes the minimum angle in the triangulation and minimizes the largest circumcircle. This max-min angle optimality was discovered by Lawson. These properties of the Delaunay triangulation in R2 do not generalize to three and higher dimensions. A useful property of the Delaunay triangulation that holds in all dimensions, including three, is the containment radius. In R3, the containment radius is defined as the radius of the smallest sphere containing the tetrahedron. This is called the min-containment sphere and note that this need not necessarily be the circumsphere of the tetrahedron. [Rajan] showed that the Delaunay triangulation in R3 minimizes the containment radius of its tetrahedral. This makes it the most compact triangulation in R3. Algorithms Many algorithms for computing Delaunay triangulations rely on fast operations for detecting when a point is within a triangle's circumcircle and an efficient data structure for storing triangles and edges. In two dimensions, one way to detect if point D lies in the circumcircle of A, B, C is to evaluate the determinant:

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Ax Bx Cx Dx The Delaunay triangulation with all the circumcircles and their centers (in red)

Figure 4.11

Ay By Cy Dy

A 2x + A 2y B2x + B2y C 2x + C 2y D 2x + D 2y

1 1 0 1 1

Eq. 4.2

Connecting the centers of the circumcircles produces the Voronoi diagram (in red)

Relationship Between Delaunay Triangles and the Voronoi Diagram

As shown in Figure 4.13 when A, B and C are sorted in a counterclockwise order, this determinant is positive if and only if D lies inside the circumcircle. The majority of Delaunay based methods exploit an incremental algorithm that starts with an initial triangulation of just a few points. The complete triangulation is generated by introducing points and locally reconstructing the triangulation after each point insertion. A particularly attractive feature of this approach is the opportunity to place new points at specified locations with the aim of retaining, or possibly improving, the quality of the mesh55. The main difficulty is the need to ensure surface integrity. Most methods allow the boundary points to be inserted into the volume triangulation unchecked, reestablishing the surface edges and faces by a series of edge/face swaps and the occasional introduction of an extra point. The left hand side of Figure Figure 4.12 Two-Three Tetrahedral swap 4.12 illustrates a simplified complex formed by two tetrahedral which share a common face. If this face is removed and an edge is inserted connecting the vertices A and B one obtains three tetrahedral (shown on the right hand side of Figure 4.12 which occupy the same 55

Bowyer A. Computing Dirichlet tessellations. Computer J 1981; 24(2):162–6.

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region of space as the two original tetrahedral. This so-called “2 to 3” swap can often be used to establish a boundary edge; the reverse operation can similarly be applied to establish a boundary face. Not all boundary edges and faces can be established by this one operation but other more complicated swapping operations are possible. When the boundary triangulation has been established within the initial volume mesh, additional points are then inserted into the triangulation in order to create a volume mesh of wellshaped tetrahedral. (see Figure 4.13 and Figure 4.10). A detailed description of this process is given in the books by [George and Borouchaki] and by [Frey and George]56. Advantages Figure 4.13 Robust and Fast way to The important advantage of triangulation techniques is the Detect if point D lies in the higher degree of automation that is achieved in the meshing Circumcircle of A, B, C process. It can be shown, for example, that a Delaunay mesh can be generated to conform to any prescribed boundary in 2D57. The situation in 3D is much more complicated and no similar mathematical guarantee exists. The method has, nevertheless, been brought to a high level of automation and current tetrahedral mesh generators will reliably create good quality isotropic meshes if they are provided with a good quality surface triangulation58-59. Delaunay triangulation is a concept that extends back well before the emergence of mesh generation60. Together with its geometric dual, the Voronoı diagram, it has proved to be a fertile

Figure 4.14

Delaunay Triangulation (white) and Voronoi Diagram (blue) – Courtesy of [Labbe])

Baker T.J., “Triangulations, mesh generation and point placement strategies”, Caughey DA, Hafez MM, editors. “Frontiers of computational fluid dynamics”, New York: Wiley, 1994, pp. 101–15. 57 Lee DT, Lin AK. Generalized Delaunay triangulation for planar graphs. Discrete Comput Geom 1986;1: 201– 58 George PL, Borouchaki H. Delaunay triangulation and meshing. Hermes; 1998. 59 Baker TJ, Vassberg JC. Tetrahedral mesh generation and optimization. 6th Iinternational conference on numerical grid generation. ISGG; 1998. p. 337–49. 60 DelaunayB. Sur la sphe`re vide, Izvestia Akademia Nauk SSSR, VII Seria. Otdelenie Matematicheskii Estestvennyka Nauk 1934; 7:793–800. 56

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construct whose applications extend from cartography to crystallography. In the seventies it attracted the attention of computer scientists and quickly became an important topic within the then emerging discipline that is now known as computational geometry61. In the early nineties computer scientists rediscovered mesh generation as an application of Delaunay triangulation although computer graphics and animation was, and still remains, the main justification for their research into triangulation problems and Delaunay triangulation. Delaunay Adaptive Refinement In early isotropic meshing techniques, Delaunay triangulations were constructed from existing point sets, generated through other means. Incorporating the Delaunay structure of the point set in the generation of the point set was pioneered in 2D by [Chew]62and [Ruppert]63. They proposed to iteratively insert the circumcenters of triangles that do not fit a given set of criteria, such as the size or shape of elements. This technique allowed to generate meshes with lower bounds on the smallest angle of triangles and was then extended to 3D domains. In 3D, although no guarantee can be made on the dihedral angles of simplexes, the radius-edge ratio of tetrahedral can be shown to be bounded. An efficient algorithm to insert a vertex in an isotropic Delaunay triangulation, now known as the Bowyer-Watson algorithm, was proposed simultaneously and independently by [Bowyer]64 and [Watson]65. The simplexes whose Delaunay ball contains the simplex are collected and removed from the triangulation, which forms a cavity. Linking the refinement point to the vertices of the border of the cavity creates the new Delaunay simplexes of the triangulation. In the context of anisotropy, the generation of points was also originally done independently from the construction of the triangulation, using for example anisotropic quad trees. [Mavriplis]66-67 first considered the idea of stretched Delaunay methods and using nodes generated from an anisotropic advancing front technique; the connectivity is set by first constructing a large isotropic mesh and then inserting vertices with the Bowyer-Watson algorithm adapted to the anisotropic setting. The stretching of the space is obtained by computing gradients of the solution. Good results were achieved, but the swapping techniques employed do not extend nicely to higher-dimensional settings. [Borouchaki et al.]68 formalized the approach of stretching spaces of [Mavriplis] through the use of Riemannian metric tensors and introduced the anisotropic Delaunay kernel, their anisotropic version of an anisotropic Bowyer Watson algorithm. Along with this new insertion algorithm, they introduced a Delaunay refinement algorithm based on edge swapping, merging and splitting techniques to generate meshes whose edges lengths are close to 1 in the metric at each of their endpoints. Many developments have sprouted from this approach: 3D mesh generation, periodic anisotropic mesh generation, metric-orthogonal mesh generation . While these algorithms produce good results and have seen much use in the context of computational fluid dynamics, theoretical results are limited for these algorithms and there are no guarantees on either the termination or the robustness of algorithms, nor on the quality of the elements produced by these techniques. A theoretically sound approach to anisotropic Delaunay triangulations was

Preparata FP, Shamos MI. Computational geometry. Berlin: Springer; 1985. Chew, L. P. Constrained Delaunay triangulations. (1989). 63 Ruppert, J. A new and simple algorithm for quality 2-dimensional mesh generation. SODA (1993). 64 Bowyer, A. Computing dirichlet tessellations. The Computer Journal 24, 2 (1981). 65 Watson, D. F. Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. The computer journal 24, 2 (1981), 167–172. 66 Mavriplis, D. J. Adaptive mesh generation for viscous flows using triangulation. Journal of computational Physics 90, 2 (1990), 271–291. 67 Mavriplis, D. J. Unstructured mesh generation and adaptivity. Tech. rep., DTIC Document, 1995. 68 Borouchaki, H., George, P. L., Hecht, F., Laug, P., and Saltel, E. Delaunay mesh generation governed by metric specifications. part I algorithms. Finite Elem. Anal. Des. 25, 1-2 (1997), 61–83. 61 62

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proposed by [Boissonnat et al.]69, who introduced the framework of locally uniform anisotropic meshes. In their algorithm, the star of each vertex v is composed of simplices that are Delaunay for the metric at v. Each star is built independently and the stars are stitched together in the hope of creating an anisotropic mesh. The star structure was first introduced by [Shewchuk]70 to handle moving vertices in finite element meshes and considering stretched stars was first proposed by [Schoen]. Two stretched stars may be combinatorically incompatible, a configuration called an inconsistency. [Boissonnat et al.] proved that inconsistencies can be resolved by inserting Steiner points, yielding an anisotropic triangulation. The algorithm works in any dimension, can handle complex geometries and provides guarantees on the quality of the simplices of the triangulation. Voronoi Diagrams The well-known duality between the Euclidean Voronoi diagram and its associated Delaunay triangulation has inspired authors to compute anisotropic Voronoi diagrams, with the hope of obtaining a dual anisotropic triangulation. The approaches of[ Labelle and Shewchuk]71 and [Du and Wang]72 aim at approximating the geodesic distance between a seed and a point of the domain by considering that the metric is constant and equal to the metric at the seed (in the case of [Labelle and Shewchuk]73) or at the point (in the case of [Du and Wang ]). Contrary to the isotropic setting, the dual of an anisotropic Voronoi diagram is not necessarily a triangulation and inverted elements can be present in the dual triangulation. The algorithms were initially introduced for two-dimensional (Labelle and Shewchuk) and surface (Du and Wang) domains and have since then been studied and extended by various authors. The approach of Labelle and Shewchuk was shown to be theoretically sound in 2D, but the approach of the proof does not extend to higher dimensions. This result was extended to surfaces by [Cheng et al.]74 by locally approximating the surface with a plane and then using a density argument similar to the proof of [Canas and Gortler]. Centroid Voronoi tessellations, which are Voronoi diagrams for which the seeds are the centers of mass of their associated Voronoi cell, are known to create elements of good quality. The famous Lloyd algorithm iteratively moves the seeds to the center of mass of their respective cell and recomputed the Voronoi diagram of this new seed set. This algorithm was modified to be used in the anisotropic Voronoi diagram of Du and Wang, but the process is computationally expensive. Restricted Delaunay Triangulation The Delaunay and Voronoi structures presented so far are built from (almost) arbitrary point sets living in Rn. It is possible to employ these structures to approximate bounded domains. The restriction of a Delaunay complex to a domain is the subcomplex (the restricted Delaunay complex) composed of the simplices of whose dual Voronoi face intersect. Restricted Delaunay triangulations were introduced by [Chew] [53] and allow to accurately capture complex geometric objects. For example, it can be shown that under the condition of good sampling of a surface, the restricted Delaunay triangulation and the domain are homeomorphic [9]. Thanks to these good properties, 69 Boissonnat, J.-D., Wormser,

C., and Yvinec, M. Locally uniform anisotropic meshing. Proceedings of the twentyfourth annual symposium on Computational geometry (2008), ACM, pp. 270–277. 70 Shewchuk, R. Star splaying: an algorithm for repairing Delaunay triangulations and convex hulls. Proceedings of the 21st annual symposium on Computational geometry (New York, NY, USA, 2005. 71 Labelle, F., and Shewchuk, J. R. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. Proceedings of the 19th annual symposium on Computational geometry (New York, NY, USA, 2003, 72 Du, Q., and Wang, D. Anisotropic centroidal Voronoi tessellations and their applications. SIAM Journal on Scientific Computing, 2005. 73 Labelle, F., and Shewchuk, J. R. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. Proceedings of the 19th annual symposium on Computational geometry (New York, NY, USA, 2003). 74 Cheng, S.-W., Dey, T. K., Ramos, E. A., and Wenger, R. Anisotropic surface meshing. Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm (2006).

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restricted Delaunay triangulations have often been used to create provably correct refinement algorithms in the case of surfaces. It was however proven by [Boissonnat, Guibas and Oudot]75 that this does not extended to higher-dimensional settings. Nevertheless, restricted Delaunay triangulations will be consistently used in the refinement algorithms considered. Figure 4.15 represents 2D Delaunay triangulation of a set of vertices (black) restricted to a curve (blue). Voronoi edges are represented in teal, and in pink if they intersect the curve. The Voronoi vertices are marked with orange circles. Restricted Delaunay edges are drawn in yellow, and restricted Delaunay triangles are drawn in green.

Figure 4.15

2D Delaunay triangulation of a set of vertices (black) restricted to a curve (blue)

Anisotropic Mesh Generation Anisotropic meshes are desirable for various applications, such as the numerical solving of partial differential equations and graphics. From an equivalent point of view, the use of meshes whose elements are stretched according to the anisotropy of the phenomenon requires a lower number of elements to achieve the same precision of the result of the simulation. When stretched elements are used, the mesh is said to be anisotropic. Additionally to providing increased accuracy in the simulations of scientific modeling, anisotropic meshes also find use in geometric modeling as they can improve the visualization of objects, and lower the number of vertices required to represent a shape or interpolate a smooth function. By requiring fewer elements, anisotropic meshes thus provide another way to accelerate mesh generation, and increase the quality and the speed of computations. Where the main improvements in the accuracy of modeling only came from increased computation power and denser meshes for isotropic meshes, anisotropic mesh generation offers another independent way to obtain faster and more accurate results. While anisotropic meshes offer many benefits, their generation is also much harder than the traditional isotropic meshes. Due to its wide range of applications, several classes of methods have been proposed, yet no solution is satisfying for all classes of domains and anisotropy. Moreover, while isotropic meshes and their generation are now well studied from a theoretical point of a view, almost all the algorithms on

Boissonnat, J.-D., Guibas, L. J., and Oudot, S. Y. Manifold reconstruction in arbitrary dimensions using witness complexes. Discrete & Comp. Geom. 42 (2009), 37–70. 75

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anisotropic mesh generation are heuristic. With this observation in mind, we seek to develop methods that are both provable, robust, and practical [Labbe]76. Case Study - Anisotropic Mesh Generation via Discretized Riemannian Delaunay Triangulations Due to its wide array of practical applications, anisotropic mesh generation has received considerable attention and several classes of methods have been proposed. In this study, the generation of anisotropic meshes using the concepts of Delaunay triangulations and Voronoi diagrams, has been investigated by [Labbe]77. First, consider the framework of locally uniform anisotropic meshes introduced by [Boissonnat, et al.]78. Despite known theoretical guarantees, the practicality of this approach has only been hardly studied. An exhaustive empirical study is presented and reveals the strengths but also the overall impracticality of the method. The ideal shape of simplex has so far been described as the regular simplex, but this is not always the case. We follow closely the development by [Labbe]79-80. The category for discretization of these are: 1 2 3

Algorithms based on the concepts of the Delaunay triangulation and the Voronoi diagram, Algorithms based on an embedding of the input domain to simplify the problem, Algorithms based on the optimization of particles.

The different approaches that we consider here are all based upon extending the notions of Voronoi diagrams and Delaunay triangulations to the anisotropic setting. We hope to benefit from the known results and theoretical soundness of the isotropic Delaunay triangulation and Voronoi diagram to generate anisotropic meshes with provable and practical meshing techniques. As all our methods are based upon the same structures, we dedicate a chapter to introducing the notions that will be used throughout this thesis. Our main chapters follow a logical progression, with each method taking more metric information into account to determine the connectivity and placement of points than the previous ones. We begin with a thorough practical investigation of the framework of locally uniform anisotropic meshes, a theoretically sound meshing technique proposed by [Boissonnat et al.]81 that is based on the idea of constructing at each point a triangulation that is well adapted to the local metric. The theoretical aspect of their approach has been extensively described, but its practicality is comparatively lesser known. We detail our implementation, which is both more robust and faster than the one previously presented in the short experiment investigation of the algorithm for surfaces, investigate the role of the numerous parameters, and give some results. Limitations of the approach are then exposed, along with our attempts to address those. In the Euclidean setting, the Delaunay triangulation of a point set can be constructed by first generating the Voronoi diagram of the point set and then computing the dual of this diagram. Anisotropic Voronoi diagrams have been considered to build anisotropic triangulations, however the dual of an anisotropic Voronoi diagram is not necessarily a valid triangulation and elements can be inverted. Different distances are possible to create such anisotropic Voronoi diagrams.

Mael Rouxel-Labbe, “Anisotropic mesh generation”, Université Côte d’Azur, 2016. Mael Rouxel-Labbe, “Anisotropic mesh generation”, Université Côte d’Azur, 2016. 78 Boissonnat, J.-D., Chazal, F., and Yvinec, M. Geometry and Topology Inference. in preparation. 79 Mael Rouxel-Labbe, “Anisotropic mesh generation”, Université Côte d’Azur, 2016. 80 M. Rouxel-Labbéa, M. Wintraeckenb, J.D. Boissonnatb, “Discretized Riemannian Delaunay triangulations”, 25th International Meshing Roundtable (IMR25). 81 Boissonnat, J.-D., Dyer, R., and Ghosh, A. Delaunay stability via perturbations. Int. J. Comp. Geom.& App .(2014). 76 77

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Along with the introduction of their anisotropic distance, [Labelle and Shewchuk]82 presented a refinement algorithm that generates a point set for which their anisotropic Voronoi diagram has a valid dual triangulation. However, their method is limited to the setting of planar domains. We give requirements on point sets such that the dual of the anisotropic Voronoi diagram is a nice triangulation and propose a refinement algorithm to generate such point set. Our proof links anisotropic Voronoi diagrams built using the distance of [Labelle and Shewchuk]83 with the framework of locally uniform anisotropic meshes, relating along the way the concepts of quasicosphericity (used in locally uniform anisotropic meshes) and of protection of a point set. The second part of this chapter introduces a refinement algorithm based on the combination of the alternative point of view of the anisotropic Voronoi diagram as the restriction of a high-dimensional power diagram to a fixed paraboloid manifold and the tangential Delaunay complex, a structure used in manifold reconstruction that is well-adapted to the high-dimensional setting. We detail our implementation, study its theoretical grounds and investigate the practicality of the algorithm. Anisotropic Voronoi diagrams studied by previous authors compute and compare distances using a fixed metric, justifying this approximation by invoking the computational and time complexity of computing geodesics in a domain endowed with a metric field. To better facilitate, Table 4.1 is the list of symbols used. Ω P G F λi , vi φ(G1,G2) g0 g dG ‖.‖G dE dg Vord(P) Del(P) DelG(P) Delg0(P) Sp Svp Svp

Domain Point set Metric Square root of a metric Eigenvalues and eigenvectors of a metric Distortion between two metrics G1 and G2 Uniform metric field Arbitrary metric field Distance with respect to the metric G Norm with respect to the metric G Distance with respect to the Euclidean metric E Geodesic distance with respect to the metric field g Voronoi diagram of P using the distance d Abstract Delaunay complex of the point set P Delaunay complex of the point set P with respect to G Delaunay complex of the point set P with respect to g0 (uniform metric field) Star of p Restricted volume star of p Restricted surface star of p Table 4.1

Nomenclature

Labelle, F., and Shewchuk, J. R. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. Proceedings of the nineteenth annual symposium on Computational geometry (New York, NY, USA, 2003), ACM Press, pp. 191–200. 83 see Previous. 82

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4.4.1.1 Anisotropic Delaunay Triangulations 4.4.1.1.1 Locally Uniform Anisotropic Meshes The Delaunay triangulation has been extensively studied and is well known to possess useful and well-defined properties [Preparata and Shamos]84. Attempts have been made to extend the notion of Delaunay triangulation to the anisotropic setting either by adapting the famous Bowyer-Watson algorithm [Mavriplis] , [Borouchaki], [Dobrzynski], [Alauzet] or as the dual of an anisotropic Voronoi diagram (see Labelle and Shewchuk85, [Du and Wang]86. However, neither class of method offer in all dimensions the theoretical guarantees nor the practical robustness that we are interested in. A theoretically sound framework for Delaunay-based anisotropic meshes was introduced by [Boissonnat et al.], called locally uniform anisotropic meshes. Locally uniform anisotropic meshes are simplicial complexes in which the star of each vertex is Delaunay for the metric attached to the vertex. The use of anisotropic stars, inspired by the works of [Shewchuk] and [Schoen], is combined with the sliver removal techniques proposed by [Li and Teng ]. These techniques are adapted to the anisotropic setting to construct a star-based refinement algorithm that cleverly selects refinement points. The algorithm works in any dimension and offers guarantees on its termination and on the quality (size and shape) of the final simplexes. While the theoretical aspect of the approach has been thoroughly studied, its practicality is comparatively not as well explored. This chapter intends to fill this gap and presents a comprehensive empirical study of the algorithm. We first recall the required theoretical basis and detail some minor changes brought to the theory after practical experimentation. We then dive in the heuristic analysis of the behavior of the algorithm and its parameters and propose some improvements. A strong focus is put on the phenomenon of inconsistencies, which turns out to be a real issue of the algorithm. We introduce an implementation of the anisotropic Delaunay refinement algorithm introduced by [Boissonnat et al.]. We specifically detail the technical issues, and their solution, that arise when aiming for efficiency. This implementation of the star set is completely new and results in significant improvements in terms of computational speed, robustness and genericity over the former implementation that was used in. The empirical study of our implementation and of the algorithm is comprehensive and produce (minor) theoretical improvements. We investigate the practicality of the algorithm, analyze its limitations and propose various fixes that attempt to remedy these limitations. 4.4.1.1.2 Metric Tensor The adaptation of meshes based on a metric tensor was introduced by [Borouchaki et al.]87-88-89 and has imposed itself as the standard way to describe and prescribe anisotropy. A metric tensor, or simply a metric, in Rn is defined by a symmetric positive definite (SPD) quadratic form, represented by a n x n matrix G. [Labbe]90. The quadratic form defines as an inner product as Eq. 4.3

〈u, v〉G = ut , Gv = 〈u, Gv〉

Preparata, F. P., and Shamos, M. Computational geometry: an introduction. Springer Science & Business 2012. Labelle, F., and Shewchuk, J. R. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. Proceedings of the nineteenth annual symposium on Computational geometry, NY, 2003. 86 Du, Q., and Wang, D. Anisotropic centroid Voronoi tessellations and their applications. SIAM Journal on Scientific Computing 26, 3 (2005), 737–761. 87 Borouchaki, H., George, P. L., Hecht, F., Laug, P., and Saltel, E. Delaunay mesh generation governed by metric specifications. part I algorithms. Finite Elem. Anal. Des. 25, 1-2 (1997), 61–83. 88 Diaz, M. C., Hecht, F., Mohammadi, B., and Pironneau, O. Anisotropic unstructured mesh adaptation for flows simulations. Internat. J. Numerical Methods Fluids 25 (1997), 475–491. 89 Hecht, F., and Mohammadi, B. Mesh adaption by metric control for multi-scale phenomena and turbulence. AIAA 35th Aerospace Sciences Meeting & Exhibit (1997). 90 Mael Rouxel-Labbe, “Anisotropic mesh generation”, Université Côte d’Azur, 2016. 84 85

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and the norm of a vector u with respect to G is then given by

‖u‖𝐺 = √〈u, u〉 = √𝑢𝑡 𝐺𝑢

Eq. 4.4 Using this norm, the distance between two points p and q in Ω can be measured in the metric G as

dG (x, y) = ‖x − y‖G = √(x − y)t G(x − y)

Eq. 4.5 One recognizes the equation of an ellipsoid with semi-axis lengths 1/√λi. Since O is an orthonormal matrix, it can also be seen as a rotation matrix and the unit sphere in the metric is thus an ellipsoid in Rn whose axes directions are given by the eigenvectors {vi} of G and whose semi-axis lengths are equal to 1/√λi in the direction vi, where λi is the eigenvalue corresponding to vi. Incidentally, this construction proves that there is a natural bijection between ellipsoids and metrics. (see Figure 4.16).

Figure 4.16

Representation of a 3D Metric with Eigenvalues λ1, λ2 and λ3 as an Ellipsoid – (Courtesy of [Labbe])

4.4.1.1.3 Distortion The notions of metrics and metric fields are introduced to convey the stretching of spaces. To create a solid theoretical framework, it is required to be able to express how differently two metrics see distances and geometrical objects. For this purpose, [Labelle and Shewchuk] introduced the concept of distortion between two metrics. We recall here their definition. Properties and limitations as well as a new alternative that remedy those shortcomings are proposed by [Labelle and Shewchuk] introduced the concept of distortion between two points p and q of Ω as

φ(p, q) = φ(Gp , Gq ) = Max{‖Fp 𝐹𝑞−1 ‖, ‖Fq 𝐹𝑝−1 ‖}

Eq. 4.6 where ∥·∥ is the Euclidean matrix norm, Observe that φ(Gp,Gq) ≥ 0 and φ(Gp,Gq) = 1 when Gp = Gq.

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4.4.1.1.4 Locally Uniform Anisotropic Meshes The Delaunay triangulation has been extensively studied and is well known to possess useful and well-defined properties [Preparata & Shamos]91. While the theoretical aspect of the approach has been thoroughly studied, its practicality is comparatively not as well explored. We first recall the required theoretical basis and detail some minor changes brought to the theory after practical experimentation. We then dive in the heuristic analysis of the behavior of the algorithm and its parameters and propose some improvements. We introduce an implementation of the anisotropic Delaunay refinement algorithm introduced by [Boissonnat et al.]92. This implementation of the star set is completely new and results in significant improvements in terms of computational speed, robustness and genericity over the former implementation that was used. The empirical study of our implementation and of the algorithm is comprehensive and produce (minor) theoretical improvements. We investigate the practicality of the algorithm, analyze its limitations and propose various fixes that attempt to remedy these limitations. Attempts have been made to extend the notion of Delaunay triangulation to the anisotropic setting either by adapting the famous BowyerWatson algorithm [Mavriplis]93, [Borouchaki]94, [Dobrzynski]95, [Alauzet]96or as the dual of an anisotropic Voronoi diagram [Labelle and Shewchuk]97, [Du and Wang]98. However, neither class of method offer in all dimensions the theoretical guarantees nor the practical robustness that we are interested in. A theoretically sound framework for Delaunay-based anisotropic meshes was introduced by [Boissonnat et al.]99, called locally uniform anisotropic meshes. Locally uniform anisotropic meshes are simplicial complexes in which the star of each vertex is Delaunay for the metric attached to the vertex. The use of anisotropic stars, inspired by the works of [Shewchuk]100 and [Schoen]101, is combined with the sliver removal techniques proposed by [Li and Teng]102-103. These techniques are adapted to the anisotropic setting to construct a star-based refinement algorithm that cleverly selects refinement points. The algorithm works in any dimension and offers guarantees on its termination and on the quality (size and shape) of the final simplexes.

91 Preparata, F. P., and Shamos, M. Computational geometry: an introduction.

Springer Science & Business, 2012. Locally uniform anisotropic meshing. Proceedings of the twentyfourth annual symposium on Computational geometry (2008), ACM, pp. 270–277. 93 Mavriplis, D. J. Adaptive mesh generation for viscous flows using triangulation. Journal of computational Physics 90, 2 (1990), 271–291. 94 Borouchaki, H., George, P. L., Hecht, F., Laug, P., and Saltel, E. Delaunay mesh generation governed by metric specifications. part Ialgorithms. Finite Elem. Anal. Des. 25, 1-2 (1997), 61–83. 95 Dobrzynski, C., and Frey, P. Anisotropic Delaunay mesh adaptation for unsteady simulations. Proceedings of the 17th International Meshing Roundtable (2008), 177–194. 96 Alauzet, F., and Loseille, A. A decade of progress on anisotropic mesh adaptation for computational fluid dynamics. Computer-Aided Design 72 (2016), 13–39. 97 Labelle, F., and Shewchuk, J. R. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. Proceedings of the 19th annual symposium on Computational geometry (New York, NY, USA, 2003), ACM Press, pp. 191–200. 98 Du, Q., and Wang, D. Anisotropic centroid Voronoi tessellations and their applications. SIAM Journal on Scientific Computing 26, 3 (2005), 737–761. 99 Boissonnat, J.-D., Wormser, C., and Yvinec, M. Locally uniform anisotropic meshing. Proceedings of the twentyfourth annual symposium on Computational geometry (2008), ACM, pp. 270–277. 100 Shewchuk, R. Star splaying: an algorithm for repairing Delaunay triangulations and convex hulls. Proceedings of the twenty-first annual symposium on Computational geometry (New York, NY, USA, 2005). 101 Schoen, J. Robust, guaranteed-quality anisotropic mesh generation. M.S. thesis, UC at Berkeley, 2008. 102 Li, X.-Y. Sliver-free Three Dimensional Delaunay Mesh Generation. PhD thesis, University of Illinois at Urbana Champaign, PhD thesis, University of Illinois at Urbana-Champaign 2000. 103 Li, X.-Y., and Teng, S.-H. Generating well-shaped Delaunay meshed in 3d. Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms (2001). 92 Boissonnat, J.-D., Wormser, C., and Yvinec, M.

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4.4.1.1.5 The Star Set Many algorithms have been devised to construct Euclidean Delaunay triangulations, but they cannot be simply extended to arbitrary metric fields. Although the computation of a curved Riemannian Delaunay triangulation is difficult, it is easy to compute the Delaunay triangulation of a set of points P with respect to the metric Gp (Delp(P)). Any metric-dependent geometric construction on a set of points P, like the Voronoi diagram or a simplex circumcircle, can therefore be obtained for the metric Gp by the following set of operations. First, compute the transformed point set. Then, compute the construction with the Euclidean norm on and transform the result back through . The triangulation is thus simply the image through the stretching transformation F−1p of the Euclidean Delaunay triangulation Del(Fp(P)) where Fp(P) = {Fppi, pi ∈ P}. As explained before, a sphere in metric space is an ellipsoid in the Euclidean space. Each simplex of a uniformly anisotropic Delaunay triangulation Delp(P) thus possesses an empty circumscribing ellipsoid, the inverse-transformed Delaunay ball from metric to Euclidean space (see Figure 4.17).

Figure 4.17

An anisotropic uniform Delaunay triangulation (orange) and the corresponding stretched

Delaunay balls and circumcenters (black circles) - (Courtesy of [Labbe]) The central idea of the framework of locally uniform anisotropic meshes is to approximate at each vertex p a given arbitrary metric field g by the uniform metric defined by extending Gp over the domain. Similarly to the way affine functions are locally good approximations of a generic continuous function, the approximation of an arbitrary metric field by a uniform metric will be accurate as long we stay in a small neighborhood. At each vertex, a Delaunay triangulation that conforms to the uniform metric field of that vertex is constructed. These independent triangulations can under some density conditions be combined to obtain a final triangulation of the domain. 4.4.1.1.6 Stars and Inconsistencies The star of a vertex p in a simplicial complex K, denoted by Sp, is defined as the sub-complex of K formed by the set of simplexes that are incident to p. The idea of considering independent stars at the vertices of a point set was first conceived by [Shewchuk] [128] to handle moving vertices in finite element meshes. This structure was also employed by [Schoen] [126], who introduced anisotropic stars whose connectivity is obtained by building an isotropic Delaunay mesh of a transformed point set. The star Sp of p ∈ P is in that case extracted from the complex Delp(P). This construction was described in the previous section and forms the core of the locally uniform anisotropic mesh framework. The collection of all the stars is called the (anisotropic) star set of P and is noted S(P). As the connectivity of each star is set according to the metric Gp at the center of the star, a given n-

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simplex has n + 1 different Delaunay balls, one with respect to the metric of each vertex of the simplex. Consequently, there are in general inconsistencies among the stars of the sites: a simplex τ appearing in the stars of some of its vertices, may not appear in the stars of all of them (Figure 4.18). If a simplex is involved in such configuration, it Figure 4.18 Two stars Sp and Sq forming an inconsistent is said to be configuration - (Courtesy of [Labbe]) inconsistent. Stars containing such simplexes are inconsistent stars and a star set with at least one inconsistent star is also said to be inconsistent. Oppositely, a star whose simplexes are consistent is said to be consistent and a star set is if all of its stars are consistent. The main idea of the algorithm is to refine the set of sites P while maintaining the set of stars S(P) until each star Sp in S(P) is composed of simplexes that are well shaped and well sized in the metric Gp, and until there are no more inconsistencies among the stars. Once a consistent star set is achieved (which is proven to happen), all the stars can be stitched into a single triangulation: a locally uniform anisotropic mesh that conforms to the specified metric field and offers guarantees on the quality of the simplexes. 4.4.1.2 Refinement Algorithm The simplest idea to refine a simplex τ in a star Sp is to insert a new site at the center cp(τ) of the Delaunay Gp-ball of τ. This technique is very common in Delaunay refinement algorithm as the Delaunay ball of the simplex is by construction not empty after the insertion of its center and thus the simplex cannot appear in the new Delaunay triangulation. This simple strategy may unfortunately lead to cascading occurrences of inconsistencies, for the same reason that the refinement of Delaunay meshes cannot remove slivers. An alternative strategy is devised, inspired by the work of [Li and Teng] to avoid slivers in isotropic meshes. The adaptation of [Li and Teng’s] techniques to the present algorithm and the development of the refinement algorithm are described in detail in the following. 4.4.1.3 Discussion on the Parameters The algorithm relies on various parameters: φ0, r0, ρ0, σ0, β, δ... It can be difficult, at first glance, to estimate their influence on the outcome in term of speed, number of vertices or quality of the produced mesh. We investigate each parameter independently. 4.4.1.3.1 Parameter φ0 The rule with the highest priority in the algorithm is the distortion rule, which bounds the maximal distortion in a simplex to be at most φ0, with the intent of increasing the odds of finding a valid solution when the Pick_valid procedure is called later in the algorithm. The value φ0 has naturally a strong impact over both the computation time and the final mesh: if φ0 is chosen too large, the Pick_valid procedure will be at first largely unsuccessful, causing many unsuccessful and costly insertions till the sampling increases and valid points are be found. Oppositely, if φ0 is chosen too

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small, simplexes might be refined even if they already satisfy all other criteria including consistency which is pointless and costly and thus undesirable. To determine what is the optimal value forφ0, we look at the number of vertices in a final mesh, the number of calls to the Pick_valid procedure and their success percentage, and the computation time for various values of φ0. All the parameters except for φ0 are kept constant and the sizing constraint r0 is chosen large enough as to only be responsible for a negligible number of vertices added. The domain is a square of side 4 and centered on c = (0 , 0) endowed with the hyperbolic shock. Results are shown in Figure 4.19 (left). For the sake of completeness, we also consider a three-dimensional domain, a cube of side 3 and centered on c = {1.5, 1.5, 1.5}, endowed with a unidimensional shock metric field. Since the goal is to compare the influence of φ0 for the refinement of cells, we disable the refinement of surface stars and construct a pure volume mesh. Results can be found in Figure 4.19 (right). These experiments provide a lot of interesting and unexpected information: • • •

The success percentage of the Pick_valid procedure increases when φ0 decreases, as the theory predicts. However, a low value of φ0 will very quickly be responsible for the insertion of an exceptionally large number of vertices (red curves). The final number of points is greater whenever φ0 is used than when it is not (blue curves). However small φ0 is, there are still inconsistencies that appear (the teal curves show the number of vertices inserted to solve inconsistencies), proving that the Pick_valid procedure is a necessary part of the algorithm. Even more interestingly, they add a relatively constant number of vertices, indicating that the distortion rules did not help.

Figure 4.19

Influence of the Parameter ψ0 in a 2D (shown on the left) and 3D Domain (shown on the right) - (Courtesy of [Labbe])

4.4.1.3.2 Parameters r0 and ρ0 The values of r0 (size criterion) and ρ0 (shape criterion) naturally affect the number of vertices as additional vertices must be inserted to satisfy these criteria. In isotropic mesh generation, the number of vertices follows roughly the square of the sizing field in 2D and the cube in 3D: if the sizing field is divided by 2, there are 4 times more vertices in 2D and 8 times more vertices in 3D. As explained before, the size parameter is usually left at 1 as the size of the elements can be directly encoded in the metric. We exceptionally scale r0 instead of the metric field in this experiment for the purpose of clarity. This relation is somewhat similar in the case of anisotropic mesh generation. Here

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again, the number of points added by the inconsistency queue (that can be roughly computed as the difference of the green and blue curve) is more or less constant, even when the number of points inserted for the size rule is important. The shape parameter ρ0 and its corresponding rule ensure that all simplexes satisfy shape requirements, which is both useful in itself but also improve the odds of finding valid solutions in the Pick_valid procedure. This parameter has, in practice, very little influence and the corresponding rule is very rarely called upon. This can be seen as a consequence from the fact that Rules 1 and 2 (distortion and size) insert the circumcenter, an already appropriate choice to create meshes of good quality. 4.4.1.3.3 Parameters β and δ Contrary to the parameters φ0 and ρ0, who serve to satisfy quality requirements on simplexes ahead of Pick_valid procedure calls, the parameters β and δ are directly involved in the Pick_valid procedure. Indeed, the parameter β controls the size of acceptable inconsistencies and δ determines the size of the picking region Pp(τ ) as its radius is given by δrp(τ ). We use a cubic domain of side 1, centered on (0.5, 0.5, 0.5) and endowed with a unidimensional shock metric field with maximal anisotropy 3. The number of Pick_valid tries is fixed at 60 but the value of δ varies. We investigate the final number of vertices in the mesh and the quality of the simplices. For planar and surface domains, the quality is estimated with the formula of [Zhong et al.]104:

Q = 4√3

A ph

Eq. 4.7 where A is the area of the triangle, p the perimeter and h the longest edge (all computed in the metric). For cells, we use the quality estimation of [Frey and George]105

V2 Q = 216√3 3 𝐴Σ

Eq, 4.8 where V is the volume of the tetrahedron, and A∑ the sum of the areas of the four facets (all compute in the metric). Both these quality measures live between 0 and 1, with 1 signaling the highest quality. Results are detailed in Table 4.2. As expected, more solutions are available in the pick valid algorithm when the picking region is enlarged, resulting in smaller meshes. However, the metric is followed more loosely and the simplexes of the meshes have lower quality. Note that in extreme cases –δ close

Table 4.2 Comparison of the number of vertices and quality of the mesh for different values of δ - (Courtesy of [Labbe]) Zhong, Z., Guo, X., Wang, W., Lévy, B., Sun, F., Liu, Y., and Mao, W. Particle-based anisotropic surface meshing. ACM Trans. Graph. 32, 4 (2013). 105 Frey, P., and George, P. L. Mesh generation: Application to finite elements. Hermes Science, 2008. 104

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to 1 – the final number of vertices starts to increase again, as the refinement points are often not creating satisfying elements with respect to other criteria, such as the shape. The parameter β is assigned the value 2.5 by default, but changing this value (within 1 to 5) has barely any influence on the outcome. 4.4.1.3.4 Parameters σ0 The parameter σ0 controls the maximal slivery of the simplexes. A majority of inconsistencies do not stem from slivery quasi-cosphericities, and thus this parameter has little influence on the outcome. By increasing its value, one simply trades the removal of inconsistencies for the removal of silvers, but the result stays the same. Furthermore, the slivery and inconsistent queue both rely on the Pick_valid procedure to insert a point and thus there is no difference in the running time of the algorithm either. 4.4.1.4 Results and Limitations 4.4.1.4.1 Uniform Metric Fields A uniform metric field associates to any point of the domain the same metric. Note that since all the metrics are identical, neighboring stars necessarily have compatible connectivity and there can be no inconsistencies. It should be noted that in the setting of uniform metric fields, an anisotropic mesh can simply be obtained by stretching the domain through the usual metric transformation (the square root of the metric), meshing isotopically in the metric space and finally stretching back the constructed isotropic Delaunay triangulation to obtain an anisotropic Delaunay triangulation which is in fact what is done in each star, but for different metrics in the general case. This setting is handled without any issue by the algorithm and our implementation, whether in the setting of planar, surface or volume domains: the algorithm terminates, all criteria are honored and we obtain edges that have a length close to 1. As the algorithm perform most computations in the metric space where triangulations are isotropic the algorithm is extremely robust and can handle anisotropy ratios up to 1017; numerical issues appear in the transformations for higher ratios, which could easily be fixed by changing number types (but no one realistically deals with such ratios). 4.4.1.4.2 Shock-Based Metric Fields on Planar Domains We depart from uniform metric fields and consider a planar domain a square endowed with various metric fields. These artificial metric fields are chosen to exhibit different phenomena: Straight shock - A region of high-anisotropy along a relatively constant direction in between two regions of lower anisotropy. This corresponds to a change in the eigenvalues of the metric. Waves along a (straight) shore is a good illustration of such phenomenon. Rotational anisotropy - A region where the anisotropy ratio is constant, but the direction of the stretching is changing. This corresponds to a change in the eigenvectors of the metric. We will sometimes say that the metric is rotating to describe such regions. These two phenomena are the primal blocks and can be combined and scaled to form any more complex or real-world metric fields. 4.4.1.4.3 Starred Our first example uses the Starred metric field, detailed in Appendix A of [Labbe]106. The anisotropy of this metric field varies between 1 and 10. The Starred metric field possesses long compared to the prescribed size of the elements – regions of straight anisotropy, and regions where the metric field is rotating (and the anisotropy ratio is lower) in between, thus providing a good first example of an arbitrary metric field. The domain is a square of side 10, centered on the origin. Figure 4.20- A square of side 10 and centered on the origin, endowed with the Starred metric field (left). The final mesh is composed of 47126 vertices and 94366 triangles. On the right, a zoom on one of the rotating regions. The algorithm terminates without any issue and the size and shape criteria are solved 106

Mael Rouxel-Labbe, “Anisotropic mesh generation”, Université Côte d’Azur, 2016.

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quickly (4499 vertices are needed). However, the resolution of inconsistencies is difficult and requires around 40000 additional vertices. While regions where the metric field is rotating are clearly suffering from inconsistencies, regions where the metric field is straight fare only slightly better and also require many vertex insertions to solve inconsistencies.

Figure 4.20

A square of side 10 and centered on the origin, endowed with the “Starred” metric field

(left) - (Courtesy of [Labbe])

4.4.1.4.4 Hyperbolic The hyperbolic shock metric field has already been used several times in previous sections and its definition is detailed. This metric field is characterized by an anisotropy ratio that varies between 1 and 15 and is interesting as its anisotropy ratio does not vary (too much) along the shock, despite the shock being shaped like a sinusoidal curve. We shall refer to the regions of the shock where the

Figure 4.21

Anisotropic Triangulation of a Rectangle Endowed with the Hyperbolic Shock Metric Field - (Courtesy of [Labbe])

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eigenvectors change rapidly as the “turns”. In these turns, the process of generating a good anisotropic mesh is difficult as the eigenvectors of the metrics are changing rapidly. As larger meshes have already been produced in other sections and very dense refinement was observed at the turns, we here zoom on one of these regions. The result is shown in Figure 4.21. The size and shape constraints are quickly satisfied and only require around 414 vertices, but the final mesh is composed of 4621 vertices. Indeed, the resolution of inconsistencies is difficult especially within the shock and many vertices are required to obtain a consistent mesh despite a relatively low maximum anisotropy ratio. Consequently, the metric field is honored (and consequently the sizing field is too), but simplexes are often much smaller and then resolution of inconsistencies much longer than what we hoped for. 4.4.1.4.5 Swirl The Swirl metric field aims to represent a whirling phenomenon. Figure 4.22 A square of side 6 and centered on the origin, Contrary to the two previous endowed with the “Swirl” metric field - (Courtesy of Labbé et al.) metric fields, the Swirl metric field has a relatively constant anisotropy ratio and is rotating almost everywhere. Figure 4.22 shows the mesh obtained by our implementation for a square of side 6 endowed with this metric field. The result exhibits the same issues as in the previous experiments: the resolution of inconsistencies is difficult and many vertices are required to solve inconsistencies after all other criteria are satisfied. 4.4.1.4.6 Curvature-Based Metrics Fields on Surfaces We now consider the setting of domains embedded in R3, and the generation of pure surface meshes. The metric field induced by the curvature of the domain is known to prescribe an anisotropy that is asymptotically optimal - a mesh whose elements follow this metric field will require the lowest number of element (out of all the meshes) to achieve a given approximation of the domain. It is thus interesting to observe the results produced by our algorithm for this specific metric field. 4.4.1.4.7 Optimization Optimization is often used to improve the quality of a triangulation. Centroid Voronoi tessellations are Voronoi diagrams whose generators are also the centroids (centers of mass) of their respective cells. The famous Lloyd algorithm107 iteratively moves the seeds to the center of mass of their respective cell and recomputed the Voronoi diagram of this new seed set. We approximate the Riemannian Voronoi center of mass of a cell V with the following formula:

107

S. Lloyd, Least squares quantization in pcm, IEEE Trans. Inf. Theo. 28 (2006) 129–137.

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cg =

∑i ci |t i |gi ∑i|t i |gi

Eq. 4.9 where |ti|gi is the area of the triangle ti in the metric gi and the ti make a partition of V. The canvas conveniently provides this decomposition of a geodesic Voronoi cell in small triangles, making the approximation of cg accurate. The formula in Eq. 4.9 does not extend to surfaces as the result of the weighted sum might not lie on the domain. In that setting, we use a process similar to [Wang et al.]108. As its Euclidean counterpart, this algorithm comes with no guarantees (not even for termination) but works well in practice. In Figure 4.23, the initial SRDT of 4000 seeds has been optimized with 100 iterations. The metric field is well captured with few elements, especially in the rotational region.

Figure 4.23 The optimized SRDT of 4000 seeds in a planar domain endowed with a hyperbolic shock induced metric field (left). On the right, a zoom on a rotational region of the metric field shows the difference between pre- (above) and post- (bottom) optimization – (Courtesy of Labbé et al.)

4.4.1.5 Discrete Riemannian Voronoi Diagrams Several authors have considered Voronoi diagrams based on anisotropic distances to obtain triangulations adapted to an anisotropic metric field. These authors hoped to build upon the wellestablished concepts of the Euclidean Voronoi diagram and its dual structure, the Delaunay triangulation, for which many theoretical and practical results are known. The computation of geodesic path lengths in any domain is a difficult task as there is generally no closed form available. The wide range of domain can be shrunk, through mesh generation, to consider only piecewise-linear X. Wang, X. Ying, Y.-J. Liu, S.-Q. Xin, W. Wang, X. Gu, W. Mueller-Wittig, Y. He, Intrinsic computation of centroidal Voronoi tessellation (CVT) on meshes, Computer-Aided Design 58 (2015) 51–61. 108

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domains. Even in this simpler setting, the computation of geodesic distances and paths is still a complex problem to which much work has been dedicated in the last decades. Despite many studies, geodesic distances still cannot be obtained exactly for most domains endowed with an arbitrary metric field. Nevertheless, we introduce a discrete structure that is, under some conditions, combinatorically equivalent to the Riemannian Voronoi diagram and whose duals are triangulations. 4.4.1.5.1 Advantages Over Isotropic Canvasses To ensure that the nerve of the Riemannian Voronoi diagram is captured in the case of an isotropic triangulation used as canvas, the (uniform) sizing field of the canvas must be small enough such that Voronoi bisectors are clearly distinct. As the anisotropy ratio increases, Voronoi cells become thinner and the number of canvas vertices required to capture the nerve rapidly grows (Figure 4.24, left and center109). On the other hand, the placement of vertices in an anisotropic canvas is by construction not uniform and does not suffer from the same issue: as the anisotropy of the metric field grows, Voronoi cells and canvas simplices become thinner in tandem. The number of canvas vertices in a Voronoi cell is thus relatively constant regardless of the anisotropy. As the star set satisfies a sizing field of 0.1r0, the canvas edges are roughly 10 times smaller than the distance between seeds. Consequently, the canvas is composed of approximately 10n more vertices than seeds, with n the intrinsic dimension of the domain. The use of an anisotropic canvas greatly decreases the computational time as the number of vertices in the canvas is drastically reduced, without any change in the extracted nerve.

Figure 4.24

Isotropic and Anisotropic Canvas Sampling - (Courtesy of [Labbe])

4.4.1.5.2 Straight Riemannian Delaunay Triangulation In the case of an isotropic canvas, increasing the anisotropy of a cell increases the number of vertices required to properly capture it (left and middle). This is not the case if the canvas can be anisotropic (left and right). 109

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Figure 4.25

Unit Sphere Endowed with the Hyperbolic Metric field - (Courtesy of [Labbe])

By definition, the Riemannian Voronoi diagram captures the metric field more accurately than other methods that typically only consider the metric at the vertices. This additional input of information allows us to construct curved Riemannian Delaunay triangulations, but also has a positive influence on the straight realization of the diagram. Figure 4.25 shows the different structures involved in our algorithm during the generation of an anisotropic meshes for the sphere endowed with a hyperbolic shock metric field110. On the left is the canvas (an isotropic triangulation here); the middle sphere shows the discrete Riemannian Voronoi diagram computed upon this canvas; finally, the right picture shows the dual of the discrete diagram, an anisotropic triangulation. Contrary to the previous approaches introduced and investigated no over-refinement is observed, including in the regions where the eigenvectors of the metric field are rotating (where the shock turns). The final mesh has slightly fewer than 4000 vertices, which is the number of vertices that was required by our locally uniform anisotropic meshes to generate a mesh of a only small region of that domain. 4.4.1.5.3 Curved Riemannian Delaunay Triangulation The large amount of additional information that is provided by the canvas allows to construct curved Riemannian Delaunay triangulations, which have not been produced before. Figure 4.26 shows the discrete diagram and the curved Riemannian Delaunay triangulation for the Chair surface endowed with a curvature-

Figure 4.26 On the left, the discrete Riemannian Voronoi diagram of 1020 seeds on the “Chair” surface, with a curvature induced metric field; the edges of the curved Riemannian Delaunay triangulation are traced in black - (Courtesy of [Labbe])

The unit sphere endowed with the hyperbolic metric field (approximately 4000 vertices). Isotropic canvas (left), discrete Riemannian Voronoi diagram (center), and straight Riemannian Delaunay triangulation (right). 110

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induced anisotropic metric field. In the computation of the curvature metric field, the value ǫ = 0.7 is used (see Section A.2.1 in Appendix A of [Labbe]111). The curvature of the simplexes is noticeable and the metric field is well captured with only few curved Riemannian simplexes. Comparatively, a straight dual would require more elements to obtain the same approximation Figure 4.27 shows the curved Riemannian Delaunay triangulation dual of the Riemannian Voronoi diagram for the hyperbolic shock metric field. We obtain an aesthetically pleasing curved mesh that conforms closely to the metric field.

Figure 4.27 Discrete Riemannian Voronoi Diagram (top) and Curved Riemannian Delaunay Triangulation (bottom) endowed with the Hyperbolic Shock Metric Field - (Courtesy of [Labbe]) 111

Mael Rouxel-Labbe, “Anisotropic mesh generation”, Université Côte d’Azur, 2016.

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4.4.1.6 Conclusion We have presented an empirical study of the locally uniform anisotropic meshes framework, introduced by [Boissonnat, et al]112. During this study, we have introduced minor changes to the theory and provided a robust and generic implementation of the algorithm. While the algorithm is simple to understand, the computation speed of a naive implementation would be unreasonable. We presented an implementation of the algorithm that is both robust and fast, which is more difficult to achieve. The theoretical requirements are demanding, but in practice both the geodesic and straight edge duals require relatively few points to become embedded triangulations, even with nontrivial, highly distorted metric fields. The RVD and its duals are shown to be particularly well suited to capture the metric field in regions where it is both anisotropic and rotational. No preprocessing or smoothing of the metric field is required, a technique that is often used and results in noticeable loss of anisotropy in this type of region113.

Octree Decomposition In 2D this procedure can be viewed as a division of the domain into a collection of rectangles followed by a division of rectangles into triangles. A rectangle can be further subdivided into four new rectangles. For a rectangle that intersects the boundary, this subdivision can be repeated until a sufficiently fine resolution has been achieved. Rectangles that intersect the boundary and are sufficiently small are then replaced by a polygon consisting of the part of the rectangle lying inside the domain together with the part of the boundary that lies inside the rectangle. Figure 4.28 (a) shows a schematic that illustrates the concept of an Octree decomposition of the space around an airfoil. A further division of rectangles and boundary polygons into triangles creates a valid triangulation of the domain (see Figure 4.28 (b)). The concept generalizes in an obvious way to three dimensions although the cutting procedure at the boundaries becomes much more complicated. The main drawback of Octree based triangulation methods is their inability to match a prescribed surface triangulation since the surface triangulation arises as a byproduct of the volume meshing procedure. The size of the

(a)

(b) Figure 4.28

Converging of an Octree Decomposition Around an Airfoil

Boissonnat, J.-D., Wormser, C., and Yvinec, M. Locally uniform anisotropic meshing. Proceedings of the twenty-fourth annual symposium on Computational geometry (2008), ACM, pp. 270–277. 113 M. Rouxel-Labbé, M. Wintraeckenb, J.D. Boissonnat, “Discretized Riemannian Delaunay triangulations”, 25th International Meshing Roundtable (IMR25). 112

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individual Octree components and hence the size of the tetrahedral elements in the near field can be tailored to match the variation in surface curvature. But the quality of elements adjacent to the boundary surface and likewise the quality of the surface triangulation can be very poor. This can be a considerable handicap since an accurate implementation of the boundary conditions often requires a good quality mesh near the boundary. For high Reynolds number Navier Stokes computations, which must capture the flow details inside thin boundary layers, the lack of a good quality mesh near a boundary causes considerable difficulties. One way to alleviate these problems is to build a good quality mesh in the near field by extrusion of hexahedra, prisms or tetrahedral off the boundary surface and then merge this extruded mesh with an Octree based mesh at a position that is some way off the boundary 114,115. It is best when creating an octree mesh to do the following: Perform volume meshing •

Improve the quality of the volume mesh using Edit Mesh options

•

Create prism layers for boundary layer near the walls

•

Improve the total mesh quality using Edit Mesh options.

The paper by [Mar´echal]116 presents advances

Figure 4.29

A close-up view of nasty cheese a well-known test-case featuring 30◦ dihedral angles – (Courtesy’s of [Mar´echal])

Karman SL, “SPLITFLOW: a 3-D unstructured Cartesian/prismatic grid CFD code for complex geometries”, AIAA 33rd aerospace sciences meeting, Reno, NV. AIAA paper 95-0853, 1995. 115 Shaw JA, Stokes S, Lucking MA, “The rapid and robust generation of efficient hybrid grids for rans simulations over complete aircraft”, International Journal Numeric Method Fluids 2003; 43:785–820. 116 Lo¨ıc Mar´echal, “Mesh Generation: Handling Sharp Features”, Gamma project, I.N.R.I.A., Rocquencourt, 78153 Le Chesnay, France. 114

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made in terms of sharp angles meshing, non- manifold geometries, based on all-hexahedral mesh. An example showing sharp feature is revealed on Figure 4.29.

Unstructured Hexahedral Meshes It is possible, in principle, to cut up a triangle into three quadrilaterals by inserting an extra point at the triangle barycenter and an extra point at the mid-point of each edge. A similar division of a tetrahedron into four hexahedra is also possible. At first sight, this would appear to be a straightforward way to create unstructured quadrilateral or hexahedral meshes from 2D or 3D triangulations. In practice, the quality of the resulting meshes is generally very poor. Alternative ways to create unstructured hexahedral meshes have been based on paving techniques117 (essentially a moving front approach), Octree decomposition together with a means of merging, or snapping, the outermost Octree hexahedra to the boundary, or by making use of the medial axis in 2D, medial surface in 3D, to produce a type of multi-block decomposition that is more amenable to meshing by paving118. The possibility of automatically generating unstructured hexahedral meshes is tantalizing and offers the prospect of providing automated mesh generation suitable for solid mechanics computations using finite element methods as well as meshes suitable for the computation of the RANS equations. Much depends on the mesh quality near solid boundaries and it remains to be seen whether any of the current approaches to hexahedral mesh generation can provide the required

Hexahedral

Hybrid

Single Block

Tetrahedral/

(Fully Structurd)

Hexahedral

Multiblock (contiguous)

Multiblock Overset (Chimera

Tetrahedral/ Primatic Cartesian (Hexahedral +assorted Polyhedral

Tetrahedral Octree Decompostion

Advancing Front

Delany

Fully unstructured Hexahedral Figure 4.30

Hierarchy of Meshing Methodologies

Zhu JZ, Zienkiewicz OC, Hinton E, Wu J, “A new approach to the development of automatic quadrilateral Mesh generation”, Inter J Number Meth Eng. 1991; 32:849–66. 118 Sheehy, DJ, Armstrong CG, Robinson DJ,”Computing the medial surface of a solid from a domain Delaunay triangulation”, Proceedings of the ACM symposium on solid modeling and applications, Salt Lake City, UT, New York: ACM Press; 1995. p. 201–12. 117

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flexibility in terms of geometry handled and the necessary quality in terms of mesh orthogonality near solid boundaries. Figure 4.30. displays different methodologies which currently available in mesh generation engines. Conversion of Triangular to Quadrilateral Meshes (2D) Another simple strategy is developing by [Lyra & de Carvalho]119 where quadrilateral mesh generated from triangular meshes. Unstructured quadrilateral meshes can be automatically generated in several different ways and do not impose serious topological restrictions on the meshes, being appropriated to deal with complex geometries, naturally allowing local non-uniform mesh refinement. Several different approaches have been proposed to generate unstructured quadrilateral meshes. These methodologies can be divided into two basic groups: ➢ those that try to generate quadrilaterals directly, ➢ and those that convert a previously generated mesh of triangles into a mesh of quadrilaterals The conversion of triangular meshes is particularly attractive because these meshes can inherit the properties of the triangular meshes, whose generators are very well developed and once it is always possible to build a triangular mesh over any arbitrary 2D domain, quadrilateral meshes can be constructed as general as the triangular ones. It also allows the use of any triangular mesh generator as a “black box”. As we generate a quadrilateral mesh using the conversion strategy, the quadrilateral mesh inherits the characteristics of the initial triangulation. For both, iso and anisotropic meshes this strategy consists of four main steps, as presented . 1. Generate a triangular mesh (either iso or anisotropic); 2. Remove an edge between two adjacent triangles, to forma quadrilateral; 3. Split all elements in the intermediate mixed mesh(triangles into three quadrilaterals and quadrilaterals into four quadrilaterals); 4. Perform some post processing steps in order to enhance mesh quality. The standard strategy of merging triangles into quadrilaterals consists in eliminating a common edge that belongs to two adjacent triangles. Following the work done by [Xie and Ramaekers (1994)] and [Alquati and Groehs (1995)], our mesh generator is such that it refrains from merging triangles that would form a non-convex quadrilateral. Besides, for anisotropic meshes, the merging process will remove a common edge between two adjacent triangles, only if the two quadrilaterals to be created satisfy a quality criteria which is controlled by two geometric parameters. The adopted procedure generates a quadrilateral mesh with edges that are approximately half of those of the corresponding triangular elements and usually this is not a serious concern, since the user can generate a coarser

Paulo Roberto M. Lyra, Darlan Karlo E. de Carvalho, “A Computational Methodology for Automatic TwoDimensional Anisotropic Mesh Generation and Adaptation”, Methodology for Automatic Two-Dimensional Anisotropic Mesh Generation and Adaptation. 119

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initial triangulation to obtain the desired mesh density. The four steps involved in the quadrilateral mesh generation can be seen in Figure 4.31 (1-4).

(1) Initial triangulation

(3) Mesh non-optimized

Figure 4.31

(2) Intermediate mesh

(4) Final quadrilateral mesh

Quadrilateral Mesh Generation

Overset Grids There are several variants of multi-block depending on whether or not continuity of mesh lines is maintained across the block boundaries. Overset methods represent one extreme where no attempt is made to match meshes from neighboring blocks. Figure 4.32 shows a combination of two overset meshes, an O-mesh around an airfoil plus an H-mesh for the far field. First suggested by [Atta]120 the

120

Atta E. Component–adaptive grid interfacing. AIAA19th aerospace sciences meeting. AIAA paper 81-0382.

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overset approach was extensively developed by [Benek et al.]121 with later contributions by [Chesshire and Henshaw]122. The lack of any constraint at the block boundaries means that mesh generation for the individual blocks is much easier. In particular, there is no a priori need to create block interfaces and this advantage has facilitated the early application of the overset approach to complicated geometries. Another advantage of permitting such a loose connection between neighboring meshes is the possibility of treating moving body problems (e.g. store separation). The penalty for these advantages lies in the need to transfer information between neighboring meshes. This requires a means of determining an appropriate Figure 4.32 Overset Mesh Combination overlap region and the development of interpolation formulae to ensure accurate data transfer. Overset meshes are also known as Chimera or overlapping meshes. An overset mesh typically containing a body of interest such as a boat or a gear, superimposed on a background mesh containing the surrounding geometry. The data is interpolated between them123. This approach allows complex motion and moving parts to be easily set up and simulated. Overset meshes typically involve a background mesh adapted to the environment and one or more overset grids attached to bodies, overlapping with the background mesh. Multiple overlapping overset regions are also possible, expanding the potential applications of this technology. Data interpolation occurs between the grids, which can move with respect to one another. They are most useful Figure 4.33 Two Counter-Rotating Objects Embedded in Two in simulating multiple or moving Overset Regions with Background Mesh – (Courtesy of Siemens) bodies, as well as parametric studies Benek JA, Buning PG, Steger JL. “A 3-D Chimera grid embedding technique”. AIAA 7th CFD conference, Cincinnati, OH. AIAA paper 85-1523, 1985. 122 Chesshire G, Henshaw WD,”Composite overlapping meshes for the solution of partial differential equations.” Journal of Computational Phys 1990; 90:1–64. 123 Siemens PLM Software www.siemens.com/plm, 2016. 121

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and optimization analyses. By allowing the overset body to move and also be replaced as many times as needed with different geometry, this technology truly brings multidisciplinary design exploration to the fingertips of engineers and designers. Composite Grids

Cartesian Grids Background and Cartesian Grid Origins Cartesian grids have been utilized in solving a variety of CFD problems from potential flows to Navier-Stokes equations124. Cartesian grids consist of a collection of non-overlapping, connected control volumes with coordinate aligned edges. Thus, the edge (or face in three dimensions) normal for all complete cells are aligned with one of the coordinate directions. Figure 4.34 shows a typical two-dimensional Cartesian grid around a curved surface. Cartesian gridding techniques have become the focus of recent research due to their ability to easily handle complex geometries in the grid generation phase. The ease with which higher order schemes can be applied and the natural connection between the Figure 4.34 Example of Cartesian Grid Near grid refinement techniques and multigrid Curved Surface – (Courtesy of NASA Ames) acceleration schemes. The difficulties in using Cartesian grids arise from the fact that the control volumes adjacent to the surfaces are not usually aligned with the surfaces and thus special techniques need to be employed to handle the non-Cartesian (cut or split) cells in these regions. Cut cells are created when the intersection of the Cartesian cell and the solid surface results in one computational volume with only a fraction of the original volume and possibly non-Cartesian aligned edges, see Figure 4.35 (a). Split cells are created when the intersection of the Cartesian cell and the solid surface results in two or more computational volumes which might have non-Cartesian aligned edges, see Figure 4.35 (b).

(a) Cut 1 Figure 4.35

(b) Cut 2

Solid Surface Over-Layer Cartesian Cell and Resulting Cut and Split Cell – (Courtesy of NASA Ames)

Stephen M. Ruffin, NASA Ames Research Center in coordination with Georgia Institute of Technology, “GSRP/David Marshall: Fully Automated Cartesian Grid CFD Application for MDO in High Speed Flows”, 2003. 124

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The original use of Cartesian grids involved solving the 2D full potential equation by [Purvis and Burkhalter]125, followed shortly afterwards by [Wedan and South]126, in which a non-body-oriented structured grid was created on which the full potential equation was solved. Their solution strategy was to use finite volume techniques in order to more easily handle the computational cells that were intersected by the solid surface. Additionally, they used linear approximations in the cut cells for the reconstruction of the wall boundary conditions which provided a simple algorithm for implementation and preserved the structure of their coefficient matrix during the solution iteration so that no extra computational costs were incurred for the cut cells. However, this did not preserve the actual body curvature and also only provided a linear approximation to the actual surface lengths and area for the cut cells, and thus could not exactly model curved surfaces. Also, little mention was made of any attempts at cell refinement to more accurately capture the surface geometry and flow features. Earlier, [Clarke et al.]127 used Cartesian grids to solve the two-dimension Euler equations (again on non-grid aligned surfaces). They attempted to more accurately model the solid surface boundary conditions by utilizing the local surface curvature in reconstructing the wall boundary conditions. They also provided more accurate modeling of the cut cell lengths and areas by using the actual surface geometry in their calculations and not linear approximations. Additionally, they noted that clustering was needed in certain critical regions in order to produce accurate results, and this was achieved by clustering entire grid lines. Cut cells that were too small (less than Figure 4.36 Example of Merge Cell Creation – (Courtesy of NASA Ames) 50% of the original cell size) were merged with neighbor cells in order to avoid time stepping problems associated with very small computational cells. [Gaffney and Hassan]128 extended this research to 3D. Figure 4.36 demonstrates the case of cell merging. Cartesian Grids Schemes While the majority of research into Cartesian grids has focused on solving the Euler equations in 2-3 D, there has been some efforts into the utilization of Cartesian grids to solve the Navier-Stokes equations. These efforts have focused on solving the full N-S equations using either the Adaptive Mesh Refinement (AMR), Immersed Boundary, Volume-of-Fluid, Reconstruction, Cut Cell Based techniques, or Coupling Body-Fitted Grid solutions of the Navier-Stokes equations with a Cartesian background grid. The Grid Coupling technique has its foundations in the idea of the viscous/inviscid coupling. Cartesian grids do not, in general, provide grids that are body aligned, however some work has been performed applying the thin-layer techniques to Cartesian grids. Hybrid Methods do exist which couple a body oriented grid solving the thin-layer Navier-Stokes equations with a background J. W. Purvis and J. E. Burkhalter. Prediction of Critical Mach Number for Store Configurations. AIAA J.,1979. B. Wedan and J. C. South, Jr. A Method for Solving the Transonic Full-Potential Equation for General Configurations. In AIAA 6th Computational Fluid Dynamics Conference, Danvers, MA, July 1983. AIAA-83-1889. 127 D. K. Clarke, M.D. Salas, and H. A. Hassan. Euler Calculations for Multi element Airfoils Using Cartesian Grids. AIAA Journal, 24(3):353-358, March 1986. 128 R. L. Gaffney, H. A. Hassan, and M.D. Salas. Euler Calculations for Wings Using Cartesian Grids. AIAA 25th Aerospace Sciences Meeting, Reno, NV, January 1987. AIAA-87-0356. 125 126

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Cartesian grid129. 4.8.2.1 Adaptive Mesh Refinement [Berger and LeVeque]130 addressed several deficiencies that existed in the established uniform grid methodologies. First, they applied the concept of Adaptive Mesh Refinement (AMR) in order to improve the accuracy in critical regions without adversely affecting the efficiency of the numerical integration scheme. The use of AMR effectively allowed the clustering of blocks of computational grids as the solution process evolved only in the region that they were needed (and not clustering entire grid lines), by using Richardson-type extrapolation error estimates to identify regions of large errors and adding grid blocks in those regions. An example of AMR is Figure 4.37 which represents a simple adapted grid for a supersonic wedge flow with four levels of adaption. As can be seen in the figure, there are more control volumes where gradients are to be expected, specifically along the surface to capture the geometry and along the oblique shock. In regions with small gradients, there is a lower density of control volumes. Also notice that in this figure there is at most a 2:1 ratio at the refinement interface, which is typical of most A MR schemes, in order to promote stability in the numerical schemes. One problem with [Berger and LeVeque's] original implementation of AMR on Cartesian grids was the problem of state variable conservation during the AMR stages. They carefully constructed conservative schemes for the inter-grid transfer to address the problem. They also used the idea Figure 4.37 Example Adaptive Grid for Supersonic Wedge Flow – of wave propagation and (Courtesy of NASA Ames) directional differencing in order to increase the stability near the small boundary cells. This helped keep the CFL of the boundary cells reasonably close to the CFL of the flow cells and allowed larger time steps to be taken with the solver remaining stable. Several researchers have extended [Berger and LeVeque's] research into areas such as multigrid Cartesian grids, higher accuracy flow solvers using more sophisticated flux approximations, time-accurate unsteady flows, and a front tracking AMR scheme that attempted to track the discontinuities (such as shocks) as the solution evolved in order to provide more accuracy in the refined mesh calculations. According to recent investigation by [Hiroshi Abe ]131, Cartesian grid method fall into two categories with the demand of accurate solutions. One keeps its structured grid nature and introduces embedding structured sub grids within the underlying coarse structured grids. Adaptive Mesh Refinement (AMR) is one of them. Figure 4.38 (a) shows an example of AMR in two dimension. The intersected cells by a circle in the underlying coarse grids are tagged in blue. The R. L. Meakin. On Adaptive Refinement and Overset Structured Grids. 13th AIAA Computational Fluid Dynamics Conference, CO, 1997. 130 M. J. Berger and R. J. LeVeque. An Adaptive Cartesian Mesh Algorithm for the Euler Equations in Arbitrary Geometries. 9th AIAA Computational Fluid Dynamics Conference, Buffalo, NY, June 1989. AIAA-89-1930-CP. 131 Hiroshi Abe, “Blocked Adaptive Cartesian Grid FD-TD Method for Electromagnetic Field with Complex Geometries”, International Conference on Modeling and Simulation Technology, Tokyo, JAPAN, 2011. 129

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blue-tagged cells are to be refined. In the AMR procedure, several embedded rectangle patches are defined so as to contain the blue tagged cells. Then, the embedded rectangle patch areas are refined.

(a) Intersecting meshes with a circle are tagged (blue) Figure 4.38

(b) 2D case of Adaptive Cartesian grid method

Schematic image of Adaptive Mesh Refinement – (Courtesy of Hiroshi Abe)

The other considers the Cartesian mesh as an unstructured collection of h-refined meshes. The data structure is not the same as structured grids but the same as unstructured grids. Adaptive Cartesian grid method was introduced as an unstructured Cartesian grid method and has shown the great

Figure 4.39

Pressure Contours in 2D Backward Step

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success in simulating complex geometries. Figure 4.38 (a) shows a case of two dimensional adaptive Cartesian grid method. Beginning with a root cell covering whole domain, the intersected cells by the circle are recursively bisected. This simple procedure finally gives Figure 4.38 (b). Figure 4.40 Flow Around a F16XL Fighter Jet using Cut Cells and AMR – Further examples (Courtesy of M. J. Aftosmis, M. J. Berger, J. E. Melton) provided a 2D backward step (see Figure 4.39), and 3D F16XL fighter jet using cut cells and AMR. (see Figure 4.40). 4.8.2.2 Immersed Boundary Methods The immersed boundary method was originally developed by [Peskin]132-133 for heart valve modeling using the Navier-Stokes equations in two dimensions. The heart valves were modeled as flexible surfaces that can propagate with the flow, subject to certain limitations such as hinge points or rigid regions on the surfaces. Instead of remeshing the computational domain as the surface is propagated, the cells that contain the surface have a body force added to their momentum equations that represents the reactive force that the body is applying to the fluid in response to the fluid surface pressure and shear stress. [Goldstein et al.]134 applied [Peskin's] work to incompressible, solid body flows using a force feedback approach. In this formulation, the surface force takes the form of a feedback loop function that acts on the surface cell to bring the surface velocity to zero by adjusting the applied forces appropriately. This approach requires an extremely small time step (CFL around 1 -3) in order for it to remain stable. In order to more accurately determine the appropriate surface forces to add to the momentum equations, [Fadlun et al.]135 developed a second-order boundary interpolation scheme for three dimensional incompressible flows by using linear interpolation to reconstruct the state information at the surface. This approach resulted in the use of larger time steps (CFL around 1.5) and better accuracy at the surface. Further advances by [Lai and Peskin]136 developed second-order methods for moving membranes. Additionally, [Kim et al.]137 developed a second-order method with both momentum and mass sources in order to improve the overall accuracy of their results. While these schemes handle the Navier-Stokes equations on Cartesian grids, they all suffer from numerical C. S. Peskin. Numerical Analysis of Blood Flow in the Heart. Journal of Computational Physics, 1977. C. S. Peskin. The Fluid Dynamics of Heart Valves: Experimental, Theoretical, and Computational Methods. Annual Review of Fluid Mechanics, 14:235-259, 1982. 134 D. Goldstein, R. Handler, and L. Sirovich. Modeling a No-Slip Flow Boundary with an External Force Field. Journal of Computational Physics, 105(2):354-366, 1993. 135 E. A Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof. Combined Immersed Boundary Finite-Difference Methods for Three-Dimensional Complex. Journal of Computational Physics, 161(1):35-60, 2000. 136 M.-C. Lai and C. S. Peskin. An Immersed Boundary Method with Formal Second Order Accuracy and Reduced Numerical Viscosity. Journal of Computational Physics, 160(12):705-719, 2000. 137 J. Kim, K. Kim, and H. Choi. An Immersed-Boundary Finite-Volume Method for Simulations of Flow in Complex Geometries. Journal of Computational Physics, 171(1):132-150, 2001. 132 133

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stability problems that typically require numerical diffusion. Also, the surface is not sharply resolved, and is typically smeared between 2 or 3 cells. This can cause problems when flow details are needed near the surface. 4.8.2.3 Volume of Fluid Methods Another approach to solving the Navier-Stokes equations on Cartesian grids is the volume of fluid method. In this method, a scalar transport equation is solved in addition to the Navier-Stokes equations. The scalar is a value between 0 and 1 that represents the volume fraction that the fluid (or gas) occupies in that cell. The typical use of this scheme is free surface flows, where the scalar represents the amount of the cell that the fluid occupies, and interfacial flows, where the scalar represents the volume fraction that a species occupies in the cell. [Hirt and Nichols]138 originally developed this method as part of an incompressible free-surface Navier-Stokes solver. In order to retain the incompressible invariance in the transport equation, strict mass conservation was required of the numerical solver. They also used a first order accurate surface reconstruction technique which causes problems resolving the interface boundaries. The volume of fluid schemes typically work well when the interface curvature is small with respect to the surface modeling. Otherwise, artificial discontinuities can develop as well as the inability to resolve the small scale features at the interfaces. Additionally, without accurate propagation of the scalar transport equation and sophisticated schemes to resolve the interface boundaries, artificial mixing can occur. 4.8.2.4 Reconstruction Schemes Another class of schemes used to solve the Navier-Stokes equations on Cartesian grids are the reconstruction based schemes. These have been proposed by [Ye et al.]139-140 and [Majumdar et al.]141. These schemes are all based around the idea of interpolating the state information to the nodes in the computational domain around the surface. [Ye et al.] have developed a two-dimensional incompressible Navier-Stokes equation solver. The solver use the cell merging technique to eliminate any surface cells that are smaller than 50% of their full size. Then, the state information for the faces of the new cell are found by utilizing a linear-quadratic two-dimensional interpolation from the surrounding cells. This technique results in a slow convergence of the pressure Poisson equation and requires acceleration techniques. This technique has been extended to moving boundaries. [Majumdar et al.] have developed two-dimensional, turbulent Reynolds Averaged Navier-Stokes solver on uniform Cartesian grids. This solver uses interpolation polynomials in one and two dimensions to reconstruct the state of the cells that are inside the body. Thus, the solution process is performed over uniform cells at the surface. The interpolation process can cause numerical instabilities due to the negative coefficients that can arise with certain interpolation polynomials. 4.8.2.5 Cut Cell Based Methods [Fryrnier et al.]142 developed the first work in the application of the full Navier-Stokes equations on Cartesian grids using the cut cell approach. The solution procedure was a straight-forward finiteC. W. Hirt and B. D. Nichols. Volume of Fluid (VOF) Method for Dynamics of Free Boundaries. Journal of Computational Physics, 39(1):201-221, 1981. 139 T. Ye, R. Mittal, H. S. Udaykumar, and W. Shyy. An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries. Journal of Computational Physics, 156(2):209-240, 1999. 140 T. Ye, R. R. Mittal, H. S. Udaykumar, and W. Shyy. A Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries. In AIAA 3rd Weakly Ionized Gases Workshop, Norfolk, VA, November 1999. 141 S. Majumdar, G. Iaccarino, and P. Durbin. RANS Solvers with Adaptive Structured Boundary Non-Conforming Grids. Annual Research Briefs 208782, Center for Turbulence Research, Stanford University, Stanford, CA, 2001. 142 P. D. Fryrnier, Jr., H. A. Hassan, and M.D. Salas. Navier-Stokes Calculations Using Cartesian Grids: I. Laminar Flows. AIAA Journal, 26(10):1181-1188, October 1988. 138

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volume approach with the Cartesian grids clustered using grid line. Their results demonstrated strong dependencies on the smoothness of the surface grid where non-smooth surface grids produced non-smooth skin-friction and surface pressure values. A large number of standard viscous flux formulations for cut cell based schemes were analyzed to ascertain their accuracy and positivity characteristics. These viscous flux formulations fell into two categories: 1. Green-Gauss reconstructions where the divergence theorem was applied to cells neighboring the face that the flux was being calculated to build the integration path, 2. polynomial based reconstructions that used a Lagrange polynomial and a set of support cells to interpolate the state variables where they were needed with the polynomial being differentiated to obtain the needed gradients. This research focused on the accuracy of the various formulations via a standard Taylor series approximation analysis and on the positivity of the formulations. The positivity is a measure of how well the discretization satisfies the local maximum Figure 4.41 Example Chimera Grid Near Curved Surface principle that holds for all homogeneous, (Courtesy of NASA Ames) second order partial differential equations (PDEs). The local maximum principle simply states that the solution to a homogeneous, second order PDE at one point is bounded by the values of its neighbors. It is a statement of the diffusive nature of second order PDEs, and thus it is a necessary requirement for any discretization of a homogeneous, second order PDE. The results of this effort were that all of the schemes demonstrated (to some degree) a competition between the accuracy of the scheme and the viscous stencil positivity for non-uniform cells, i.e. any attempt to improve the accuracy/positivity adversely effected the resulting positivity/accuracy. The resulting numerical analysis was performed for low to moderate Reynolds number flows. Cases where the surface was predominantly aligned with the coordinate directions showed excellent agreement with theoretical values, but when the body was not aligned with the coordinate directions (thus, the surface had cut cells of varying volume fractions of the uncut cells) large oscillations occurred in the results due to the sensitivity of the viscous stencil to the grid smoothness (for both cut cells and coarse/fine cell interfaces). Another impediment to utilizing this scheme for high Reynolds number flows was the large number of control volumes needed to adequately resolve the viscous regions. Even with AMR this became prohibitively large for even moderately complex geometries. In addition to the viscous flux formulation results, AMR was applied to Coirier's solution strategies with a positive effect, but without fully eliminating the viscous stencil sensitivity on the cut cell smoothness. Another approach that was discussed was the use of embedded, body oriented grids to capture the boundary layers, but no numerical results were given.

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4.8.2.6 Chimera Grid Schemes The use of a collection of grids to cover the computational domain is known as chimera gridding. Typically, a body-oriented structured grid is used around each component of the solid surfaces. Each of these structured grids are then overlaid onto a background Cartesian mesh. Figure 4.41 shows an example of a two-dimensional chimera grid collection around a simple curved surface. Notice that there is no simple mapping of cells in the body oriented grid and the background Cartesian grid. This feature is one of the drawbacks to chimera gridding schemes, but it is only a performance penalty when the grid needs to be generated during initialization and after any AMR processes. The development of chimera gridding schemes were not solely founded in the viscous/inviscid coupling problems, but chimera gridding schemes were applicable to that use. Throughout the history of chimera gridding there have been a number of motivations for their investigation such as increasing grid point resolution near solid bodies, overcoming structured gridding issues associated with modeling complex geometries. [Atta]143 developed one of the first uses of chimera grids for the full potential equation in two-dimensions using a finite difference formulation. A uniform Cartesian grid was used for the background grid and a body-fitted 0-type structured grid was used around the body. The two grids were coupled via boundary information exchanges during the iteration process. First, the solution around the body fitted grid was converged through an outer iteration using a Dirichlet boundary condition imposed on the outer boundary. Next, the outer grid was converged using a Neumann boundary condition on the inner boundary, utilizing the solution information from the body solution. This information was then used to converge the body fitted grid once again. This cycle continued until the solution approached steady-state. This procedure required each grid (body and background) to have at least one complete cell inside the domain of the other, with the inner grid having an extent of between 1 and 3 chord lengths in all directions. Significant effort was needed to minimize the overlapping region in order to achieve optimal performance. [Atta] later extended this methodology to three-dimensions as well as more complex configurations. [Steger et al.]144 developed a finite-difference chimera grid scheme that could handle a much larger variety of configurations compared to Atta's work. While limited to twodimensions, they presented results for an airfoil-flap, cascading blades, a non-lifting bi-plane and an inlet with center body configuration. All of these configurations were handled automatically by their solver with little changes to the standard finite-difference formulations. State variables were exchanged between grids through interpolations which can cause performance penalties in the initialization stages when the connectivity is being constructed, but they addressed this by using the "stencil-walk" search pattern, where the cells that are used for the interpolation of one cell are assumed to be close to the cells that are needed for the interpolation of that cell's neighbors. 4.8.2.7 Hybrid Grid Schemes Another approach that was related to the chimera grid approach was the use of unstructured grids between the body surface and the background Cartesian mesh, as opposed to the overlaying of these grids. These schemes were usually referred to as hybrid grid techniques. Figure 4.42 demonstrates an example hybrid grid around a curved surface in two dimensions. One application of a hybrid scheme known as SPLITFLOW, by Karman145 and enhanced by [Domel and

E. Atta. Component-Adaptive Grid Interfacing. In 19th Aerospace Sciences Meeting, St. Louis, MO, January 1981. AIAA-81-0382. 144 J. L. Steger, F. C. Dougherty, and J. A. Benek. A Chimera Grid Scheme. InK. N. Ghia and U. Ghia, editors, Advances in Grid Generation, Presented at the Applied Mechanics, Bioengineering, and Fluids Engineering Conference, volume 5, pages 59-69. The Fluid Engineering Division, ASME, Houston, TX, June 1983. 145 S. L. Karman, Jr. SPLITFLOW: A 3D Unstructured Cartesian/Prismatic Grid CFD Code for Complex Geometries. In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, January 1995. AIAA. AIAA-95-0343. 143

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Karmen]146, used Cartesian grids for the majority of the computational domain, and prismatic grids to resolve the boundary layers. Standard Cartesian grid cutting techniques were used at the interface between the prismatic grids and the Cartesian grid. The prismatic cells were grown from the surface triangulation using a marching layers technique. The difficulties was addressed that could arise in the prismatic-Cartesian technique near convex regions, overlapping regions, and other regions where the prismatic marching technique needed to be modified to create viable grids. Other Related Method Similar to the reconstruction method is the class of finite element solution techniques called element-free Galerkin methods. Originally developed by [Belytschko et al.]147 for elasticity and heat conduction problems, it is currently being investigated for its applicability to fluid dynamics because of its automated handling of grid generation. The basic premise of this method is the use of polynomial curve fits to approximately represent the data surrounding the node Figure 4.42 Example Hybrid Grid Near Curved Surface – of interest. Typically, a least-squares (Courtesy of NASA Ames) error minimization is used due to the larger number of data points surrounding the node than the number of unknowns in the curve fit. Most implementations demonstrate oscillations near sharp gradients ( especially with higher-order interpolation functions) with more research needed to developing effective limiters. Another scheme related to the reconstruction method that is the grid-less method. This method uses a cloud of points to reconstruct a polynomial curve fit (similar to the element-free Galerkin method) using a least-squares error minimization. These curve fits are then used to calculate the derivatives required to solve the Navier-Stokes equations in differential form. The number of calculations per node is higher than for other techniques due to the large number of least-squares fits that are required. Unfortunately, this scheme does is not conservative and requires numerical dissipation in order to obtain a solution. Other researchers have extended this work, but without addressing the conservation problem. 4.8.2.7.1 Composite Grid Approach Composite grid generation approach is based on meshing of given arbitrary domain by geometric union of lower level grids built in more primitive domains. Advantages of such approach are the simplicity of meshing domains with complicated geometry and convenient definition of appropriate mesh refinement. Furthermore, resulting grid is partly structured and this feature can be utilized for building robust numerical solution schemes. The methodology includes three basic steps: 1. constructing structured prototype grids, N. D. Domel and S, T-. Karman, Jr. Splitfow: Progress in 3D CFD with Cartesian Omni-tree Grids for Complex Geometries. AIAA 38th Aerospace Sciences Meeting & Exhibit, Reno, NV, January 2000. AIAA-2000-1006. 147 T. Belytschko, Y. Y. Lu, and L. Gu. Element-Free Galerkin Methods. International Journal for Numerical Methods in Engineering, 37(2):229-256, January 1994. 146

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2. mapping these grids to non-regular geometry (if necessary) and 3. final superposition of low level grids into the final one. The procedure are outlined in Figure 4.43 and discussed in details148.

Figure 4.43

Basic Superposition Example – (Courtesy of Kalinin, Mazo and Isaev)

E I Kalinin1, A B Mazo1 and S A Isaev, “Composite mesh generator for CFD problems”, 11th International Conference on "Mesh methods for boundary-value problems and applications" IOP Publishing, IOP Conf. Series: Materials Science and Engineering 158 (2016) 012047 doi:10.1088/1757-899X/158/1/012047. 148

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Discussion It is generally accepted that a boundary conforming mesh is desirable to achieve accurate solutions from any numerical solver. If one is willing to sacrifice this requirement then mesh generation becomes a much simpler task. No approach beats regular structured grids in terms of efficiency and accuracy. Thus, there have been a number of efforts to use such grids for complex geometries which are called Cartesian grid approach. An early example of a non-aligned Cartesian mesh can be found in the work of [Carlson]149. Difficulties arise at the boundary where the Cartesian mesh intersects the boundary surface. Although finite difference methods can be derived to interpolate the boundary conditions onto the nearest mesh points, it is difficult to ensure solution accuracy. If extra points are inserted, however, where mesh lines intersect the surface then it is possible to create a boundary conforming mesh. In this respect, boundary conforming Cartesian methods are seen to be closely related to the Octree based triangulation methods. In fact, the elements obtained from the Octree and its intersection with the boundaries is precisely the elements that make up the Cartesian mesh. Conversely, any Cartesian mesh can be converted into an Octree type triangulation by splitting all elements into tetrahedral (or triangles in 2D). Most of the elements in a Cartesian mesh will be hexahedra although the elements adjacent to the surface can be expected to assume a variety of polyhedral shapes depending on the way in which an Octree hexahedron intersects any given region of the boundary surface. A Cartesian mesh is therefore well suited for use by a finite volume or finite element method that can accept arbitrarily shaped elements. This approach has been developed extensively by [Aftosmis et al.]150. Given the close affinity between Cartesian meshes and Octree based triangulations it is to be expected that they share the same advantages and limitations. In particular, the problems of correctly finding the intersection between the Cartesian/Octree mesh and the boundary surface, identifying the element shapes for the intersected Cartesian cells and adequately refining the mesh near small boundary features, are substantial. Cartesian mesh methods also suffer

Figure 4.44

Example of Cartesian Grid on a Generic Airplane – (Source: Richard Smith 1996)

Carlson LA. Transonic Airfoil Analysis and Design Using Cartesian Coordinates. AIAA 2nd computational fluid dynamics conference, Hartford, CT, June 1975.p. 175–83. 150 Aftosmis MJ, Berger MJ, Melton JE. ,”Robust and efficient Cartesian mesh generation for component-based 149

Geometry”, AIAA J 1998; 36:952–60.

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from the drawback that the surface discretization is not known beforehand and it is therefore often difficult to ensure good surface mesh quality. On the plus side, since the surface discretization is a by-product of the volume discretization, it is possible to generate meshes around highly complex geometries without the need for carefully crafted surface meshes. In fact, the surface definition can be obtained directly from the CAD description provided there is a utility to determine the intersection of given line with the surface. Cartesian and Octree based mesh generation methods thus circumvent the need for the prior creation of a surface mesh, a significant advantage if a fast turnaround time in going from design prototype to flow solution is desired. Figure 4.44 shows a Cartesian grid on a generic airplane configuration.

Trimmed (SAMM) Cells This was a novel idea which was generated by [Wayne Oaks]151 of CD-Adapco® for better capturing geometry surfaces. The idea was to use the trimmed or deformed hexahedral cells (see Figure 4.45) or better known as SAMM (SemiAutomatic Meshing Methodology) to generate a sub-surface, then fill the gap with between sub-surface and real surface, with prismatic layer and the core volume with regular hexahedral cells. If there is any unresolved cells, then use the mesh quality button to fix that. Unfortunately, some meshing designers complained of too many unresolved cells. To that end, CD-Adapco added new features and fixes to decrease the amount Figure 4.45 Meshing Types in SAMM of unresolved cells. Figure 4.45 shows a SAMM approach for meshing a manifold showing step by step procedures. It is most useful in modeling external aerodynamic flows due to its ability to refine cell in a wake region, unsteady, and turbulent fluid caused by boundary layer separation.

Polyhedral Cells As a new comer in the field of Mesh Generation, polyhedral cells merits special attention in CFD community. Polyhedral meshes, as the term implies, means many faces. It consist of cells of 12 and 14 faces (although the number of faces is unrestricted). This means that they fill space in close to the most efficient way possible. A polygonal face is defined by a list of vertex labels. The ordering of vertex labels defines the face normal (orientation) using the right-hand rule. A polyhedral cell is defined by a list of face labels that bound it. In Figure 4.46 (a) cell center is marked by P, face center and face includes by f, face normal Sf and neighboring cell center by N. Face center and cell volume are calculated using a decomposition into triangles or pyramids152. Cell Decomposition It remains to choose an appropriate decomposition of a polyhedron into tetrahedral; two methods used in OpenFOAM® are shown in Figure 4.46 (b-c). A cell is decomposed by introducing a point in W. Oaks, S. Paoletti, “Polyhedral Mesh Generation”, adapco Ltd, 60 Broadhollow Road, Melville 11747 New York, USA. 152 Hrvoje Jasak, ˇZeljko Tukovi´c, “Automatic Mesh Motion for the Unstructured Finite Volume Method”, ISSN 1333–1124, UDK 532.5:519.6. 151

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its centroid and building tetrahedral above the triangular decomposition of a face. The two methods proposed here are the cell decomposition, Figure 4.46 (b), where additional points are introduced only in cell centers; and the cell-and-face decomposition, Figure 4.46(c), where points are introduced in both face and cell centers. In the first method, the number of algebraic equations in the matrix equals the sum of cell and point count, while the second method introduces an equation for each face, giving a considerable increase in the number of unknowns153. For a given resolution level, a mesh consisting of polyhedral cells has fewer faces than a mesh of any other cell type. As the lowestorder polyhedron, tetrahedral are often deformed during meshing to look more like wedges or slivers. Some of these point upstream, letting fluid flows hit their very oblique surfaces. The disparity between the small inflow area and large outflow areas leads to excessive numerical diffusion. Figure shows manifold and piston cylinder head meshing using polyhedral. (a) typical polyhedral cells

Figure 4.46

(b) Cell Decompostion

(c) Cell and Face Decompostion

Typical Polyhedral Cell and their Decomposition

Mesh Duality A different approach in generating polyhedral meshes, which does not suffer by the aforementioned restrictions, comes with the introduction of indirect mesh generation methods. These are based on the principle of duality transforms, which define a mapping from entities of an input mesh, which is referred to as primal, to a destination mesh, referred to as dual. The main mapping process dictates that the vertices of a dual mesh are generated at the centers of the primal cells154. This relation is unique, leading to a one-to-one correspondence of the two counterpart meshes, while it is also characterized by inverse applicability. This means that the original primal mesh can be obtained back, if the same mapping is applied to the dual mesh. This property can be applied for Voronoi tessellations, as well. The dual counterpart of a Voronoi mesh is a Delaunay triangulation, which is defined as a partitioning scheme, such that no vertex is inside the Figure 4.47 Polyhedral meshing circumcircle of any triangle (Figure 4.47). The using Delaunay triangulation implementation of Delaunay triangulation algorithms is 153

See previous.

154 H. Ledoux, “Computing the 3D voronoi diagram robustly: An easy explanation”, In Voronoi Diagrams in Science

and Engineering, 2007. ISVD '07. 4th International Symposium on, pages 117{129, July 2007.

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relatively simple and can be of complexity O (n log n), following Ruppert's algorithm155. As the duality property can be applied both ways, it is then possible to obtain a Voronoi mesh, by applying a duality transform on a previously generated Delaunay triangulation, considering the circumcenters of the primal tetrahedral as generator vertices. In 3D space, an equivalent mesh generation method would require a tetrahedral primal mesh that complies with the Delaunay criterion. Delaunay partitioning is known to maximize the minimum angle of all formed simples, which leads to well-conditioned tetrahedral. However, in order to obtain a valid dual mesh, a far stricter criterion needs to be filled: that of well centered tetrahedral, meaning that the circumcenter of a primal cell needs to be located within its volume156. This is something that is not always possible, as tetrahedral at the boundaries may be very at, having their circumcenters outside the model's domain, while the Delaunay criterion still remains fulfilled. Situations like this are especially encountered at sharp concavities of the geometric model, and several suggestions have been made in order to overcome this issue. Other possibilities include non-Delaunay tetrahedral meshes, hexahedral or even mixed meshes, which are further discussion. Finally, the advantage of indirect mesh generation lies in the fact that efficient algorithms can be implemented in order to obtain topologically involved dual meshes, based on primal meshes with simple topology. Furthermore, the primal meshes, themselves, can be created following equally efficient and wellstudied algorithms. This approach leads into an elective two-step mesh generation, rather than an expensive, direct one. Methodology Given a triangular mesh in 2D, such as that of (Figure 4.48), a polygonal mesh is formed, following the principle that a dual cell will be formed around every primal vertex. In the interior of the domain, this one-to-one correspondence between primal and dual entities extends to other types as well, with one dual edge per primal edge and a dual vertex for every primal face. However, generation of polygonal faces on the boundary demands for additional dual edges and vertices, at specific locations of the boundary that denote the classification of primal entities as significant. An slightly modified approach of the generic polygonal mesh generation, as previously described, is used to obtain a variation known as median meshes. This method differentiates itself by considering as significant every existing primal edge, thus creating dual vertices at the midpoints of primal edges lying in the interior as well. These dual vertices become, consequently, vertices of the dual faces formed around primal vertices in the interior, however the resulting polygons are characterized by highly concave shapes (Figure 4.48). Concave polygons are in general non-desirable in computational methods, due to their poor numerical properties. Additionally, they are also known for posing further difficulties in geometric computations, making the mesh generation itself problematic, whenever using plain topological relations is not adequate. It is, subsequently, clear that median meshes are not an optimal choice for numerical simulations. However, the concept behind median mesh generation provides a useful basis for meshing curved boundary surfaces of three-dimensional models, where the exact geometry of the primal mesh needs to be preserved.

Treatment of Boundary Layer The poor numerical properties of tetrahedral meshes have dictated the generation of a thin layer of prismatic, pentahedral elements at the boundary. With this common practice, analysts have been able to partially overcome the inability of tetrahedral to capture the details of a flow at regions close to Paul-Louis George and Houman Borouchaki, “Delaunay Triangulation and Meshing - Application to Finite Elements”, Editions HERMES, 1998. 156 Rao V. Garimella, Jibum Kim, and Markus Berndt, “Polyhedral mesh generation and optimization for nonmanifold domains”, Proceedings of the 22nd International Meshing Roundtable, pages 313-330. Springer International Publishing, 2014. 155

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the boundary . The appropriate generation of a corresponding boundary layer for polyhedral meshes, or even the need for one, are to be examined in future studies. An interesting, however, by-product of the polyhedral mesh generation method, is the automatic formation of a prismatic boundary layer. This effect comes as a result of connecting the centers of primal cells, in order to form dual entities, which, at the boundary, forms dual cells of approximately half the thickness of their primal counterparts. The phenomenon can be intensified by generating cascading dual meshes, having as a starting point an initial primal mesh. This is made possible, given the observation that the dual counterpart of a general bounded polyhedral mesh tends to resemble the primal, tetrahedral mesh, with the exception at the boundaries. This correspondence emerges in a similar way that the dual counterpart of a Voronoi tessellation is a Delaunay triangulation/tetrahedralization, and vice versa. Therefore, for each generation of meshes, the dual mesh that is obtained serves as the primal mesh for the next iteration. It can, then, be observed that for each generation, the boundary layer gets approximately half the thickness of that of the input mesh. Since two iterations are needed in order to cascade from a polyhedral mesh to a tetrahedral dominant and back to a polyhedral one, the formed boundary layer will conclude to a thickness of a 1/4 factor. It is, however, apparent that with such an approach it is difficult to control the properties and thickness of the formed boundary layer and the application of this method seems of limited use. Figure 4.49 and Figure 4.50 are displaying the involving concepts for 2D and 3D.

Domain Mesh Stretching in Unstructured Environment The drive towards full Navier-Stokes solvers has necessitated the development of stretched grid generation techniques in order to resolve the thin boundary layers, wakes, and other viscous regions characteristic of high-Reynolds number viscous flows . Proper boundary-layer resolution usually requires mesh spacing several orders of magnitude smaller in the direction normal to the boundaries than in the stream-wise direction, resulting in large cell aspect-ratios in these regions. In [Babushka & Aziz] it is shown how the accuracy of a two-dimensional finite-element approximation on triangular elements degrades as the maximum angle of the element increases. Therefore, stretched obtuse triangles that contain one large angle and two small angles are to be avoided, while stretched right-angle triangles, with one small and two nearly right angles, are preferred. Delaunay triangulations, which maximize the minimum angles of any triangulation, tend to produce equiangular triangulations and are thus ill suited for the construction of highly stretched triangular

Figure 4.48

Dual surface Triangulation resulting in Polyhedron

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elements. One of the earliest approaches for generating highly stretched triangulations for viscous flows makes use

2D

Figure 4.49

Boundary Layer Prisms Generated on a Cascade of a 2D Triangulation and Dual Polyhedron

of a Delaunay triangulation performed in a locally mapped space [Mavriplis]157-158, [Vallet et al]159, [Castro-Diaz et al 1995]. By defining a mapped space based on the desired amount and direction of stretching, an isotropic Delaunay triangulation can be generated in this mapped space that, when mapped back to physical space, provides the desired stretched triangulation. Difficulties with such methods involve defining the stretching transformations and determining suitable point distributions for avoiding obtuse triangular elements. An alternative to the above approaches is to generate a locally structured or semi structured mesh in the regions where high stretching is required. One approach [Nakahashi 1987, Ward & Kallinderis 1993] attempts to preserve the mesh structure in the direction normal to the boundary up to a Mavriplis, DJ. ”Adaptive mesh generation for viscous flows using Delaunay triangulation”, Journal computational. Phys. 1991. 158 Mavriplis, DJ, ”Unstructured and adaptive mesh generation for high-Reynolds number viscous flows”Proceedings of the International Conference on Numerical Grid Generation: Computational Fluid Dynamics and Related Fields, 3rd Barcelona, Spain, ed. AS Arcilla, J Hauser, PR Eisman, JF Thompson, pp. 79–92. New York: North-Holland, 1991. 159 Vallet MG, Hecht F, Mantel B., “Anisotropic control of mesh generation based upon a Voronoi type method”. 157

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specified distance away from the boundary, after which fully unstructured isotropic meshing techniques are employed. Special care must be taken in this case to avoid mesh cross-overs in regions of concave curvature and to ensure a smooth transition between the structured and unstructured region of the mesh. Another strategy [L¨ohner]160; [Pirzadeh]161; [Connell & Braaten 1995] consists of generating a semi structured mesh, where the “stack” of mesh cells emanating from each individual boundary face may terminate independently from those at other boundary faces, as shown. Termination of these “advancing-layers” [Pirzadeh]162 is triggered when the local cell aspect-ratio approaches unity, or when cross-over with other cells is detected, such as in concave corners. The remaining region is then gridded with a conventional isotropic unstructured mesh generation approach. The resulting structured or semi structured meshes can either be conserved as local structured entities of quadrilaterals in two dimensions and prisms in three dimensions (since the surface grid is generally assumed to be triangular), or the different element types may be divided into triangles or tetrahedral in two or three dimensions, respectively.

3-D

Figure 4.50

Concept of cascading for boundary layer in 3D

L¨ohner R.,” Matching semi-structured and unstructured grids for Navier-Stokes calculations”, AIAA 1993. Pirzadeh S.,”Viscous unstructured three dimensional grids by the advancing-layers method”, AIAA, 1994. 162 Pirzadeh S. 1994, AIAA J. 32(8):1735–37. 160 161

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Spatial (Field) Discretization The previous section describes techniques for generating suitable meshes about arbitrary geometric configurations. Once such a mesh has been generated, it serves as the basis for the spatial discretization of the governing fluid dynamic equations. Unstructured mesh discretization techniques can generally be classified as finite-volume or finite-element strategies, depending on whether a discrete integral or Variational viewpoint is adopted, although many common discretization may be simultaneously interpreted from both viewpoints. For finite-volume formulations, a distinction between vertex-based and cell centered schemes can be made. In a vertex based scheme, the flow variables are stored at the vertices of the mesh. The discrete equations usually involve stencils that are defined by the nearest or next-to-nearest neighbors of each mesh point. In a cell-centered scheme, the flow variables are stored at the centers of the cells or elements of the mesh. The stencils of the discrete equations usually involve the nearest or nextto-nearest neighboring cells. The equivalence between vertex-based and cell-centered schemes can be demonstrated with the concept of a dual mesh. If a dual mesh point is created at each cell center and dual Figure 4.51 Dual Mesh for Mixed Triangular-Quadrilateral mesh edges are drawn by joining Unstructured Mesh neighboring cell centers, the cellcentered scheme can be seen to be equivalent to a vertex-based scheme operating on the dual mesh. Figure 4.51 illustrates the dual mesh for a mixed quadrilateral and triangular mesh in two dimensions and associated control-volumes with edge-based fluxes for a vertex-based scheme. For a purely quadrilateral or hexahedral mesh, the dual mesh also contains quadrilateral or hexahedral elements, and the number of vertices and edges in the original and dual meshes is identical. For triangular or tetrahedral meshes, the number of dual mesh vertices is larger than the number of original mesh vertices. This is due to the fact that a triangular mesh contains twice as many triangles as vertices (neglecting boundary effects), and a tetrahedral mesh five to six times more elements than vertices. On the dual mesh, the degree of a vertex (number of incoming edges) is fixed and equal to three for triangular elements, or four for tetrahedral elements, whereas on the original mesh, the degree of each vertex is variable. Similar relationships hold for other elements such as prisms and pyramids. Thus, cell-centered and vertex-based schemes operating on the same grid result in vastly different discretization when non-quadrilateral or non-hexahedral elements are present. In particular, a cellcentered scheme can be expected to result in a much larger number of unknowns while generating relatively simple stencils of fixed size. The vertex-based scheme, on the other hand, will result in a smaller number of unknowns with larger variable-size stencils. Because of the larger number of unknowns, cell-centered schemes generally incur larger overheads than vertex-based schemes on equivalent grids. However, there is evidence to suggest that they also provide more accurate solutions on equivalent grids. The question of whether the additional overheads are offset by the increase in accuracy is still an open one, especially since vertex-based schemes operate on more complex stencils (more fluxes per unknown) than cell-centered schemes.

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Considerations for the Navier-Stokes Equation The above discussion on discretization is principally concerned with the purely convective terms that arise in the context of inviscid flows. With the exception of the finite-element methods described above, where convection and diffusion terms can be treated in a unified manner, additional viscous terms must be discretized for the Navier-Stokes equations. These terms are diffusive in nature and generally take the form of second differences. Stable second-order discretization can therefore be constructed using simple central differences. For cell-centered schemes, one approach consists of first computing the gradients at the mesh vertices and then averaging these to the cell faces in order to compute second derivatives at cell centers (Frink163). For vertex based schemes, a similar two-pass procedure based on finite-volume central difference arguments can be used to construct the viscous term discretization. For simplicial meshes, gradients can first be computed on the mesh elements, which can then be interpolated to the control-volume faces in order to form second differences at the mesh vertices. This type of discretization can be derived more formally using a Galerkin finiteelement procedure and assumes linear variation of the flow variables on the mesh elements. The resulting discretization produces a nearest-neighbor stencil and may be implemented as a single loop over the edges of the mesh, rather than as a two-pass procedure that computes intermediate cellbased gradients; [Barth]164 and [Mavriplis]165. For non-simplicial meshes, Galerkin finite-element formulations using bilinear or trilinear variations on quadrilateral or hexahedral elements can be constructed. The resulting stencils, however, are no longer compact, as they involve vertices within mesh elements that are not connected by a mesh edge, such as diagonally opposed vertices in hexahedral elements [Braaten & Connell166]. An alternative strategy for discretizing viscous terms for vertex-based schemes is to employ the vertex-based gradients already computed in the context of second-order upwind schemes (using a Green-Gauss integration around the vertex-based control volumes, for example), instead of the element-based gradients described above. A vertex-based second difference can then be computed by integrating these gradients themselves around the control-volume boundaries [Luo et al 1993]. This approach enables the viscous term discretization to be assembled on meshes of arbitrary element types using the same data structures as required for the upwind convection terms. The principal drawback of this method is that it results in a large stencil that involves neighbors and next to-neighbors, which on a structured mesh reduces to a stencil of size 2h, where h represents the mesh spacing. This not only reduces overall accuracy but is also ineffective at damping odd-even oscillations in the numerical scheme. For high-Reynolds-number flows of practical interest, the Reynolds-averaged form of the Navier-Stokes equations is generally employed, which requires the use of additional turbulence-modeling equation(s). Although algebraic models can be implemented on unstructured grids [Mavriplis]167, they were conceived for simple wall-bounded flows and are thus ill suited for flows over complex geometries. The current practice is to employ two-equation models of the K-ε or K-ω type, or simpler one-equation eddy viscosity models [Baldwin & Barth 1992], [Spalart & Allmaras]168. These equations contain convective, diffusive, and source terms that Frink NT. 1994. Recent progress toward a three dimensional unstructured Navier-Stokes flow solver. AIAA Pap. 94-0061 164 Barth TJ. 1992. Aspects of unstructured grids and finite-element volume solvers for the Euler and NavierStokes equations. Von Karman Inst. Lect. Ser., AGARD Publ. R-787 165 Mavriplis DJ. 1995b. A three-dimensional multigrid Reynolds-averaged Navier-Stokes solver for unstructured meshes. AIAA J.33(3):445–53 166 Braaten ME, Connell SD. 1996. Three dimensional unstructured adaptive multigrid scheme for the Navier-Stokes equations. AIAA J. 34(2):281–90 167 Mavriplis DJ. 1991a. Algebraic turbulence modeling for unstructured and adaptive meshes. AIAA J. 29(12):2086–93 168 Spalart PR, Allmaras SR. 1992. A one-equation turbulence model for aerodynamic flows. AIAA Pap. 92-0439 163

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can be discretized in a manner analogous to the discretization of the flow equations. Turbulence modeling equations often result in stiff numerical systems, and care must be taken to devise schemes that preserve positivity of the turbulence quantities at all stages of the solution process. A common practice is to discretize the convective terms using a first order upwind strategy, from which a positive scheme that obeys a maximum principle can be obtained.

Unstructured Quadrilateral Mesh Generation Because commercial aircraft are built with thin-walled structures, their structural performance is well-modeled using shell-element meshes [Hwang and Martins]169. However, creating these meshes for the full aircraft configuration can be challenging and presents a bottleneck in the design process, especially in a configuration-level design space. The aim is to presents an algorithm that automatically creates unstructured quadrilateral meshes for the full airframe based on just the description of the desired structural members. The approach consists in representing each node in the mesh as a linear combination of points on the geometry so that the structural mesh morphs as the geometry changes, as it would, for example, in aero-structural optimization. The algorithm divides the aircraft skin into 4-sided domains based on the underlying B-spline representation of the geometry. It meshes each domain independently using an algorithm based on constrained Delaunay triangulation, triangle merging and splitting to obtain a quadrilateral mesh, and elliptical smoothing. Examples of full configuration structural meshes are provided, and a mesh convergence study is performed to show that element quality can be maintained as the structural mesh is refined. Here, presented an automatic unstructured quadrilateral mesh generation algorithm for aircraft structures that uniquely satisfies the four requirements mentioned above. The algorithms starts with a B-spline surface geometry representation and a list of requested structural members defined in terms of parametric locations on the surfaces. It then splits the geometry into domains, meshes each domain independently using Constrained Delaunay triangulation (CDT) as well as merging and splitting operations, and then applies Laplacian smoothing as a final step. Geometry Representation The only requirements on the geometry representation are that it is continuous and watertight. Representing the geometry using untrimmed B-spline surfaces, though this is not the only choice with which the structural mesh generation algorithm would work. B-splines are piecewise polynomials used frequently in computeraided design because of their favorable mathematical properties: compact support for a desired order and smoothness, and flexibility in terms of the number of control points and polynomial degree. B-spline surfaces are tensor products of B-spline curves that maintain the advantages of smoothness and sparsity. Figure 4.52 illustrates how a conventional wing-bodytail aircraft geometry can be constructed

Figure 4.52 Conventional configuration geometry (a), final structural mesh (Courtesy of Hwang & Martins)

J. T. Hwang and J. R. R. A. Martins, “An unstructured quadrilateral mesh generation algorithm for aircraft structures”. Aerospace Science and Technology, 59:172182, 2016. doi:10.1016/j.ast.2016.10.010. 169

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with 4-sided B-spline surfaces170. Local Mesh Generation Algorithm In general, 2D quad meshing algorithms fall under three general categories: domain-decomposition, advancing-front, and triangulation-based methods. The first two recursively splitting the domain through heuristic algorithms and marching out from boundaries, respectively are not suitable for the current problem because of the line constraints imposed by the structural members intersecting the skin. Two additional ideas that have been successful are topology clean-up and smoothing. There has been work dealing with line constraints in structural mesh generation for marine engineering. The local mesh generation algorithm consists of six stages, as illustrated in Figure 4.53. The figure shows a domain for illustrative purposes, containing a vertical edge extending from the top to the bottom of the domain, two diagonal edges intentionally chosen to form a triangular region, and a shorter edge that is floating by itself near the center of the domain. The six stages are as follows:

1. Initial domain: We start with a 4-sided domain representing a single B-spline surface, with the internal members intersecting this surface pre-determined.

2. Discretization: We discretize the boundaries and the interior of the domain. The boundaries are simply discretized using a global parameter representing the requested resolution. This guarantees that the bounding edges shared by two neighboring domains always agree on the boundary nodes because all domains use the same resolution parameter. The interior of the domain is populated with a grid of points that are spaced based on the sizes of the element

Figure 4.53 170

Same as previous.

The Six Steps of the Unstructured Quad Meshing Algorithm

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3. 4. 5. 6.

boundaries, as measured from the preview mesh. Triangulation: We perform CDT on the domain while respecting the edges from the boundaries and from the intersecting structural members. Quad-dominant mesh: From the triangulation, we obtain a quad-dominant mesh by ranking all potential merges of adjacent triangles based on how close the angles would be to 90 degrees. The triangles are merged according to this ranking until no possible merges remain. Fully-quad mesh: We split all quads into four smaller quads and all triangles into three quads using its centroid to obtain a fully quad mesh. Smoothing: We perform an elliptical smoothing operator (Laplacian) as the last step.

Other noticeable sources regarding quad meshing are by [Remacle et al,]171, and [Verma & Tim Tautges]172.

Connectivity Information and Data Structure We must identify what information is required to identify the cell and all the neighbors of the cell in the computational mesh. We can choose to locate the arbitrary points anywhere we want for the unstructured grid. A point insertion scheme is used to insert the points independently and the cell connectivity is determined. This suggests that the point be identified as they are inserted. Logic for establishing new connectivity is determined once the points are inserted. Data that form grid point that identifies grid cell are needed. As each cell is formed it is numbered and the points are sorted. In addition the neighbor cell information is needed. Therefore, a brief discussion on data structures is useful because the success of most discretization methods ultimately depends on how efficiently they may be implemented173. Traditionally, finite-element methods have relied on element-based data structures, where for each element of the mesh a list of the forming vertex addresses is stored (i.e. four vertices for tetrahedral, eight vertices for hexahedra, etc.). For many fluid dynamics problems, the discretization, which are typically thought of as summations of fluxes, can be implemented more effectively using an edge-based data structure. For a vertex-based scheme, this corresponds to storing, for each edge of the mesh, the addresses of the two vertices on either end of the edge. For a cell-centered scheme, the relevant entity is the dual edge that joins two neighboring cell centroids and pierces the face common to these two cells. The discretization may be evaluated by computing a flux on each edge, which is then added to and subtracted from the respective control volumes on each end of the edge. In order to compute this flux, a face area must be stored for each edge, which corresponds to the area of the dual control-volume face associated with the mesh edge in the vertexbased scheme, and to the area of the cell face pierced by the dual edge in the cell-centered scheme. The use of edge-based data structures results in lower memory overheads and increased computational throughputs because redundant computations are eliminated and the gather-scatter required for vectorization in supercomputers is minimized. Furthermore, because sets of edges can be used as building blocks for arbitrarily shaped elements, hybrid meshes with mixed element types may be handled by a single edge-based data structure, at least for inviscid flows. For viscous flows, the Galerkin finite-element discretization of the diffusion terms on simplicial meshes results in a nearest-neighbor stencil and thus may be implemented using an edge-based data structure 174. J.-F. Remacle, J. Lambrechts, B. Seny, E. Marchandise, A. Johnen and C. Geuzaine, “Blossom-Quad: a nonuniform quadrilateral mesh generator using a minimum cost perfect matching algorithm”, Int. J. Numerical Meth. Eng. 2010; 00:1-6 172 Chaman Singh Verma and Tim Tautges, “Jaal: Engineering a high quality all-quadrilateral mesh generator”, Argonne National Laboratory, Argonne IL, 60439. 173 Wikipedia. 174 Mavriplis DJ, “Unstructured mesh generation and adaptively”, VKI Lect. Ser. Computational Fluid Dynamics, 26th, VKI-LS 1995. 171

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However, this is not the case for non-simplicial meshes, since the resulting stencils involve vertices that are not connected to the center vertex by a mesh edge (such as diagonally opposed vertices in hexahedral elements). In these situations, the element-based data structure must be retained175. An alternative is to resort to the thin-layer approximation of the viscous terms on non-simplicial meshes, which can be implemented exclusively along edges176. The limitations of this approach are obvious, although it is justifiable for highly stretched prismatic or hexahedral meshes, where stream wise resolution has been sacrificed for efficiency. An interesting property of the edge-based data structure is that it can provide an interpretation of the discrete operator as a sparse matrix. For nearest neighbor stencil discretization, all points in the stencil are joined to the center point by a mesh edge. The discretization operator can be written as a sparse matrix, where each nonzero entry in the matrix corresponds to a stencil coefficient or edge of the mesh. For systems of equations, the edges correspond to nonzero block matrix entries in the large sparse matrix. This interpretation has implications for the implementation of implicit and algebraic multigrid solution schemes 177-178. One of the disadvantages of the edge-based data structure is that it requires a preprocessing operation to extract a unique list of edges from the list of mesh elements and to compute the associated edge coefficients. For unsteady flows with dynamic meshes, this preprocessing must be performed every time the mesh is altered, although this may be done locally. Additionally, for dynamic grid cases, the element data structures are generally required for performing mesh motion or adaptation, since edge lists represent a lower-level description of the mesh179.

Braaten ME, Connell SD., “Three dimensional unstructured adaptive multigrid scheme for the Navier-Stokes equations”, AIAA J. 34(2):281–90, 1996. 176 Mavriplis DJ, Venkatakrishnan V.,”A unified multigrid solver for the Navier-Stokes equations on mixed element meshes”, .AIAA Pap. 95-1666, 1995. 177 Venkatakrishnan V, Mavriplis DJ., ”Implicit solvers for unstructured meshes”, J. Computational. Phys. 105(1):83–91, 1993. 178 Mavriplis DJ, Venkatakrishnan V., “A unified multigrid solver for the Navier-Stokes equations on mixed element meshes”, AIAA Pap. 95-1666, 1995. 179 Annual. Rev. Fluid Mech. 1997.29:473-514, journals.annualreviews.org by Pennsylvania State University. 175

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5 Hybrid Meshes A hybrid grid contains a mixture of structured portions and unstructured portions. It integrates the structured meshes and the unstructured meshes in an efficient manner. Those parts of the geometry that are regular can have structured grids and those that are complex can have unstructured grids. These grids can be nonconformal which means that grid lines don’t need to match at block boundaries180. In recent years due to accuracy consideration while capturing the physics (sunlayer in boundaries), and in the same time added flexibility in domain discretization (automated meshing using tetrahedral, polyhedral, etc.), received lots of attention. Figure 5.1 shows a Hybrid mesh obtained from a STL surface Figure 5.1 Hybrid Grid and Steady State Solution using an OpenFOAM© meshing module and solution for a steadystate incompressible turbulent flow.

Accuracy Consideration It is known that the truncation error of a finite volume discretization depends on the shape of the control volume. In particular, a trapezoidal approximation for a vertex based method, though nominally second order, becomes first order accurate unless the control volume possesses central symmetry181. For a vertex based discretization, the control volume associated with a given point typically corresponds to the boundary of the collection of elements incident at that point. For a cell centered discretization it is the element boundary that functions as the control volume. On a structured mesh of hexahedra, one can generally expect central symmetry at all mesh points unless there are extreme distortions in the mesh. A planar triangulation will have central symmetry if the centered discretization it is the element boundary that functions as the control volume. On a structured mesh of hexahedra, one can generally expect central symmetry at all mesh points unless there are extreme distortions in the mesh. A planar triangulation will have central symmetry if the triangles are all equilateral resulting in hexagonal control volumes for vertex based schemes. In an anisotropic layer of highly stretched triangles central symmetry can only be achieved if the mesh maintains a structured appearance (e.g. advancing layers) and all the diagonal edges are oriented in the same direction. In a tetrahedral mesh, however, it appears impossible to achieve central symmetry under any circumstances. The possible loss of second order accuracy in truncation error for the Euler equations is localized and the first order error terms tend to cancel each other when averaged over several elements. As a result, the global solution error should remain second order182. It is possible, however, that this canceling of first order truncation error does not occur viscous regions where second order derivatives of the flow variables play a significant role. If this is the case then it will be necessary to maintain central symmetry of the control volumes in boundary layer 180

From Wikipedia, the free encyclopedia.

181 Roe PL. Error estimates for cell-vertex solutions of the compressible Euler equations. ICASE Report No, 1987.

Giles MB. Accuracy of node-based solutions on irregular meshes. Eleventh international conference on numerical methods in fluid dynamics. Williamsburg, VA, June 1988. 182

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regions for Navier Stokes computations. Comparing Mesh Type for Viscous Accuracy After the emergence of so many different mesh types it is reasonable to assume that mesh generation has reached a plateau and that the future is unlikely to expand the choice of element types or meshing methods. Figure 5.2 lays out a subjective view of the suitability of different meshing methods for the computation of high Reynolds number Navier Stokes solutions. The structured Multi-block methods achieve good viscous accuracy but are time consuming to apply. Tetrahedral meshes with anisotropic elements in boundary layer regions are easier to create but their accuracy is suspect. Overset methods lie on the diagonal and thus represent a compromise between ease of use and their purported solution accuracy for viscous flows. The best meshing types should lie near the diagonal and be as far away from the origin as possible. It seems likely that the trade-off between accuracy and ease of use will shift so that perhaps one of the meshing methods will stand out as clearly superior in meeting the dual Figure 5.2 Comparison of different mesh types for RANS requirements of solution accuracy and Computations ease of application. In the best of all possible worlds one might hope that all mesh generation methods would one day meet this goal. At the time of writing it appears that composite multi-block meshes of hexahedra offer the best accuracy for RANS computations but the lack of an algorithm for automated block decomposition renders these meshes time consuming to create. At the other extreme, approach offers essentially fully automated mesh generation but the poor quality of mesh elements near boundary surfaces severely limits the accuracy of these mesh types, particularly for RANS computations. Overset meshes of hexahedra represent a compromise that lies between these two extremes; they are more complicated to set up than tetrahedral meshes and computations on overset meshes are arguably less accurate than comparable computations on composite multi-block hexahedral meshes. Effect of Prismatic Extrusion Sub-Layer in Viscous Layer If prism shaped elements are used in the viscous layer there will be central symmetry provided that there is good triangle quality in the lateral direction parallel to the boundary surface. By combining prismatic elements in viscous regions with a tetrahedral mesh for the inviscid part of the flow field one might expect to achieve solutions of the Navier Stokes equations that match the accuracy of computations on structured hexahedral meshes. Since the prism layer is unstructured in the lateral direction there is much more flexibility in handling complex geometries and a greater opportunity to achieve a high level of automation in the mesh generation process than would be the case with purely hexahedral elements. For these reasons, hybrid meshes of prisms and tetrahedral have considerable appeal as the best compromise to achieve accuracy in RANS computations, while permitting ease of mesh generation for complex configurations. Hybrid meshes consisting of prisms and tetrahedral were first proposed by [Nakahashi] and then developed extensively by

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[Kallinderis et al]. There is little hard evidence to support the contention that using prisms in the boundary layer region is more accurate than using tetrahedral, or that hybrid meshes achieve the same accuracy as composite multi-block meshes made up of hexahedra. In fact, the limited evidence that is available from an analysis of the results presented at the second drag prediction workshop tends to contradict both these beliefs. The extrusion layer is usually extruded with linear or exponential Figure 5.3 Constructions of Hybrid mesh stretching functions for desired spacing of viscous flow calculation. [Moxey, et al.]183 proposed an isoperimetric approach, whereby a mesh containing a valid coarse discretization comprising of high-order triangular prisms near walls is refined to obtain a finer prismatic or tetrahedral boundary-layer mesh.

Meshing Tools in CD-Adapco® There are different methodologies developed for Hybrid meshes. Each has its own merits and of course who you talking to. One such method developed by [Star-CD©]184 where their automated meshing is involved. The essence of the method is Inside-Out where most of applications are interior domain. By creating a sub-surface along actual surface, the interior mesh, composed of Hex and a transition layer (Tetrahedron), is filled. Once the interior is done, an extrusion layer will extrude from sub-surface to the surface with prismatic cells as depicted in Figure 5.3. The automated meshing where sub-surface mesh (orange) is clearly visible. Same procedure can be applied using the (Advanced Layer) on a T-Section for Figure 5.4 Predominantly polyhedral meshing predominantly polyhedral cells and boundary prisms as shown in Figure 5.4. A detailed description of the tools available, from CAD data to sub-models, is provided in Figure 5.6 in particular reference to advanced layer generation and automatic meshing. Other relevant details of automatic meshing is provided in Figure 5.5.

D. Moxey∗, M.D. Green, S.J. Sherwin, J. Peir´o, “An isoperimetric approach to high-order curvilinear boundarylayer meshing”, Computer Methods Applied Mechanics Engineering, 283 (2015) 636–650. 184User Guide STAR-CCM+ Version 8.06. 2013. 183

Sub-Models

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Figure 5.6

Meshing tools in CD-adapco

Trimmed cells (surface) with Hexahedral Core

All Tetraheadral Cells

Hybrid Meshing

Cell Layer Extusion for Trimmed Cells

Figure 5.5

Combined Volume and Extrusion Layer Meshes

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A Novel Methodology for Extrusion Layer Meshing A configuration of prismatic elements in boundary layers created by marching a surface triangulation on viscous walls along certain directions is a typical mesh for viscous-flow simulations [Zheng et al.]185. The quality of the resulting elements and the reliability of the meshing procedure thus highly depend on the computing strategy used to determine the marching directions. Here, we propose to compute a field of marching directions governed by Laplacian equations. This new approach can ensure the smooth transition of marching directions, and thereby lead to more desirable element shapes. To demonstrate the effectiveness of the proposed method for computing the normal vectors, a comparison was made with the traditional geometric method.

Figure 5.7

Meshes Generated by a) Proposed Algorithm and b) Leading Commercial Vendor

To generate the boundary-layer mesh, it first generated an anisotropic tetrahedral mesh around the model and then formed the boundary-layer mesh by combining three tetrahedral elements into a prism. In this software, the vertex normal direction is initially defined as the area weighted average of the face normal of the manifold of adjacent triangles, and is then smoothed and adjusted as necessary to guarantee the mesh quality and avoid collisions. Figure 5.7 a presents a cut-out view of the boundary-layer mesh generated by the proposed method near the intake of the engine. It can be seen that the normal at nodes of a same layer change in a smooth way, whereas the counterparts generated using a commercial code change more sharply (see Figure 5.7-b), and stretched elements can be observed in this region. To evaluate the quality of the generated prismatic elements, the scaled-aspect-ratio quality measure proposed in the literature was adopted in this study. It has been reported that this quality measure in effect combines the measures of triangle shapes and edge orthogonality186.

Mesh Refinement Mesh movement or R-refinement (to be discussed later) has applications in both steady and unsteady flows. For example, R-refinement may be useful for optimizing the normal resolution of the mesh in a developing boundary layer or for clustering points around shock waves [Palmerio187, Yao Zheng, Zhoufang Xiao, Jianjun Chen, and Jifa Zhang, “Novel Methodology for Viscous-Layer Meshing by the Boundary Element Method”, AIAA Journal Vol. 56, No. 1, January 2018. 186 See Previous. 187 Palmerio B.,” An attraction-repulsion mesh adaption model for flow solution on unstructured grids”, Computational Fluids 23(3):487- 506. 185

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Castro-Diaz et al]. For transient problems with moving boundaries, R-Refinement is indispensable. Difficulties center on conserving a minimum degree of grid quality under severe deformations. In areas where grid deformation becomes unacceptable, local reconnection through swapping techniques can easily be performed, but only for simplicial meshes [Kennon et al, Baum et al], [Venkatakrishnan & Mavriplis]188. In practice, combinations of H and R-Refinement are often employed [Baum et al.]189, [Castro-Diaz et al]190 for such cases. Central to the implementation of any solution-adaptive scheme is the ability to detect and assess solution error (P-Refinement). The construction of a suitable refinement criterion represents the weakest point of most adaptive strategies. The main problem is that an exact characterization of the error requires a knowledge of the solution itself, which is obviously impractical. Most error estimates are based on the assumption that the solution is smooth and asymptotically close to the exact solution. This is often not the case for fluid dynamics problems, which are governed by nonlinear hyperbolic partial-differential equations. Solutions may contain discontinuities, and downstream flow features often depend on adequate resolution of upstream flow features. Most refinement criteria are heuristically based on (undivided) gradients and/or second differences of the various flow variables. Conservative criteria (i.e. over-refining) are often employed to compensate for the inability to accurately characterize the true solution error. Because adaptive meshing results in different mesh topologies for each simulation, even when the geometry of the problem is unchanged, parametric studies (typically used in design processes) are complicated by the requirement to distinguish between grid-induced and physical solution variations. Nevertheless, for problems with disparate length scales, adaptive meshing is often indispensable for resolving small flow features, and their full potential awaits the development of more well-founded adaptive criteria. Mesh adaptation, often referred to as Adaptive Mesh Refinement (AMR), refers to the modification of an existing mesh so as to accurately capture flow features. Generally, the goal of these modifications is to improve resolution of flow features without excessive increase in computational effort. We shall discuss in next chapter on some of the concepts important in mesh adaptation. Mesh adaptation strategies can usually be classified as one of three general types: R-refinement, Hrefinement, or P-refinement as depicted in Figure 5.8.

(b) H-refinement

(a) R-refinement Figure 5.8

(c) Hanging Node

Adaptive Mesh Refinement Types

Venkatakrishnan V, Mavriplis DJ.,”Implicit method for the computation of unsteady flows on unstructured grids”, Proc. AIAA CFD Conf., 12th, San Diego. AIAA Pap. 95-1705-CP 189 Baum JD, Luo H, L¨ohner R., ”A new ALE adaptive unstructured methodology for the simulation of moving bodies”, AIAA Pap. 94-0414. 190 Castro-Diaz MJ, Hecht F, Mohammadi B.,”Anisotropic unstructured mesh adaptation for flow simulations”, Int. Mesh Roundtable, 4th, Albuquerque, NM, pp. 73–85. 188

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R-refinement This is the modification of mesh resolution without changing the number of nodes or cells present in a mesh or the connectivity of a mesh. The increase in resolution is made by moving the grid points into regions of activity, which results in a greater clustering of points in those regions. The movement of the nodes can be controlled in various ways. On common technique is to treat the mesh as if it is an elastic solid and solve a system equations (subject to some forcing) that deforms the original mesh. Care must be taken, however, that no problems due to excessive grid skewness arise. (See Figure 5.8 (a)). H-refinement The modification of mesh resolution by changing the mesh connectivity. Depending upon the technique used, this may not result in a change in the overall number of grid cells or grid points. The simplest strategy for this type of refinement subdivides cells, while more complex procedures may insert or remove nodes (or cells) to change the overall mesh topology. In the subdivision case, every "parent cell" is divided into "child cells". The choice of which cells are to be divided is addressed in Figure 5.8 (b). For every parent cell, a new point is added on each face. For 2D quadrilaterals, a new point is added at the cell centroid also. On joining these points, we get 4 new "child cells”, (3 in tetrahedral). Thus, every quad parent gives rise to four new off springs. The advantage of such a procedure is that the overall mesh topology remains the same (with the child cells taking the place of the parent cell in the connectivity arrangement). It is easy to see that the subdivision process increases both the number of points and the number of cells. An additional point to be noted is that this type of mesh adaptation can lead to what are called "hanging nodes." In 2D, this happens when one of the cells sharing a face is divided and the other is not, as shown in Figure 5.8 (c). For two quad cells, one cell is divided into four quads and other remains as it is. The highlighted node is the hanging node. This leads to a node on the face between the two cells which does not belong (properly) to both of the cells. The node "hangs" on the face, and one of the cells becomes an arbitrary polyhedron. In the above case, the topology seemingly remains same, but the right (undivided) cell actually has five faces. Figure 5.9 show a mesh modeling supersonic flow around a space shuttle in

Figure 5.9

An H-refinement mesh about a Shuttle-like body (left) and Computed CP (right)

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which h-method adaptivity has been employed to optimize the mesh structure to produce accurate simulation of flow features important in assessing the performance of the design such as the profiles of pressure distribution shown191. 5.3.2.1 Isotropic vs. Anisotropic Meshing There are two ways of H-Refinement: Isotropic and Anisotropic. Anisotropic is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. In isotropic refinement, new points are added in both the directions, say x and y. In anisotropic refinement, the division takes place in one predominant direction (see Figure 5.10). Thus, in short, an isotropic refinement for a quad would produce four new off springs, while anisotropic refinement would only generate two. Anisotropic refinement is made use of, when the user knows that the flow feature is predominantly to be resolved in one direction, for e.g. Boundary Layers. However, there are situations where an anisotropic refinement alone may not be satisfactory, such as a shock-boundary layer interaction.

Figure 5.10

Isotropic vs. Anisotropic Meshing

P-refinement A very popular tool in Finite Element Modelling (FEM) rather than in Finite Volume Modelling (FVM), it achieves resolution by increasing the order of accuracy of the polynomial in each element (or cell). In AMR, the selection of "parent cells" to be divided is made on the basis of regions where there is appreciable flow activity. It is well known that in compressible flows, the major features would include Shocks, Boundary Layers and Shear Layers, Vortex flows, Mach Stem, Expansion fans and the like. It can also be seen that each feature has some "physical signature" that can be numerically exploited. For e.g., Shocks always involve a density/pressure jump and can be detected by their gradients, whereas boundary layers are always associated with rotationally and hence can be detected using curl of velocity. In compressible flows, the velocity divergence, which is a measure of compressibility is also a good choice for shocks and expansions. These sensing parameters which can indicate regions of flow where there are activity are referred to as Error Indicators and are very popular in AMR for CFD. The spectral order p of the approximation is raised or lowered to control error. In finite element methods or boundary element methods, the order p corresponds to the degree of the polynomial shape function used over an element. Just as refinement is possible by Error Indicators as mentioned above, certain other issues also assume relevance. Error Indicators do detect 191

The National Academies Press, “Research Directions In Computational Mechanics”, 1991.

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regions for refinement, they do not actually tell if the resolution is good enough at any given time. In fact the issue is very severe for shocks, the smaller the cell, the higher the gradient and the indicator would keep on picking the region, unless a threshold value is provided. Further, many users make use of conservative values while refining a domain and generally end up in refining more than the essential portion of the grid, though not the complete domain. These refined regions are unnecessary and are in strictest sense, contribute to unnecessary computational effort. It is at this juncture, that reliable and reasonable measure of cell error become necessary to do the process of "coarsening", which would reduce the above-said unnecessary refinement, with a view towards generating an "optimal mesh". The measures are given by sensors referred to as Error Estimators, literature on which is in abundance in FEM, though these are very rare in FVM. Control of the refinement and/or coarsening via the error indicators is often undertaken by using either the 'solution gradientt' or 'solution curvature'. Hence the refinement variable coupled with the refinement method and its limits all need to be considered when applying mesh adaptation.

Mesh Modification Operators The tetrahedral region of the grid is locally refined by means of a constrained Delaunay refinement algorithm combined with a circumcenter point placement strategy192. Any inconsistency between the circumradius of a tetrahedron and a desired point density triggers the point insertion procedure. This iterative cell refinement is repeated until the cell circumradii are consistent with a prescribed point spacing. Coarsening Triangulation Regions Coarsening of the tetrahedral region is also permitted through an edge collapse procedure. In regions where the grid is distorted or where solution errors are negligible, edges may be selected for removal. All cells incident to the deleted edge are removed from the mesh, the adjacent cells are redefined, and the two nodes of the edge are collapsed to a single vertex. The prismatic and

Figure 5.11 192 Cavallo,

Coarsening by Edge Collapsing – Courtesy of [Cavallo]

P.A., Sinha, N., and Feldman, G.M., “Parallel Unstructured Mesh Adaptation For Transient Moving Body And Aeropropulsive Applications”, Combustion Research and Flow Technology, Inc. (CRAFT Tech), Pipersville, PA.

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hexahedral regions of the grid may be refined through cell subdivision procedures. As the boundary of the tetrahedral region is refined the adjacent prism layers are also modified. This is accomplished by splitting edges at the tet/prism interface, and propagating this subdivision down to the wall through all of the layers. In addition, a procedure is in place to refine entire layers of prisms if an improved boundary layer resolution is desired. The algorithm loops on all the edges marked for coarsening at a certain mesh level, and an attempt is made at collapsing it to one of its end vertices, M01 or M02. Before physically performing the collapsing, checks are performed to ensure that the collapsing is topologically and geometrically possible, and that the quality of the created mesh regions is above a predetermined threshold. Finally, the target vertex for the collapsing, say M01, that produces the best triangulation with respect to a given mesh quality measure is chosen. A limit can be placed on the longest edge to collapse, in order to avoid the creation of excessively large elements. In the case of collapsing to M01, the algorithm proceeds by deleting all the mesh regions connected to M02, creating a polyhedral cavity within the mesh. The edge collapsing is then completed connecting all the faces of the cavity to M01 in order to form the new mesh regions. The procedure is illustrated in Figure 5.11. 5.4.1.1 Case Study - Numerical Testing for Engine Nacelle In this example we consider the model of an engine inlet with a center body. The initial CFD mesh of

Figure 5.12

Hierarchy of Successively Coarser Meshes Obtained by Uniform Coarsening for the Nacelle Model for four levels

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119,861 tetrahedra was generated with the Finite Octree mesh generator193. A 4-level multigrid was then generated by means of uniform coarsening marking all the edges for DE-refinement, obtaining coarser levels of 24,619, 6,477 and 1,819 tetrahedra, respectively. We set a limit of 1600 to the largest dihedral angle generated during coarsening, while we targeted for optimization all the regions with at least ne angle above 1450. The coarser meshes are presented in Figure 5.12 (a), (b), (c) and (d). As opposed to the refinement procedure, an improvement of the mesh quality cannot be expected % 1 since coarsening introduces constraints by deleting degrees of freedom. Nonetheless, the coarsening procedure was able to reduce the number of elements by a factor or approximately 74 in three levels. The final mesh has 1,710 elements, where 96% of the them have a largest dihedral angle below the value of 1450. The usage of mesh (d) for numerical computations is clearly limited, since it gives a poor discretization of the complex curved model, but the goal of this example is mainly to show that we are able to control the quality of the meshes even for a large coarsening ratio. Selection or the right coarse mesh for a specific problem depends strongly on the type of analysis to perform and it is not investigated here. The effect of locally improving the mash using the retriangulation procedures during coarsening was investigated. The initial mesh was redefined six times without optimization after each edge collapsing. The final coarse mesh is denoted by 6,512 elements. In this case, for facilitating the coarsening process, the constraint on the largest dihedral angle was relaxed from 1600 to 1750. 5.4.1.2 Coarsening With/Without Local Re-Triangulation The ratio of coarsening is given in Figure 5.13. The diagram clearly indicates that coarsening without local retriangulation is not able to produced more than one or two coarser meshes. In fact, due to the increased number of badly shaped elements after each coarsening, constraints are introduced in the process and most of the attempted edge collapsing fail. In contrast, coarsening using local retriangulation after each edge collapsing is able to maintain nearly constant, slightly increasing coarsening ratio. This gives the possibility to use the coarsening procedure to create a coarse mesh as coarse as it is necessary. Refinement of Triangulation Region The refinement scheme adopted in this work is also edge based194, in the sense that edges marked for refinement are split. The algorithm implements all possible subdivision patterns corresponding to all possible configurations or marked

Figure 5.13

Coarsening ratio for coarsening with and without local Retriangulation.

Hierarchy of successively coarser meshes obtained by uniform coarsening for the nacelle model. (a) Level 1: initial base mesh (119,861 tetrahedra); (b) Level 2: first DE-refined mesh (24,619 tetrahedra); (c) I.evel 3: second DE-refined mesh (6,477 tetrahedra); (d) Level 4: third DE-refined mesh (1,819 tetrahedra) 194 H.L. De Cougny and MS. Shephard, 'Local modification tools for adaptive mesh enrichment and their parallelization', Scientific Computation Research Center, RPI, submitted to Comp. Meth Appl. Mech. 193

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edges, to allow the maximum flexibility in how mesh refinement is accomplished. A limit can be placed on the shortest edge to split, in order to avoid excessive refinement. In the presence of curved newly generated vertices classified on model boundary must be properly placed on the true geometric boundary. This snapping procedure is critical, in the sense that it the mechanism through which refinement of a given mesh improves the geometric approximation that the mash gives of the geometry. This operation relies on the interaction of the refinement procedure with the geometric modeler storing the geometric information, and the classification information of the mesh entities. The snapping of a undefined vertex can produce invalid regions of negative volume or of poor quality. In this case, the retriangulation procedure explained in the following is applied and repeated until the vertex can be successfully snapped. For supporting the computation of the restriction and prolongation operators, a double link is stored from the vertex to the edge and back, together with the value of the split location along the edge. This is realized "on the fly" during refinement of each marked edge. Clearly, not even a local search is needed in this case. Please consult [Bottasso et al.]195 for further info. 5.4.2.1 Local Re-Triangulation Local retriangulation algorithms are an important aspect of any automated mesh modification procedure, their goal being the control and the improvement of the quality of a mesh with respect to a given criterion. The optimizing procedures implemented in this work are edge removal and multiface removal, which do not change the number of vertices, and edge collapsing and splitting of one or more edges, faces or a region, which remove or add vertices Edge removal deletes an edge from a mesh by introducing one or more faces, depending on the number of regions surrounding the edge. Face or mufti-face removal represents the dual operation, removing one or more faces by introducing a new edge. Figure 5.14 gives an example of these swaps for the simplest configuration, a 3 to 2 swap of the three elements [M01, M02, M0α, M0β] , [M02, M03, M0α, M0β] and [M03, M01, M0α, M0β] surrounding edge [M0α , M0β]. The 3 to 2 swap is performed by introducing a new face [M01, M02, M03]

Figure 5.14

3 to 2 and 2 to 3 Swap

Carlo L. Bottasso, Ottmar Klaas, Mark S. Shephard, “Data Structures and Mesh Modification Tools for Unstructured Multigrid Adaptive Techniques”, Article in Engineering With Computers · January 1998. 195

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and by deleting the edge {M0α, M0β], yielding two new elements and [M01, M02, M03, M0β]. The reverse operation (2 to 3 swap) is given by deleting the face [M01 , M02; , M03] and introducing the edge [M0α M0β]. As previously explained, edge collapsing removes an edge by merging two vertices and removing all regions connected to that edge. The splitting procedures introduce new vertices on one or more edges of a region. The new configuration is then given by applying the corresponding refinement subdivision pattern. The procedures are region based, in the sense that the algorithm tries to improve all regions which violate a given mesh quality criterion. Given the impact that large dihedral angles usually have on the condition number of the discrete problem that approximates the set of PDE's to be solved, we typically use dihedral angles as optimization targets. As a first. step, the dihedral angles of all elements considered for optimization are calculated. Each element violating a user defined threshold value is put into a linear list. Depending on the configuration of each element in the list (number of dihedral angles above the threshold value, topological or geometrical constraints, etc.), a suitable subset of the above mentioned optimization procedures is applied to eliminate that element in favor of improved elements. The procedures might fail for a specific element if the resulting configuration is topologically or geometrically not valid, or if they lead to a degradation of the quality of any element involved in that local retriangulation. In this case, or if the element is improved but the largest dihedral angle is still above the threshold value, the element is considered for improvement in a second pass, after all elements have been processed. Since the neighborhood of the elements that failed in the first pass may have been modified, it is possible that they can be fixed in a second pass. The procedure is repeated until a given threshold value is reached or no further improvement can be achieved. The local retriangulation algorithm is used to improve the meshes produced by the refinement and coarsening procedures. Since the refinement is an edge based operation that takes into account all passible subdivision patterns, refining an element is a localized procedure that does not affect the neighborhood of that element, and consequently the local retriangulation can be performed after the refinement procedure is completed. The situation is different when coarsening is considered. The coarsening procedure itself tends to give a mesh of poor quality, since collapsing of an edge has a strong costly negative impact on the dihedral angles of the surrounding elements. prevent losing control of the mesh quality, especially when multiple coarsening steps are performed, it. is necessary to introduce a threshold value to be satisfied by the largest dihedral angle in each of the newly generated elements. However, this usually represents a strong constraint and prevents a large coarsening ratio. It is therefore advantageous to improve the neighborhood of a to-be-removed element before picking the next edge for collapsing. This can be done by sending a list of regions connected to the target vertex of the edge collapsing to the local retriangulation procedure, after each edge collapsing is performed. Such a locally improved mesh makes the next edge collapsing more likely to lead to acceptable elements. Local procedures, such as the ones here considered can only lead to optima with respect to the quality of the triangulation. Nonetheless, these tools have been proven to be valuable in controlling the degradation of the mesh quality during its adaptive modification. They also find application in the context of curved model boundaries, when snapping of newly created vertices can create invalid or poorly shaped regions. In this case, local retriangulation tools can be used for eliminating those regions that prevent snapping196 . Refinement of Hexahedral Region (Near Wall) The refinement of the hexahedral region of the grid is accomplished using the pattern formation

H.L. De Cougny and MS. Shephard, 'Local modification tools for adaptive mesh enrichment and their parallelization', Scientific Computation Research Center, RPI, 'Eoy, NY, submitted to Comp. Meth Appl. Mech. 196

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procedure of [Biswas and Strawn]197, which employs a parent-child data structure to split the cells. Each hexahedral cell is then split according to a pattern, to generate 2:1, 4:1, or 8:1 sub-divisions. This cell subdivision creates buffer cells, which are tetrahedral, pyramid, or prismatic elements used to transition between different levels of hexahedral refinement, thus ensuring a conforming mesh with no hanging nodes. An initial point spacing ρ0 is defined for each vertex as the average edge length for all edges incident to the node. A larger grid spacing produces an iterative coarsening of the mesh. Using the mesh deformation measure, we modify the local grid spacing to be

ρ=

ρ0 σ where τ = min τ σ max

Eq. 5.1

And τ represent the dilatation of the tetrahedron in each of three principal directions, and are equivalent to the singular values of the transformation matrix. Note that the more deformed the cell, the larger the prescribed spacing, and hence an increased amount of coarsening will be performed. This improves the likelihood of the distorted cell being removed. Conversely, the enrichment procedures are invoked by specifying a smaller spacing. After the coarsening phase, an appropriate gradation of cell size is restored by solving a Laplace equation for ρ, using the boundary mesh spacing as Dirichlet boundary conditions. An approximate solution is obtained by summing the difference in the point spacing for all edges N incident to the node using a relaxation technique.

ρ

n +1

(

ε N n n = ρ +  ρk − ρ N k =1 n

)

Eq. 5.2

Prescribing new point spacing also drives solution-based coarsening and refinement. A variation on the solution error estimate developed in two dimensions by198 has been implemented in three dimensions for arbitrary mesh topologies. The method is based on forming a higher order approximation of the solution at each mesh point using a least squares approach. The difference between the higher order reconstruction from incident nodes and the current solution forms the error measurement. If the current mesh is sufficiently fine to support the spatial variation in the solution, the estimated error will be low, allowing coarsening to take place. Conversely, a high degree of error indicates additional refinement is needed. 5.4.3.1 Improvement to Near-Field Grid Generation Procedure (Hexahedral) The successful drag prediction workshop series set the focus of its fourth gathering (DPW4) in the blind prediction of drag and moment coefficients of the NASA common research model (CRM) transonic wing-body-tail configuration199. One of the main objectives of DPW4 is to evaluate the performance of state-of-the-art Navier-Stokes codes, thus this study documents some of the steps undertaken at the Institute of Aerodynamics and Flow Technology of DLR, to prepare the contribution to the DPW4. To identify possible CFD areas needing additional research and development, both standard procedures were used in this study and advanced methodologies, such 197 Biswas,

R., and Strawn, R.C., "Tetrahedral and Hexahedral Mesh Adaptation for CFD Problems", NAS Technical Report NAS-97-007, 1997. 198 Ilinca, C., Zhang, X.D., Trépanier, J.-Y., and Camarero, R., “A Comparison of Three Error Estimation Techniques for Finite-Volume Solutions of Compressible Flows, “Computer Methods in Applied Mechanics and Engineering, Vol. 189, pp. 1277-1294, 2000. 199 Simone Crippa, “Application of Novel Hybrid Mesh Generation Methodologies for Improved Unstructured CFD Simulations”, 28th AIAA Applied Aerodynamics Conference - CFD Drag Prediction Workshop Results, 2010.

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as new grid generation methods and advanced turbulence models. The input values required by the advancing layer grid generation process, first layer spacing and expansion ratio, have to be chosen wisely. It is important to resolve the same physical region on the various grid levels with the same element types. A similar near-field extent normal to the walls, guarantees that the transition location from hexahedral/prismatic elements to tetrahedra is similar between the grid levels. Having the element type transitions in the same physical region allows to capture in a self-similar way, on all grid levels, eventual discretization errors. The relations between the grid levels in terms of first layer spacing and number of wall-normal layers should follow the scaling factor given above. Given the requirement for a self-similar total layer thickness, scaled first layer spacing and scaled total amount of layers, leaves only the expansion ratio as variable to be determined. The geometric series for the total layer thickness (H) is n

1 − qn+1 H = ∑ a. q = a 1−q n

i=1

Eq. 5.3 where the total number of layers is N = n + 1, the expansion ratio is q and the first layer spacing is a. Keeping the total layer thickness between two grid levels constant (H1 = H2), results in

1 − 𝑞1𝑛+1 1 − 𝑞2𝑛+1 a1 = a2 1 − q1 1 − q2

Eq. 5.4 Hereby the relation between a1 and a2, as well as N1 and N2, is set by the scaling factor ∛3; for example, with grid level 2 being finer than grid level 1, follows a2 = a1/∛3 and N2 = N1 / ∛ 3. Starting with a sensible value for the expansion ratio q1 and a total amount of layers N1, the only unknown in Eq. 5.4 is q2, which can be computed iteratively. For the DPW4 grid-convergence family, the values for the coarse and _ne levels are derived from the medium grid. The first layer spacings given in the gridding guidelines are used, as the scaling factor of 1.5 is sufficiently close to ∛3. The resulting values for the near-field mesh are summarized in Table 5.1.

Table 5.1

Near-Field Grid Details.

Note that for full consistency, the number of wall-normal layers with constant spacing should also be scaled by ∛ 3, but neglecting this was not deemed of primary influence to the results. Furthermore, note that the expansion ratios of Error! Reference source not found. fulfill the requirement given in t he gridding guidelines only for the medium and fine grids. The expansion ratio of the coarse grid is larger than the defined, maximal value of 1.25. A comparison of the three final grids is shown in Figure 5.15, where the self-similar relation between the three levels is visible in the highlighted region. In wall-tangential direction, the factor of approx. 1.5 can be recognized by the cascade of 2, 3

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and 4.5 quadrilateral elements. A similar wall-normal total layer extent is also recognizable.

Figure 5.15

Comparison of Coarse, Medium and Fine Grids: lateral view on fore-body with Symmetry.

Discretization Improvement Through Chimera Technique for Sharp Corners A majority of contributions detected small separated regions at the trailing edge of the wing and tail plane junctions with the body. The existence or absence of the separation bubbles was found to be neither coupled to a solver type (unstructured or structured) nor a specific physical modelling approach. The results on the Solar grids from two participants, using in total four different turbulence models, do not feature these small separations. An insight gained through the adjoint dissipation error evaluation, is that the wing-body and tail plane-body junction regions are not discretized sufficiently well. The span-wise field cut at x =1400 inch (wing-body) reveals that the contracted near-field mesh leads to very large tetrahedra in the concave corners that are not adequate to resolve the edge of the boundary layer, see Figure 5.16. A simple solution to this problem is not known,

Figure 5.16

local dissipation error of drag coefficient on field cut-plane at x=1400 inch; isometric/downstream view

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thus this problematic region was not fully fixed for the contribution to DPW4. The near-field layer contraction in these concave regions cannot be completely excluded from the Solar grid generation process, thus a solution is sought on the solver side. Since recently,9 the TAU code has the capability to compute on chimera (overset) grids with overlapping viscous boundaries. To make use of this capability, a fully hexahedral grid was generated with a C-H topology around the complete wing airfoil and some of the wake at the wing root. The grid spacing in the overlap/interpolation region is similar to the medium Solar grid, but the resolution at the wing-body junction is improved due to the chosen H-topology, as opposed to an O-grid topology. The five million elements, hexahedral grid is referred to as SolarChimera5, whereas the initial medium grid plainly Solar. A comparison between the Solar and SolarChimera5 grids at the wing-body junction is shown in Figure 5.17.

Figure 5.17

Comparison of SolarChimera5 and Solar Grid at x =1454 inch plane; Viscous Wall Surface in Dark grey, _eld cut in white.

Case Study 1 - Hybrid Unstructured Meshes for Common Research Model (CRM & JSM) via ANSA® In this work an unstructured CRM Common Research Model meshing approach is taken GMGW Geometry and Mesh Generation Workshop using the commercial software HLPW High Lift Prediction Work ANSA®, developed by BETACAE Systems [Skaperdas and JAXA Japan Aerospace Exploration Agency Ashton]200. This describes the JSM JAXA high-lift configuration Standard Model meshing process within ANSA MAC Mean Aerodynamic Chord as well as an analysis of the STEP Standard for the Exchange of Product unstructured grids, similar to the tools provided in CDTable 5.2 Abbreviations Adapco®. It is produced for the 1st AIAA Geometry and Meshing Workshop and 3rd AIAA High-Lift Workshop. Particular focus is made on the process to generate suitable grids for various CFD codes including OpenFOAM and StarCCM+. Some of the Abbreviations is been provided in Table 5.2 for clarity.

Vangelis Skaperdas, and Neil Ashton, “Development of high-quality hybrid unstructured meshes for the GMGW-1 workshop using ANSA”, AIAA, January 2018. 200

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Geometry and Mesh Generation Background The AIAA Drag Prediction and High Lift Prediction (DPHLP) provide an opportunity for engineers in the aerospace sector to present and exchange information on the latest CFD methods and tools and to directly compare these methods on open-source geometries. The 1st Geometry and Mesh Generation Workshop took place under the umbrella of the 3rd High Lift Prediction Workshop. It was the first of its kind to focus specifically on the details of preparing high fidelity mesh models for CFD simulations. The aim of was to assess the current state of the art in geometry pre-processing & mesh generation technology and to bring together meshing specialists to discuss challenges and possible solutions. The models studied were NASA’s Common Research Model and JAXA’s high-lift configuration Standard Model. The JSM model was not studied in the workshop but given its inclusion in HLPW-3, the same mesh procedure was applied to both geometries. Both geometries are available in high lift configuration with slats and flaps extracted while the JSM model is also available with an optional engine nacelle and pylon. The main goal for participating in HLPW-3 was to assess the open-source CFD code OpenFOAM. Many aerospace specific CFD codes are restricted for national security reasons e.g., NASA CFD codes are typically not available to researchers in the UK or Greece. Whilst a number of commercial CFD codes are routinely used for industrial aerospace simulations, the ability to implement custom turbulence models, numerical schemes and algorithms means that these are not ideal for research and collaboration. Open source CFD codes like OpenFOAM, SU2 and Saturne have grown in popularity in recent years as a growing movement of international collaboration that is improved by the ease of sharing . Whilst OpenFOAM has its own mesh generation utility; SnappyHexMesh, a Cartesian-prismatic unstructured generation tool, the experience of the authors have shown that it is not suitable for low y+ grids and the region between the prismatic and Cartesian is often subject to severe non-orthogonality and large cell size jumps. For this reason an alternative mesh generator is used, which is capable of generating high-quality grids that represent the kind of unstructured grids that are typically used by the aerospace industry; ANSA® 17.1, a pre-processor from BETA-CAE System. Geometry Handling The geometries for the CRM and JSM high-lift models were downloaded from the HLPW-3 website201. A multitude of CAD file formats were available and for this project the STEP format was selected for both CRM and JSM models. 5.5.2.1 The CRM Model The STEP file of the HL-CRM model is in inches. It has a MAC of 275.8 inches and it represents a full scale aero plane model. No clean-up was required as the geometry had already been cleaned. Based on the GMGW/HLPW-3 meshing guidelines document, a hemi-spherical domain, suitable for imposition of far field boundary conditions, with a radius of 100 times MAC was created in ANSA and connected to the half symmetric airplane model. The raw CAD model was separated only in two zones, the slat and the whole remaining model. In order to facilitate meshing and pre-processing, we separated the model in 17 zones as revealed in Figure 5.19. Two versions of the CRM model are available202. One where the inboard flap is unconnected referred to as “gapped”) and one where the gaps between the inboard flap and outboard flap and main fuselage were sealed (referred to as “sealed”). In the latter case the worst proximity areas are removed, facilitating layers generation.

“HLPW-3 Geometry.” [Online]. Available: https://hiliftpw.larc.nasa.gov/Workshop2/geometries.html%0D. Vangelis Skaperdas, and Neil Ashton, “Development of high-quality hybrid unstructured meshes for the GMGW-1 workshop using ANSA”, AIAA, January 2018. 201 202

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5.5.2.2 The JSM Model The JSM model is designed in mm and it represents the actual wind tunnel model with a MAC of 529 mm. Geometry was read into ANSA without any topological problems. A hemi-spherical domain was created with a radius of 100 times MAC. The model was separated in ANSA in 13 zones to facilitate meshing and preprocessing as shown in Figure 5.21. Similarly to the HL-CRM model, the JSM model is available in two variants, without engine nacelle, as well as with engine Figure 5.18 JSM Model with Engine Nacelle nacelle, as shown in Figure 5.18 using the HLPW-3 notation for cases. The geometry of the JSM model has a particularity, which the HL CRM model does not have, that would eventually lead to problems in the generation of layers203. Those areas can be easily identified in ANSA through a quick test layers generation run for one layer. Such areas are identified, marked and can optionally be excluded from layers generation, although for the case of this study we wanted to avoid any area of the model without layers, as that would result to solution instability and error. Three such areas were found in the JSM geometry as highlighted in Figure 5.20. It was

Figure 5.19

Computational Domain of the HL-CRM Gapped Flaps Model

Vangelis Skaperdas, and Neil Ashton, “Development of high-quality hybrid unstructured meshes for the GMGW-1 workshop using ANSA”, AIAA, January 2018. 203

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therefore decided to perform some small local geometry modifications204. The size of these geometrical additions is limited to around one or two local element lengths so as not to cause a significant disturbance to the flow field. It is believed that the benefits of allowing for good quality layer generation in these regions surpass any side-effects from deviating from the original CAD geometry.

Figure 5.21

Figure 5.20

Computational Domain and Separation of Zones of the JSM Model with Engine Nacelle

Three Locations of Problematic Areas of the JSM Geometry for the Generation of Boundary Layers

Vangelis Skaperdas, and Neil Ashton, “Development of high-quality hybrid unstructured meshes for the GMGW-1 workshop using ANSA”, AIAA, January 2018. 204

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Surface Meshing ANSA provides the user with full control and automation in meshing though the Batch Mesh tool. Batch Mesh is a template driven meshing tool that consists of meshing scenarios for surface meshing, layers generation and volume meshing. Each scenario consists of several sessions and each session contains different zones of the model and the corresponding meshing characteristics. The contents of each session can be assigned manually, or via standard name convention filters of the zones of the model (like name contains, name starts with, name ends with etc.). Using filters allows more automation, since for every new design variant, the sessions can be populated automatically. One of the advantages of Batch Mesh tool is the fact that a scenario can be defined once, saved and then run several times for every new geometry, ensuring automation and mesh consistency for every variant. The CFD meshing algorithm in ANSA creates a high quality surface mesh, controlled by the following settings for each session of the Batch Mesh: ➢ ➢ ➢ ➢ ➢ ➢ ➢

Mesh type (triangular or quad) Target curvature refinement Growth rate of mesh on flat areas Minimum and maximum length Assigned length on all identified feature lines (different for convex and concave ones) Proximity refinement via prescribed length to gap ratio Target quality criteria threshold values

In addition to these mesh controls, the user can create Size Boxes to limit the maximum length of the surface and the volume mesh in different areas. Figure 5.22 displays the Batch Mesh surface meshing scenario that contains eight sessions, each with different zone contents and mesh specifications for the Medium JSM model. Ten Size Boxes were used, each one with a larger maximum length limit the further away of the aircraft. Size Boxes case be cylindrical or hexahedral and can also be manipulated by the user to take various curved forms aligned to the flow field where necessary. The only feature currently missing from Batch Mesh tool is the creation of anisotropic mesh, usually at the leading and trailing edges. The reason for this is that anisotropic mesh is not used in the automotive industry where ANSA usage was built on, while if it offers a great advantage for aerospace meshing. The main difference between these two industries with respect to meshing, is that in the aerospace industry the dimensions of the wings are much larger, while they require very fine

Figure 5.22

Batch Mesh setup for the JSM Model with Size Boxes for Local Mesh Control

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curvature refinement. The flow gradients are considerably larger in the chord wise direction than in the span wise direction and as a result anisotropic mesh provides the most efficient refinement method. For the same level of curvature refinement on a typical wing geometry, an isotropic mesh may require at least three times more elements than an equivalent anisotropic mesh. In addition, in the aerospace industry the total height of the boundary layer elements is considerably larger than that used in the automotive industry. In Figure 5.23 the advantage of anisotropic meshing is obvious as in highly convex areas like leading and trailing edges, starting from a span wise anisotropy allows the layer elements to improve in quality with every new step. In the end, the top cap of the layers is perfectly isotropic and this is the best basis for the remaining pyramid and tetra meshing to follow. In contrast, when starting from an isotropic mesh, the mesh quality of the layers deteriorates with each step. For the case of the isotropic surface mesh, the top cap does not have a good quality and this makes the remaining volume meshing process harder. Therefore, the surface meshing of all the

Figure 5.23

Resulting Layers for Isotropic Surface Mesh (Top) and Anisotropic (Bottom)

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models was performed with the following combination of manual and Batch Mesh automatic operations: ➢ Identification and manual meshing of all trailing edges with map quad mesh of specific rows of elements. ➢ Automatic Batch Meshing of all remaining surfaces of the model. ➢ Manual imprint of anisotropic mesh patterns away from the trailing edges and along all leading edges. Note how the anisotropic mesh dies out near the ends with the span wise imposed nodal distribution, in order to smoothly transit to isotropic mesh. The first surface scenarios for both HLCRM and JSM models were performed for the medium mesh size according to the meshing guidelines for rows of elements across the trailing edges and mesh resolution. From the medium meshes we generated the coarse and fine versions by simply scaling up and down respectively the assigned element length of all sessions of Batch Mesh scenario. Of course when dealing with unstructured

Figure 5.24

Close ups of Coarse CRM Gapped Flap Model with Comparison of Tridiagonal Dominant (Top) vs. Quad Dominant (Bottom) Surface Mesh

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mesh and especially with anisotropic features and Size Boxes, it is not easy to determine a priori the scale length factor in order to achieve the desired volume cell count. This can only be done for structured hex meshes, where cell number and edge length scale directly. Therefore, after certain trial and error runs, we ended up with a length scale multiplying factor of 1.2 to 1.25. This resulted in volume cell count changes of 1.6 to 1.8 between the different levels of refinement. Using the batch mesh and simply changing the mesh type of all sessions, we also generated a quad dominant surface mesh for the coarse CRM case. Figure 5.24 displays the two variations of surface mesh, tridiagonal dominant and quad dominant. The quad dominant mesh has 30% fewer shell elements for the same mesh resolution. The only problem that may arise with a quad dominant mesh is that due to the fact that the near wall layers have extreme aspect ratios, there may be curved areas of the model where these quads may also have considerable warping. The combination of warping and high aspect ratio may lead to problems in the solution. At the time of the HLPW-3 the quad dominant meshes were not prepared, so no simulations were performed for them. Currently ANSA development work for the next version focuses on the integration of anisotropic meshing of leading and trailing edges inside Batch Mesh tool, thereby eliminating any manual work for the user. Volume Meshing Volume meshing is also a part of the Batch Mesh. Two scenarios were created, one for layers and one for volume meshing. The scenarios were setup once for the CRM and JSM models and then with simple modifications in their parameters (growth rate, max size etc.) were executed automatically in order to generate the final volume meshes. 5.5.4.1 Extrusion Layers Generation The generation of layers is the most demanding part of the meshing process as the there are many factors that should be considered: very high aspect ratio elements whose quality is difficult to control, large total boundary layer height, resulting in proximity issues, especially around the areas of the flaps and slats, where the gaps are small. ANSA layers generation algorithm is very robust and controllable, with characteristics like: ➢ Generation of hex or penta layers from quad or triangle surface mesh. ➢ Generation of initial layers without growth for better refinement of the near wall region. ➢ Generation of initial layers without vector smoothing ensuring high orthogonality near the wall. ➢ Advanced smoothing algorithm to overcome problems of layers extrusion in concave areas. ➢ Generation of layers with different growth rate, number and first height from different zones of the model. ➢ Local element squeezing and collapsing at proximities to avoid intersections and bad quality. ➢ Local collapsing when a target aspect ratio is reached, ensuring a nice volume with the tetra mesh to be connected. Layers squeezing and collapsing modes work in combination. The user can specify a maximum aspect ratio that the elements can attain when squeezed in order to overcome proximities. If this limit is exceeded then ANSA switches to local layer collapsing. Collapsing works for both penta and hex elements. Depending on the number of nodes that need to be collapsed in a certain area, pyramid and tetra elements are created out of the original penta and hex elements. No collapsing of course must take place at the first layers as that would result to very skewed tetras and pyramids. As the layers grow thicker however, collapsing does not compromise the quality of the resulting elements. In contrast to the recommended meshing guidelines of the workshop, the first layer height was kept constant throughout the mesh refinement study. The reason for this is that the initial simulations

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that were performed showed that a y+ value of just below 1 was achieved with these values, so there was no reason to change this parameter, as it was set to its optimum value. During the mesh refinement study the growth rate of the layers was reduced from 1.25 for the coarse mesh to 1.1 for the fine mesh. The first two layers were kept with constant height, as prescribed by the meshing guidelines. The number of layers was increased for the finer meshes in order to maintain the same total boundary layer height. In addition, the number of layers differed between the main wing and the fuselage. The reason for this was that the surface mesh length on the wing was smaller than that of the fuselage. As a result we needed more layers from the fuselage in order to reach a layer thickness with aspect ratio just below 1 so as to have a nice volume ratio between the last layer and the first tetra or pyramid to connect on it205. Hex layers growing from quad surface mesh are also checked in every step for warping. In certain cases, due to local squeezing at proximities, if the top cap warping exceeds a user specified threshold, ANSA splits the hex into two pentas, so as to avoid having highly warped quads that would have a negative effect in the pyramid generation for the tetra meshing. For further details regarding the meshing, consult the206.

Figure 5.25

Volume Mesh of the JSM

5.5.4.2 Tetra Meshing After layers generation is completed, the third and final Batch Mesh scenario is executed. It consists of the automatic detection of the boundaries of the external volume, and its meshing with tetra elements using the TetraRapid volume meshing algorithm. This algorithm is a hybrid advancing front Delaunay, optimized for speed, smooth size variation and robust surface capturing even in aerospace applications where map anisotropic high aspect surface mesh is present and size variations exist from the surface of the model to the far field boundary of up to one hundred thousand orders of magnitude. Even in such cases, and despite of the fact that it runs on a single thread, it manages to generate high quality tetra mesh at the speed of one to three million tetras per minute,

Vangelis Skaperdas, and Neil Ashton, “Development of high-quality hybrid unstructured meshes for the GMGW-1 workshop using ANSA”, AIAA, January 2018. 206 See Above. 205

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depending on the complexity of the domain and the presence of additional refinement Size Boxes. The growth rate of the volume mesh was set to 1.15. Figure 5.25 shows the medium JSM mesh. Sample CFD Results Full details of the results are given in [Ashton et al.]207 however a brief sample of the results are provided here to demonstrate the performance of the meshes generated. Simulations were undertaken in STAR-CCM+ and OpenFOAM in order to assess the accuracy and robustness of OpenFOAM against a popular commercial code. In OpenFOAM a segregated pressure-based solver (rhoPimpleFoam) is used with local time-stepping to accelerate steady-state convergence. Second order upwind schemes were used for the both momentum and turbulence quantities with a greengauss scheme for the gradient operators. In STAR-CCM+ a fully implicit compressible density based scheme is used with the Roe scheme for the flux. A green-gauss scheme with a min-mod limiter was used for the gradient calculations and a second order upwind scheme was used for both the momentum and turbulence quantities. 5.5.5.1 CRM The flow conditions for the CRM geometry are a Reynolds number of Re = 3.2 x 106 and a Mach number of M = 0.2, however as the geometry is full-scale the flow parameters are adjusted to achieve the required Reynolds number. The viscosity is computed using Sutherlands Law and the density is based upon the ideal gas law. Simulations are conducted at 8 & 16 degrees angle of attack and for both STAR-CCM+ and OpenFOAM the Spalart-Allmaras turbulence model is used. Figure 5.26 displays the Lift and Drag coefficients for the CRM geometry at 8 degree for STAR-CCM+ and OpenFOAM. There is a clear mesh refinement trend for both codes, suggesting that even finer meshes would be required to reach a mesh converged solution for the lift. Given the finest mesh is 269 M cells, it is likely that a meshes up to a billion cells might be required. The agreement between OpenFOAM and STAR-CCM+ is within 0.5% for the lift coefficient and less than 2% for the drag coefficient. They show the same outboard flap separation with the size and position being almost identical. This less than 2% difference to a popular commercial CFD code would suggest that

Figure 5.26

CL and CD for CRM Geometry at 8 degree AoA using OpenFOAM and STAR-CCM+

N. Ashton, M. Fuchs, C. Mockett, and B. Buda, “EC135 Helicopter Fuselage, ”Go4Hybrid: Grey Area Mitigation for Hybrid RANS-LES Methods”, C. Mockett, W. Haase, and D. Schwamborn, Springer International Publishing, 2018, pp. 2013–2015. 207

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OpenFOAM is a competitive tool for engineering analysis, which reflects the recent findings of [Ashton et al.]208. 5.5.5.2 JSM Close agreement between OpenFOAM and STAR-CCM+ was observed for the CRM geometry, however without experimental data it is not possible to assess the accuracy. The JAXA Standard Model (JSM) high-lift model is similar to the CRM but has a detailed experimental data set making it ideal to assess the accuracy of OpenFOAM and STAR-CCM+. The flow conditions are a Reynolds number of Re =1.9 x106 and a Mach number of M = 0.172. The viscosity is computed using Sutherlands Law and the density is based upon the ideal gas law. Simulations are conducted at 4.36, 10.47, 14.54, 18.58, 20.59, 21.57degrees angle of attack and all simulations use the Spalart Allmaras turbulence model. Figure 5.27 shows the lift and drag coefficient throughout the angle of attack range using OpenFOAM and STAR-CCM+ for the geometry (no nacelle). It can be seen that there is again close agreement between STAR-CCM+ and OpenFOAM, with only changes becoming clear in the post-stall region. At 4.36 degree, the flow is completely attached in both CFD and the experiment, which is reflected in the close agreement between CFD and experimental for the lift in the lower angle of attack range. At 18.57 degree, just before stall, the agreement in the total lift is close, however the flow structures start to exhibit slightly too much stall in the outboard wing section. By 21.57 degree where the flow is now stalled, the agreement is close but actually for the wrong reason. Whereas the experimental flow-vis shows both separation at the root and the most outboard region of the wing, the CFD (both STAR-CCM+ and OpenFOAM) show almost no separation at the root and much larger separation at the outboard of the wing. The total amount of separation is roughly similar which explains why the lift and to a lesser extent the drag follow the experimental values. Given that all simulations were undertaken with the Spalart-Allmaras model with no corrections for curvature nor anisotropy of the flow it is not surprising that the results do not perfectly correlate with the experiment. More details results for these cases are shown in [Ashton et al.]209. The results from both the CRM and JSM have shown that both STAR-CCM+ and OpenFOAM on ANSA generated grids can

Figure 5.27

Lift and Drag Coefficients for the JSM Geometry using OpenFOAM and STAR-CCM+

N. Ashton, M. Fuchs, C. Mockett, and B. Buda, “EC135 Helicopter Fuselage,” in Go4Hybrid: Grey Area Mitigation for Hybrid RANS-LES Methods, vol. 134, C. Mockett, W. Haase, and D. Schwamborn, Eds. Cham: Springer International Publishing, 2018, pp. 2013–2015. 209 N. Ashton and V. Skaperdas, “Verification and Validation of OpenFOAM for High-Lift Aircraft Flows,” Submitted to AIAA Journal, 2017. 208

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perform well for complex full aircraft geometries and in the case of the JSM, match experimental values up to the stall region. The next steps are to properly assess the code for transonic flows, which is the typical flow regime for industrial aerospace simulations.

Case Study 2 - A 3D Hybrid Grid Generation Technique and a Multigrid/Parallel Algorithm Based on Anisotropic Agglomeration Approach210 Statement of the Problem A hybrid grid generation technique and a multigrid/parallel algorithm are presented by [Laiping et al.]211, for turbulence flow simulations over the (3D) complex geometries. The hybrid grid generation technique is based on an agglomeration method of anisotropic tetrahedrons. Firstly, the complex computational domain is covered by pure tetrahedral grids, in which anisotropic tetrahedrons are adopted to discrete the boundary layer and isotropic tetrahedrons in the outer field. Then, the anisotropic tetrahedrons in the boundary layer are agglomerated to generate prismatic grids. The agglomeration method can improve the grid quality in boundary layer and reduce the grid quantity to enhance the numerical accuracy and efficiency. In order to accelerate the convergence history, a multigrid/parallel algorithm is developed also based on anisotropic agglomeration approach. Introduction, Background and Contributors Many grid generation techniques, such as multi-block or patched structured grids212-213 overlapping or chimera grids214 and unstructured grids215 have been proposed in the last decades. More recently, mixed or hybrid grids including many different cell types have gained popularity, because they integrate the advantages of both structured and unstructured meshes to improve efficiency and accuracy. For example, hybrid (prism/tetrahedral) grids216-217, mixed grids including (tetrahedral/prism/pyramid/hexahedral) cells218, adaptive cartesian grid methods219-220, and (cartesian/tetrahedral/prismatic) grids221 have been used in many applications. It is relatively easier to use unstructured grids over complex configurations, even for viscous flow simulations, where the anisotropic tetrahedrons are used in boundary layer. Generally, the anisotropic tetrahedrons can be Zhang Laiping, Zhao Zhong, Chang Xinghua, He Xin, “A 3D hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration approach”, Chinese J of Aeronautics, 2013. 211 See Previous. 212 Jochem H, Peter E, Yang X, Cheng ZM. Parallel multiblock structured grids. Thompson JF, Soni BK, Weatherill NP, editors. Handbook of grid generation. CRC Press; 1999 chapter 12. 213 Sebastien E. Numerical simulation and drag extraction using patched grid calculations. AIAA Paper 2003. 214 Benek A, Buning PG, Steger JL. A 3-D Chimera grid embedding technique. AIAA Paper 1985-1523; 1985. 215 Weatherill NP. Unstructured grids: procedures and applications. In: Thompson JF, Soni BK, Weatherill NP, editors. Handbook of grid generation. CRC Press; 1999 chapter 26. 216 Kallinderis Y, Khawaja A, McMorris H. Hybrid prismatic/tetrahedral grid generation for complex geometries. AIAA J 1996;34(2):291–8. 217 Pirzadeh S. Three-dimensional unstructured viscous grids by the advancing-layers method. AIAA J 1996;34(1):43–9. 218 Coirier WJ, Jorgenson PCE. A mixed volume grid approach for the Euler and Navier–Stokes equations. AIAA Paper 96-0762; 1996. 219 Coirier WJ, Powell KG. Solution-adaptive Cartesian cell approach for viscous and inviscid flows. AIAA J 1996;34(5):938–45. 220 Wang ZJ, Chen RF. Anisotropic solution-adaptive viscous Cartesian grid method for turbulent flow simulation. AIAA J 2002;40(10):1969–78. 221 Zhang LP, Yang YJ, Zhang HX. Numerical simulations of 3D inviscid/viscous flow fields on Cartesian/unstructured/prismatic hybrid grids. Proceedings of the fourth Asian CFD conference, Mianyang, Sichuan, China; September 2000. 210

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automatically generated by an advancing front method222. However, the enormous total grid number will reduce the efficiency of the viscous flow simulations over complex geometries. More importantly, the forfeiture of orthogonality will influence the simulation accuracy of boundary layer. Therefore, the prism grids, even unstructured hexahedral grids, may be a better choice in the boundary layer. The traditional prism grid generation method is the advancing layer method in which the prism grids are generated layer-by-layer in the normal direction from the surface triangular grids on the solid wall. Alternatively, the idea of solving the hyperbolic equations to generate structured grids has been introduced to generate prism grids223-224. However, for some realworld configurations, these methods will fail in the concave and/or convex regions, because the marching vector may be invisible from some of the nodes in its node-manifold225. Examples include the trailing edge of an airfoil, the tip of a sharp nose, the wing-body conjunction, the tail of a store and the nacelles of aircraft. So it is still difficult to automatically generate viscous grids in the boundary layer. Since the anisotropic tetrahedrons can be generated fully automatically, we can agglomerate them into prisms in the boundary layer and then improve the grid quality of the pure anisotropic tetrahedron grids. That is the basic idea of present work. On the other hand, the computation efficiency is another key issue for turbulence flow simulations over complex configurations, because the total grid number may be several ten M, even up to hundreds of M, for a real-life aircraft. The high aspect ratio grids in boundary layer will bring about very strong stiffness during time-iteration, resulting in lower converging efficiency. The multigrid algorithm is an effective method to improve the efficiency. After [Fedorenko]’s development of the method in the 1960s226, it was discovered, further developed and popularized by [Brandt] in the 1970s227. Multigrid was applied to the transonic small-disturbance equation by [South and Brandt]228 and to the full potential equation by [Jameson]229. Subsequently, the idea of agglomeration multigrid has been extended to unstructured grids [Smith]230, [Lallemand et al.]231, [Venkatakrishnan & Mavriplis]232, and also [Mavriplis]233-234. Despite considerable progress towards improving the convergence performance of multigrid algorithm based on cell-vertex finite volume schemes, the performance of these methods for viscous flow simulations is not satisfying for cell-centered finite volume schemes. The key issue is how to generate high-quality coarser grids using the agglomeration approach. In other words, how to ensure the ‘‘convex’’ property for the coarser grids, especially in the boundary layer. Lohner R, Parikh P. Generation of three-dimensional unstructured grids by the advancing front method. Int J Numerical Method Fluids, 1988. 223 Chan WM, Steger JL. Enhancements of a three-dimensional hyperbolic grid generation scheme. Appl Math Computing 1992. 224 Matsuno K. High-order upwind method for hyperbolic grid generation. Computational Fluids 1999. 225 Kannan R, Wang ZJ. Overset adaptive Cartesian/prism grid method for stationary and moving-boundary flow problems. AIAA J 2007. 226 Fedorenko R. The speed of convergence of one iterative process. USSR Comput Math Phys 1964;4(3):227–35. 227 Brandt A. Multi-level adaptive solutions to boundary value problems. Math Comput 1977;31(138):333–90. 228 South JJC, Brandt A. Application of a multi-level grid method to transonic flow calculations. Adamson Jr TC, Platzer MF, editors. Transonic flow problems in turbomachinery. Washington: Hemisphere; 1977. 229 Jameson A. Acceleration of transonic potential flow calculations on arbitrary meshes by the multiple grid method. Proceedings of the AIAA fourth computational fluid dynamics conference. Virginia, 1979. 230 Smith WA, Multigrid solutions of transonic flow on unstructured grids. In: Baysal O, editor. Recent advances and applications in computational fluid dynamics. In: Proceedings of the ASME winter annual meeting, 1990. 231 Lallemand M, Steve H, Dervieux A. Unstructured multi gridding by volume agglomeration: current status. Com. Fluids 1992. 232 Venkatakrishnan V, Mavriplis D. Agglomeration multigrid for the three-dimensional Euler equations. AIAA 1995. 233 Mavriplis DJ. Unstructured grid techniques. Annual Rev Fluid Mech 1997;29(1):473–514. 234 Mavriplis DJ. Viscous flow analysis using a parallel unstructured multigrid solver. AIAA J 2000.. 222

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The work of 235-236 gave some inspirations. They agglomerate the grids in boundary layer with a normal-direction restriction. This idea can be extended to improve the coarser grid quality in boundary layer. In this paper, a hybrid grid generation technique is presented for turbulence flow simulations over 3D complex configurations, which is based on an anisotropic agglomeration of pure tetrahedral grids. Firstly, pure unstructured grids are generated over a given complex geometry, and anisotropic tetrahedral elements with high aspect ratio are adopted in the boundary layer. Then, the anisotropic tetrahedrons are agglomerated to generate the prismatic grids in the boundary layer, while the isotropic tetrahedrons in the outer flow field keep alone. To validate the method, the hybrid grids over some complex geometries are generated, including the DLR-F6 wing-body configuration, a fighter and a human body, which demonstrate the robustness of the present hybrid grid generation technique. Furthermore, a multigrid computing algorithm based on semi-structured agglomeration method is developed to improve the convergence performance and couple with the parallel computing based on computational domain decomposition. The semi-structured agglomeration means that the agglomeration is mainly limited to the normal direction of the solid wall to keep the orthogonality of hybrid grids in the boundary layer. This multigrid computing algorithm matches the present hybrid grid generation technique, because both of them are based on the anisotropic agglomeration approach. Some typical cases are tested to validate the robustness and efficiency of the present multigrid computing method for viscous flow simulations over complex geometries. The numerical results are compared with the experimental data and other numerical results, which demonstrate the efficiency and accuracy of the present method. Hybrid Grid Generation Technique based on Anisotropic Agglomeration Approach As mentioned in the introduction, despite considerable progress towards facilitating the grid generation process itself, the high-quality grid generation over 3D complex real-world configurations, especially for turbulence flow simulations, is still an open issue for producing accurate CFD solutions and, thus, require further attention. Fortunately, the unstructured grid generation method is currently at a stage of maturity that allows discretization of complex, 3D, realworld configurations with relative ease and a reasonable amount of time and effort. Generally, the pure unstructured grids mean triangles in 2D and tetrahedrons in 3D. Thanks to many advances by a number of researchers in the science/art of grid generation, this crucial step no longer represents an obstacle for the routine use of CFD in the context of large-scale (industrial) applications. Some pieces of commercial grid generation software are available in the market, such as Gridgen, ICEMCFD, etc. Also, there are some in-house grid generation software, such as VGrid in NASA and Centaur in Europe. The unstructured grids can be generated by the advancing front method237-238, Delaunay method239 and/or the modified Quadtree/Octree methods240. Actually, in the commercial grid generation software, the integrated strategy is adopted to improve the grid quality and the grid generation efficiency. For viscous flow simulations, the anisotropic tetrahedrons are generally adopted in the boundary layer. However, the enormous total grid number will reduce the efficiency. Daniel G, Sreenivas K. Parallel FAS multigrid for arbitrary Mach number, high Reynolds number unstructured flow solver. AIAA Paper 2006-2821; 2006. 236 James L, Nishikawa H. A critical study of agglomerated multigrid methods for diffusion on highly stretched grids. Comput Fluids 2011;41(1):82–93. 237 Jochem H, Peter E, Yang X, Cheng ZM. Parallel multiblock structured grids. Thompson JF, Soni BK, Weatherill NP, editors. Handbook of grid generation. CRC Press; 1999 chapter 12. 238 Sebastien E. Numerical simulation and drag extraction using patched grid calculations. AIAA Paper, 2003. 239 Waston DF. Computing the n-dimensional Delaunay tessellation with application to voronoi polytopest. Com. J 1981;24(2):167–72. 240 Merry MA, Shephard MS. Automatic three-dimensional mesh generation by the modified-Octree technique. Int J Numer Methods Eng 1984;20(11):1965–90. 235

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More importantly, the forfeiture of orthogonality will influence the simulation accuracy of boundary layer. A possible better choice is to generate prisms in the boundary layer, which is the main advantage of the so-called hybrid grids. However, the prism grid generation is still not a routine task due to the geometric complexity. Since the pure anisotropic tetrahedrons can be generated automatically and easily using the available commercial grid generation software, is it possible to generate prisms based on the anisotropic tetrahedrons? The answer is possible, because the prisms can be cut into three tetrahedrons inversely. Actually, in most of the commercial grid generation software (for example, Gridgen), the anisotropic tetrahedrons are generated by refining the temporary prisms. The reason of generating pure anisotropic tetrahedrons is that it is difficult to ensure the unified prism structure in the whole boundary layer due to the geometric complexity, although it is relatively easier for simple configurations. Following this idea, we develop a hybrid grid generation technique based on the anisotropic agglomeration. 5.6.3.1 Prism Grid Generation Method Based on Anisotropic Agglomeration Approach The prism grid generation is the key step in the present hybrid grid generation technique. The details are listed as follows, including the volume agglomeration and the interface agglomeration. 5.6.3.1.1 Volume Agglomeration 1 Extract the geometric characteristics of all the cell interfaces and label each cell as anisotropic or isotropic. 2 Agglomerate two anisotropic cells into a pyramid. 3 Agglomerate the third anisotropic cell and the pyramid into a prism. 4 For all the non-agglomerated anisotropic cells, find out a neighbor prism and agglomerate them into a polyhedron. The purpose of the Step 2 and Step 3 is to agglomerate three anisotropic tetrahedrons into a prism. But for the real world configurations, some isolated anisotropic tetrahedrons may exist in the concave and/or convex regions. If we allow these cells to exist, the ratio of the volume of two neighboring cells may be 1:3, so the smoothness of grids in boundary layer is not satisfied. Hence, the agglomeration of Step 4 combines the isolated tetrahedrons into the neighboring prisms to improve the smoothness of grids in boundary layer (The volume ratio of two neighboring cells is about 3:4). In practice, only a small number of isolated anisotropic tetrahedrons are found. See (Figure 5.28). 5.6.3.1.2 Interface Agglomeration After the volume agglomeration, the interface agglomeration is carried out to reduce the number of interfaces between two neighboring prisms. The two triangles shared by two neighboring prisms are agglomerated into a quadrilateral (see Figure 5.28 Step 4). The above hybrid grid generation technique has some distinguished properties: ➢ Once the pure tetrahedral grids have been generated, the hybrid grids can be generated fully automatically, without any user interference. Generally speaking, the tetrahedral grids may also be generated automatically using advancing front method or Delaunay method. ➢ The grid quality, especially in the boundary layer, is much better than that of pure unstructured grids, which is very crucial for viscous flow simulations. ➢ The smoothness of grids from the boundary layer to the outer flow field is much better.

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Step 1

Step 2

Step 3

Step 4

Figure 5.28

Interface Agglomeration Procedure Wing – Courtesy of [Laiping et al.]

Multigrid/Parallel Algorithm In order to further accelerate the convergence history, the multigrid computing approach is adopted and further improved in this section. The basic idea of the multigrid method is to carry out early iterations on a fine grid and then progressively transfer these flow field variables and residuals to a series of coarser grids. On the coarser grids, the low frequency errors become high frequency ones and they can be easily eliminated by a time stepping scheme. The flow equations are then solved on the coarser grids and the corrections are then interpolated back to the fine grid. The process is repeated over a sufficient number of times until satisfactory convergence on the fine grid is achieved. In this paper, the V-type cycle is adopted for easy implementation. Multi-Level Coarser Grid Generation Based on Anisotropic Agglomeration Approach Before multigrid computing, a series of coarser grids should be generated. As mentioned in the introduction, the agglomeration approach from the initial finest grids is the most popular method. Traditionally, the ‘isotropic’ agglomeration approach is adopted in inviscid flow simulations, but it is not suitable for viscous flow simulations because the isotropic agglomeration may generate ‘singular’ coarser polyhedron grids due to the isotropic randomicity, as illustrated in Figure 5.29 (b) in a 2D case (whose initial fine grid is shown in Figure 5.29 (a)). The word ‘singular’ means that the agglomerated grids may be an ‘L’ shape or even ‘U’ and ‘S’ shape, and the geometric centers of these singular coarser grids will be located out of the cell faces themselves. This kind of ‘singular’ situation will deteriorate in 3D cases, which will result in failure of multigrid computing, because the interpolation operator should be carried out during V-type iterations. If we use an anisotropic agglomeration approach, the quality of coarser grids will be improved very well (see Figure 5.29

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(c)) to benefit the viscous flow simulation with multigrid computing algorithm.

(a) Initial Hybrid Grid Figure 5.29

(b) Coarsen grid by Isotropic Allogmeration

(c) Coarsen grid by Anisotropic Allogmeration

Initial Hybrid Grids and Coarsen Grids Wing – Courtesy of [Laiping et al.]

The concept of anisotropic agglomeration is introduced by Mavriplis241 for cell-vertex finite volume method and further developed by Refs.242-243 for cell-centered finite volume method. However, for arbitrary hybrid grids, we still meet some problems because the multigrid iteration is so sensitive to the shape of multi-level coarsen grids. If the quality of coarser grid is not good enough, the accelerating performance of multigrid iteration will be attenuated. In order to improve the quality of coarser grid by agglomeration, an improved anisotropic agglomeration approach is developed in this paper. The details are listed as follows: 1 2

Check the cell property (isotropic or anisotropic cells). The checking criterion is the same as that of Step 1 in hybrid grid generation above. Agglomerate the surface triangles in a pseudo-2D manner. The surface triangles are agglomerated with a node-based agglomeration approach. In order to ensure the smoothness and the quality of coarser surface grids, the following two criteria are considered: • •

3

If two triangles are located on the two separated sides of a sharp edge (For example, the trailing edge of a wing, the joint-line of fuselage and wing, see [Laiping et al.]244), the two triangles cannot be agglomerated. If there are some isolated non-agglomerated triangles after first-round agglomeration, they should be agglomerated into the neighboring cells to improve the smoothness.

Agglomerate the anisotropic prism grids layer-by layer with an analogy ‘advancing layer’ method. Advancing in the normal direction from each agglomerated surface coarser grids (Two initial layers are integrated into one layer), then the prism grids in the boundary layer are agglomerated layer-by-layer to ensure the semi-structured property as the initial finest

Mavriplis DJ. Viscous flow analysis using a parallel unstructured multigrid solver. AIAA J 2000. Daniel G, Sreenivas K. Parallel FAS multigrid for arbitrary Mach number, high Reynolds number unstructured flow solver. AIAA Paper 2006-2821; 2006. 243 James L, Nishikawa H. A critical study of agglomerated multigrid methods for diffusion on highly stretched grids. Computational Fluids 2011;41(1):82–93. 244 Zhang Laiping, Zhao Zhong, Chang Xinghua, He Xin, “A 3D hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration approach”, Chinese J of Aeronautics, 2013. 241 242

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4

grids. Agglomerate the isotropic grids in the outer flow field. In this step, we use a node-based agglomeration approach, which means agglomerating all the non-agglomerated cells connecting to a node. However, there are some special or ‘singular’ cases during agglomeration, see [Laiping et al.]. For these cases, the agglomeration is not permitted.

Applications and Discussions Based on the above method, a cell-centered flow-solver named USTAR was developed by [Laiping et al.]. The code is written by C++ language and can be run in different computer platforms, such as desk-top computer, PC-cluster or work-station. In order to save the CPU time during multigrid iteration, the inviscid model and the first-order scheme are adopted only on the coarser grids. 5.6.6.1

Subsonic Turbulence Flow over 2D 30P30N Airfoil The first test case is subsonic turbulence flow over the 2D 30P30N airfoil. The main purpose of this simple test case is to validate the multigrid computing algorithm. The initial finest hybrid grids and the second-level and the forth-level coarsening grids are shown in Figure 5.31. The incoming flow conditions are the Mach number M∞ = 0.2, the angle of attack α = 19.0 degrees, the Reynolds number Re = 9.0 x 106. The SST two-equation turbulence model is adopted in this simulation. The numerical results (pressure coefficient Cp distribution on the solid wall) Figure 5.30 CP Distribution on Solid Wall Wing are plotted in Figure 5.30, in which the Courtesy of [Laiping et al.] results by single-level (non-multigrid) and forth-level multigrid approach are compared with the experimental data245. Consult the [Laiping et al ]246 for complete analysis.

(a) Initial Hybrid Grid Figure 5.31

(b) Second-Level

(c) The Thrid-Level

Initial Hybrid Grids and Coarsening Grids over 30P30N Airfoil Wing – Courtesy of [Laiping et al.]

Spaid FW, Lynch FT. High Reynolds number multi-element airfoil flow field measurements. AIAA 1996. Zhang Laiping, Zhao Zhong, Chang Xinghua, He Xin, “A 3D hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration approach”, Chinese J of Aeronautics, 2013. 245

246 246

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5.6.6.2 Transonic Turbulence Flow over ONERA M6 Wing The second application is the transonic turbulence flow simulation over ONERA M6 wing, a typical validation case. The incoming flow conditions are M∞ = 0.8395, α= 3.06 deg, Re= 1.172 x 107. For this case, the Spalart-Allmaras one equation turbulence model is employed. The initial finest hybrid grids and the second coarser grids are shown in Figure 5.32- (a) and (b), and the close-up view near the boundary layer before and after agglomeration is plotted in Figure 5.32–(c). Note that the quality of the coarser grid in boundary layer is very well (keeping the semi-structured grid structure). Pressure coefficient distribution on three different cross-sections. The calculated pressure coefficient distributions on three cross-sections (g = z/b =20 %, 80 % and 90 %, respectively, where b is the span length of the wing in z-direction). The agreement with the experimental data247 much better than those by inviscid flow simulation (see [Laiping et al.]248).

(a) Initial Hybrid Grid Figure 5.32

(b) Second-Level

(c) Close Up View of the Boundary

Initial Hybrid Grids and Coarsening Grids over ONERA M6 Wing – Courtesy of [Laiping et al.]

5.6.6.3 Transonic Turbulence Flow over DLR-F6 Wing-Body Configuration The last application is transonic turbulence flow simulation over the DLR-F6-WBNP configuration. It is the typical test case in DPW-II. The computational conditions are chosen as M∞ = 0.75, Re = 3.0 x 106. The angle of attack is set from 3.0 to 1.5 deg. For this case, the SST turbulence model is applied. The initial hybrid grids are shown in Figure 5.33. The model is downloaded from the second drag prediction workshop (DPW-II)249 which is a fuselage-wing-nacelle-pylon conjunction configuration. Figure 5.33 Top (b) shows the surface triangular grids near the wing-nacelle-pylon. Figure 5.33 (Middle) shows the hybrid grids over DLR-F6-WBNP at a longitudinal cross-section. The number of initial pure unstructured grids is 17.43 M, including 34.96 M faces and 2.95 M nodes. After agglomeration, the number of hybrid grids is about 7.8 M, including 20.80 M faces. The total number of cells and faces is reduced by 50% and 33%, respectively. The close-up views near the nacelle-pylon are shown in Figure 5.33 (Bottom). The initial and the coarsening grids on surface and symmetric plane are shown in Figure 5.35. The close-up view near the boundary layer is given in Figure 5.35 (c). The total number of the initial hybrid grids over the half-model is about 7.8 M. So the multigrid/parallel computing strategy is adopted for this case. The computational domain is decomposed into 32 sub-domains (see Figure 247 Schmitt V, Charpin F. Pressure distributions on the

ONERA-M6-Wing at transonic Mach numbers, experimental data base for computer program assessment. Report of the Fluid Dynamics Panel Working Group 04, 1979. 248 Zhang Laiping, Zhao Zhong, Chang Xinghua, He Xin, “A 3D hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration approach”, Chinese J of Aeronautics, 2013. 249 Second AIAA CFD Drag Prediction Workshop. ; 2003.

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5.35 (a)) using the METIS approach. Figure 5.34 displays the aerodynamic force coefficients for different angles of attack, in which the experimental data and the numerical results by other solvers

Figure 5.33

Close-up Views of Hybrid Grids After Agglomeration Wing – Courtesy of [Laiping et al.]

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(USM3D, FUN3D, NSU3D)250-251 are also plotted. In the same figure, the results on pure unstructured grids before agglomeration is presented (marked as ‘Unstructured’). Note that the present numerical results (marked as ‘Hybrid Grid’) are in good agreement with the experimental data, which are slightly better than others’ numerical results. The drag polar on hybrid grids is much better than that on pure unstructured grids. Furthermore, the CPU time is saved greatly because the number of hybrid grid is only half of the pure unstructured grids. These results demonstrate that the hybrid grid technique is superior indeed to the pure unstructured grid approach. When M∞ = 0.75 and α = 1.0 degree, the pressure coefficient distributions at three typical sections (see Figure 5.36 (a)) are

Figure 5.34

Aerodynamic Force Coefficients for Different Angles of Attack (M∞ = 0.75) Wing – Courtesy of [Laiping et al.]

Mavriplis DJ. Drag prediction of DLR-F6 using the turbulent Navier-Stokes calculations with multigrid. AIAA Paper 2004-397; 2004. 251 Sclafani AJ, Dehaan MA. OVERFLOW drag prediction for the DLR-F6 transport configuration: a DPW-II case study. AIAA Paper 2004-393; 2004. 250

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shown in Figure 5.36 (b)–(d), where z/b =15.0 %, 33.1 % and 63.8 %, respectively. The present results are marked as USTAR. The results by others252-253 are also plotted in the same figures. It can be seen that the present results are very similar to the best results by USM3D. The flow separation pattern on the leeward surface of the wing is shown in Figure 5.36 (a), meanwhile the

(a) Initial hybrid grids (after METIS decomposition)

(b) Coarsening grids on the surface and symmetric plane Figure 5.35

(c) Close-up view of coarsening grids

Hybrid Grids over DLR-F6-WBNP Configuration Wing – Courtesy of [Laiping et al.]

Mavriplis DJ. Drag prediction of DLR-F6 using the turbulent Navier-Stokes calculations with multigrid. AIAA Paper 2004-397; 2004. 253 Lee-Rausch EM, Mavriplis DJ. Transonic drag prediction on a DLR-F6 transport configuration using unstructured solvers. AIAA Paper 2004-554; 2004. 252

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results by UG3254 and the experimental oil-flow pattern are shown in Figure 5.36 (b) and (c). The size of the separation zone is still larger than those by experiment255 but is slightly better than that by UG3. For additional info please consult [Laiping et al.]256.

Figure 5.36

CP Distributions at Three Typical Sections (M = 0.75, α = 1.0 deg) Wing – Courtesy of [Laiping et al.]

Yamamoto K, Ochi A. CFD sensitivity of drag prediction on DLRF6 configuration by structured method and unstructured method. AIAA Paper 2004-398; 2004. 255 Yamamoto K, Ochi A. CFD sensitivity of drag prediction on DLRF6 configuration by structured method and unstructured method. AIAA Paper 2004-398; 2004. 256 Zhang Laiping, Zhao Zhong, Chang Xinghua, He Xin, “A 3D hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration approach”, Chinese J of Aeronautics, 2013. 254

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Concluding Remarks An effective hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration are presented for viscous flow simulations over complex configurations. (1) The hybrid grid generation technique can improve the quality of grids in boundary layer over the pure unstructured grids. For real-world complex geometries, the prism grids in boundary layer can be automatically generated from the initial pure tetrahedral grids. (2) The multigrid/parallel computing algorithm based on anisotropic agglomeration can improve the convergence performance, especially for high Reynolds number viscous flow simulations. (3) Applications for complex 3D configurations have demonstrated the robustness of present method. In the future work, we will extend this method to unsteady flow simulations.

Recent Advances in Hybrid Mesh Generation and Literature Survey257 Parallel Consideration There is a need for an ever increasing resolution of flow field computations requiring more grid points, therefore, it is essential for faster grid generation algorithms. Each processor generates the surface grids for a number of surface patches. [L¨ohner]258 reports on the use of a parallel advancing front algorithm for volume grid generation. The front is dynamically divided into boxes by use of the octree algorithm. For each processor the grid is generated for a number of boxes. The number of boxes is much larger than the number of processors. Another alternative is to first generate a surface grid in parallel and then divide the surface grid and volume into smaller regions, as in [Larwood et al]259. The volume grid in these regions is generated in parallel. Parallel grid generation by the Delaunay algorithm has been presented in [Weatherill et al.]260. Local Remeshing [Ito et al.]261 describe a method where it is possible to introduce new components into an existing grid without regenerating the entire grid. The grid needs only to be regenerated in the region where the new component has been introduced. This speed up the time for the grid generation significantly. A similar extension of the grid generator TRITET with routines for the generation of locally remeshed grids has been done. This local remeshing works also for hybrid grids, but the original prismatic layers must be kept. Figure 5.37 (a) shows the cavity in a grid where the grid around a rotated airfoil is inserted into a grid around another airfoil. The cavity is formed by a set of inner and outer boxes. Inside the cavity a grid is generated by the advancing front algorithm. The merged grid is shown in Figure 5.37 (b). An application of this technique together with the flow solver Edge, see [Eliasson]262, for store separation computations are reported in [Berglind]263 and [Berglind et al.]264.

Lars Tysell, “Hybrid grid generation for viscous flow computations around complex geometries”, Technical Reports from Royal Institute of Technology, Department of Mechanics, Stockholm, Sweden, 2010. 258 L¨ohner, R., A parallel Advancing Front Grid Generation Scheme. Paper AIAA 2000. 259 Larwood, B., Weatherill, N. P., Hassan, O. & Morgan, K. 2003, Domain Decomposition Approach for Parallel Unstructured Mesh Generation. International Journal for Numerical Methods in Engineering, 2003. 260 Weatherill, N. P., Hassan, O., Morgan, K., Jones, J. & Larwood, B., “Towards Fully Parallel Aerospace Simulations on Unstructured Meshes”. International Journal for Numerical Methods in Engineering, 2001. 261 Ito, Y., Murayama, M., Yamamoto, K., Shih, A. & Soni, B., Development of a Grid Generator to Support 3-D Multizone Navier-Stokes Analysis. Paper AIAA-2008-7180. 262 Eliasson, P., Nordstr¨om, J., Peng, S-H. & Tysell, L.,”Effect of Edge-based Discretization Schemes in Computations of the DLR F6 Wing-Body Configuration”. Paper AIAA-2008-4153. 263 Berglind, T., Numerical Simulation of Store Separation for Quasi-Steady Flow. FOI-R-2761-SE, FOI, Swedish Defense Research Agency, 2009. 264 Berglind, T., Peng, S-H. & Tysell, L., FoT25: Studies of Embedded Weapons Bays - Summary Report. FOI-R-2775SE, FOI, Swedish Defense Research Agency, 2009. 257

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(a) Before Merging

(b) After Merging

Figure 5.37

Local Remeshing

Background Mesh The use of stretched grids in the advancing front method, which was reported in [Tysell]265, has also been reported in [Ghidoni et al.]266. The methods are similar but the specification of the background grid differs from the smoothing technique used since the algorithm used here only modify the background grid cell size for a node if it is larger than allowed by the prescribed expansion from the nodes in the background grid connected to the node. The technique is similar to the one used for isotropic cell sizes reported in [Kania & Pirzadeh]267, where surface curvature dependent grids also are generated. A way to improve the initial background grid by introducing extra background grid nodes using background boxes is given in [Tysell]268. This play somewhat the same role as the concept of bounding boxes and volume sources introduced in [Pirzadeh]269. A method to achieve an even better background grid by first generating a coarse volume grid and then interpolating the surface cell size specification into this grid and finally using this grid as a new background grid is given in [Tysell ]270. This is an alternative to the octree background grid generation method given in [McMorris & Kallinderis]271. For further details, please consult the work by [Tysell]272.

Tysell, L., CAD Geometry Import for Grid Generation. Proceedings of the 11th ISGG Conference on Numerical Grid Generation, International Society of Grid Generation (ISGG), Montreal, Canada, 2009. 266 Ghidoni, A., Pelizzari, E., Rebay, S. & Selmin, V., 3D Anisotropic Unstructured Grid Generation. International Journal for Numerical Methods in Fluids, 51, pp. 1097-1115, 2006. 267 Kania, L. & Pirzadeh, S., A Geometrically-derived Background Function for Automated Unstructured Mesh Generation. Paper AIAA-2005-5240. 268 Tysell, L. 2009, CAD Geometry Import for Grid Generation. Proceedings of the 11th ISGG Conference on Numerical Grid Generation, International Society of Grid Generation (ISGG), Montreal, Canada. 269 Pirzadeh, S. 2008, Advanced Unstructured Grid Generation for Complex Aerodynamic Applications. Paper AIAA-2008-7178. 270 Tysell, L. 2009, CAD Geometry Import for Grid Generation. Proceedings of the 11th ISGG Conference on Numerical Grid Generation, International Society of Grid Generation (ISGG), Montreal, Canada. 271 McMorris, H. & Kallinderis, Y. 1997, Octree-Advancing Front Method for Generation of Unstructured Surface and Volume Meshes. AIAA Journal, 35 (6), pp. 976-984. 272 details c 272 Lars Tysell, “Hybrid grid generation for viscous flow computations around complex geometries”, Technical Reports from Royal Institute of Technology, Department of Mechanics, Stockholm, Sweden, 2010. 265

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Boundary Viscous Meshes & Sharp Corners In [Aubry & L¨ohner]273 the same prismatic grid generation algorithm as was given in [Tysell] to compute the most visible normal vector to a surface is described. A problem in prismatic grid generation is the quality of the grid in corners of the surface. [Sharow et al.]274 suggest this can be solved by generating extremely fine surface grids at corners, having a cell size of about the height of the first cells in the prismatic layer. A drawback of this approach is of course that the number of cells will increase considerably. [Soni et al.]275 use the concept of a semi-structured topology in the near surface regions, generated by a parabolic grid generation algorithm. In this concept corners can be handled by excluding or introducing nodes from one layer to the next layer. Khawaja et al. (1999) introduce the concept of varying number of prismatic layers, where different surface nodes can have different number of layers. This concept has also been used in [Tysell ]276. For some surface nodes there may not be possible to define a visible normal vector. Thus a prismatic layer cannot be generated at these nodes. A remedy is to use multiple normal vectors at these nodes. The use of multiple normal also improve the grid quality at sharp convex corners like wing trailing edges. In [Steinbrenner & Abelanet]277 difficult regions, especially concave regions, is handled by collapsing cells. Thus, the number of nodes in one layer may be less than in the previous layer. This kind of technique can only be used for layers consisting of stretched tetrahedra instead of prisms. Further details can be obtained at [Tysell]278. Procedures for Mesh Generation In all the methods above the stretched cells close to the boundary have been generated first and the isotropy tetrahedra have been generated in a second step. The methodology has been depicted in Figure 5.38 using a Centrum™ generated hybrid grid279. [Ito & Nakahashi]280 and [Karman]281 use a different strategy where a grid consisting of isotropy tetrahedra has been generated first. In a second step this grid is pushed away from the boundary and the gap is filled with prismatic cells. A drawback with this method is that it will likely be a jump in cell size at the interface between the prismatic layer and the isotropy tetrahedra, since it is difficult to push the initial tetrahedra far enough without inversion to get a sufficient height of the prismatic layer. [L¨ohner & Cebral]282 present a method where an isotropic tetrahedral grid is generated first and then refined with stretched tetrahedra close to the boundary. The use of prismatic layers along imaginary surfaces in

Aubry, R. & L¨ohner, R., Geration of Viscous Grids with Ridges and Corners. Paper AIAA-2007-3832. D., Lou, H. & Baum, J. 2001, Unstructured Navier Stokes Grid Generation at Corners and Ridges. Paper AIAA-2001-2600. 275 Soni, B., Thompson, D., Koomullil, R. & Thornburg, H. 2001, GGTK: A Tool Kit for Static and Dynamic GeometryGrid Generation and Adaptation. Paper AIAA-2001-1164. 276 Tysell, L. 2007, The TRITET Grid Generation System. Proceedings of the 10 th ISGG Conference on Numerical Grid Generation, International Society of Grid Generation (ISGG), Heraklion, Crete, Greece. 277 Steinbrenner, J. & Abelanet, J. 2007, Anisotropic Tetrahedral Meshing Based on Surface Deformation Techniques. Paper AIAA-2007-0554. 278 details c278 Lars Tysell, “Hybrid grid generation for viscous flow computations around complex geometries”, Technical Reports from Royal Institute of Technology, Department of Mechanics, Stockholm, Sweden, 2010. 279 This geometry represents airflow over a transonic transport aircraft, complete with a pylon and flow through nacelle attached to the wing. A hybrid mesh was generated using CENTAUR containing both prisms in the boundary layer and tetrahedra in the interior. A symmetry plane was used so that only half the geometry needed to be modeled. 280 Ito, Y. & Nakahashi, K., Improvements in the Reliability and Quality Un-structured Hybrid Mesh Generation. International Journal for Numerical Methods in Fluids, 2004. 281 Karman, S., Unstructured Viscous Layer Insertion Using Linear-Elastic Smoothing. AIAA Journal, 2007. 282 L¨ohner, R. & Cebral, J. 2000, Generation of Non-Isotropic Unstructured Grids via Directional Enrichment. International Journal for Numerical Methods in Engineering, 49 (1-2), pp. 219-232. 273

274 Sharow,

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the flow field in order to catch shocks is presented in [Shih et al.]283. Pointwise™ is advising following step for their mesh generation, specifically airplane geometry284. The procedure can be summarized as: ➢ Advancing Front Ortho surface mesh generated automatically on watertight solid model ➢ Generates boundary aligned isotropic triangles ➢ Anisotropic triangles grown off slat, wing, flap LE using T-Rex ➢ Structured diagonalized meshes created on slat, wing, flap TE surfaces to comply with TE point req. ➢ Spacings applied to key locations in surface mesh. ➢ Chordwise/spanwise number of grid points increased to reduce area ratio/aspect ratio to reasonable levels. ➢ Manually correct problem areas that were geometry limited. ➢ Grow anisotropic tetrahedra off surface mesh based on refinement level growth rate and wall spacing. ➢ Insert equilateral tetrahedra into remainder of volume using modified Delaunay method.

Figure 5.38

Hybrid Mesh on a Wing-Body-Pylon-Nacelle Configuration – Courtesy of Centrum®

Dynamic Mesh The spring analogy method in [Batina]285 for grid deformation easily gives grid inversion for larger deformations. This has been improved by [Farhat et al. ]286 by the introduction of torsion springs in addition to the tensions springs. In [Acikgoz & Bottaasso]287 springs has been introduced also Shih, A., Ito, Y., Koomullil, R., Kasmai, T., Jankun-Kelly, M., Thompsson, D. & Brewer, W. 2007, Solution Adaptive Mesh Generation using Feature Aligned Embedded Surface Meshes. Paper AIAA-2007-0558. 284 Carolyn D. Woeber , “Pointwise Unstructured and Hybrid Mesh Contributions to GMGW-1”, 1st Geometry and Mesh Generation Workshop Denver, CO June 3-4, 2017 285 Batina, J., Unsteady Euler Algorithm with Unstructured Dynamic Mesh for Complex-Aircraft Aerodynamic Analysis. AIAA Journal, 29 (3), pp. 327-333, 1991. 286 Farhat, C., Degand, C., Koobus, B. & Lesoinne, M. 1998, An Improved Method of Spring Analogy for Dynamic Unstructured Fluid Meshes. Paper AIAA-1998-2070. 287 Acikgoz, N. & Bottasso, C. 2006, A new Mesh Deformation Technique for Simplicial and Non-Simplicial Meshes. Paper AIAA-2006-0885. 283

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between nodes and edges and not only between nodes as in the previous methods. This greatly improves the capability to prevent the generation of inverted cells. Still there can be cells of bad quality, especially close to the boundaries, since only translation of the boundary has been taken into account. In [Martineau & Georgala]288 the rotation has been taken into account by rigid movement of the nodes close to the boundary and use of the spring analogy away from the boundary. In [Samareh]289 the use of quaternions has been introduced. Quaternions is an extension of complex analysis to three-dimensional space, which are ideal for modeling rotations. Since the method in [Tysell]290 solves a 4th order partial differential equation it is possible to have two boundary conditions on the boundary. In the original paper the derivative of the displacement was set to zero on the boundary. Later an option has been introduced to take the local rotation around each boundary node into account. Another modification of the algorithm is that in the discrete minimization problem the volume of the tetrahedra is retained when solving the bi-harmonic equation, otherwise the movement will be too rigid for the small tetrahedra close to the boundaries. If the deformation is governed by the Laplace equation only the original formulation is kept. In [Nielsen & Anderson]291 a structural mechanic analogy is used by solving the equation for isotropic linear elasticity. A similar approach is used in [Sheta et al.]292 where the structural Navier equation is applied. Both methods show good results, but they probably suffer by the fact that they both solve a 2nd order partial equation, thus only one boundary condition can be applied. In [Liu et al. ]293 a fast algebraic method is presented based on the deformation of a temporary grid generated from the boundary nodes only by a Delaunay algorithm. The deformation of the grid is computed by interpolation in the temporary Delaunay grid. The deformation of the boundary must be done in such small steps that the Delaunay grid is not inverted. After each step the Delaunay grid may be regenerated. For large 3D grids this may take a considerable time, and the method does not account for surface rotation. Another new algebraic method is the use of radial basis functions, see [Jakobsson & Amignon]294 and [Allen & Rendall]295. Adaptation In [Peraire et al.]296 the use of the Hessian297, which measures the second derivatives of the flow quantities, is introduced in order to generate directionally adapted grids. One drawback with the use

Martineau, D. & Georgala, J., A Mesh Movement Algorithm for High Quality Generalized Meshes. AIAA-2004. Samareh, J. 2002, Application of Quaternions for Mesh Deformation. Proceedings of the 8th International Conference on Numerical Grid Generation in Computational Field Simulations, pp. 47-57, International Society of Grid Generation (ISGG), Honolulu, Hawaii, USA. 290 Tysell, L. 2002, Grid Deformation of 3D Hybrid Grids. Proceedings of the 8th International Conference on Numerical Grid Generation in Computational Field Simulations, pp. 265-274, International Society of Grid Generation (ISGG), Honolulu, Hawaii, USA. 291 Nielsen, E. & Anderson, K. 2001, Recent Improvements in Aerodynamic Design Optimization on Unstructured Meshes. Paper AIAA-2001-0596. 292 Sheta, E., Yang, H. & Habchi, S. 2006, Solid Brick Analogy For Automatic Grid Deformation For Fluid-Structure Interaction. Paper AIAA-2006-3219. 293 Liu, X., Qin, N. & Xia, H. 2006, Fast Dynamic Grid Deformation Based on Delaunay Graph Mapping. Journal of Computational Physics, 211 (2), pp. 405-423. 294 Jakobsson, S. & Amignon, O. 2005, Mesh Deformation using Radial Basis Functions for Gradient Based Aerodynamic Shape Optimization. Technical Report FOI-R-1784-SE, FOI, Swedish Defense Research Agency. 295 Allen, C. & Rendall, T. 2007, Unified Approach to CFD-CSD Interpolation and Mesh Motion using Radial Basis Functions. Paper AIAA-2007-3804. 296 Peraire, J., Peiro, J. & Morgan, K. 1992 Adaptive Remeshing for Three Dimensional Compressible Flow Computations. Journal of Computational Physics, 103 (2), pp. 269-285. 297 In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The 288 289

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of the Hessian is that the grid can only be adapted to one selected flow quantity. It is in most cases not possible to find one single flow quantity that works for all cases, or even for all regions of one case. One remedy to this has been presented in [Castro-Diaz et al.]298 by computing the Hessian for several flow quantities and then compute the combined Hessian by so called metric intersection. This way to compute the metric intersection is not optimal, since only the combined eigenvalues are computed, while the eigenvectors are arbitrarily chosen from one of the flow quantities. Thus, the way to compute the combined metric from several flow quantities presented in [Tysell et al. (1998)] is done in a more rigorous way, but the drawback may be that only the first derivatives of the flow quantities are used instead of the second derivatives. Another way to compute the metric intersection has been presented in [Frey & Alauzet]299. The Hessians can be represented by a set of ellipsoids and in that paper the intersection is computed by computing the largest ellipsoid inscribed in all intersected ellipsoids. A rigorous way to compute this metric intersection has recently been presented in [McKenzie et al. ]300], where each Hessian is introduced successively by transforming the Hessian to a space where the current transformation is represented by a sphere. In this space the intersection is easy to compute. Both for first and second derivative adaptive sensors limits of the cell sizes must be set in regions of flow discontinuities, where the cell size otherwise would become indefinitely small, and in regions where the flow is varying very slowly, where the cell sizes would become too large. One drawback by using the Hessian is also that the cell sizes tends to be indefinitely large where the flow is varying linearly. I has been shown in [Venditti & Darmofal]301 where they compare results for flow around a multiple airfoil configuration, that a Hessian based method gives too large cell sizes in regions of linearly varying flow compared to an adjoint based adaptation method. In the paper they propose a combination of the two methods, where the cell sizes are taken from the adjoint computation, whereas the directional stretching are taken from the Hessian. In [Remaki et al. ]302 the metric tensor is computed by taking a weighted sum of the Hessian and the gradient tensor, in order to get better grid resolution also in areas of linearly varying flow quantities. The use of the Hessian and combinations of grid cell split, merge, swapping and node movement is used in [Xia et al.]303 and [Dompierre et al.]304 for directional h-adaptation. In these papers the method has been applied in two dimensions. The same method has been used to present three-dimensional results in [Park & Darmofal]305. This method is faster than the total remeshing used in [Peraire et al.]306 and Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". 298 Castro-Diaz, M., Hecht, F., Mohammadi, B. & Pironneau, O., Anisotropic Unstructured Mesh Adaption for Flow Simulations. International Journal for Numerical Methods in Fluids, 25 (4), pp. 475-491, 1997. 299 Frey, P. & Alauzet, F. 2005, Anisotropic Mesh Adaptation for CFD Computations. Computer Methods in Applied Mechanics and Engineering, 194 (48-49), pp. 5068-5082. 300 McKenzie, S., Dompierre, J., Turcotte, A. & Meng, E. 2009, On Metric Tensor Representation Intersection and Union. Proceedings of the 11th ISGG Conference on Numerical Grid Generation, International Society of Grid Generation (ISGG), Montreal, Canada. 301 Venditti, D. & Darmofal, D. 2003, Anisotropic Grid Adaptation for Functional Outputs: Application to TwoDimensional Viscous Flows. Journal of Computational Physics, 187 (1), pp. 22-46. 302 Remaki, L., Nadarajah, S. & Habashi, W. 2006, On the a Posteriori Error Estimation in Mesh Adaptation to Improve CFD Solutions. Paper AIAA-2006-0890. 303 Xia, G., Li, D. & Merkle, C., Anisotropic Grid Adaptation on Unstructured Meshes. Paper AIAA-2001. 304 Dompierre, J., Vallet, M., Bourgault, Y., Fortin, M. & Habashi, W. 2002, Anisotropic Mesh Adaptation: Towards User-independent Mesh-independent and Solver-independent CFD. Part III. Unstructured Meshes. International Journal for Numerical Methods in Fluids, 39 (8), pp. 675-702. 305 Park, M. & Darmofal, D. 2008, Parallel Anisotropic Tetrahedral Adaptation. Paper AIAA-2008-0917. 306 Peraire, J., Peiro, J. & Morgan, K. 1992 Adaptive Remeshing for Three-Dimensional Compressible Flow Computations. Journal of Computational Physics, 103 (2), pp. 269-285.

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[Tysell et al.]307 but appears to give grids of less good quality. In [Pirzadeh]308 adaption by remeshing is done by only doing local remeshing, where the grid needs to be adapted. In this way the time for remeshing is reduced. Further details can be obtained at [Tysell]309. Special Issues Two verdures offer special difficulties and field complexities which may be encounter during mesh generation session: 5.7.8.1 ➢ ➢ ➢ ➢ ➢ ➢

Centaur © Singular points Small scales Small angles Disparate length scales Different scales orientation / directionality Mega Geometries - Mega geometries pose additional requirements to the grid generator. Some common issues are the existence of thousands of panels and "dirty" CAD. Solutions to these issues include: • Automatic Setup • Automatic identification of similar parts (e.g. pipes) • Auto CAD Cleaning

Figure 5.39

Meshing Aircraft Landing & Takeoff – Courtesy of Centaur©

Tysell, L., Berglind, T. & Eneroth, P., Adaptive Grid Generation for 3D Unstructured Grids. Proceedings of the 6th International Conference on Numerical Grid Generation in Computational Field Simulations, pp. 391-400, International Society of Grid Generation (ISGG), Greenwich, UK, 1998. 308 Pirzadeh, S. 2000, A solution-Adaptive Unstructured Grid Method by Grid Subdivision and Local Remeshing, Journal of Aircraft, 37 (5), pp. 818-824. 309 details c 309 Lars Tysell, “Hybrid grid generation for viscous flow computations around complex geometries”, Technical Reports from Royal Institute of Technology, Department of Mechanics, Stockholm, Sweden, 2010. 307

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➢ Mega Meshes - Mega meshes are required in applications such as LES, multi-stage turbomachinery, aircraft take-off and landing, etc. (see Figure 5.39). Requirements that have to be met for the generation of "mega" meshes include: • Optimum meshes (minimum number of elements for given accuracy - hybrid mesh approach) • Parallel / Multi-Core grid generation • Robustness of grid generation 5.7.8.2 ➢ ➢ ➢ ➢ ➢

Pointwise© Some meshing guidelines could not be retained & create a volume mesh of reasonable quality Recommended grids have at least 2 layers of constant cell spacing normal to viscous walls. Achieving consistent cell sizes and spacings across gaps. Mesh quality quality/characteristics Medium grid level ~ 3-4 hours per iteration.

Listing of Available Meshing Software Table 5.3 shows the list of currently available grid generation software (please be advised that some vendors and features might have been updated yet).

0

0 - Mesh Software 1 – Structured 1 2 3

3DGRAPE ANSA CFD-GEOM CSCMDO EAGLEVIEW GAMBIT GEMS GENIE++ GRID* GRIDGEN GridPro ICEM-CFD IGG INGRID Hyper Mesh

yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes

-----yes yes -----yes yes ---------------yes yes yes yes -----yes

Table 5.3

-----yes yes ----------yes ---------------yes yes yes yes -----yes

2 - Un-Structured 0 1

Hyper Mesh MACGS MBGRID MEGACADS NGP PEGSUS Pro-Star RAGGS RAPID SAUNA TGRID TIGER UNISG VGRID CENTAUR

yes yes yes yes yes -----yes yes yes yes -----yes yes -----yes

Currently Available Grid Generation Software

3 - Hybrid 2

yes ---------------yes Chimera yes yes -----yes yes ----------yes yes

3

yes ---------------yes -----yes ----------yes --------------------yes

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6 Adaptive Mesh (Unstructured) The case for structured adaptive grid is covered before, therefore, will not be mentioned here, therefore, this section is intended for unstructured adaptive meshing. Aside from the treatment of complex geometries, the second main advantage of unstructured meshes is the ease with which solution-adaptive meshing may be implemented310. Since no inherent structure is assumed in the representation of the mesh, mesh points may be added, deleted, or displaced, and the mesh connectivity may be locally reconfigured in the affected regions. The goal of mesh adaptation is the determination of the optimum mesh-point distribution that results in equipartition of the error for each individual simulation. The character of the problem to be solved dictates the requirements of the mesh adaptation strategy. For example, steady-state problems usually involve a small number of adaptation phases as part of a lengthy solution process. Therefore, relatively sophisticated adaptation strategies can be employed, such as, in the extreme case, complete mesh regeneration. Mesh refinement procedures are most important here, while de-refinement has only a minor effect and can often be omitted for steady-state cases. For transient problems, mesh adaptation must be performed every several time steps, and thus efficiency is much more important than optimality. Mesh refinement and de-refinement are both essential for transient cases, as well as mesh movement for cases with moving boundaries. Furthermore, the accuracy of interpolation from the original mesh to the refined mesh affects the solution accuracy (unlike the steady-state case), and thus accurate transfer schemes are required. Delaunay-based mesh generation techniques can easily be extended to incorporate adaptive refinement capabilities [Mavriplis]311-312; [Weatherill]313. Once a solution has been obtained on an initial mesh, new points can be added in regions where high errors are detected. These new points can be triangulated into the mesh using the Bowyer-Watson point insertion algorithm. Alternatively, if a Non-Delaunay mesh is employed, new points may be inserted through element subdivision, and the connectivity of the resulting mesh may be optimized through several face-edge swapping passes based on any appropriate criterion. Rule-based hierarchical element subdivision is a very effective adaptive technique, particularly for unsteady flows, where efficiency and accuracy of interpolation between successive grids are important considerations [Stoufflet 1987, L¨ohner & Baum 1992, Rausch1992, Braaten & Connell 1996]. The essential approach consists in recursively subdividing mesh elements where large solution errors are detected, but conforming to a set of well-defined subdivision rules, which are necessary to prevent the formation of degenerate element shapes and connectivity. Storing the hierarchy of the recursive process enables de-refinement to be implemented by simply retracing the subdivision history. Hierarchical subdivision is also applicable to non-simplicial hybrid meshes by constructing a library of subdivision rules for each element type.

Adaptive Meshing by Subdivision In the interest of developing a single strategy for adapting simplicial as well as mixed element meshes, a hierarchical element subdivision approach has been adopted314. As described before, one must ensure that a compatible refinement pattern is obtained on all elements of the mesh if a valid refined meshes obtained. This technique can be applied to fully tetrahedral meshes, as well as to any Mavriplis, DJ:” Unstructured Grid Techniques”, Annual. Rev. Fluid. Mech. 1997 by Annual Reviews Inc. DJ., “Accurate multigrid solution of the Euler equations on unstructured and adaptive meshes”, AIAA J. 28(2):213–21 312 Mavriplis DJ., “Adaptive mesh generation for viscous flows using Delaunay triangulation”, J. Comp. Phys. 1990. 313 Weatherill NP, Hassan O, Marcum DL, “Calculation of steady compressible flow fields with the finite-element method”, AIAA Pap.93-0341 314 314 Mavriplis D.J., ”Adaptive Meshing Techniques For Viscous Flow Calculations On Mixed Element Unstructured Meshes”, NASA Contract No. NAS1-19480, May 1997. 310

311 Mavriplis

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hybrid mesh containing mixtures of tetrahedral, pyramids, prisms and hexahedra. The resulting meshes can be employed by the multigrid solver described in315 without modification. In order to implement this technique on mixed element meshes, the various allowable subdivision types for each element type must be defined. The hierarchical rules required to prevent the degeneration of the grid quality with successive adaptation levels must also be constructed. For tetrahedral elements, the subdivision rules have already been well formulated in the literature. We allow only three basic subdivision types: A tetrahedron may be divided into 2 children, 4 children, or 8 children. The two former cases result in anisotropic refinement, while the last case produces an isotropic refinement. In order to prevent the degeneration of grid quality, any anisotropic children may not be refined further. If any such cells require refinement, they are removed, the parent cell is isotopically refined, and the resulting isotropic children may then be further refined. When limiting the possible refinement types. This is achieved by adding refinement points along the all appropriable edges on all elements which are flagged as having non-valid refinement pattern. Since the addition of a refinement point to an edge affects all elements which contain the edge, the process is applied iteratively, until all resulting elemental refinement are valid and no further points are required. The isotropic regiment of a hexahedra element results in eight similar but smaller hexahedral elements. However, anisotropic refinement of a hexahedral element results in children which may consist of hexahedra, pyramids, prisms and tetrahedral. By applying the same hierarchical rules as described for tetrahedral meshes we can ensure that lese elements will never be refined further. Instead, if further refinement ill these regions is desired, such elements are deleted and that parents refined into eight smaller hexahedra. Thus, for fully hexahedral meshes, additional element may only be the boundaries between refined and non-refined regions, or more generally, between two regions which differ by one refinement level. The task of implementing adaptive mesh subdivision elements other than tetrahedral consists in defining the minimum number of allowable subdivision types. On the one hand, it, is desirable to limit the number of subdivision types for complexity reasons. On the other hand, a minimum number of subdivision types must be implemented to allow for compatible subdivision types to be attained on all elements without, incurring excessive additional refinement316.

Adaptive Mesh Refinement (AMR) [Berger and LeVeque]317 addressed several deficiencies that existed in the established uniform grid methodologies. First, they applied the concept of Adaptive Mesh Refinement (AMR) in order to improve the accuracy in critical regions without adversely affecting the efficiency of the numerical integration scheme. The use of AMR effectively allowed the clustering of blocks of computational grids as the solution process evolved only in the region that they were needed (and not clustering entire grid lines), by using Richardson-type extrapolation error estimates to identify regions of large errors and adding grid blocks in those regions. An example of AMR is Figure 6.1 which represents a simple adapted grid for a supersonic wedge flow with four levels of adaption. As can be seen in the figure, there are more control volumes where gradients are to be expected, specifically along the surface to capture the geometry and along the oblique shock. In regions with small gradients, there is a lower density of control volumes. Also notice that in this figure there is at most a 2:1 ratio at the

l). J. Mavriplis and Venkatakrishnan. A unified multigrid solver for the Navier-Stokes equations on mixed element meshes. AIAA Paper 95-1666, June 1995. 316 Mavriplis D.J., ”Adaptive Meshing Techniques For Viscous Flow Calculations On Mixed Element Unstructured Meshes”, NASA Contract No. NAS1-19480, May 1997. 317 M. J. Berger and R. J. LeVeque. An Adaptive Cartesian Mesh Algorithm for the Euler Equations in Arbitrary Geometries. 9th AIAA Computational Fluid Dynamics Conference, Buffalo, NY, June 1989. AIAA-89-1930-CP. 315

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refinement interface, which is typical of most A MR schemes, in order to promote stability in the numerical schemes. One problem with [Berger and LeVeque's] original enactment of AMR on Cartesian grids was the problem of state variable conservation during the AMR stages. They carefully constructed conservative schemes for the inter-grid transfer to address the problem. They also used the idea of wave propagation and directional differencing in order to increase Figure 6.1 Example Adaptive Grid for Supersonic Wedge Flow the stability near the small boundary cells. This helped keep the CFL of the boundary cells reasonably close to the CFL of the flow cells and allowed larger time steps to be taken with the solver remaining stable. Generalities Several researchers have extended [Berger and LeVeque's] research into areas such as multigrid Cartesian grids, higher accuracy flow solvers using more sophisticated flux approximations, timeaccurate unsteady flows, and a front tracking AMR scheme that attempted to track the discontinuities (such as shocks) as the solution evolved in order to provide more accuracy in the refined mesh

(a) Intersecting meshes with a circle are tagged (blue) Figure 6.2

(b) 2D case of Adaptive Cartesian grid method

Schematic image of Adaptive Mesh Refinement

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calculations. According to recent investigation by [Abe]318, Cartesian grid method fall into two categories with the demand of accurate solutions. One keeps its structured grid nature and introduces embedding structured sub grids within the underlying coarse structured grids. Adaptive Mesh Refinement (AMR) is one of them. Figure 6.2 (a) shows an example of AMR in two dimension. The intersected cells by a circle in the underlying coarse grids are tagged in blue. The blue-tagged cells are to be refined. In the AMR procedure, several embedded rectangle patches are defined so as to contain the blue tagged cells. Then, the embedded rectangle patch areas are refined. The other considers the Cartesian mesh as an unstructured collection of h-refined meshes. The data structure is not the same as structured grids but the same as unstructured grids. Adaptive Cartesian grid method was introduced as an unstructured Cartesian grid method and has shown the great success in simulating complex geometries. Figure 6.2 (a) shows a case of two dimensional adaptive Cartesian grid method. Beginning with a root cell covering whole domain, the intersected cells by the circle are recursively bisected. This simple procedure finally gives Figure 6.2 (b). The actual data structure (quad-tree) of a two dimensional adaptive Cartesian grid method is shown in Figure 6.3. The case shows a quad-tree structure. In this two dimensional case a tree node (a cell) may have four child nodes. In the case of three dimension, a node may have eight children. Level of a node is defined as the depth of the nest in the tree structure. The root node is specified as ’level 0’ (Figure 6.3)319.

Figure 6.3

Octree Data Structure of Adaptive Cartesian Grid Method

Cell Division for a Geometry First, we describe the strategy to decide a cell to be divide into smaller cells which is described in320. The strategy gives a criteria which determines whether a intersected cell by triangular facets is to be divided or not. Next, a method which is a fast algorithm to find intersecting triangular facets with respect to a cell is described.

Hiroshi Abe, “Blocked Adaptive Cartesian Grid FD-TD Method for Electromagnetic Field with Complex Geometries”, International Conference on Modeling and Simulation Technology, Tokyo, JAPAN, 2011. 319 Each black circles indicates leaf nodes in the tree structure and they correspond to the cells as is shown with the numbers. 320 M.J. Aftosmis. Solution adaptive Cartesian grid methods for aerodynamic flows with complex geometries, 1997. 318

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6.2.2.1 Division Criteria 3D geometry is often provided as a CAD file or STereo Lithography (STL) data . Both data may consist of a set of triangulated facets of the geometry’s surface. We need criteria to decide the cells to be divided in order to resolve the geometry through the triangulated facets. We adopt a curvature detection strategy321. Suppose a cell is intersected by triangular facets Ti . ni is the normal vector of the facets (Figure 6.4). Angle variation Vj can be defined as,

Vj = Max(nkj − Min (nkj )

k ∈ Ti (j = x, y, z)

Eq. 6.1 Ti. The angle which indicates the curvature of the facets is given by,

cos (θi j ) =

Vj ⃗| |V

Eq. 6.2 If θ in a cell exceeds a predefined angle threshold, then the cell is tagged for division. This procedure for division is very simple and robust. One can have adaptive Cartesian cells automatically. Further examples provided a 2D backward step (see Figure 6.5).

Figure 6.5

Figure 6.4 Schematic 2D view of angular variation of normal vector of triangles within cut-cell i. (a) is small variation case and (b) is large variation case.

Pressure Contours in 2D Backward Step

Uniform AMR The simplest refinement anyone can think of is to divide all cells in the domain. This is referred to as "Uniform Refinement". Although it does improve the solution vastly, it is easy to realize that we are going for a huge unwanted effort in doing so. For e.g., in the far field region of an airfoil, cell division 321

See Previous.

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is not bringing in any improvement because the flow such as a shock-boundary layer interaction. To achieve the goal of mesh adaptation, the refinement is done at "selected" regions alone based on certain criterion. This is referred to popularly as AMR or Adaptive Mesh Refinement. It is to be remarked that AMR does not only encompass division of cells into smaller ones (Refinement), but also the agglomeration of smaller cells into a larger one (De-Refinement or coarsening), when the need arises. 6.2.3.1 Transient Inviscid Flow Considers the solution of an internal transient inviscid supersonic flow (Lyra et al.)322. The geometry consists in a wind tunnel with a step and the inflow boundary condition consists of a uniform Mach 3.0 flow with angle of attack 0°. At the right boundary the flow is let free to leave the domain and along the walls, reflecting boundary conditions are applied. During the transient adaptive procedure several adapted meshes are generated along the time integration according to the error analysis. Figure 6.6 shows some selected meshes: mesh, is the third mesh generated during the transient adaptive process, and meshes and are meshes generated before and after the time when the shock starts to be reflected from the top boundary. The mesh refinement is clearly following the physical features of the flow. The adaptive algorithm try to obtain an “optimal” mesh for a pre-defined number of elements. The target number of elements for this analysis was 1000 and a limited aspect ratio of 4 was considered. The number of elements Figure 6.6 Selected Initial Meshes for the Transient generated in the meshes shown was 620, Adaptive Procedure (Meshes 3, 20, 27 and 29) 977, 994 and 1010, and the corresponding number of nodes was 638, 1007, 1029 and 1047, showing that the procedure obeyed well the imposed constraints. Case Study 1 - An Adaptive Hybrid Mesh Generation Method for Complex Geometries An adaptive hybrid mesh generation method is described by [Cameron Thomas Druyor JR.]323 to automatically provide spatial discretization suitable for 2D solver applications. This method employs a hierarchical grid generation technique to create a background mesh, an extrusion-type method for inserting boundary layers, and an unstructured triangulation to stitch between the boundary layers and background mesh. This method provides appropriate mesh resolution based on geometry Lyra, P.R.M., de Carvalho, D.K.E., Willmersdorf, R.B. and Almeida, R.C.C., 2002, “Transient Adaptive Finite Element Analysis of Compressible Flows”, WCCM'2002-Proceedings of the 5th World Conference on Computational Mechanics, Vienna-Austria. 323 Cameron Thomas Druyor JR, “An Adaptive Hybrid Mesh Generation Method for Complex Geometries”, A Thesis Submitted to the faculty of the University of Tennessee, Chattanooga in Partial Fulfillment of the Requirements for the Degree of Master of Science in Computational Engineering, 2011. 322

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segments from a file, and has the capability of adapting the background mesh based on a spacing field generated from solution data or some other arbitrary source. By combining multiple approaches to the grid generation process, this method seeks to benefit from the strengths of each, while avoiding the weaknesses of each. 6.2.4.1 Mesh Stitching The hybrid method, such as AMR, generates several different meshes which must be assembled into a single mesh before exporting to a mesh file that can actually be used by a solver. There are two main elements to accomplishing this task, removing elements from the background mesh that are not to be part of the final mesh, and bridging the gap between the viscous and inviscid meshes. 6.2.4.1.1 Removal of Background Mesh Elements Removal of the unnecessary elements of the inviscid mesh is done in two stages. First, all elements that violate a boundary segment, or come close to violating a segment within a tolerance, are removed from the mesh. This partitions the mesh into contiguous blocks that are wholly inside, or outside the computational domain. The remaining voxels are marked in or out with a recursive flood-fill algorithm. It is important to note that flooding the contiguous blocks recursively can result in a stack over flow for large meshes because of the number of function calls that get pushed onto the call stack. There are two options for avoiding this situation: increase the maximum stack size, or develop a replacement routine that applies the recursive algorithm without making recursive function calls. The proposed method takes the second approach, utilizing a queue style structure. The unmarked cell checks to see if its neighbors have been marked. Each neighbor that is not yet marked is marked and then pushed onto the queue. Then, while the queue is not empty, the first element of the queue is popped off. This element checks for neighbors that have not yet been marked, marks them, and pushes them onto the queue. When there are no voxels (i.e., cell) left in the queue, the algorithm searches for another unmarked voxel to start the process again, until there are no unmarked voxels left. 6.2.4.2 Triangulation Once the voxel removal is complete, there is a gap between the viscous region and the background mesh (in the absence of a viscous mesh, there is a gap between the geometry and the background mesh). This region must be filled with cells before a valid mesh can be created. This is done in two steps. First a list of unique nodes and boundaries for the region to be triangulated are created. The boundaries consist of the exposed edges of the voxel front, the outermost edges of the viscous mesh, and any exposed geometry segments, and the unique node list contains each point that is part of any of those edges. Once these are packaged up properly, they are passed to a Delaunay triangulation method written by Dr. Steve Karman, which returns a list of triangles that fill the gap region. 6.2.4.3 Test Cases 6.2.4.3.1 30P30N Multi-Element Airfoil The proposed method has a robust viscous layer production method that can create viscous layers without introducing negative elements caused by crossed normal. To showcase this, the 30P30N airfoil was chosen. shows the far field view of the mesh, demonstrating that it spans the computational domain and that the square far field boundaries are preserved. Zooming in on the slat element, Figure 6.7 shows that viscous layers are created on the slat and the leading edge of the wing. Taking a closer look at the bottom trailing edge of the slate shows that the expected issues are present at the sharp point, but the mesh is valid in the region. Note that the elements in the stitching region are generally of high quality and appropriate size; it is only the areas where the viscous elements are skewed that the cell size gradation changes drastically between the viscous elements and the stitching elements. Further discussion are available in[Cameron Thomas Druyor JR.].

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Figure 6.7

30P30N Multi-Element Airfoil & close up of slat

6.2.4.3.2 2D Fuel Cell Slice The geometry for this test case is a 2 dimensional slice of a hypothetical fuel cell. It was chosen to showcase the proposed method's ability to handle many different geometries at once, and also to demonstrate that it can be used to generate grids for applications other than airfoils. Figure 6.8 shows a far field view which shows the configuration of the rods.

Figure 6.8

2D Fuel Cell Slice & Zoomed

Case Study 2 – Unstructured Mesh Adaptation for 2D Airfoil324 In order to be able to examine shock-dominated processes at high spatial resolutions without David A. Venditti and David L. Darmofal, “Grid Adaptation for Functional Outputs: Application to Two Dimensional Inviscid Flows", Journal of Computational Physics 176, 40–69 (2002). 324

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incurring heavy computational penalties, even in two dimensions, it is desirable to use some form of mesh adaptation. Adaptive codes make use of the local flow solution itself to determine where the high spatial mesh resolution is required and then employ some strategy to increase the grid resolution in those regions. This enables the high spatial mesh resolution to be concentrated around the important flow features (e.g. shocks, vortices and so on) rather than being wasted on parts of the computational domain where the flow activity is relatively unimportant. One increasingly popular approach to mesh adaptation is so-called refinement, where additional mesh nodes and elements are inserted into the computational domain in regions where the greater resolution is required and then removed from the mesh when the higher mesh node density is judged to be redundant. The process of mesh adaptation is invoked automatically in response to some dynamically evolving flow solution criteria, with the regions of mesh Figure 6.9 Grid Adaption using Supersonic Flow for an refinement corresponding to regions of Airfoil (bow shock) significant flow activity where it is desirable to have increased spatial resolution (see Figure 6.10 and Figure 6.9). How these criteria are chosen has important consequences for the overall operation of the adaptive solver. The complete adaptive solver may be thought of as consisting of three parts, • • •

mesh data structure, adaptation algorithm and flow integration algorithm, where these objects are organized as adaptation data structure integration

Thus the adaptation and integration operations can be thought of as two distinct processes that are applied to the central data structure. The connectivity outlined above is sufficient to completely specify a given mesh, but it does not contain the connectivity required to construct the adaption hierarchy. For this some additional information is required, which in the case of elements and edges consists of storing parent and child addresses. The bisection of a parent edge by the addition of a node to the mesh results in the creation of two child edges. To extend the, sequence of events involved in a complete adaption and integration of the mesh is shown at first, mesh elements are flagged for adaption. This results in each mesh edge

Figure 6.10

NACA 0012 Transonic test case: M∞ = 0.8, α=1.25

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being targeted either for refinement, de-refinement or no action data structure to include numerical parameters such as the flow variables. 6.2.5.1 Adaption Control Mechanism The Euler flow solver is combined with the adaptive algorithm by flagging regions of the mesh with (low) high density gradients for de-refinement, with the calculation of local flow gradients being performed across element faces. Where the face normal density gradient falls below or exceeds a chosen tolerance, the edges on the face are flagged to de-regime. In addition, a `safety layer' of refinement flagging is employed to ensure the full capture of solution discontinuities, which is the principal concern for this application. Likewise, a maximum mesh refinement depth is also specified. Coupling the adaption algorithm to the solution integrator is managed in a similarly straightforward manner. Figure 6.10 depicts a NACA airfoil in a transonic flow, while Figure 6.9 shows the same geometry when placed in supersonic flow with a bow shock. Case Study 3 – Parallel Implementation of Unstructured Mesh Refinement of Duct Flow The simultaneous alteration of the decomposed domains of an unstructured mesh presents a number of challenges325. In parallel, each processor operates on its own partition, concurrent with and independent of the others. Previous work in parallel mesh refinement326-327 demonstrated methods in which adaptation was performed on each processor, and patterns for cell subdivision were exchanged across inter-processor boundaries, ensuring a conforming mesh. Coarsening the interprocessor boundary was not a concern, nor was the possible motion of the mesh boundaries. Therefore the first issue that arises in parallel adaptation is how to treat the inter-processor boundaries. Rather than modify these faces, the inter-processor boundaries are shifted using a cell migration technique. The inter-processor faces and adjacent cells then become interior faces and interior cells, which may be readily modified through a second adaptation pass. In the second pass only the former inter-processor boundary region needs to be coarsened, refined, or smoothed, as the remainder of the mesh is already consistent with the prescribed point spacing illustrates the twopass approach for solution-based coarsening and refinement of supersonic flow entering a duct. The original mesh partitions, shown in Figure 6.11-(1), are independently coarsened and refined to produce the adapted mesh of (2). Note that the inter-processor boundaries are not modified, which leaves a region of the mesh that still requires adaptation. Several layers of cells are migrated from the right processor to the left, as seen in (3). The inter-processor boundary is now to the right of its original location. A second coarsening and refinement pass treats the former inter-processor faces and adjacent cells, producing the final adapted grid of (4). A consequence of the cell migration approach is that the shape and extent of the decomposed domains change. The cell migration process may introduce new pairs of adjacent domains that did not initially communicate. Similarly, pairs of processors that once shared common nodes, edges, and faces may become disconnected as a result of cell migration. Updating the inter-processor communication schedule proceeds in two stages. First, the current communication lists are updated for each pair of adjacent domains. If no common nodes are found between two domains, the communication is removed from the cycle. The second stage involves checking for any new communication pairs introduced as a result of migration. In addition, one can no longer refer to the original decomposed grid to obtain data for solution transfer or for establishing point spacing after Cavallo, P.A., Sinha, N., and Feldman, G.M.,” Parallel Unstructured Mesh Adaptation For Transient Moving Body And Aeropropulsive Applications”, Combustion Research and Flow Technology, Inc. (CRAFT Tech), PA. 326 De Keyser, J., and Roose, D., “Run-Time Load Balancing Techniques for a Parallel Unstructured Multi-Grid Euler Solver with Adaptive Grid Refinement”, Parallel Computing, Vol. 21, pp. 179-198, 1995. 327 Flaherty, J.E., Loy, R.M., Shephard, M.S., Szymanski, B.K., Teresco, J.D., and Ziantz, L.H., “Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws”, Journal of Parallel and Distributed Computing, Vol. 47, pp. 139-152, 1997. 325

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the coarsening phase. This issue is remedied by recomposing the global grid arrays at the start of the mesh adaptation process, such that the list of global vertex coordinates, solution vectors, and computed point spacing may be readily available to all processors. (1) Original

(2) First Adaption Pass

(3) Cell Migration into inter processor boundary

(4) Second Addaption pass

Figure 6.11

Two-Pass Approach for Parallel Coarsening and Refinement.

Case Study 4 – Generic Transonic Store Release328 The next application considered is the separation of a finned store from a wing/pylon configuration at Mach 1.2. Inviscid flow is assumed for this tetrahedral grid. A constant ejection force is applied over the first 0.1 seconds of the simulated separation. After the initial ejection stroke, the motion of the store is provided by general 6-degree-of-freedom (6-DOF) equations of motion using the current integrated surface pressure distribution. Gravitational acceleration is also included. Figure 6.12 provides an overview of the simulation. A total of ten

Figure 6.12 Store position, orientation, and surface pressures at selected points in trajectory

P.A., Sinha, N., and Feldman, G.M.,”Parallel Unstructured Mesh Adaptation For Transient Moving Body And Aeropropulsive Applications”, Combustion Research and Flow Technology, Inc. (CRAFT Tech), PA 18947. 328 Cavallo,

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adaptations were performed at regular intervals. The unstructured grid is comprised of approximately 2.7 M cells, and is decomposed on 16 processors. In this image, the store is colored by the current pressure distribution at each of the four instants shown, and the black lines indicate the changing inter-processor boundaries on the store surface resulting from cell migration and load rebalancing. As it translates, the store yaws nose away from the symmetry plane and pitches nose down. The surface pressure distribution reflects the changing local angle of attack and sideslip angle of the store. As the distance between the store and pylon surfaces increases, the mesh distortion becomes less severe. With each successive adaptation, the deformation measure reduces to a minimum value greater than the previous minimum. This indicates that mesh movement may likely be applied for a longer period of time before adaptation is warranted. Such strategies and tradeoffs are yet to be investigated. The evolution of the unstructured mesh as the store falls away is depicted in Figure 6.13 where inter-processor boundaries are highlighted in red. Through the adaptive coarsening and refinement procedures, overall mesh quality is maintained, and an appropriate cell distribution is provided as the distance between the store and pylon increases. Although Figure 6.13 illustrates a slice through the mesh, one can readily see the migration and rebalancing of the inter-processor boundaries329 as left(before), right(after) redistribution. To improve the partitioning through high aspect ratio cells, FLUENT® recently device a partitioning method based on grouping of the Laplace coefficients as shown in Figure 6.14. In addition, it improves convergence rate for cases with highly

Figure 6.13 Adapted Mesh Partitioning During Store Dispense P.A., Sinha, N., and Feldman, G.M., ”Parallel Unstructured Mesh Adaptation For Transient Moving Body And Aeropropulsive Applications”, Combustion Research and Flow Technology, Inc. (CRAFT Tech), Pipersville, PA. 329 Cavallo,

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stretched cell.

Figure 6.14

Inter-Processor Partitioning Based on Laplace Coefficients

Case Study 5 - Adaptive Hybrid Mesh Refinement for Multiphysics Applications330 We have developed methods for optimizing meshes that are comprised of elements of arbitrary polygonal and polyhedral type. We present in this research the development of r-h hybrid adaptive meshing technology tailored to application areas relevant to multi-physics modeling and simulation. Solution-based adaptation methods are used to reposition mesh nodes (r-adaptation) or to refine the mesh cells (h-adaptation) to minimize solution error. The numerical methods perform either the r-adaptive mesh optimization or the h-adaptive mesh refinement method on the initial isotropic or anisotropic meshes to equidistributional weighted geometric and/or solution error function. We have successfully introduced r-h adaptivity to a least-squares method with spherical harmonics basis functions for the solution of the spherical shallow atmosphere model used in climate modeling. In addition, application of this technology also covers a wide range of disciplines in computational sciences, most notably, time-dependent multi-physics, multi-scale modeling and simulation. 6.2.8.1 Adaptive Hybrid Mesh Optimization The principle objective of this paper is to present an overview of current meshing efforts and development at Oak Ridge National Laboratory. Our capability is geared for generating highquality adaptive meshes for petascale applications. In this work, we have researched and developed tools and algorithms for the generation and optimization of adaptive hybrid meshes using finitevolume discretization approach. The hybrid mesh approach attempts to combine the advantages of both structured and unstructured meshing strategies. The prismatic and hexahedral elements are used in regions of high solution gradients, and tetrahedra are used elsewhere with pyramids used at the boundary between these two element categories to provide a transition region. In addition, the polyhedral/icosahedral meshes are often the best choice of solving symmetric computational problems (e.g., inertial confinement fusion and climate modeling). They have the properties of producing symmetric higher order orthogonal meshes and do not introduce artificial geometric interfaces. The geometry of the mesh and its symmetries are matched to the analytical and numerical methods used to solve the governing equations. Furthermore, hexahedral/prismatic layers close to wall surfaces exhibit good orthogonality and clustering capabilities characteristic of structured mesh generation approaches. The mesh example demonstration in Figure 6.15 showcase the generation 330 Ahmed Khamayseh and Valmor de Almeida, “Adaptive Hybrid Mesh Refinement for Multiphysics Applications”,

Journal of Physics: Conference Series 78 (2007) 012039.

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of hybrid surface and volume meshes on symmetric multi-region geometries331. The geometry of the mesh and its symmetries are matched to the analytical and numerical methods used to solve the governing equations.

Figure 6.15

Hybrid Icosahedra Surface Mesh (left) and Multi-Material Hybrid Volume Mesh (right) – (Courtesy of Khamayseh and Almeida)

The principle objective of this paper is to present an overview of current meshing efforts and development at Oak Ridge National Laboratory. Our capability is geared for generating highquality adaptive meshes for petascale applications. In this work, we have researched and developed tools and algorithms for the generation and optimization of adaptive hybrid meshes using finitevolume discretization approach. The hybrid mesh approach attempts to combine the advantages of both structured and unstructured meshing strategies. The prismatic and hexahedral elements are used in regions of high solution gradients, and tetrahedra are used elsewhere with pyramids used at the boundary between these two element categories to provide a transition region. In addition, the polyhedral/icosahedral meshes are often the best choice of solving symmetric computational problems (e.g., inertial confinement fusion and climate modeling). They have the properties of producing symmetric higher order orthogonal meshes and do not introduce artificial geometric interfaces. The geometry of the mesh and its symmetries are matched to the analytical and numerical methods used to solve the governing equations. Furthermore, hexahedral/prismatic layers close to wall surfaces exhibit good orthogonality and clustering capabilities characteristic of structured mesh generation approaches. The mesh example demonstration in Figure 6.15 showcase the generation of hybrid surface and volume meshes on symmetric multi-region geometries332. The geometry of the mesh and its symmetries are matched to the analytical and numerical methods used to solve the governing equations. Our approach to the meshing problem is to utilize tools and technologies developed by the center of Interoperable Technologies for Advanced Petascale Simulations (ITAPS) into our integrated geometry, meshing and adaptivity server (GMAS). The ITAPS center is one of the mathematics Enabling Technologies Centers (CET) in the Department of Energy's Scientific Discovery through Advanced Computing (SciDAC) program. The center's focus is on developing advanced scalable interoperable software associated with geometry, mesh, and field manipulation. It also provides the 331 332

In geometry, an icosahedra is a polyhedron with 20 faces. In geometry, an icosahedra is a polyhedron with 20 faces.

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necessary meshing tools to reach new levels of understanding through the use of high-fidelity calculations based on multiple coupled physical processes and multiple interacting physical scales. GMAS is a code intended to integrate scientific software and provide geometry, meshing and adaptivity services for PDE solvers of coupled Multiphysics applications without exposing details of the underlying libraries. GMAS is currently used to handle multiple meshes for multiple PDE solvers for a given geometry in a coupled application, and it provides the basic infrastructure to allow the application to evaluate fields over multiple meshes. It has been used in Multiphysics applications to provide meshing services for a neutron transport simulation code and a solvent extraction fluid flow code in development. The following example (Figure 6.16) exhibits coarse and refined anisotropic meshes generated using GMAS. The fine mesh is used to capture boundary layer flow and heat flux and the coarse mesh is needed for neutronics in the coolant channels of high-temperature test reactor (HTTR).

Figure 6.16

HTTR Multi-Material Geometry, Initial Coarse Mesh (left), Refined Mesh (right) ) – (Courtesy of Khamayseh and Almeida)

6.2.8.2 Hybrid Adaptive Meshing Our ongoing meshing research and development concentrates on hybrid mesh adaptation strategies,

Figure 6.17

Orography field (left), r-adaptivity (center) and h-adaptivity (right) for climate modeling ) – (Courtesy of Khamayseh and Almeida)

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along with mesh optimization. In certain Multiphysics applications, the size of mesh at a given location should be selected to resolve the smallest physics length scale at that point. Too few mesh elements result in a locally incorrect solution; whereas, too many mesh cells slow the calculation needlessly. The quality of the solution also depends on other mesh characteristics, such as, element shapes and connectivity, smoothness and “impedance” requirements, element orthogonality, anisotropic elements to match anisotropic physics, and boundary representation requirements. We have developed a hybrid finite volume-based mesh generator for r-adaptivity with certain emphasis on climate modeling. We employ conformal mapping to derive the elliptic PDEs models for the optimization and adaptation of hybrid surface meshes. However, an algebraic method is used in the case of combined h-p adaptivity wherein the degree p of the polynomial basis functions can be adapted to the features of the field quantities. The following demonstrated examples (Figure 6.17) exhibit the generation of r-h adapted hybrid surface meshes for climate modeling. It has been shown that mesh adaptation can reduce simulation error in prediction of the dynamics of the climate system. Applications to this capability also include field transformation and mapping across multiple meshes. In particular, the generation and smooth adaptive grid transformations for resolving orography (earth surface height) and fine-scale processes in climate modeling. Orography plays an important role in determining the strength and location of the atmospheric jet streams. Its impact is most pronounced in the numerical simulation codes for the detailed regional climate studies. In addition, orography is a crucial parameter for prediction of many key climatic dynamics, elements, and

Figure 6.18 Coupled orography field transfer with h-adaptivity. Planar orography field (top), h-adapted surface mesh (bottom-left) and closeup view of the mesh (bottom-right)

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moisture physics, such as rainfall, snowfall, and cloud cover. The phenomenon of climate variability is sensitive to orographic effects and can be resolved by the generation of finer meshes in regions of high altitude. Resolving orography produces a more accurate prediction of wetter or dryer seasons in a particular region. And moreover, orography defines the lower boundary in general circulation models. The following example (Figure 6.18) shows a very dense orography filed on planar uniform mesh with two kilometer gridded resolution (Figure 6.18- top). The initial field data size was two gigabits and it was obtained from http://www.ngdc.noaa.gov/mgg/topo/globe.html. We have successfully introduced h-p adaptivity to a least-squares method with spherical harmonics basis functions for field mapping and mesh adaptation. The end mesh (Figure 6.18-bottom) is much coarser at the sea level (50-kilometers) and finer at the high altitude regions (1-kilometers) with only a fraction of the original field data size. Moreover, the orography field was globally preserved to very small accuracy. For a full detailed presentation of this adaptive meshing approach we refer the reader to 333-334. 6.2.8.3 Meshing and Load Balancing. The advent of petascale computing creates new opportunities for representation of realistic geometries via meshing at an unprecedented fidelity. Fine meshes also have a beneficial impact on the accuracy of the PDE’s solution if they can be generated with sufficient quality. One central effort is the generation of meshes with large number of elements that truly represent complex multi-region geometries and adapt (h-adaptivity) to areas of steep solution gradient, notably at the walls of the vortex generator. Such very large meshes can only be created with the help of parallel computing. Our approach is to generate an initial mesh that resolves the surface geometry at any practical tolerance in a single processor while keeping the interior volume mesh coarse. In practice, such meshes can only be of unstructured type and there is a risk that many elements will become singular

Figure 6.19

Meshing and Partitioning of Centrifugal Contactor ) – (Courtesy of Khamayseh and Almeida)

Kahamyseh A., de Almeida V.F. and Hansen G. “Hybrid Surface Mesh Adaptivity for Shallow Atmosphere Simulation”, ORNL/TM-2006-28. 334 de Almeida V. F., Khamayseh A. K. and Drake J. B. “An h-p Adaptive Least-Squares Cartesian Method with Spherical Harmonics Basis Functions for the Shallow Atmosphere Equations ORNL/TM-2006-26. 333

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or of unacceptable aspect ratio. Next, we leverage tools from the ITAPS center into GMAS to partition the mesh, distribute the data, refine/improve the quality of the distributed mesh in parallel, and balance the load of the new mesh. Existing functionally of GMAS already provides the partition and distribution of the data (Figure 6.19), we are currently concentrating on developing of the parallel mesh refinement, data distribution and load balancing. 6.2.8.4 Conclusions. The paper presents an overview of the current tools being developed by ITAPS and GMAS for the generation, optimization, and adaptation of hybrid meshes. In addition, the research in this paper is involved in the development of r-h-p adaptive technology tailored to application areas relevant to other simulation fields. The r-h-p adaptive meshing approach and its underlying methods can be attractive to many application areas when solving three-dimensional, multi-physics, multi-scale, and time-dependent PDE's. This method builds on r-h-refinement/coarsening, p-refinement, interpolation, and error estimation applied to climate modeling and astrophysics simulation. For additional information, please refer to [Khamayseh and Almeida]335.

Strategies for Driving Mesh Adaptation in CFD336 Discretization error occurs in every (CFD) solution and is often one of the main contributors to the overall uncertainty in a CFD prediction. It is formally defined as the difference between the exact solution to the discrete equations and the exact solution to the governing partial differential equations. Discretization error is the most difficult type of numerical error to estimate and is usually the largest of the numerical error sources, which also include iterative error, round-off error, and statistical error (where relevant). There are a number of different approaches for estimating discretization error, but they all rely on the underlying numerical solution (or solutions) being in the asymptotic range with regards to either the truncation error or the discretization error. In addition to the importance of estimating the discretization error, we also desire methods for reducing it. Applying uniform mesh refinement (required for extrapolation-based discretization error estimation such as Richardson extrapolation) is not the most efficient method for reducing the discretization error. Since uniform refinement, by definition, uniformly refines over the entire domain, it generally results in meshes with highly refined cells/elements in regions where they are not needed. For 3D CFD applications, each time the mesh is refined by grid doubling in each coordinate direction, the number of cells/elements increases by a factor of eight. Thus uniform refinement for reducing discretization error can be extremely expensive. Targeted local refinement, or mesh adaptation, is a much better strategy for reducing the discretization error. There have been several extensive reviews of mesh adaption approaches for CFD (e.g., see [Baker]337 and [McRae]338); however much of this work has focused on methods for actually performing the adaption rather than the approach for driving the mesh adaptation. This paper examines several different criteria for driving a mesh adaptation scheme. 6.3.1.1 Feature-Based Adaption The most widely-used approach to grid adaptation employs feature-based methods. These methods often use solution gradients, solution curvature, or even identified solution features to drive the adaptation process. Feature based adaptation often results in some feature being over-refined while Ahmed Khamayseh and Valmor de Almeida, “Adaptive Hybrid Mesh Refinement for Multiphysics Applications”, Journal of Physics: Conference Series 78 (2007) 012039. 336 Christopher J. Roy, “Strategies for Driving Mesh Adaptation in CFD (Invited)”, AIAA 2009-1302. 337 Baker, T. J., “Mesh Adaptation Strategies for Problems in Fluid Dynamics,” Finite Elements in Analysis and Design, Vol. 25, 1997, pp. 243-273. 338 McRae, D. S., “r-Refinement Grid Adaptation Algorithms and Issues,” Computer Methods in Applied Mechanics and Engineering, Vol. 189, 2000, pp. 1161-1182. 335

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other features are not refined enough. In some cases, gradient-based refinement can actually increase the solution error. An example of the failure of feature-based adaptation is given by [Dwight]339 for the inviscid transonic flow over an airfoil using an unstructured finite-volume discretization. Figure 6.20 (a) shows the discretization error in the drag coefficient as a function of the number of cells. Uniform (global) adaptation shows second order convergence on the coarser grids, then a reduction to first order on the finer grids, likely due to the presence of the shock discontinuities. Adaptation based on solution gradients initially shows a reduction in the discretization error for the drag coefficient, but then subsequent adaptation steps show an increase in the discretization error. The adjoint-based artificial dissipation estimator gives the best results. The adapted grids for the latter two cases are given in Figure 6.20 (b-c). While the gradient-based adaptation refines the shock

(a)

(b) Gradient-based Adaptation

Figure 6.20

(c) Adjoint-Based Artificial Dissipation Estimator

Discretization Error in the Drag Coefficient for Transonic Flow over an Airfoil (Reproduced from Dwight)

Dwight, R.P., “Heuristic A Posteriori Estimation of Error due to Dissipation in Finite Volume Schemes and Application to Mesh Adaptation,” Journal of Computational Physics, Vol. 227, No. 5, 2008, pp. 2845-2863. 339

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waves on the upper and lower surface as well as the wake, the adjoint-based adaptation also refines near the surface and in the region above the airfoil containing acoustic waves that impinge on the trailing edge. 6.3.1.2 Discretization Error and Recovery-Based Adaption Since it is the discretization error that one wishes to reduce with mesh adaptation, on the surface it might appear the discretization error (or its estimate) would serve as an appropriate driver for the adaption process. However, as will be shown in Section V discretization error is not an appropriate mesh adaption criterion. In the finite element method, an error indicator that is frequently used for mesh adaptation is gradient recovery or reconstruction such as the Zienciwicz-Zhu4 error estimator. The main idea is that gradients of the finite element solution are compared to the gradients found from post-processing patches of neighboring elements. Larger mismatches between these two gradient computations serve as an indicator of larger errors in the gradients. This approach relies on the super-convergent properties of the finite element method which states that for sufficiently smooth solutions, the nodal values of the finite element solution converge at a higher rate than those at other locations.5 In this sense, recovery-based adaption is thus similar to adaption based on the discretization error and is not the ideal driver for mesh adaption. Furthermore, recovery-based adaptation, while possible for the finite element method, may not be feasible for other discretization methods which are not super-convergent. 6.3.1.3 Adjoint-Based Adaption Another promising method for grid adaptation is the adjoint approach. Adjoint methods hold the promise of estimating the local contribution of each cell or element to the discretization error in any solution functionals of interest (e.g., lift, drag, and moments), and can thus provide targeted mesh adaption depending on the goals of the simulation. The main drawback for adjoint methods is their complexity and code intrusiveness, as evidenced by the fact that adjoint-based adaption has not yet found its way into commercial CFD codes. An example of adjoint-based mesh adaption in CFD is given by Venditti and Darmofal6-8 who successfully applied the method in finite-volume form to inviscid and viscous flow over airfoils at various Mach numbers. While adjoint methods hold much future promise in the area of mesh adaption, they are beyond the scope of the current paper. 6.3.1.4 Truncation Error-Based Adaption In broad terms, the truncation error is the difference between the partial differential equation and its discrete approximation. As will be discussed in later, the truncation error provides the contribution of the local element discretization (cell size, skewness, etc.) to the discretization error. As such, the truncation error is a good indicator of where mesh adaptation should occur. The general concept behind truncation error-based adaption is to equidistributional the truncation error over the entire domain to reduce the total discretization error. For simple discretization schemes, the truncation error can be computed directly. For more complex schemes where direct evaluation of the truncation is difficult, an approach for estimating the truncation error is needed. Section VII discusses two approaches for estimating the truncation error. Furthermore, in the finite element method, a class of error estimators has been developed that rely on the residual. This residual is found by inserting the finite element solution, which is made up of basis functions and corresponding coefficients, into the original partial differential equation. As will be shown in Section VII, this residual can be found in a similar manner as one approach for estimating the truncation error. Thus we expect there to be a close relationship between residual-based adaption in the finite element method and truncation error-based adaption with more general discretization schemes. Current Approach for Performing Mesh Adaptation Local solution adaption can be conducted by moving points from one region to another (r-adaption), selectively refining/coarsening cells (h-adaption), or increasing/decreasing the order of accuracy of

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the method (p-adaption). For general unstructured grid methods the h-adaption approach is the most popular, while for structured grid methods the r-adaption approach is most often used. padaption has not found widespread use for CFD problems340. In addition to mesh refinement, other issues that should be considered when adapting a mesh are mesh quality and the alignment of the mesh with key solution features. Since the focus here is on methods for driving mesh adaption and not for performing adaption itself, we limit ourselves to a simple approach of r-adaption in one dimension based on a linear spring analogy. Extensions to handle multiple dimensions are possible based on a torsional spring, which serves to prevent skewing of the multi-dimensional cells. First, a mesh adaptation function φi is created based on solution features (e.g., gradients, curvature), discretization error, truncation error, etc., where i denotes the mesh node point (ordered from 1 to N). A weighting function is then created from the mesh adaptation function as:

Wi = |φi |q

Eq. 6.3 Where q is an exponent that is set to unity in the present work. This weighting function is used to drive the mesh adaption process, with smaller values denoting a region for mesh coarsening and larger values a region for refinement. This weighting function is then passed through a smoothing algorithm to promote smoothness of the mesh adaptation. This smoothing algorithm includes 10 passes of the following smoothing operation for all interior points:

Wi =

Wi+1 + 4Wi + Wi−1 6

Eq. 6.4 Once this weighting function has been determined, it is used to determine a spring constant for linear springs connecting the nodes: ki+1/2 = (Wi + Wi+1)/2. The new nodal location can then be found according to:

k i−1/2 (xi − xi−1 ) = k i+1/2 (xi+1 − xi )

Eq. 6.5 Nodes with high weighting functions will have higher spring constants and thus promote refinement in that region. The new nodal locations are solved for using an explicit iterative approach according to:

xim+1

=

m m k i−1/2 xi−1/2 + k i+1/2 xi+1/2

k i−1/2 + k i+1/2

Eq. 6.6 The weighting functions are interpolated as the nodes are moved, thus ensuring that the weighting functions are fixed in space for a given mesh adaptation step. Four different techniques will be used to drive the mesh adaptation process. The first two are feature-based and adapt the mesh based on either the local solution gradients or solution curvature. The third technique adapts the mesh based on the discretization error and the fourth technique adapts the mesh based on the leading truncation error term. These four techniques are summarized below. 1. Solution Gradient: This feature-based adaptation technique sets φi = (∂u/∂x)i 2. Solution Curvature: This feature-based adaptation technique sets φi = (∂2u/∂x2)i

Baker, T. J., “Mesh Adaptation Strategies for Problems in Fluid Dynamics,” Finite Elements in Analysis and Design, Vol. 25, 1997, pp. 243-273. 340

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3. Discretization Error (DE): This approach uses the true discretization error in the solution to drive the adaptation process, although the estimated discretization error (e.g., from Richardson extrapolation) could be used in cases where the exact solution is not available : φ i = ui - ũ i 4. Truncation Error (TE): This approach uses the leading truncation error terms to drive the adaptation process as discussed in the next section. In this approach, the partial derivatives of the solution and the transformation metrics are replaced by second-order accurate finite difference approximations. Examples of these different weighting functions applied to steady-state Burgers equation for a Reynolds number of 32 are shown in Figure 6.21 for q = 1. Clearly each approach will produce meshes with different adaptation characteristics. Steady-state Burgers equation for Reynolds number341. The left figure depicts the exact solution and numerical solution using 33 uniformly spaced nodes, while right shows an example weighting functions for the region -2 ≤ x ≤ 0 based on solution gradients, solution curvature, discretization error (DE), and truncation error (TE).

Figure 6.21

Steady-State Burgers Equation for Reynolds Number 32

Case Study - Mesh Adaption Results for 1D Burgers Equation (Re = 32) In this section, different methods for driving the mesh adaption are analyzed as well as the case without adaption (i.e., a uniform mesh). The four different methods for driving the mesh adaptation are: adaption based on solution gradients, adaption based on solution curvature, adaption based on the discretization error (DE), and adaption based on the truncation error (TE). Numerical solutions to steady-state Burgers equation for Reynolds number = 32 are given in Figure 6.22(left) for the uniform mesh and the four mesh adaption approaches, all using 33 nodes. The final local node spacing is given in Figure 6.22 (right) for each method and shows significant variations in the vicinity of the viscous shock.

Dwight, R.P., “Heuristic A Posteriori Estimation of Error due to Dissipation in Finite Volume Schemes and Application to .Mesh Adaptation,” Journal of Computational Physics, Vol. 227, No. 5, 2008, pp. 2845-2863. 341

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Figure 6.22

Adaption Schemes Applied to Burgers Equation Left) numerical solutions and right) local nodal spacing Δx.

A Solution-Based Adaptive Redistribution Method for Unstructured Meshes342 We propose an unstructured mesh redistribution method without using skewed elements for steadystate problems. The regions around solution features are indicated by a sensor function. The medial axes of the strong feature regions are calculated so that elements can be clustered around the most important solution features efficiently. Two approaches, a discrete surface-based approach using a Delaunay triangulation method and a mathematical-representation approach using least square fitting, are shown to calculate the medial axes. Remeshing of an initial volume mesh is performed around the medial axes using an advancing front method and/or an advancing layer method. Two examples are shown to present how our approach works. Introduction & Literature Survey In order to obtain accurate results for highly complex flow fields, meshes must be clustered near the areas where the solution gradients are high. This is an arduous task the engineer must perform prior to the completion of the calculation. The meshes can be clustered in two ways; either a very fine mesh is generated, or some solution-based mesh clustering is performed. The first approach can be very expensive in terms of computational costs. Although surface meshes can be adapted geometrically based on surface curvature and local volume thickness343, it is often difficult to choose adaptation criteria for volume meshes before numerical simulations. The second approach can be achieved by mesh adaptation. There are three mesh adaptation approaches: mesh refinement/de-refinement344-

Yasushi Ito, Alan M. Shih, Roy P. Koomullil and Bharat K. Soni, “A Solution-Based Adaptive Redistribution Method for Unstructured Meshes”, Dept of Mechanical Engineering University of Alabama at Birmingham, Birmingham, AL, U.S.A. 343 Ito, Y., Shum, P. C., Shih, A. M., Soni, B. K. and Nakahashi, K., “Robust Generation of High-Quality Unstructured Meshes on Realistic Biomedical Geometry,” International Journal for Numerical Methods in Engineering, 2006. 344 Cavallo, P. A. and Grismer, M. J., “Further Extension and Validation of a Parallel Unstructured Mesh Adaptation Package,” AIAA Paper 2005-0924, 43rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 2005. 342

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345,

mesh redistribution346 and the combination of these347-348. Since structured meshes are not flexible for adding or deleting nodes locally, the mesh redistribution approach is widely used to move nodes toward solution features while the connectively of the mesh is maintained. Although solution features are adapted by unstructured meshes relatively easily, there are two issues needed to be addressed. One is the maintenance of valid elements. Hanging nodes can be created during a mesh refinement process. Local refinement of hybrid meshes for viscous flow simulations, which contain regular elements such as tetrahedra, prisms, pyramids and hexahedra, is difficult without creating low-quality elements to eliminate hanging nodes. To overcome this issue, an

approach using generalized elements is promising349.

The other issue is the quality of resulting refined meshes. Stretched elements may affect solution accuracy and cause a stiffness problem in numerical simulations. [Mavriplis]350 reports spanwise grid stretching, which is widely used in aircraft CFD simulations, may have substantial repercussions on overall simulation accuracy even at very high levels of resolution351. Since typical refinement and redistribution algorithms for unstructured meshes create highly stretched tetrahedra around solution features, the validation of the simulation process may be required. If a refined mesh does not have elements that have too small or too large angles even near solution features, we do not need to worry about these issues. Here, we propose a solution-based redistribution method for unstructured volume meshes. The structured mesh redistribution methods only allow nodes to move towards solution features, while maintaining the mesh connectivity. In our unstructured mesh redistribution method, a mesh is remeshed around the solution features detected. The main objective here is to extract strong solution features as smooth surfaces based on sensor values and then to create high quality elements around them. The entire domain can be re-meshed with the embedded surfaces using an advancing front method with tetrahedra and an advancing layer method with prisms or hexahedra if needed. Alternatively, elements around the feature surfaces are removed from the initial volume mesh and only the resulting voids are re-meshed to reduce the required CPU time. Two examples are shown to present how our approach works. Feature Detection After a numerical simulation using an initial mesh, the next step is the detection of solution features. The location of solution features is indicated by the weight function by [Soni et al.]352 or the shock

Senguttuvan, V., Chalasani, S., Luke, E. and Thompson, D., “Adaptive Mesh Refinement Using General Elements,” AIAA Paper 2005-0927, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 2005. 346 Soni, B. K., Thornburg, H. J., Koomullil, R. P., Apte, M. and Madhavan, A., “PMAG: Parallel Multiblock Adaptive Grid System,” Proceedings of the 6th International Conference on Numerical Grid Generation in Computational Field Simulation, London, UK, 1998, pp. 769-779. 347 Shephard, M. S., Flaherty, J. E., Jansen, K. E., Li, X., Luo, X., Chevaugeon, N., Remacle, J.-F., Beall, M. W. and O’Bara, R. M., “Adaptive Mesh Generation for Curved Domains,” Applied Numerical Mathematics, 2005. 348 Suerich-Gulick, F., Lepage, C. and Habashi, W., “Anisotropic 3-D Mesh Adaptation for Turbulent Flows,” AIAA Paper 2004-2533, 34th AIAA Fluid Dynamics Conference and Exhibit, Portland, OR, 2004. 349 Senguttuvan, V., Chalasani, S., Luke, E. and Thompson, D., “Adaptive Mesh Refinement Using General Elements,” AIAA Paper 2005-0927, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 2005. 350 Mavriplis, D. J., “Grid Resolution Study of a Drag Prediction Workshop Configuration Using the NSU3D Unstructured Mesh Solver,” AIAA Paper 2005-4729, 23rd Applied Aerodynamics Conference, Toronto, 2005. 351 Mavriplis, D. J., “Grid Resolution Study of a Drag Prediction Workshop Configuration Using the NSU3D Unstructured Mesh Solver,” AIAA Paper 2005-4729, 23rd Applied Aerodynamics Conference, Toronto, 2005. 352 Soni, B. K., Koomullil, R., Thompson, D. S. and Thornburg, H., “Solution Adaptive Grid Strategies Based on Point Redistribution,” Computer Methods in Applied Mechanics and Engineering, Vol. 189, Issue 4, 2000. 345

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sensor by [Lovely and Haimes]353. The weight function is calculated based on the conserved variables and indicates the regions of important flow features. It is defined at each element as follows:

W=1+

W1 W2 W3 ⨁ ⨁ Max(W1 , W 2 , W 3 ) Max(W1 , W 2 , W 3 ) Max(W1 , W 2 , W 3 ) nq

qiξk qiξk W = ∑ [ i ∕[ ] |q | + ε |qi | + ε k

⨁i=1

] Max k=1,2,3

Eq. 6.7 where Wk (k = 1, 2, 3), qiξk and qi are x, y and z components of normalized gradient, the kth component of the gradient calculated using i-th variable and the average variable at the centroid of the element, respectively. The symbol ⊕ represents the Boolean sum, which for two variables q1 and q2 is defined as

q1 ⊕ q 2 = q1 + q 2 − q1 q 2

Eq. 6.8 The shock sensor is based on the fact that the normalized Mach number Mn = 1 at a shock.

M𝑛 =

𝑉 ∇𝑝 =1 𝑎 |∇𝑝|

Eq. 6.9 where a, V and ⩢ p are the speed of sound, velocity vector and pressure gradient, respectively. Extraction of Solution Feature Surfaces To adapt high-quality elements around strong solution features, the next step is recognition of feature surfaces. Although this approach may need more meshing steps than a typical mesh redistribution method, much better quality elements can be generated around the solution feature surfaces. Marcum and Gaither propose a pattern recognition algorithm in 2D and mention the difficulty of extending it to 3D354. Although our approach needs user interaction during the process it enables feature surface extraction. The direct extraction of solution feature surfaces is difficult from the initial mesh and solution data. At least two steps are needed. First, regions around the solution features are specified by selecting a certain sensor value. Although elements can be subdivided in the entire regions, the number of elements in the resulting mesh may become too big. The regions can be very thick if an initial volume mesh is coarse at the solution feature locations. To avoid this problem, the medial axis (also known as skeleton) of each region is extracted in the following step. Elements are clustered around the medial axes. Two approaches can be considered to extract solution feature surfaces. One is a discrete surfacebased approach. A medial axis is extracted from a triangulated closed surface using Delaunay triangulation355. Triangulated iso-surfaces at a certain sensor value can be calculated easily and robustly, which enclose regions around solution features. For example, the shock features are surrounded by the iso-surfaces at Mn = 1. A Delaunay tetrahedral mesh can be obtained from a triangulated iso-surface. The center of the circum-sphere of each tetrahedron is considered to Lovely, D. and Haimes, R., “Shock Detection from Computational Fluid Dynamics Results,” AIAA Paper 19993285, 14th AIAA Computational Fluid Dynamics Conference, Norfolk, VA, 1999. 354 Marcum, D. L. and Gaither, K. P., “Solution Adaptive Unstructured Grid Generation Using Pseudo-Pattern Recognition Techniques,” AIAA Paper 97-1860, 13th AIAA CFD Conference, Snowmass Village, CO, 1997. 355 Sheehy, D. J., Armstrong, C.G. and Robinson, D. J., “Shape Description by Medial Surface Construction,” IEEE Transactions on Visualization and Computer Graphics, Vol. 2, Issue 1, 1996, pp. 62-72. 353

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represent the medial axis. The quality of the resulting medial axes depends on the smoothness of the iso-surfaces. However, iso-surfaces are not usually smooth, and they may have bumps and holes due to truncation errors in the entire simulation process. User interaction is often required to fix the resulting surfaces. The other approach is a mathematical-representation approach. A medial axis can be estimated using least square fitting directly from the nodes on an iso-surface. Least square fitting methods often minimize the vertical offsets from a surface function instead of the perpendicular offsets to simplify an analytic form for the fitting parameters. Consequently, the least square fitting does not estimate the surface function well when the region defined by an iso-surface is thick. Although a set of coordinates of nodes near a solution feature is needed as an input for a least square fitting method, the connectivity of the nodes is not required. Therefore, we define a solution feature as a set of nodes based on the following process: 1 2 3 4 5

Select nodes of a volume mesh that have a certain range of sensor values. Also select nodes that are one-ring neighbors of the nodes in Step 1 to eliminate noise due to truncation errors. Number each cluster of selected nodes, which can be defined as their connectivity, if the mesh has more than one solution features. Calculate distance from the closest boundary at each selected node. The boundary is represented by the selected nodes that have at least one unselected node as their one-ring neighbor. The distance is defined as the number of edges from the boundary. The nodes that have local maxima of the distance values are considered to form medial axes.

The coordinates of the nodes in Step 5 are fitted to functions, such as a plane, quadric and cone, using a least square fitting method. Local mesh size can be considered to be the error range of a data point. The reciprocal of the local mesh size is used for weighing. Suppose that a cluster of selected nodes xmj (j = 1, 2,…, nm) is fitted to a function z = f (x, y). nm

2

zj − f(xj , yj ) E = ∑( ) lj j=1

Eq. 6.10 where lj is the maximum edge length connected to node j. E should be minimized. The resulting function should be trimmed to define a surface in the computational domain. 6.4.3.1 Case Study 1 - NACA0012 Wing-Section Figure 6.23 (a) shows a mesh around a NACA0012 wing (50 K nodes). An inviscid flow simulation is carried out at a freestream Mach number of 0.799 and an angle of attack, α = 2.26 deg. Figure 6.23 (b) and (c) illustrate pressure coefficient (Cp) distribution and weight function value distribution based on Eq. 6.7 the results indicate a shock on the wing. In this example, the shock location is estimated using the discrete surface-based approach based on Delaunay triangulation (Figure 6.24 (a)). First, an iso-surface of a weight function value of 0.2 is extracted and smoothed using Visualization Toolkit (VTK). It is sometimes difficult to generate an expected medial axis as single surface using existing algorithms even if a solution feature is simple. Once the feature surface is computed, the surface mesh generation algorithm is applied to create a high quality mesh on it. Elements of the initial volume mesh, Vn0, near the solution feature are removed and the void is re-meshed using the advancing front method. Figure 6.24 (c) shows the resulting volume mesh (110 K nodes; Vn1). As a result, a high quality redistributed mesh is produced with alignment to the major flow feature. Figure 6.24 (d) shows another redistributed volume mesh

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after the second simulation cycle (130 K nodes; Vn2). Figure 6.25 illustrates hybrid meshes for the same wing geometry to perform viscous flow simulations and Cp distribution. The shock location is estimated using the same approach from the initial hybrid mesh, Figure 6.25 (a), and then the entire domain is re-meshed with the embedded surface (Figure 6.25(b)). To avoid creating skewed elements around the intersection between the wing upper surface and the embedded surface, the near-filed mesh around the wing is generated first. The embedded surface close to or within the nearfield mesh is trimmed automatically, and then the rest of the domain is filled with tetrahedral elements. (a) Tetrahedral Mesh

(b) Cp Distribution (M = 0.799, α = 2.26 Deg.

(c) Weight Function value distribution

Figure 6.23

NACA0012 wing-Section Adaption

Once the feature surface is computed, the surface mesh generation algorithm is applied to create a high quality mesh on it. Elements of the initial volume mesh, Vn0, near the solution feature are removed and the void is re-meshed using the advancing front method. Figure 6.24 (c) shows the resulting volume mesh (110 K nodes; Vn1). As a result, a high quality redistributed mesh is produced with alignment to the major flow feature. Figure 6.24 (d) shows another redistributed volume mesh after the second simulation cycle (130 K nodes; Vn2). Figure 6.25 illustrates hybrid meshes for the same wing geometry to perform viscous flow simulations and Cp distribution. The shock location is estimated using the same approach from the initial hybrid mesh, Figure 6.25 (a), and then the entire domain is re-meshed with the embedded surface (Figure 6.25(b)). To avoid creating skewed elements around the intersection between the wing upper surface and the embedded surface, the near-filed mesh around the wing is generated first. The embedded surface close to or within the nearfield mesh is trimmed automatically, and then the rest of the domain is filled with tetrahedral domain

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is re-meshed with the embedded surface (Figure 6.25(b)). To avoid creating skewed elements around the intersection between the wing upper surface and the embedded surface, the near-filed mesh around the wing is generated first. The embedded surface close to or within the near-field mesh is trimmed automatically, and then the rest of the domain is filled with tetrahedral elements.

(a) Isosurface of a weight function value of 0.2 at theshock location

(c) Redistribution #1(110 K nodes) – elements around the shock are replaced with finer elements Figure 6.24

(b) Initial mesh (50 K nodes)

(d) Redistribution #2 (130 K nodes)

Extraction of a Flow Feature & Redistributed Volume Meshes

6.4.3.2 Case Study 2 - Capsule Model Figure 6.26 (a) shows an initial tetrahedral mesh around a re-entry capsule model and Mach number distribution on a cross-section. The bow shock in front of the capsule becomes steady, but the flow solution is not fully converged. The shape of the outer boundary is a hemisphere so that the mesh can be used for flows at different angles of attack. Iso-surfaces of a certain weight function value can be extracted as triangulated surfaces (Figure 6.26 (b)), the medial axes of which are considered to represent the most important locations. An approach to obtain medial axes using a Delaunay triangulation method from the triangulated surfaces can be considered. However, it is difficult to obtain medial axes automatically as smooth surfaces as discussed in the previous example.

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Although the iso-surfaces shown in Figure 6.26 (b) are smoothed using a Laplacian method, many holes and small features prevent extracting smooth medial axes. The other approach using the least square fitting method is more appropriate in this case.

Figure 6.25

Hybrid Meshes for the NACA0012 Wing-Section and Cp Distribution (-1.0 to 1.0) (a) Initial Hybrid Mesh; (b) Redistributed Hybrid Mesh

(b) Flow features on extracted isosurfaces at a weight function value of 0.05

(a) Initial Grid

(c) Redistributed mesh for a capsule model and Mach numbe distribution on a cross-section (M=1.0-4.0) Figure 6.26

(d) Redistributed mesh for a capsule model using anisotropic elements

Adaptive Remeshing of Capsule

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After a user specifies one of the template functions, such as a cone, quadratic and quartic, and the z axis of the function, a corresponding medial axis is obtained as a mathematical function. The bow shock in front of the capsule is fitted to a quadratic, and the shock from the aft of it is fitted to a cone. It can be shown shows that the obtained surfaces and the iso-surfaces for reference on the symmetry plane. The least square fitting method estimates the medial axes well. One of the disadvantages using unstructured meshes is that flow features diverge quickly. This approach enables us to estimate missing flow features. Figure 6.26 (c) shows a redistributed mesh, which has 0.74 M nodes. In this case, the entire mesh is regenerated because the shape of the outer boundary is changed to remove extra elements. The initial mesh shown in Figure 6.26 (a) can be used for cases at different angles of attack, but it has 0.89 million nodes. In addition, the elements around the shocks in the far field are coarse. The initial mesh gives a carbuncle phenomenon on the bow shock, while the redistributed mesh gives better result. One of the solution feature surfaces shown in Figure 6.26 (b) fits the bow shock well. The most notable advantage of the surface-based mesh redistribution method is that anisotropic nonsimplicial elements can be used around the feature surfaces to avoid creating skewed elements. Figure 6.26 (d) shows a redistributed hybrid mesh, which has 0.62 M nodes, based on the same numerical result. Prismatic layers are placed around the bow shock.

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7 Dynamic Meshing The moving-mesh provides a capability of tackling flow simulations where the domain shape changes during the simulation. In such cases, the computational mesh needs to adapt to the time-varying shape of the domain and preserve its validity and quality. The mesh motion solver support which calculates the internal point motion based on the prescribed motion of the boundary. The performance of the method is preserved through the choice of decomposition of cells, the bounded discretization and the use of iterative solvers356. We covered the dynamic mesh before a little bit with Adaptive Mesh Refinement (AMR) where it was characterized it as H, P, and R-Methods. To recap: • • •

H-Method - It involves automatic refinement or coarsening of the spatial mesh based on a posteriori error estimates or error indicators. The overall method contains two independent parts, i.e. a solution algorithm and a mesh selection algorithm. P-Method - the adaptive enrichment of the polynomial order. R-Method - The R-Method is also known as Moving Mesh Method (MMM). It relocates grid points in a mesh having a fixed number of nodes in such a way that the nodes remain concentrated in regions of rapid variation of the solution.

Where most of adaptive refinements is using R-Methods, with key ingredients which includes Interpolation of time dependent mesh equation357. In the Dynamic Mesh, the computational mesh is moved to follow the changing shape of the boundary by moving its points in every step of the transient simulation. The main difficulty in this case is maintaining the mesh validity and quality without user interaction where the performance will be quantified by speed, accuracy, robustness, and stability358.

Type of Mesh Motion Several deforming mesh algorithms have been presented in literature, with various approaches to defining mesh motion. The most popular method to date is the spring analogy359. Here, all point-topoint connections within the mesh are replaced by linear springs and point motion is obtained as a response to boundary displacement. However, this approach proved to lack robustness, particularly for arbitrarily unstructured (polyhedral) meshes. A review of merits and limitations of the spring analogy and its variants is given by [Blom]360. Other approaches to creating a robust mesh motion solver include the use of Laplacian smoothing 361 with the constant and variable diffusivity and the Pseudo-Solid Equation (static equilibrium equation for small deformations of a linear elastic solid)362 in Arbitrary Lagrangian-Eulerian (ALE) codes. In an effort to simultaneously control the

Hrvoje Jasak, ˇZeljko Tukovi´c, “Automatic Mesh Motion for the Unstructured Finite Volume Method”, ISSN 1333–1124, UDK 532.5:519.6. 357 Tao Tang, “Moving Mesh Methods for Computational Fluid Dynamics”, Contemporary Mathematics, 1991. 358 Hrvoje Jasak, ˇZeljko Tukovi´c, “Automatic Mesh Motion for the Unstructured Finite Volume Method”, ISSN 1333–1124, UDK 532.5:519.6. 359 Batina, J. T., “Unsteady Euler airfoil solutions using unstructured dynamic meshes”, AIAA Journal 28 (8) (1990). 360 Blom, F. J., “Considertions on the spring analogy, International journal for numerical methods in fluids”, (2000). 361 L¨ohner, R., Yang, C., “Improved ALE mesh velocities for moving bodies”, Communications in numerical methods in engineering 12 (1996), pp. 599–608. 362 Johnson, A. A., Tezduyar, T. E., “Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces”, Computer methods in applied mechanics and engineering 119 (1994). 356

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position of moving boundary and mesh spacing next to it, [Helenbrook]363 proposes the use of a biharmonic equation to govern mesh motion. Other methods includes Dynamic-Overset meshing, dynamic re-meshing using Radial-Basis Functions (RBF), Delaunay Method and mesh motion and dynamic re-meshing using a generalized grid interface (GGI)364. The Radial Basis Function (RBF) method and Delaunay Method which have been used widely in fluid-structure interaction. An analysis of dynamic-meshing techniques was one by quantifying the accuracy, robustness, stability, and speed of each one and while dynamic re-meshing via solution of a Laplace equation was robust and GGI was the fastest, Overset meshing was found to be the most stable and the most general technique for complex geometries and motions365. RBF proved to be too computationally expensive and unrealistic for 3D problem.

Mesh Deformation The mesh deformation problem can be stated as follows. Let D represent a domain configuration at a given time t with its bounding surface B and a valid computational mesh, as shown in Figure 7.1. During a time interval Δt, D changes shape into a new configuration D′. A mapping between D and D′ is sought in such a way that the mesh on D forms a valid mesh on D′ with a minimal distortion of control volumes. In this study, the displacement vector u is chosen as the dependent variable in the mesh motion problem. Thus, the point position in the deformed configuration is calculated as r´ = r + u where r ∈ D and r′ ∈ D′ are point position vectors.

Figure 7.1

Mesh Deformation Problem

Finite Volume in Dynamic Mesh With respect to dynamic meshes, we start with scalar transport equation as: ∂

⏟(ρφ) +

∂t Transient

∂

∂

∂φ

(ρuj φ) = ∂x (Γφ ∂x ) + S⏟ φ ∂x j ⏟j ⏟j Source ⏟ Convection Diffusion Transport

Helenbrook, B. T., “Mesh deformation using the bi-harmonic operator”, International journal for numerical methods in engineering 56 (2003), pp. 1007–1021. 364 OpenFOAM ®. 365 Hrvoje Jasak, ˇZeljko Tukovi´c, “Automatic Mesh Motion for the Unstructured Finite Volume Method”. 363

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Eq. 7.1 Where ϕ is the general scalar quantity, ρ is the fluid density, and u i is the flow velocity vector. Furthermore, Γ is the diffusion coefficient and S is the source term. After integrating over a Control Volume and applying the divergence theorem, we obtain the integral form

      ( )   t (ρ  ) + . ρ u j  − . ΓQ x j  − S  dV = 0 V      V t (ρ  ) dV + A (ρ u j  ). dA = A Γ  . dA + V S dV

Eq. 7.2

d (ρ  ) dV +  ρ (u i − u g ) . dA =  Γ  . dA +  S dV dt V A A V Here, ug is the grid velocity of the moving mesh, and A is used to represent the boundary of the control volume V. The unsteady term (first term) could be written as

d (ρV) n +1 − (ρV) n ρ dV = dt v Δt

→ V n +1 = V n +

dV Δt dt

Eq. 7.3

Where dV/dt is the volume time derivative of the control volume. In order to satisfy the grid conservation law, the volume time derivative of the control volume is computed from Face Face δV dV j =  u g .dA =  u gj.A j =  dt Δt j j V

Eq. 7.4

With δVj is the volume swept out by the control volume face j over the time step Δt. In the case of the sliding mesh, the motion of moving zones is tracked relative to the stationary frame. Therefore, no moving reference frames are attached to the computational domain, simplifying the flux transfers across the interfaces366. In the sliding mesh formulation, the control volume remains constant, therefore, dV/dt = 0 and Vn+1 = Vn





d (ρ ) n +1 − (ρ ) n V ρ dV = dt v Δt

Eq. 7.5

Dynamic Mesh Techniques Laplacian Mesh Morphing A computationally robust dynamic-mesh technique used in CFD is Laplacian Mesh morphing, which solves a Laplace equation to move the mesh. [Bos]367 used this method along with solid body rotation stress to compare with radial basis function interpolation while studying insect flight. [Bos] found 366 367

FLUENT 6.3 User’s Guide. Frank Martijn BOS, “Numerical simulations of flapping foil and wing aerodynamics”, PhD thesis, 2009.

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that the Laplace equation method was not able to maintain high mesh quality around the boundary of a rectangle when it rotates, shown in Figure 7.2-(a). The cell skewness in the domain is highest near the body, while the remaining mesh is relatively unaltered. However, the method remained quite robust even with the complex geometry and high amplitude mesh motion. The mathematical representation for mesh motion via solution of a Laplace equation is

∇ . (k ∇𝐮i,mesh ) = 0

,

k=

1 𝑙2

Eq. 7.6 Where ui,mesh is the velocity of points in the mesh and k is a distance function that minimizes the mesh distortion, and l is the distance to the moving boundary. The body is rotated or transformed in some manner described by the user and the points on the body are moved based on a coordinate transformation. The points surrounding the body are moved based on the Laplace equation above. There is a modest amount of error introduced before the cells surrounding the body are moved368. Pseudo-Solid Equation While the Laplace equation only allows direction-decoupled transfinite mapping, the pseudo-solid equation also allows rotation. However, this comes at a relatively high price: the pseudo-solid equation couples the components of the motion vector due to rotation. The choice here is either an increase in storage associated with the block solution of all displacement components or an iterative segregated solution method.

[.μu + μ(u)T + λtr(u)I] = 0

Eq. 7.7

7.4.2.1 Case Study – Motion of a Cylinder369 The case consists of a circle moving in a channel in 2D. An identical setup and a triangular mesh has been used by [Baker]370 with the pseudo-solid equation, and [Helenbrook]371 on the bi-harmonic equation. A polygonal mesh used for the test, where D is the cylinder diameter, the height of the channel is 2D and average mesh size is 0.15D. The first test consists of the determination of the maximum displacement of the cylinder in one step without mesh inversion when the outside boundary remains fixed. Mesh quality is determined in terms of the non-orthogonality angle αf. For reference, on the initial polygonal mesh αf,max = 18.45◦ and αf,mean = 0.34◦. The deformed meshes obtained using the Laplace and Pseudo-Solid mesh motion equations for one step maximum cylinder displacement are shown in Figure 7.2 (a-b). Maximum achievable single-step cylinder displacement is Δmax = 0.636D for the Laplace equation and Δmax = 0.995D for the pseudo-solid equation. In transient simulations, the mesh is moved in a number of time-steps. This situation will be examined by repeating the above test, but with the prescribed cylinder motion of 0.15D per timestep until the mesh becomes invalid. This equates to the effective Courant number of unity, based on the boundary motion velocity. Figure 7.2 (c-d) shows that this approach allows a considerably higher deformation, because it handles the inherent non-linearity of the mesh motion problem. It is Gina M. Casadei, “Dynamic-Mesh Techniques for Unsteady Multiphase Surface-Ship Hydrodynamics”, A Thesis in Mechanical Engineering, Pennsylvania State University, December 2010. 369 Hrvoje Jasak, ˇZeljko Tukovi´c, “Automatic Mesh Motion for the Unstructured Finite Volume Method”, ISSN 1333–1124, UDK 532.5:519.6. 370 Baker, T. J., Mesh modification for solution adaptation and time evolving domains, 7th International Conference on Numerical Grid Generation in Computational Field Simulations, Whistler, British Columbia, Canada, 2000. 371 Helenbrook, B. T., Mesh deformation using the biharmonic operator, International journal for numerical methods in engineering 56 (2003), pp. 1007–1021. 368

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interesting to notice that the Laplace and pseudo-solid equations allow the same cylinder displacement Δmax = 1.2D, contrary to the previous test. On the other hand, the increased cost of solving the pseudo-solid equation compared to the Laplace equation does not seem to be justified with the higher allowed single-step mesh deformation372.

Figure 7.2

Cylinder Motion in 2D

Biharmonic Equation The algorithm is based on the solution of the biharmonic Equation for the deformation field, see [Tysell]373. The use of the biharmonic equation has also been reported by [Helenbrook]374. The main advantage of this grid deformation algorithm is that it can handle large deformations, especially close to the boundaries. It also produces a smooth deformation distribution for cells which are very skewed or stretched. This is necessary in order to handle the very thin cells in a prismatic layer. The algorithm can handle skewed cells since the finite element method is used for the solution of the biharmonic equation. The disadvantage is that the method is slow, compared to algebraic methods, since a set of partial differential equations needs to be solved. Figure 7.3 (b) shows the deformation of the initial twin airfoil grid in Figure 7.3 (b).

Tukovi´c, ˇZ. “Finite volume method on domains of varying shape (in Croatian)”, Ph.D. thesis, Faculty of mechanical engineering and naval architecture, University of Zagreb, 2005. 373 Tysell, L., Grid Deformation of 3D Hybrid Grids. Proceedings of the 8th International Conference on Numerical Grid Generation in Computational Field Simulations, pp. 265-274, International Society of Grid Generation (ISGG), Honolulu, Hawaii, USA, 2002. 374 Helenbrook, B., Mesh Deformation Using the Biharmonic Operator. International Journal for Numerical Methods in Engineering, 56 (7), pp. 1007-1021, 2003. 372

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(a) Before Figure 7.3

(b) After Mesh Deformation via Bi-Harmonic Equations

The biharmonic surface grid projection algorithm used for h-refinement may also be used for the generation of the initial surface grid. This algorithm is better to handle surface patches with poor parameterization and internal surface discontinuities than bicubic splines. The algorithm has later been modified in [Tysell]375. The latest improvements are the use of more edge swapping in order to get a more regular mixed grid and also the setting of a fix position of some nodes close to or on the curves defining the surface patch. The initial position of the nodes in the mixed grid can be computed using a tensor-product patch definition. The algorithm is then used to adjust the position of the nodes in order to get a smooth surface grid. Radial Basis Function376 A common problem in CFD is maintaining high mesh quality during large transformations and rotations, as shown in the Laplace equation method as described before. One mesh technique that can handle large mesh deformations is based on the interpolation of Radial Basis Functions (RBF). This technique can offer superior mesh motion in terms of mesh quality on average but can be computationally expensive. It is critical when using RBF that the mesh quality remains high. If the worst mesh quality is too low, the simulation will diverge. However, if the mesh quality remains high, the simulation will remain stable, accurate and efficient. Bos 377 studied the wing performance for flapping wings of insects at small scales. The RBF method can handle this motion by interpolating the displaced boundary nodes on the surrounding mesh. Bos also studied the difference between using the Laplace equation with variable diffusivity, solid body rotation stress equation and RBF. The skewness and non-orthogonality values were compared for all cases and the RBF showed higher mesh quality for both skewness and non-orthogonality.

Tysell, L., CAD Geometry Import for Grid Generation. Proceedings of the 11th ISGG Conference on Numerical Grid Generation, International Society of Grid Generation (ISGG), Montreal, Canada, 2009. 376 Gina M. Casadei, “Dynamic-Mesh Techniques for Unsteady Multiphase Surface-Ship Hydrodynamics”, A Thesis in Mechanical Engineering, Pennsylvania State University, December 2010. 377 Frank Martijn BOS. Numerical simulations of flapping foil and wing aerodynamics. PhD thesis, 2009. 375

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Figure 7.4

Mesh Deformation via Laplace & RBF Methods

Figure 7.4-(b) clearly displays that the RBF deforms around the rotating rectangle, unlike the Laplacian mesh motion (Figure 7.4-(a)) which has highly skewed cells around the rectangle. The high mesh quality is more preserved in regards to RBF. However, RBF requires much more computational effort between iterations during the mesh update scheme, which is a huge downfall to this method. The interpolation function s(x) as defined below describes the displacement of all computational mesh points by summing a set of basis functions: Nb

(

)

s(x) =  γ j x − x b j + q(x) j=1

where x b j = [ x b j , y b j , z b j ]

Eq. 7.8

are boundary value displacemen t

Furthermore, q is a polynomial, Nb is the number of boundary points, ϕ is a given basis function as a function of the Euclidean distance x. One of the first steps in solving equation is to evaluate the interpolation function s(x) in the known boundary points in equation, s(xbj) = Δxbj where Δxbj contains the known discrete values of the boundary point displacements. Generalized Grid Interface The Generalized Grid Interface (GGI) refers to a grid on either side of two connected surfaces, where the grid connectors do not have to match. GGI connections allow non-matching of nodes which can be beneficial for many reasons. The main advantage to this meshing technique is that it does not have to adapt the topology of the mesh at the interface between two non-conformal meshes. Each of the GGI regions can be different sizes, but cannot have overlap regions. One paper studies the use of a GGI for the application of turbomachinery378. GGI can be described as using weighed interpolation to 378 M. Beaudoina and H. Jasak. “Development of a Generalized Grid Interface for Turbomachinery simulations with

OpenFOAM”. Open Source CFD International Conference, 2008.

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evaluate and transmit flow values over patches in the mesh. These flow values are controlled from the master patch to the shadow patch through a set of finite volume method discretization reasoning. The downfall to this method is that even with minimal error in the master patch variable, unacceptable discretization error can occur379. The GGI weighting factors relate to the percentage of surface intersection between two overlapping faces. The GGI method uses the Sutherland-Hodgman algorithm in OpenFOAM® to compute the master and shadow face intersection surface area. This algorithm must be used with convex polygons only, which could cause problems with complicated geometries where non-convex polygons are present. Inaccuracies can also occur at the border between a rotating and fixed part of the mesh due to possible gaps between the faces. The objective here is to mimic behavior of sliding interface without changing the mesh. (See Figure 7.5).

Figure 7.5

GGI interface

The GGI design rationale with respect to examples like Turbomachinery is: • 379

Apart from “fully overlapped” cases, turbomachinery meshes contain similar features that

Same as previous.

196

should employ identical methodology, but are not quite the same. • Non-matching cylices for a single rotor passage. • Partial overlap for different rotor-stator pitch. • Mixing plane to perform averaging instead of coupling directly. • •

Component coupling requires data manipulation (copy, transform, average). In such cases, the behavior is closer to a coupled boundary condition, but the numeric is similar to sliding interface380.

Overset Methods One method of grid generation that has many advantages is an overset-grid approach. This process involves constructing several blocks that are overlapping, made up of structured or unstructured grids. Partial differential equations are solved on each component and boundary information is then exchanged between these grids based on interpolation381. The unused grid points are cut from the solution known as hole points. The points that are overlapped between grids are known as fringe points. The interpolation points are identified as the points that interpolate between the overlapped grids to obtain a solution382. Figure 7.6-(a) shows an example of a boundary layer grid and a background grid. Figure 7.6-(b-c) displays an overset grid arrangement showing hole, interpolated and active points correlates for a ship motions using dynamic overset grids383. This method uses rigid overset grids that move with relative motion at large amplitude motions. The code Suggar is used to obtain interpolation coefficients between the grids at each time step that the grid is moved. The overset grid comparison with experimental data for sink age, trim, and resistance showed good comparison proving that this meshing technique allows accurate computations of ship flows in motion.

(b) Overset Mesh Motion with no rotation

(c) Overset Mesh Motion with 45°Rotation

(a) Initial Overset mesh Figure 7.6

Overset Method

Hrvoje Jasak, “General Grid Interface Theoretical Basis and Implementation”, Wikki Ltd, United Kingdom. S.E. Sherer and J.N. Scott. “High-order compact finite-difference methods on general overset grids”. Journal of Computational Physics, 210(2):459–496, 2005. 382 P.M. Carrica, R.V. Wilson, R.W. Noack, and F. Stern, “Ship motions using single phase level set with dynamic overset grids”, Computers and Fluids, 36(9):1415–1433, 2007. 383 See previous. 380 381

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Delaunay Method Delaunay method used in mesh motion divide into four steps: First, we generate the Delaunay graph according to the geometry boundary. The Delaunay criterion is that the circum-circle for triangles or the circum-for tetrahedron should not include other points except the points which construct the triangular or tetrahedron. If the geometry boundary is convex, this mesh can always proceed and unique. Usually, wing surface and far boundary points take as geometry to generate the Delaunay triangular. After the Delaunay triangular generation, all spatial nodes should locate in its Delaunay triangular. For a point p, search the triangular which contain this point and calculate the surface or volume coordinates as

ej =

Sj S

j = 1,2,3

ej =

Vj V

j = 1,2,3,4

Eq. 7.9

The key step in Delaunay method is moving the Delaunay triangular base on boundary deformation. All the connectivity and vertex index should be kept. If the deformation is too large, triangular deformation may failure. In this case, split the deformation into two shell steps and go back to step 1 regenerating the Delaunay triangular. At last, relocate the spatial node. According to the surface or volume coordinate at step2, turn them into Cartesian at the moved triangular. As equation : 4

xp =  ei xi i =1

Eq. 7.10

7.4.7.1 Case Study - Airfoil Rotation In this case, NACA 0012 airfoil will rotate about its back edge 30 degrees. All the fluid uses a structure mesh with lowest quality 0.7. The surface girds are quad with 200 nodes and spatial grids are

(a) mesh before deformation

(b) initial Delaunay triangular

(c) Delaunay triangular after deformation

(d) dynamic mesh by Delaunay

Figure 7.7

Delaunay Method of Dynamic mesh

198

hexahedron with 40325 nodes. Figure 7.7-(a) demonstrates the mesh before deformation. For Delaunay method, first step is generating Delaunay triangular according to geometry boundary as Figure 7.7-(b) displayed. Then, compute the surface coordinates of spatial nodes in Delaunay triangular. After the geometry deformation, the Delaunay triangular will deform as Figure 7.7-(c). By keeping the surface coordinates unchanged, we relocate the spatial nodes in Cartesian coordinates. Figure 7.7-(d) show the mesh after deformation384. Spring Analogy Other technique is the spring analogy, where the mesh nodes are connected through tension springs, where the stiffness is related to the length of the edge. This approach tends to produce highly deformed meshes with collapsed or negative volume and is incapable of reproducing solid body rotation. The tension spring model has been improved by attaching torsion springs to each vertex where the stiffness is related to the angle. Six Degrees of Ferndom (6 DOF) 7.4.9.1 Transitional Deformation The grid was deformed in the y-and zdirections using a spring analogy technique. The 6DOF solver uses the object's forces and moments in order to compute the translational and angular motion of the center of gravity of an object. The governing equation for the translational motion of the center of gravity is solved for in the inertial coordinate system. As an example, Figure 7.8 shows the mesh after the translational deformation for a wing. The original un-deformed mesh is shown in grey color, and the deformed mesh is shown in red. In this case the tip deformation along the y-axis is 20% of the wing semispan385.

Figure 7.8

Mesh before and after the translational deformations

7.4.9.2 Rotational Deformation The rotational grid can be obtained by multiplying the original grid with the matrix on R as:

JIA Huana, SUN Qin b, “A Comparison of Two Dynamic Mesh Methods in Fluid –Structure interaction”, School of Aeronautics, Northwestern Polytechnic University, Xi‘an china. 2nd International Conference on Electronic & Mechanical Engineering and Information Technology (EMEIT-2012). 385 Joaquin Ivan Gargoloff, “A Numerical Method For Fully Nonlinear Aero-elastic Analysis”, Dissertation, Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of Doctor of Philosophy, 2007. 384

199

 Cθ C ψ C θ Sψ - Sθ   R =  S SθCψ - C Sψ S SθSψ + C Cψ S Cθ  C Sθ C ψ + S Sψ C SθSψ − S C ψ C Cθ   

Eq. 7.11

where, in generic terms, CX = Cos(X) and SX = Sin (X) . The angles φ , ϴ, and ψ are Euler angles that represent the following sequence of rotations: • • •

rotation about the x-axis (e.g., roll for airplanes) rotation about the y-axis (e.g., pitch for airplanes) rotation about the z-axis (e.g., yaw for airplanes)

Figure 7.9 illustrates an transitional (y axis) + rotation about x axis.

Undeformed

Figure 7.9

Deformed

Mesh Before and After the x-axis Rotational Deformation

Dynamically Adaptive Mesh Refinement (DAMR) While our adaptive mesh can effectively resolve very singular functions we still need to provide a mechanism for dynamically adjusting the mesh to possible rapid changes of time dependent solutions. There are several methods to obtain a moving mesh. Here, we adopt the so-called moving mesh PDE approach386-387 in which a time-dependent PDE is introduced to determine the motion of the mesh. Both the moving mesh PDE (MMPDE) and the underlying physical equations are solved simultaneously or alternately. This approach has the advantage of avoiding interpolation between old and new grids which is necessary in the static methods. Interpolation may introduce too much numerical smoothing in problems in which the resolution of small scales is important and 386 W. Huang, Y. Ren, and R. D. Russel, Moving mesh methods

based on moving mesh partial differential equations, J. Comput. Phys. 113, 279 (1994). 387 W. Huang and R. D. Russell, Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J. Sci. Comput. 20(3), 998 (1999).

200

thus, desirably, it should be avoided. Recently [Huang and Russell]388 have introduced a very robust class of MMPDEs derived from the gradient flow equations associated with the mesh variational principle. Here, we apply the same idea directly to our proposed mesh equations (Eq. 7.6). A standard method to solve is to consider the equations

𝐱 𝝉 = ∇. (k∇𝐮i,mesh )

Eq. 7.12 Where τ is an artificial time. Then, beginning with an initial guess, we march in “time” to steady state. Any discrete marching scheme to solve (Eq. 7.12) can be regarded as an iterative method to solve the nonlinear system (Eq. 7.6). At τ = 0, we can find the solution up to steady state to obtain a mesh that adapts well to the initial data. With this initial adaptive mesh, the solution u can be updated (using the underlying PDE) one time step. Then a new mesh is obtained using the updated u in the monitor function. However, since u changes only very little in one time step, it is not necessary to solve again (Eq. 7.12) all the way to steady state. Besides, the initial mesh is already a very good initial guess. Thus, it is natural to march only one time step in (Eq. 7.12) (or equivalently to do only one iteration) at a time. In other words, taking τ as the actual time. Therefore, we proceed solving the moving mesh and the underlying PDEs alternately one time step at a time389. Case Study - Dynamically Adaptive Mesh Refinement FDTD: A Stable and Efficient Technique for Time-Domain Simulations390 The Finite-Difference Time-Domain (FDTD) technique has been extensively employed in the modeling of microwave and optical structures, due to its simplicity and versatility. However, these FDTD qualities are partially compensated by the stability and numerical dispersion limitations on the choice of the cell size and the time step of the method, that render its application to complex and/or electrically large structures computationally expensive. In the past, a variety of static sub-gridding techniques have been proposed, aimed at accelerating the conventional FDTD technique for structures with localized fine geometric features. According to such approaches, local mesh refinement is pursued in a priori defined regions of a computational domain, as dictated by physical considerations. For example, the presence of metallic edges, or high dielectric permittivity inclusions, would call for a locally dense mesh, embedded in a coarser global one. The use of local mesh refinement typically results in significant computational savings compared to the conventional FDTD, despite the fact that its implementation is Figure 7.10 Spiral Inductor Geometry where P1 and P2 associated with additional interpolation denote port 1 and Part 2 and extrapolation operations in both space W. Huang and R. D. Russell, Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J. Sci. Comput. 20(3), 998 (1999). 389 Hector D. Ceniceros and Thomas Y. Hou, “An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions”, Journal of Computational Physics 172, 609–639 (2001). 390 Yaxun Liu and Costas D. Sarris, “Dynamically Adaptive Mesh Refinement FDTD: A Stable And Efficient Technique For Time-Domain Simulations”, Department of Electrical and Computer Engineering University of Toronto, Toronto, ON, M5S 3G4, Canada. 388

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and time. However, static mesh refinement ignores the dynamic nature of time-domain field simulations. In fact, techniques such as FDTD and the transmission line matrix (TLM) essentially register the evolution of a broadband pulse propagating in a device under test, along with its retro-reflections. Hence, a localized discontinuity in a simulated domain is only illuminated for a (potentially small) fraction of the total simulation time, during which a local mesh refinement around it is needed. Therefore, static mesh refinement, which is widely employed in frequency-domain simulations and has been incorporated in commercial finite-element tools, is only a sub-optimal solution to the mesh refinement problem in the framework of time-domain analysis. Recently, the AMR technique was coupled with FDTD to produce a dynamically mesh adaptive FDTD algorithm, that was successfully applied to microwave integrated circuit and optical waveguide problems. Instead of applying local mesh refinement in a priori defined regions of a computational domain, the dynamic AMR-FDTD uses sub-grids, which are adaptively defined according to the spatial-temporal evolution of field distributions. As a result, significant execution time savings, up to two orders of magnitude, are attainable for large-scale open-domain problems. In this paper, the dynamic AMR-FDTD approach is explained and realistic applications, demonstrating the salient features of the method are provided. 7.5.1.1 Numerical Results To demonstrate the dynamic AMR-FDTD algorithm, the geometry of a spiral inductor of Figure 7.10 is analyzed. The AMR-FDTD method uses a 60×40×10 mesh and 8192 time steps. A reference FDTD

(a) Time-step 100

(c) Time-step 300 Figure 7.11

(b) Time-step 200

(d) Time-step 400

Vertical field evolution and associated mesh refinement in the microstrip spiral inductor, simulated by a two-level dynamic AMR-FDTD

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simulation of a 120×80×20 mesh is used for comparison. The mesh refinement process is illustrated in Figure 7.11, which shows the effective vertical electric field wave front tracking achieved by the algorithm. Despite the highly resonant nature of the spiral inductor, which necessitates the use of a significant number of time steps for the extraction of the S-parameters, significant execution time savings (of about 80%) have been achieved. The time-domain results for this case study demonstrate the absence of late-time instability in the AMR-FDTD. In fact, the number of AMR-FDTD child meshes converges to zero over time, implying that only the root mesh is still present at a late stage of the code. Therefore, no spatial or temporal interpolation operations, which are the primary sources of instabilities in adaptive mesh FDTD codes, are applied then. This is an additional advantage of using a dynamically adaptive instead of a statically adaptive mesh in time-domain simulations. For further info, please refer to development in [Liu and Sarris]391.

8 Assessment of Mesh

Types

Yaxun Liu and Costas D. Sarris, “Dynamically Adaptive Mesh Refinement FDTD: A Stable And Efficient Technique For Time-Domain Simulations”, Department of Electrical and Computer Engineering University of Toronto, Toronto, ON, M5S 3G4, Canada. 391

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It is an old age question. Which is better? Structured or Unstructured meshes? Hexahedra (Polyhedral) vs Tetrahedral? It is a lengthy and heated debate which cannot be fully covered here. The answer may be depends in the case itself. In general, most people dislike unstructured meshes because of lack of direct control over the mesh and they produce more data points and cells than their structured ones, therefore, requiring more CPU. On the other hand, they mostly automated, easier to produce. An example using a backward step in a duct by means of all three types of meshing. The result clearly Figure 8.1 Backward facing step in a duct using indicates that polyhedral meshing in Polyhedral, Hexahedral and Tetrahedral cells superior. Figure 8.1 displays residual comparison for three cell types. Polyhedral cells got the best results, then hexahedral cells closely second, with tetrahedral counting as third. But be aware that this is a modest highly anisotropic flow, and could not be used as a decisive criteria.

Structured vs. Unstructured A major mesh generation vendor (Pointwise®) argues that Structured Meshing is not going away, mainly due to Control and Quality as the reasons are. Here are several factors they view including: Time and Memory You can fill the same volume with fewer hexes than Tets, thereby lowering the cell count and your CFD computation time and memory usage. Structured grids generally have a different topology than unstructured grids, so it is difficult to make a direct cell count comparison. At its simplest, each hexahedron can be decomposed into 5 Tetrahedral that share its edges, giving a 5:1 reduction in cell count for the same flow field resolution. The benefit to reducing cell count becomes very apparent when generating a mesh with a wide variation in resolved length scales; you will use many more Tets than you would Hex cells. Resolution Flow of a fluid will often exhibit strong gradients in one direction with milder gradients in the transverse directions (e.g. boundary layers, shear layers, wakes). In these instances, high quality cells are easily generated on a hex grid with high aspect ratio (on the order of one thousand or more). It is much more difficult to generate accurate CFD solutions on highly stretched tetrahedral. (Plus, not all stretched Tet are equal depending on the maximum included angles.) Alignment CFD solvers converge better and can produce more accurate results when the grid is aligned with the predominant flow direction. Alignment in a structured grid is achieved almost implicitly because grid lines follow the contours of the geometry (as does the flow), whereas there's no such alignment in an unstructured mesh. Definable Normal Application of boundary conditions and turbulence models work well when there is a well-defined computational direction normal to a feature such as a wall or wake. Transverse normal are easily defined in a structured grid. To demonstrate, following example is used.

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Effect of Cell Topology in Truncation Error In general, structured mesh are more aligned with flow gradient, therefore, producing less truncation errors, as depicted in Figure 8.2392. In contrast, tetra cells, have angle with the flow direction as shown in previously. It is suffice to say that the argument is not as simple as case in 2D here, like so many things in CFD where no clear cut definitions are involved (an art?). While some application like Turbomachinery (with clear flow direction) prefer hex meshes, others depend on tet meshes. Basically it comes down to whom you talking and case in hand. While unstructured meshes offer better flexibility for today’s application, then Figure 8.2 Effect of truncation error on Hex and Tet cells lack strict control of structure and overheads of data base. Also the availability of resources should be factor. With advent of Polyhedral cells and their relatively ease of use some vendors such as CD-Adapco© and others are keen to pursue their use (economical concerns?). In fact CD-Adapco© been one of the pioneers of poly cell, argued against the Tet meshing, and for poly meshing, in a blog in titled “Natures answer to Meshing”. That may be true since most of nature is composed of different (patch work) topology. With hummer, it argues that “So how is it that honeybees (average brain size 1g) manage to out mesh those CFD engineers (average brain size 1250g) who still religiously rely on tetrahedral meshing”? With exception of course. (See the smart bee in Figure 8.3)393. But seriously, the case for/or against structures vs unstructured Figure 8.3 Average Bees Being Smarter than meshes with particular attention to Poly versus Tet CFD Engineer? (Courtesy of Stephen Ferguson) cells are argued in two folds as discussed below. Polyhedral vs. Tetrahedral The argument against the Tet cells versus Poly cells are relatively straight forward. Beside cell the count, a major advantage of polyhedral cells is that they have many neighbors (typically of order 10), so gradients can be much better approximated (using linear shape functions and the information from nearest neighbors) than is the case with tetrahedral cells. Two of the most common methods for gradient calculation are the Green-Gauss and the Least Squares approaches394. Polyhedral cells are

Sideroff, C,. “Multi-Block Structured Meshing and Pre-Processing for OpenFOAM Turbomachinery Analysis”. Stephen Ferguson, CD-Adapco Blog, “Nature’s Answer to Meshing”, 2013. 394 Emre Sozer, Christoph Brehm and Cetin C. Kiris, “Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers”, American Institute of Aeronautics and Astronautics. 392 393

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also less sensitive to stretching than tetrahedral. Smart grid generation and optimization techniques offer limitless possibilities: cells can automatically be joined, split, or modified by introducing additional points, edges and faces. Indeed, substantial improvements in grid quality are expected in the future, benefiting both solver efficiency and accuracy of solutions. Polyhedral cells are especially beneficial for handling recirculating flows. Tests have shown that, for example, in the cubic lid-driven cavity flow, many fewer polyhedral are needed to achieve a specified accuracy than even Cartesian hexahedra (which one would expect to be optimal for rectangular solution domains). In fact for a hexahedral cell, there are three optimal flow directions which lead to the maximum accuracy (normal to each of the three sets of parallel faces); for a polyhedron with 12 faces, there are six optimal directions which, together with the larger number of neighbors, leads to a more accurate solution with a lower cell count. Although tetrahedral are the simplest form of volume elements and tetrahedral meshes are able to Figure 8.4 Polyhedral cells vs Tetrahedral cells approximate any arbitrarily shaped continuum with a remarkable level of detail. Automated tetrahedral mesh generation methods have been well studied and developed, providing currently the only robust solution for meshing complex geometries in 3D, making them a standard choice of major CFD codes. However, despite the fact that tetrahedral present several geometric assets, such as planar faces and well defined face and volume centroids, they suffer from certain disadvantages that make analysts deem them inferior to hexahedra. Tetrahedral elements cannot provide reasonable accuracy, as soon as they become too elongated, which is often the case in boundary layers or sharp corners of the domain. Furthermore, they have only four neighbors making them not an optimal choice for CFD, as computation of gradients at cell centers can become problematic. It is, therefore, not unusual during simulations serious numerical stability issues to appear, additionally to the reduced accuracy, and problematic convergence properties to dominate the analysis. Figure 8.4 indicate the pro and con of Polyhedral cells vs Tetrahedral ones. 8.1.5.1 Boundary Prismatic Cells An issue for general unstructured cells, are boundary cells. Several remedies exist in order to overcome those disadvantages. A boundary layer, formed using prismatic elements along walls, is able to balance, up to a certain degree, the negative effects in accuracy and stability (see Figure 8.5left). Furthermore, advanced discretization methods combined with very fine meshes can result to accurate solutions and good convergence properties. This, however, demands for increased memory usage and computing time, while it makes the analysis code more complicated. Recently, an alternative option to tetrahedral meshes has emerged, suggesting the use of polyhedral elements

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instead395. Polyhedral over the same level of automatic mesh generation as tetrahedral do, while they are able to overcome the disadvantages adherent to tetrahedral meshes (see Figure 8.5-right). A major advantage of polyhedral occurs from the fact that they are bounded by many neighbors, making approximation of gradients much better that tetrahedral. Furthermore, they are much less sensitive to stretching and, since their typically irregular shape is not a restriction for several CFD codes, they over the possibility of post-processing and optimization without the strict geometric criteria that are necessary for optimizing tetrahedral, or even hexahedral meshes. On the negative side, polyhedral are usually of much more complex geometry than regular volumes, and, depending on the generation method, it cannot always be guaranteed that they are convex, or, even more, that their faces are planar. The topology of polyhedral meshes is, typically, also complex, preventing the implementation of efficient and easy to maintain generation algorithms from being straightforward. As a further consequence, polyhedral meshes require a considerable amount of adjacency relations, in comparison to tetrahedral and hexahedral meshes, making them candidates for resource expensive solutions. All the above set the basis for an interesting field of exploration in volume meshing. Previous studies on the subject have shown promising results, however polyhedral meshing is still far from becoming a standard practice in CFD simulations. Some explanations for this may be its limited adoption from analysis codes and the fact that polyhedral are not an appropriate solution for every type of analysis. It should be mentioned that, currently, polyhedral meshes attract more attention in fields such as Computer Graphics and Medical Imaging, where in 3D volume rendering is of specific interest. However, the few researches dedicated to exploring polyhedral mesh generation for CFD remain active, making constant progress towards more efficient methods and high quality meshes.

Figure 8.5

Boundary prims cells for tetrahedral (left) and polyhedral (right) cells – (Courtesy of CDAdapco)

M. Peric, “Simulation of flows in complex geometries: New meshing and solution methods”, NAFEMS seminar: Simulation of Complex Flows (CFD). 395

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Accuracy Assessment of Gradient Calculation Methods 396

An unstructured mesh, commonly depicted as consisting of tetrahedral elements, can be considered a superset encompassing any valid cell geometry including hexahedral, tetrahedral and arbitrary polyhedral. A survey of gradient reconstruction methods for cell-centered data on unstructured meshes is conducted within the scope of accuracy assessment. Formal order of accuracy, as well as error magnitudes for each of the studied methods, are evaluated on a complex mesh of various cell types through consecutive local scaling of an analytical test function. The tests highlighted several gradient operator choices that can consistently achieve 1st order accuracy regardless of cell type and shape. The tests further offered error comparisons for given cell types, leading to the observation that the “ideal” gradient operator choice is not universal. During implementation of an unstructured solver, a choice about where to place the discrete data must be made. Alternative choices include vertices, face-centers, cell-centers or a combination. While advantages and drawbacks of each approach have been heavily debated in the CFD community397-398 the current feel is that no clear “best” choice emerges. Gradient operator choice for an unstructured solver has a strong impact on accuracy, efficiency and robustness. While all of these are crucial factors, we limit our scope to the analysis of accuracy alone. Geometric Properties For detail calculation of geometric properties such as Polygonal Face Area and Centroid as well as Polyhedral Volume and Centroid, the reader should refer to [Sozer et al]399. Literature Survey [Aftosmis, et al.]400 investigated the behavior of linear reconstruction techniques on unstructured meshes. Their chief concern was the behavior of limiters and the effect of element types (triangular vs. Quadrilateral) for CFD solutions, particularly for high aspect ratio or irregular elements. They did however, investigate the least squares (LSQR) and Green-Gauss methods for gradient calculation. The methods behaved similarly for regular meshes whereas the LSQR was found to be more tolerant to mesh distortions. [Mavriplis]401 examined the LSQR procedure for gradient reconstruction, observing that the method produced accurate gradients for isotropic meshes but the accuracy deteriorated for highly stretched meshes in the presence of curvature. In the latter case, they found the Green-Gauss reconstruction method to be more accurate. [Shima, et al.]402 devised an LSQR method where they incorporate weights based on face areas, attempting to inherit benefits of the Green-Gauss method for stretched meshes. While they note accuracy improvements, they still resort to a hybrid approach where Green-Gauss method is used for thin and distorted mesh regions403. A comprehensive survey of unstructured mesh gradient methods, in the context of computer graphics, is conducted by [Correa, Emre Sozer, Christoph Brehm and Cetin C. Kiris,”Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers”, American Institute of Aeronautics and Astronautics. 397 Diskin, B., Thomas, J., Nielsen, E., Nishiwaka, H., and White, J., “Comparison of Node-Centered and CellCentered Unstructured Finite-Volume Discretization: Viscous Fluxes,” AIAA Journal, Vol. 48, No. 7, 2010. 398 Diskin, B. and Thomas, J. ,“Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretization: Inviscid Fluxes,” AIAA Journal, Vol. 49, No. 4, 2011, DOI: 10.2514/1.J050897. 399 Emre Sozer, Christoph Brehm and Cetin C. Kiris, “Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers”, American Institute of Aeronautics and Astronautics. 400 Aftosmis, M., Gaitonde, D., and Tavares, T., “Behavior of Linear Reconstruction Techniques on Unstructured Meshes,” AIAA J., vol. 33, no. 11, pp. 2038-2049, 1995. 401 Mavriplis, D., “Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured Meshes,”, AIAA Paper 2003-3986, 2003. 402 Shima, E., Kitamura, K., and Fujimoto, K., “New Gradient Calculation Method for MUSCL Type CFD Schemes in Arbitrary Polyhedra”, 48th AIAA Aerospace Sciences Meeting, Jan 4-7, Orlando, FL, 2010. 403 Shima, E., Kitamura, K., and Haga, T., “Green-Gauss/Weighted-Least-Squares Hybrid Gradient Reconstruction for Arbitrary Polyhedra Unstructured Grids,” AIAA Journal, Vol. 51, No. 11, 2013, DOI: 10.2514/1.J052095. 396

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et al.]404. They focus on cost and performance in volume rendering with respect to mesh resolution, element shapes, neighborhood size and scalar field complexity. They find the inverse weighted regression method to provide the highest accuracy for irregular meshes and the Green-Gauss method to perform poorly for badly shaped elements. Gradient Calculation The difficulty in calculating gradients in an unstructured mesh stems from the lack of a consistent and inherent connectivity. The stencil for gradient calculation, as well as the corresponding coefficients vary cell-by-cell and are costly to compute. Hence, those are typically pre-computed and stored. Two of the most common methods for gradient calculation are the Green-Gauss and the Least Squares approaches. Both have several common variations, some of which are explained in the following sections. 8.2.3.1 Green-Gauss Gradient Method The Green-Gauss method represents an intuitive, sound basis for gradient calculation. According to the Green-Gauss theorem, average gradient of a scalar φ in a closed volume V can be obtained by

  dV =   nˆ dA , V

 =

A

1  nˆ dA V  A

Eq. 8.1

Where ň is the surface unit normal vector and A is the surface area. For a 2nd order scheme with midpoint quadrature, the Green-Gauss method takes on the following discrete form for a polyhedral:

 =

1 N faces f nˆ f Af V f =1

Eq. 8.2

Where Nfaces is the number of faces and φ͞f is the average of the scalar over the face f. Up to this point, average gradient of a linear function at the polyhedral cell centroid. The potential errors are introduced through the particular choice of a face averaging method to obtain φ͞f . Several common alternatives in this regard are discussed below.

Figure 8.6

GG simple face averaging

8.2.3.2 GG-Simple Face Averaging Simple average of the cell center values at the left and right sides of the face is taken as the face center value (see Figure 8.6).

f =

c-left + c-right 2

Eq. 8.3

As the most basic approach, the simple averaging method is still commonly used due to its attractive properties of straightforward and cheap implementation. In fact, its usage is usually implied when

Correa, C., R., H., and K., M., “A Comparison of Gradient Estimation Methods for Volume Rendering on Unstructured Meshes,” IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 3, March, 2011. 404

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cell-centered Green-Gauss gradient method is referred to without mention of the associated face averaging method. 8.2.3.3 GG-Inverse Distance Weighted (IDW) Face Interpolation Another popular approach to face averaging, IDW method utilizes the entire neighbor stencil around face f (see Figure 8.7). The result show difference in 90 degree bend, as shown in Error! Reference source not found.. It is clearly demonstrates the advantage of Hex cell over Tet in this example.. This also indicates a good agreement with experimental data using the hybrid mesh (Tet + Prism). As you clearly see the argument about which mesh to use is far from over and depending to whom you talking. It appears that case against the Tet cells is relatively straight, but Poly cells and Hybrid meshing is still debatable. With most vendors, have capability of offering both structured as well as unstructured meshes at will and table below displays the type of meshing available. Where the sum is carried out over the entire stencil and di = ri −rf represents the distance between the stencil point and the current face center. IDW has shortcomings when the neighboring cell centers are not evenly distributed around the face but clustered in certain directions; a scenario likely to occur for regions with poor grid quality or at the interfaces of different cell types in mixed cell type meshes. In addition to this non-isotropy, potentially large Figure 8.7 GG Inverse Distance Weighted (IDW) discrepancies in stencil distances may occur for Face Interpolation regions with poor grid quality or at the interfaces of different cell types in mixed cell type meshes. In addition to this non-isotropy, potentially large discrepancies in stencil distances may skew the weights significantly, a phenomena made worse by the usage of squared distances. Nonetheless, the squared weighting seems to be the most common approach taken with IDW and hence we chose to adopt it for our evaluation. Variations of this method involving different weightings (e.g. volume, inverse distance…) are possible but this does not change the fundamental flaws explained above. Other methodologies considered are: • • • •

GG-Weighted Least Squares (LSQR) Face Interpolation GG-Weighted Tri-Linear Face Interpolation (WTLI) Least Squares (LSQR) Gradient Method Curvilinear Gradient Method

Detailed information regarding these methods and more are available in405.

Emre Sozer, Christoph Brehm and Cetin C. Kiris, “Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers”, American Institute of Aeronautics and Astronautics. 405

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i

N

f =

d i =1 N

2 i

Eq. 8.4

1

d i =1

2 i

Visual Inspection The gradient order of accuracy (GOA) for various aforementioned methodologies are presented in Figure 8.8 (a-f) for x-coordinates. Note that for a 2nd order scheme, a gradient operator of at least order 1 is needed. Figure (a) shows the distribution of GOA for the LSQR scheme with compact stencil. The operator is able to produce at least 1st order accuracy in the entire field. The cells with a uniformly spaced stencil that are aligned with x or y directions exhibit 2nd order accuracy in the corresponding directions. This is due to perfect cancellation of 1st order errors and it breaks down

(a) LSQR (b) GG Simple

(c) GG IDW

(d) GG WTLI

(e) GG LSQR

(f) Curvlinear

Figure 8.8

Methodologies for various Gradient Order of Accuracy

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as soon as mesh uniformity is lost. The cancellation leading to the increased order now needs to be satisfied in a wider stencil. The Green-Gauss method with Simple averaging as well as IDW fails to achieve 1st order accuracy as seen in Figure 8.8 (b-c) respectively. Both of these methods neglect to incorporate directionality of their stencils and neither is linear-exact, i.e. can’t reproduce gradient of a linear function exactly. The WTLI and LSQR face interpolation methods for the Green-Gauss gradient yields very similar results (see Figure 8.8 (d-e)). The curvilinear gradient results, as shown in Figure 8.8 (f), are much like those of the LSQR with compact stencil. This is not surprising as both methods are linear-exact and both utilize compact stencils. Note that the curvilinear gradient operator has the most compact stencil of the alternative methods in scope here, utilizing at most 4 points (for 2D). Results Based on L2 Norm While order of accuracy is a crucial property to inspect, it is pertinent to look at the actual error levels as several of the gradient operator choices were demonstrated to satisfy 1st order accuracy. Figure 8.9 shows L2 error norms with respect to the refinement level. First we would like to clear the peculiar behavior of the Green-Gauss method with simple and IDW face averaging. It seem to approach a 1st order convergence rate before stalling at a fixed error L2 Norm (vertical) vs Refinement level level. This is due to the aforementioned inconsistency as they converge, in a 1st order manner, to a gradient value that is not consistent with the exact Global L2 norm value. Note here that without a for x-direction deep enough convergence study, vs various gradient this issue could have been method overlooked, leading to a false conclusion that these methods are 1st order accurate. The rest of the operators are all linear-exact, and consequently they all consistently exhibit 1st order accuracy as it was apparent from the GOA distributions shown earlier on the test mesh. The error norms shown in Figure 8.9 now reveal that the Figure 8.9 Global Error Norms for x-Direction Gradient for Various Gradient Methods Green-Gauss methods (with WTLI or LSQR face averaging) yield significantly larger errors compared to the curvilinear or the LSQR methods. Within the latter group, the LSQR compact has slightly lower error then the LSQR extended while the curvilinear method places in between. It is possible to inspect errors for individual cells of various cell types. For regular cell types (square, equilateral triangle and right triangle), all the gradient operators are able to produce at least 1st order accuracy. In fact, the square cell type stencil yields 2nd order accuracy for each method. The curvilinear method produces a notably smaller error for this case. For irregular stencils, which are of greater practical interest, we start observing the familiar result of convergence stalling for the inconsistent schemes, namely the Green-Gauss method with simple or IDW face averaging. Discarding the special case of the square stencil, the LSQR gradient operator consistently produces the lowest errors except for the cases of thin triangles and thin quadrilaterals (commonly encountered in boundary layer regions of CFD meshes). For the thin cells, the trend reverses and the

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LSQR method yields the largest errors while the consistent Green-Gauss methods perform the best. Note that the thin cells mentioned here were sampled near the curved boundary region of the test mesh. Whereas the Green-Gauss method exhibited mediocre performance elsewhere, its favorable behavior in the crucial boundary layer type meshes demonstrates its appeal. The errors associated with the curvilinear method were erratic, yielding the best result for the square cells and placing among the lower error range elsewhere with two exceptions; the thin quadrilateral and the arbitrary polyhedral where it exhibited the largest errors. This suggests that a smarter logic for stencil reduction (in 2D, down-selection of 4 stencil points) needs to be developed. Otherwise, we consider this method promising, considering that it has the most compact stencil, hence the lowest computational cost. Concluding Remarks A detailed accuracy study of gradient calculation methods for cell-centered unstructured data is presented. Necessity of the linear-exactness property for 1st order gradient accuracy, and consequently a 2nd order scheme, is emphasized. A straightforward, yet novel, approach utilizing local curvilinear transformation is proposed. The curvilinear method offers the most compact gradient stencil among those studied here. No clear “best” method emerged but strengths and shortcomings of the investigated methodologies for different cell types are exposed. Gradient operators with compact stencils, namely LSQR compact and curvilinear, generally exhibited lower errors. LSQR compact scheme caused stability issues for the solution of the inviscid standing vortex problem on the random triangulated mesh. The curvilinear scheme, on the other hand, had an erratic behavior for different cell types, yielding overall low error levels but exhibiting a large error for a sample arbitrary polyhedral cell. This suggests that the method could benefit from development of a smarter stencil reduction logic (to down-select 4 points from the available stencil in 2D). The GreenGauss method stood out with lower errors for thin triangular or quadrilateral cell types, such as those found in typical boundary layer meshes, which is it is a very attractive quality for CFD solvers.

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9 Case Studies Involving Comparisons of Structured vs. Unstructured Meshes Case Study 1 – Flow through Pipe with 90 degree Bend These last two items are why boundary layers are best modeled in an unstructured mesh with prism layers as they provide structure in the direction away from the wall. Further, another parametric study done by (Fluent®) as turbulent flow over a pipe with 90 degree bend and Reynolds number of ReD = 43000 and turbulent model of Realizable k-ε406. Axial velocity contour are compared at the 90 degree bend. The result show difference in 90 degree bend, as shown in Figure 9.1. It is clearly demonstrates the advantage of Hex cell over Tet in this example. This also indicates a good agreement with experimental data using the hybrid mesh (Tet + Prism, See Figure 9.2). As you clearly see the argument about which mesh to use is far from over and depending to whom you talking. It appears that case against the Tet cells is relatively straight, but Poly cells and Hybrid meshing is still debatable. With most vendors, have capability of offering both structured as well as unstructured meshes at will and table below displays the type of meshing available.

Figure 9.1

406

Comparison of Hex (16 K Cells) and Tet (440 K Cells) for a Pipe with 90 Degree Bend

Fluent, “Meshing and CFD Accuracy”, CFD Summit, June 2005.

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Figure 9.2

Results of Hex vs Tet Meshes as well as Hybrid Mesh in a Pipe with 90 Degree Bend

Case Study 2 - Comparison of Hexa-Structured and Hybrid-Unstructured Meshing Approaches for Numerical Prediction of the Flow Around Marine Propellers407 Here, we present a comparison between hex structured and hybrid-unstructured meshing approaches for the numerical prediction of the flow around marine propellers working in homogeneous flow (Open Water Conditions)408. The objective was to verify if the accuracy of the predictions based on structured meshes is significantly better than predictions based on hybrid meshes. The study was performed on two five-bladed propellers in model scale. Simulations were carried out with a commercial RANS solver, using a moving frame of reference approach and employing the SST (Shear Stress Transport) two equation turbulence model. Computational results from both meshing approaches were compared against experimental data. The thrust and torque coefficients were used as global quantities. Circumferentially averaged velocity components and root-mean square values of the turbulent velocity fluctuations, available for one of the propellers, were used to indicate the local flow field. The computational results of global quantities for both meshing approaches were very close to each other and in line with experimental data. Also the local values of the flow were in line with the experimental data, except for turbulent velocity fluctuations which were under predicted, especially in the case of the hybrid approach, where higher diffusivity was observed. The overall results suggest that for the prediction of the propulsive performances of marine propellers, at model scale, there are no significant differences, in term of accuracy, between structured and hybrid meshes but for a detailed study of the flow, the structured mesh seems to offer a better resolution.

Mitja Morgut, Enrico Nobile, “Comparison of Hexa-Structured and Hybrid-Unstructured Meshing Approaches for Numerical Prediction of the Flow Around Marine Propellers”, First International Symposium on Marine Propulsions smp’09, Trondheim, Norway, June 2009 408 See Previous. 407

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Introduction & Contributions The flow around a marine propeller is one of the most challenging problems in (CFD), where for decades it has been investigated numerically using conventional methods based on the potential theory. More recently, due to the improvements of computer performances, (RANS) solvers are becoming the practical tool as demonstrated by [Abdel-Maksoud et al.]409, [Chen and Stern]410, [Stanier]411, [Watanabe et al.]412, and [Rhee and Joshi]413, to name a few. Even though the capabilities of CFD technologies are improving, a fundamental role in a successful CFD simulation is still played by the mesh quality and type. Generally speaking, simulations carried out using Hybrid-Unstructured (HU for brevity) meshes i.e. tetrahedral with prisms or hexahedral layer on solid surfaces, are less accurate then simulations carried out using Hexahedral Structured (HS) grids. On the other hand the effort to generate HU meshes is generally lower than that needed to generate HS meshes. As a matter of fact the hybrid mesh generation is semi-automatic, whilst the structured mesh generation is nonautomatic, and requires a significant amount of work. It can also be very difficult to generate a good quality HS mesh for complex geometries: this is the case of modern marine propeller due to complex shapes, strong twisting of the blade and stagnation point on hub close to the propeller. Moreover the structured meshing approach can be very difficult to apply to the study of the propeller-hull interaction with the currently available commercial codes. Recently also the section of applied physics of the Department of Naval Architecture, Ocean and Environmental Engineering (DINMA) of the University of Trieste, in collaboration with the office of Naval Architecture of Fincantieri - Cantieri Navali S.P.A has developed a CFD procedure for the prediction of the flow around marine propellers working in open water conditions based on the use of the commercial RANS solver ANSYS-CFX 11 and the commercial meshing tool ANSYS ICEM-CFD The procedure has already been validated, using HS meshes. But the long time needed for the generation of HS meshes suggested to investigate the possibility of using hybrid meshes in place of structured meshes. For this reason a comparison between structured and hybrid meshing approaches is carried out. The study is performed on two five-bladed propellers in model scale. One is called Propeller A, propriety of Fincantieri, and the other is propeller DTMB (David Taylor Model Basin) P5168 [Chesnakas & Jessup]414. The study is made by comparing numerical results from both approaches with the available experimental data. For both propellers thrust coefficient and torque coefficient are compared for a wide range of advance ratios. In the case of propeller P5168 the comparison includes circumferentially averaged velocity components, and turbulent quantities, in a plane downstream of the propeller mid plane respectively. For both propellers and meshing approaches the numerical results are in line with experimental data, but the hybrid meshes seem to introduce more diffusivity in the solution than structured meshes.

Abdel-Maksoud, M., Menter, F., and Wuttke, H.. ‘Viscous flow simulations for conventional and high skew marine propellers’. Ship Technology Research, 45:64 – 71. 1998. 410 Chen, B. and Stern, F. ‘Computational fluid dynamics of four-quadrant marine-propulsor flow’. Journal of Ship Research, 43(4):218 – 228., 1999. 411 Stanier, T. ‘The application of ’rans’ code to investigate propeller scale effects’. Proc. 22nd Symposium on Naval Hydrodynamics, Washigton, D.C., USA, 1999. 412 Watanabe, T., Kawamura, T., Takekoshi, Y., Maeda, M., and Rhee, S. H. ‘Simulation of steady and unsteady cavitation on a marine propeller using a rans cfd code’. 5th International Symposium on Cavitation, CAV2003, Osaka, Japan, 2003. 413 Rhee, S. H. and Joshi, S. ‘Computational validation for flow around marine propeller using unstructured mesh based navier-stokes solver’. JSME International Journal, Series B, 48(3):562 – 570, 2005. 414 Chesnakas, C. and Jessup, S. ‘Experimental characterization of propeller tip flow’. Proc. 22nd Symposium on Naval Hydrodynamics, Washington, D.C., USA, 1998. 409

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Propeller Models Here, the two five-bladed propeller models visible in Figure 9.3 considered. One is the propeller propriety of Fincantieri. While the other is propeller P5168, designed at David Taylor Model Basin. P5168 is a controllable pitch propeller with diameter D = 0.4027m and experimental measurements were carried out at the David Taylor 36inch, Variable Pressure Water Tunnel415. For simulations both propellers were placed on the cylinder, simulating the hub, and the axis of the propellers were coincident with the direction of the free Figure 9.3 Design of Propellers, (left) Propeller P5168, stream. (right) Propeller A – (Courtesy of Morgut & Nobile) Numerical Method In this study only one blade is considered since, the flow around a marine propeller working in uniform flow can be considered periodic with respect to the blades when the hydrostatic pressure is assumed constant. In the given conditions the computational domain used in this study is 72o segment of cylinder covering only one blade. Moreover as a propriety related to the above mentioned conditions and also to capability of the ANSYS-CFX 11 to run different zones of the domain with either rotor or static frame of reference the cylinder is split as revealed in Figure 9.4 in a rotating part called Rotating, and in a stationary part called Fixed. The dimensions of computational domains of both propellers are listed in Table 9.1, where D is diameter of the propeller. The variable Lmid,

Figure 9.4 415

See Previous.

Computational Domain– (Courtesy of Morgut & Nobile)

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presented but not visible in Figure 9.4, is the length in direction of uniform flow of rotating. To simulate the flow around a rotating propeller the following boundary conditions were set. On the Inlet boundary, velocity components of uniform stream with the given inflow speed Rotating Fixed were imposed, while the turbulence intensity A P5168 A P5168 was set to 1% of the mean flow. On the Outlet Hmid 0.70 D 0.57 D boundary the static pressure was set to zero. Lmid 0.14 D 0.75 D On the outer surface and on the part of the hub L1 2.0 D 1.5 D included in Fixed free-slip boundary L2 6.0 D 5.0 D conditions were set. On the blade surface and H2 1.8 D 1.4 D on the part of the hub included in Rotating noslip boundary conditions were set. On the Table 9.1 Dimensions of Domains – (Courtesy of periodic boundaries (sides of the domain) Morgut & Nobile) rotational periodicity was ensured. As turbulence model, the two equation SST (Shear Stress Transport) model with the automatic treatment of wall functions was employed. Meshing All the meshes used in this study were generated using the commercial meshing tool ANSYS-ICEM CFD 11. For both propellers the Fixed part was discretized only with a unique structured mesh, while Rotating was discretized with both meshing approaches. Moreover in the case of propeller P5168 were used for Rotating two meshing regimes (coarse, fine). The number of nodes of meshes of propeller A and propeller P5168 are visible in Table 9.2 and Table 9.3. Since ANSYS-CFX 11 employs the node-centered finite volume method, (More precisely a Control Volume-based Finite Element Method - CVFEM) the number of nodes was chosen Fixed Nodes Rotating Nodes as a parameter of congruence. Type Hexa Hexa Hybrid For that reason, Grid1 and 223820 784914 Grid 1 Grid2, Grid3 and Grid5, Grid4 785344 Grid 2 223820 and Grid6 have a similar number of nodes, Table 9.2 Grids for Propeller A– (Courtesy of Morgut & Nobile) respectively. To generate structured meshes of both propellers, Fixed and especially Rotating were decomposed in a large number of blocks and proper nodes distributions were used to control dimensions and quality of the cells. The single hybrid meshes were instead Fixed Nodes Rotating Nodes generated with two Hexa Hexa Hybrid Type successive steps. First 229437 348810 Grid 3 Coarse surface meshes and 229437 711932 Grid 4 Fine volume tetrahedral 229437 340400 Grid 5 Coarse meshes were created 229437 741378 using the robust Octree Grid 6 Fine method. Then in order Table 9.3 Grids for Propeller P5168 – (Courtesy of Morgut & Nobile) to resolve the turbulent boundary layer on the solid surfaces, with the similar resolution to the one used with structured meshes, layers of prism were placed around the hub and blade. In the case of propeller A, 6 layers were generated and in the case of propeller P5168, 15 layers were placed. The average values of y+ on solid surfaces (hub, blade) of propeller A and propeller P5168 were 20 and 15 respectively. The y+ was defined as y+ = μT y/ν where μT = (τw/ρ)1/2 is friction velocity, y is normal distance from the wall, ν is kinematic

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viscosity, ρ is density and τw is wall shear stress. In the case of propeller P5168 during the refinement, the height of the first node off the solid surfaces was kept unchanged. For propeller P5168 the structured mesh of Fixed is visible in Figure 9.5 (a). Structured and hybrid meshes on the blade and hub surfaces are depicted in Figure 9.5 (b-c).

(a) Hexa Mesh of part Fixed, Propeller P5168 Figure 9.5

(c) Surface mesh, Hybrid, Propeller P5168

(b) Surface mesh, Hexa Fine, Propeller P5168

Meshing for Propeller P5168– (Courtesy of Morgut & Nobile)

Results To study the influence of the grid on the quality of the prediction of the flow around a marine propellers, numerical data were compared with available experimental data. Propeller A was used as a preliminary study. Comparison was carried out only on global quantities of the flow while for Propeller P5168 comparison was made, analogous as (Rhee and Joshi, 2005), also on the local values of the flow in a downstream location x/R=0.2386 measured from the propeller mid plane, where R is the radius of the propeller and x is the distance. The global values considered were thrust coefficient KT , torque coefficient KQ and efficiency η defined as:

KT =

T ρn2 D4

,

KQ =

Q ρn2 D5

,

η=

J KT 2π K Q

Eq. 9.1 where T[N] is the thrust, Q[Nm] is the torque, η[rps] is the rotational speed of propeller, D[m] is the diameter of the propeller, ρ[kg/m3] is the density of the fluid. J=V/nD is the advance coefficient, where V[m/s] is the velocity of uniform flow. Circumferentially averaged velocity components, and root-mean square values of turbulent velocity fluctuations were selected as local flow values. The root mean square of turbulent velocity fluctuations q was defined as

q = √2k

Eq. 9.2 where k is the turbulent kinetic energy. In the following graphs and contours all local flow values are non-dimensionalized by velocity of the uniform flow V. Relative percentage errors present in the next tables are defined as

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ϵ(K T )% =

K T,Num − K T,Exp K T,Num − K T,Exp × 100 , ϵ(K T )% = × 100 K T,Exp K T,Exp

Eq. 9.3 where KT,exp , KQ,Exp are experimental data and K Q,Num, K T,Num are numerical values. 9.2.5.1 Propeller A In the case of propeller A the simulations were carried out for a wide range of advance ratios. From Table 9.4 and Figure 9.6 it is seen that numerical results of different meshing a approaches, are very close to each other and also in line with the experimental data, especially within the range J = 0.1 - 1.0. Moreover differences between results obtained using different meshes are less than 4%. The relative percentage errors, within the range J = 1.1 - 1.2 but especially at J = 1.2, are very height for both meshing approaches as expected, because thrust and torque are both almost null. Table 9.4

Figure 9.6

Results of Propeller A– (Courtesy of Morgut & Nobile)

KT , KQ and η Curves of Propeller A – (Courtesy of Morgut & Nobile)

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9.2.5.2 Propeller P5168 In the case of propeller P5168 the simulations were carried out at four advance coefficients, following experimental setup (Chesnakas and Jessup, 1998) showed in Table 9.5, where N = 60n is the rotational speed of propeller in rpm. In this case, the simulations were carried out first using coarser grids and then using finer grids. The relative percentage differences of the computed values in KT and KQ on grids of different resolution are showed in Table 9.6. It is noteworthy that differences of KT are higher than differences of KQ especially for the structured meshing approach. The relative percentage differences are defined as:

λ(K T )% = Eq. 9.4

J 0.98 1.10 1.27 1.51 Table 9.6

J 0.98 1.10 1.27 1.51

N(rpm) 1200 1450 1300 1150

V(m/s) 7.89 10.70 11.08 11.73

Table 9.5 Experimental setup of Propeller P5168

K T,Fine − K T,Coarse K T,Fine − K T,Coarse × 100 , λ(K T )% = × 100 K T,Coarse K T,Coarse

λ(KT)% Hexa Hybrid 1.37 1.34 1.64 1,60 1.36 2.25 7.89 7.59

λ(KQ)% Hexa Hybrid 0.55 0.98 0.63 1.23 0.31 1.56 0.91 3.88

ε(KT)% Hexa Hybrid 1.65 3.30 0.65 2.60 6.70 8.61 18.84 23.19

ε(KQ)% Hexa Hybrid 1.35 3.49 2.31 4.88 4.28 6.92 5.73 10.83

Relative Percentage Differences of Computed Values Between Finer and Coarser Mesh for propeller P5168 – (Courtesy of Morgut & Nobile)

The results obtained using finer grids are presented and discussed in the following part. First the comparison of KT and KQ is presented. From Figure 9.7 and Table 9.6 it is visible that computed values of KT and KQ on both meshes are all slightly overestimated but they compare well with experimental data, except for J = 1.51. Results obtained using hexastructured mesh are better than those obtained using hybrid-unstructured mesh but the differences in computed values, within the range J = 0.98 - 1.27, are lower than 3%. See [Morgut and Nobile]416 as they depict respectively for structured and hybrid mesh the

Figure 9.7

KT and KQ curves of Propeller P5168 – (Courtesy of Morgut & Nobile)

Mitja Morgut, Enrico Nobile, “Comparison of Hexa-Structured and Hybrid-Unstructured Meshing Approaches for Numerical Prediction of the Flow Around Marine Propellers”, First International Symposium on Marine Propulsions smp’09, Trondheim, Norway, June 2009. 416

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circumferentially averaged velocity components in axial (Vx), tangential (Vt) and radial (Vr) direction vs non dimensioned radial coordinate (r/R) for various J, where r is the radial distance from the centerline of the hub. From these figures it is visible that the predicted trends of velocity components of both structured and hybrid meshes are very similar and differences are hard to detect. Moreover the axial and tangential velocity components compare well with the experimental data. The radial components, instead, are not so close to the experimental data, but their value are lower and therefore also the experimental uncertainty are larger. It is however noteworthy that within the range r/R = 0.6 -1.0 even though computed values are under predicted they seem to have the same trends as the experimental data. A comparison of contours of the root-mean square values of turbulence velocity fluctuations q on the plane x/R = 0.2386 downstream of the propeller mid plane, for J = 1.1 is presented in [Morgut and Nobile]417. From a qualitative point of view the contours agree well with experimental data, but from a quantitative point of view it is clear that the magnitude of turbulence kinetic energy is under predicted especially on the hybrid-unstructured mesh where is clearly visible the effect of excessive numerical diffusion. It seems therefore that, at least at model scale, the differences, between hexa-structured and hybrid structured meshes do affect the accuracy in the predictions of the turbulence quantities but the effect, for global quantities is modest. It is hard - or even impossible to extrapolate this conclusions to real scale, given the different qualitative and quantitative character of turbulent phenomena. Conclusions In this study a comparison between hexa-structured and hybrid-unstructured meshing approaches for the prediction of the flow around a marine propellers working in uniform flow was carried out. The study was performed on two five-bladed propellers in model scale. Hexa-structured and hybridunstructured meshes used for comparison were generated with the commercial meshing tool ANSYSICEM CFD 11. The simulations were carried out with the commercial RANS solver ANSYS-CFX 11, using the moving frame of reference approach and employing the SST (Shear Stress Transport) two equation turbulence model. Computational results from both meshing approaches were compared against the experimental data. In the case of propeller A the comparison was made only on global values while for propeller P5168 the comparison was carried out also on local values of the flow field. The numerical values of the thrust and torque coefficients computed using structured and hybrid meshes are both in line with the experimental data. The performance curves computed using structured meshes are slightly better than those predicted using hybrid meshes. The differences in computed values, using different meshing approaches and except for the extreme operational conditions, are less than 4% for propeller A and less than 3% for propeller P5168. Also the velocity profiles of propeller P5168, computed using different meshing approaches are in line with the experimental data, especially for axial and tangential components, (See [Morgut and Nobile]418). The overall results suggest, that for the numerical prediction of propulsive performances the use of hybrid meshes might be an adequate choice at least at model scale. They can offer a similar accuracy to the one of structured meshes and moreover they need a less effort to be generated. On the other hand, at the model scale and for the CFD code employed, the hybrid meshes do not seem to be the preferred choice for a detailed investigation of the flow field since they introduce and excessive diffusion in the solution.

Mitja Morgut, Enrico Nobile, “Comparison of Hexa-Structured and Hybrid-Unstructured Meshing Approaches for Numerical Prediction of the Flow Around Marine Propellers”, First International Symposium on Marine Propulsions smp’09, Trondheim, Norway, June 2009. 418 Mitja Morgut, Enrico Nobile, “Comparison of Hexa-Structured and Hybrid-Unstructured Meshing Approaches for Numerical Prediction of the Flow Around Marine Propellers”, 1st International Symposium on Marine Propulsions, Trondheim, Norway, June 2009. 417

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Case Study 3 – Structure & Unstructured Hybrid Meshing and its effect on Quality of Solution on Turbine Blade Applications Automatic robust unstructured hybrid meshing is indispensable for the success in design optimization419. In addition, it is important to maintain the mesh quality for deformation of geometry throughout the optimization process for the reliability of optimal design. Mesh adaptation is useful to capture the flow feature which can highly affect flow properties. Therefore, the present hybridmeshing technique with adaptation is applied for various turbomachinery components to validate its robustness. In addition, a turbine blade is used to compare the effects of mesh for the optimization. Results ➢ Effect of mesh quality for design optimization with large deformation of turbine blade is investigated by using structured mesh, unstructured hybrid mesh without adaptation, and unstructured hybrid mesh with adaptation (Figure 9.8 (a-c)). ➢ Flow around turbine blade is computed by structured and hybrid meshes (Figure 9.8 (de)). Because of different mesh topology and quality, the flow is totally different. In the figures, structured mesh can capture the wake region well compared to hybrid mesh without mesh adaptation.

(a) Structure

(d) Structured Solution Figure 9.8

(b) Hybrid

(c) Hybrid with Adaptation

(e) Hybrid Without Adaptation Solution Flow Around Turbine Blade – (Courtsy of Sasaki et al.)

Daisuke Sasaki, Caleb Dhanasekaran, Bill Dawes, Shahrokh Shahpar, “Efficient Unstructured Hybrid Meshing and its Quality Improvement for Design Optimization of Turbomachinery”, European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006. 419

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Case Study 4 - Evaluation of Structured vs. Unstructured Meshes for Simulating Respiratory Aerosol Dynamics420 In simulating biofluid flow domains, structured hexahedral meshes are often associated with high quality solutions. However, extensive time and effort are required to generate these meshes for complex branching geometries. In this study, conducted by [Samir Vinchurkar & Worth Longest]421, to evaluates potential mesh configurations that may maintain the advantages of the structured hexahedral style while providing significant savings in grid construction time and complexity. Specifically, the objective here is to evaluate the performance of unstructured hexahedral, prismatic and hybrid meshes (prismatic + Tetrahedral) based on grid convergence and local particle deposition fractions in a bifurcating model of the respiratory tract. A grid convergence index (GCI) has been implemented to assess the mesh-independence of solutions in cases where true grid halving is not feasible. Structured hexahedral, unstructured hexahedral and prismatic meshes were found to provide GCI values of approximately 5% and nearly identical velocity fields. In contrast, the hexahedral–tetrahedral hybrid model resulted in GCI values that were significantly higher in comparison to the other meshes. The resulting velocity field for the hybrid configuration differed from the hexahedral and prismatic solutions by up to an order of magnitude at some locations. Considering the deposition of 10 μm particles in the planar configuration, all meshes considered provided relatively close agreement (2–20% difference) with an available experimental study. For all particle sizes considered, local and total deposition results for the structured and unstructured hexahedral meshes were similar. In contrast, the prismatic and hybrid geometries resulted in significantly higher deposition rates when compared to the hexahedral meshes for particles less than 10 μm. As a result, only the unstructured hexahedral mesh was found to provide overall performance similar to the structured hexahedral configuration with the advantage of a significant savings in construction time. These results emphasize the importance of aligning control volume gridlines with the predominant flow direction in bio fluid applications that involve long and thin internal flow domains. Bifurcation Model, Boundary Conditions, and Contributions The geometry selected to evaluate the mesh styles of interest is a double bifurcation model representative of respiratory generations G3–G5 (see Error! Reference source not found.). This m odel is generated from the ‘‘Physiologically Realistic Bifurcation’’ (PRB) geometry specified by [Heistracher & Hofmann]422. For the PRB geometry, [Heistracher and Hofmann]423 provide a complete mathematical description of a single symmetric or asymmetric bifurcation based on a set of 11 geometric parameters and two sigmoid functions. Specific parameters for the double bifurcation model of generations G3–G5 employed in this study are identical to the values used in the work of [Heistracher and Hofmann] and the localized particle deposition measurements of [Oldham et al.]424. The inlet diameter of G3 in the model is 0.56 cm. Further geometric details of this configuration have been reported in [Longest and Vinchurkar]425. In this study, grid convergence, Samir Vinchurkar, P. Worth Longest, “Evaluation of hexahedral, prismatic and hybrid mesh styles for simulating respiratory aerosol dynamics”, Computers & Fluids, 2008. 421 See Previous. 422 Heistracher T, Hofmann W. Physiologically realistic models of bronchial airway bifurcations. J Aerosol Sci 1995;26:497–509. 423 See Previous. 424 Oldham MJ, Phalen RF, Heistracher T. Computational fluid dynamic predictions and experimental results for particle deposition in an airway model. Aerosol Sci Technol 2000;32:61–71. 425 Longest PW, Vinchurkar S. Effects of mesh style and grid convergence on particle deposition in bifurcating airway models with comparisons to experimental data. Med Eng Phys 2007;29:350–66. 420

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velocity fields, and local particle deposition profiles will be evaluated for an in-plane configuration, as implemented in the experimental study of [Oldham et al.]. For comparison, local deposition patterns will also be considered in an out-of-plane model where the second bifurcation has been rotated by an angle of 90 degrees. The steady inspiratory flow rate employed in the PRB model results in an inlet Reynolds number of 1788. For respiratory generations G3–G5, this is consistent with an inhalation flow rate in the trachea of 60 l/min and represents a state of heavy exertion. The flow rate in generation G3 is 125 ml/s, as specified in the experimental study of [Oldham et al]. Both inlet velocity and initial particle profiles are expected to have a significant impact on the flow field and particle deposition locations. For comparisons to in vitro deposition data, these profiles may be largely influenced by upstream effects in the experimental particle generation system. [Longest and Vinchurkar] have shown that upstream transition to turbulence results in a relatively blunt initial velocity field and particle profile at the model inlet. However, the flow within the PRB model can be approximated as laminar. As such, fully-developed blunt turbulent profiles of velocity and initial particle distributions have been assumed at the model inlet. Within the model, laminar flow is assumed. Outlet flow is assumed to be evenly divided between the left and right symmetric branches, i.e., homogeneous ventilation. Gravity has been included in the flow field and particle trajectory calculations of the PRB model with the gravity vector oriented in the negative z-direction, i.e., normal to the plane of the bifurcation, to remain consistent with the experiments of [Oldham et al]. Mesh Types For comparisons to the structured hexahedral base case, three unstructured mesh styles have been considered. Unstructured meshes are defined as having at least one face or block surface on which the gridlines do not remain continuous. For the structured hexahedral mesh, four-sided blocks are used (Figure 9.9 a), which allows the gridlines to remain continuous within each block. For the unstructured hexahedral mesh, blocks with one triangular face have been implemented. On the triangular cross-sectional faces, the gridlines become discontinuous at the center of each triangle (Figure 9.10 b, Slice 1). As a result, this blocking structure produces an unstructured hexahedral mesh. The other unstructured meshes considered include prismatic and hybrid styles (Figure 9.10). All meshes were created using the integrated solid modeling and meshing program Gambit 2.2.

Figure 9.9

Geometric Blocking Used (a) Structured Hexahedral (178 Blocks) and (b) Unstructured Hexahedral (80 Blocks) – (Courtesy of Samir Vinchurkar & Worth Longest)

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9.4.2.1 Structured The structured base case mesh consists of six-sided hexahedral elements arranged in a system of interconnected rectangular blocks. The blocks have been arranged in a butterfly blocking design which minimizes control volume distortions while aligning a higher percentage of elements with the local flow direction (Figure 9.9 a). Moreover, mesh density is increased near the wall and near the bifurcation points. This multi-block structure is difficult to develop because gridlines may be distorted, but must remain continuous throughout the geometry. Designing a high quality blockstructured meshing configuration for a geometry with multiple branches in which hexahedral elements largely align with streamlines is a user intensive non-trivial task. 9.4.2.2 Unstructured As with the structured mesh, the unstructured hexahedral configuration requires the creation of sub-blocks within the geometry. However, the unstructured hexahedral design allow for two faces on each block to have a non-continuous grid (Figure 9.10 b). Furthermore, blocks with one pair of triangular faces may be accommodated. As a result, the planes forming these blocks may pass entirely

Figure 9.10 Four Meshing Styles of the PRB Model (a) Structured Hexahedral, (b) Unstructured Hexahedral, (c) Prismatic, and (d) Hybrid – (Courtesy of Samir Vinchurkar & Worth Longest)

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through the geometry (Figure 9.9 b). These planes are much easier to construct than the planes in the structured hexahedral configuration that only partially bisect the geometry. In addition, the blocking structure for the unstructured hexahedral mesh reduces the number of required blocks by over 50% (Figure 9.9). Once the geometry is divided into the required blocks, non-continuous meshes are created on cross-sectional surfaces. These meshed faces are then swept through the geometry in the axial direction to generate the volumetric mesh. As a result, this mesh style retains the advantage of aligning mesh elements in the predominate direction of flow. The prismatic mesh consists of five-sided elements which are composed of two triangles joined together by a longitudinal section of three rectangular faces. Generation of this mesh style requires four-sided faces to be constructed on the surface of the PRB (Figure 9.10 c). The prismatic elements are arranged such that their triangular faces fill the axial slices (Figure 9.10 c). This allows for the rectangular sections of each prismatic element to be aligned with the direction of predominate flow. In order to improve the accuracy of the tetrahedral mesh style, an unstructured hexahedral– tetrahedral hybrid mesh has been created (Figure 9.10 d). As with the prismatic mesh, four-sided faces are required on the surface of the PRB geometry. These faces are used to construct structured quadrilateral surface meshes, which form the basis for a layer of near-wall hexahedral cells. The hexahedral elements are intended to better resolve the flow field near the walls where velocity gradients are typically highest. The inner core of the flow field is then meshed with randomly oriented tetrahedral elements. A layer of prismatic elements is used to join the hexahedral and tetrahedral cells. In this configuration, the thin near-wall layer of hexahedral elements is aligned with the predominate direction of flow. However, it is not possible for the randomly oriented tetrahedral elements, which comprise a majority of the flow field, to be aligned with the axial flow direction426. Governing Equations Flow conditions in the meshes considered are assumed to be isothermal, incompressible, laminar and steady. Furthermore, the particle concentrations are assumed to be sufficiently dilute such that momentum coupling effects of the dispersed phase on the fluid can be neglected, i.e., a one way coupled flow. The governing equations for the respiratory airflow of interest include the conservation of mass and momentum as:

∇. 𝐮 = 0

,

∂𝐮 1 + (𝐮. ∇)𝐮 = (−∇p + ∇. 𝛕) ∂t ρ

Eq. 9.5 Where u is the velocity vector, p is the pressure, ρ is the fluid density, τ the shear stress tensor is given by

𝛕 = μ[∇𝐮 + (∇𝐮)T ]

Eq. 9.6 and μ is the absolute viscosity. Hydrodynamic inlet and boundary conditions, in addition to the noslip wall condition, were selected to match the experimental conditions of interest. To approximate a uniform outflow distribution, equally divided mass flow was specified. Furthermore, flow field outlets were extended far downstream such that the velocity was normal to the outlet plane, i.e., fully developed flow profiles with no significant radial velocity component. One-way coupled trajectories of monodisperse 1–10 μm aerosols have been calculated on a Lagrangian basis by integration of an appropriate version of the particle trajectory equation for comparison to the experimental results of The hybrid style consists of tetrahedral elements throughout the interior surrounded by three layers of hexahedral control volumes on the surface. The internal block divisions have been shown in the cross-sectional slices of the structured and unstructured hexahedral meshes. 426

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[Oldham et al.]. Characteristics of the 1–10 μm aerosols of interest within this model include a particle density ρp = 1.06 g/cm3, a density ratio α = ρ/ρp ≈ 10-3, a Stokes number St =ρpd2p CCU/18μD ranging from 0.003 to 0.26, and a particle Reynolds number Rep = ρ|u -v| dp/μ ≤ 10. The appropriate equations for spherical particle motion under the conditions of interest are expressed as

dvi Dui f (u − vi ) + g i (1 − α) + fi,lubcrication =α + dt Dt τp i

and

dxi = vi (t) dt

Eq. 9.7 In the above equations, vi and ui are the components of the particle and local fluid velocity, respectively. The ratio of fluid to particle density is represented as α = ρ/ρp, and gi denotes gravity. The characteristic time required for particles to respond to changes in the flow field, or the momentum response time, is τp = CCρpd2p/18μ, where CC is the Cunningham correction factor. The pressure gradient or acceleration term is often neglected for aerosols due to small values of the density ratio. However, it has been retrained here to emphasize the significance of fluid element acceleration in biofluid flows. The drag factor f, which represents the ratio of the drag coefficient to Stokes drag, is based on the expression of [Morsi and Alexander]427 :

f=

CD Rep Rep a2 a3 = + 2) (a1 + 24 24 Rep Rep

Eq. 9.8 where the ai coefficients are constant for smooth spherical particles over the range of Reynolds number considered, i.e. 0 ≤ Rep ≤ 10. The effect of the lubrication force, or near-wall drag modifications, are shown in Eq. (3a) but are expected to be reduced for the aerosol system of interest in comparison to liquid flows due to near-wall non-continuum effects. As such, this term has been neglected for the simulations considered here. Due to the significant size of the particles considered and the dilute concentrations, Brownian motion and particle-to-particle collision effects have been neglected. The Cunningham correction factor has only been applied for 1 lm aerosols based on the expression of [Allen and Raabe]428. Inlet particle profiles have been specified to be consistent with the local mass flow rate associated with the blunt velocity profile considered. That is, the mass flow rate of particles on a finite ring, m p,ring, at the inlet is given by r2

ṁp,ring ~ṁring = ∫ ρu(r)2πdr r1

Eq. 9.9 where r1 and r2 define the extent of the ring and u(r) is the inlet velocity profile. Initial particle velocities were assumed to match the local fluid velocities. Further details describing the specification of initial particle profiles are discussed in [Longest and Vinchurkar]429. Numeric Method To solve the governing mass and momentum conservation equations in each of the geometries and for each mesh style, the CFD package Fluent 6.2 has been employed. User-supplied FORTRAN and C Morsi SA, Alexander AJ. An investigation of particle trajectories in two-phase flow systems. J Fluid Mech 1972;55(2):193–208. 428 Allen MD, Raabe OG. Slip correction measurements of spherical solid aerosol particles in an improved Millikan apparatus. Aerosol Sci Technol 1985;4:269–86. 429 Longest PW, Vinchurkar S. Effects of mesh style and grid convergence on particle deposition in bifurcating airway models with comparisons to experimental data. Med Eng. Phys 2007;29:350–66. 427

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programs have been employed for the calculation of initial particle profiles, particle deposition locations, grid convergence, and post-processing. All transport equations were discretized to be at least second order accurate in space. For the convective terms, a second order upwind scheme was used to interpolate values from cell centers to nodes. The diffusion terms were discretized using central differences. To improve the computation of gradients for the tetrahedral elements of the hybrid mesh, face values were computed as weighted averages of values at nodes, which provides an improvement to using cell-centered values for these meshes. Nodal values for the computation of gradients were constructed from the weighted average of the surrounding cells, following the approach proposed by [Rauch et al.]430. A segregated implicit solver was employed to evaluate the resulting linear system of equations. This solver uses the Gauss–Seidel method in conjunction with an algebraic multigrid approach to solve the linearized equations. The SIMPLEC algorithm was employed to evaluate pressure–velocity coupling. The outer iteration procedure was stopped when the global mass residual had been reduced from its original value by five orders of magnitude and when the residual-reduction-rates for both mass and momentum were sufficiently small. To ensure that a converged solution had been reached, residual and reduction-rate factors were decreased by an additional order of magnitude and the results were compared. The stricter convergence criteria produced a negligible effect on both velocity and particle deposition fields. To improve accuracy, CGS units were employed, and all calculations were performed in double precision. To further improve resolution in the particle deposition studies, geometries were scaled by a factor of 10 and the appropriate non-dimensional parameters were matched. To determine grid convergence and establish grid independence of the velocity field solutions, successive refinements of each mesh style have been considered. For each refinement, grid convergence is evaluated using a relative error measure of velocity magnitude between the coarse and fine solutions:

εi = |

ui,coarse − ui,fine | ui,fine

Eq. 9.10 A vector of relative error values was determined for 1000 consistent points located in the region of the bifurcation. The root-mean-square of the relative error vector was used to provide an initial scalar measure of grid convergence for the points considered 1/2

εrms

2 ∑1000 i=1 εi =( ) 1000

Eq. 9.11 Rigorously, grid convergence measures should be based on refining the grid by a factor of two, i.e., grid halving. However, dividing hexahedral elements by a factor of two in three dimensions is often not practical due to the significant increase in the number of control volumes. As such, relative error values must be adjusted to account for cases in which grid reduction factors less than r = 2 are employed. To extrapolate εrms values to conditions consistent with true grid halving, the Grid Convergence Index (GCI) has been suggested by [Roache]431. This method is based on Richardson extrapolation and can be applied as

GCI = Fs Eq. 9.12

εrms rp − 1

Rauch RD, Batira JT, Yang NTY. Spatial adaption procedures on unstructured meshes for accurate unsteady aerodynamic flow computations. Technical Report AIAA-91-1106, 1991. 431 Roache P. Computational fluid dynamics. Albuquerque: Hermosa; 1992. 430

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In the above equation, r represents the grid refinement factor and p is the order of the discretization method. Based on second-order discretization of all terms in space, p = 2 for the systems of interest. Refinement of the meshes was performed to maintain a constant reduction value in the three coordinate directions. The associated r value has been calculated as the ratio of control volumes in the fine and course meshes

Nfine 1/3 r=( ) Ncoarse

Eq. 9.13 To limit errors arising from the extrapolation procedure, r values of approximately 1.5 or greater have been considered. A factor of safety FS equal to 3 has been selected to provide a GCI value equal to the εrms value when r = 2 and p = 2. Therefore, the GCI value represents a scaled version of εrms to account for mesh refinement factors less than 2. Particle trajectories were calculated within the steady flow fields of interest as a post-processing step. The integration scheme employed to solve Eq. 9.7 was based on the trapezoid rule with a minimum of 10 integration steps in each control volume. Doubling the number of integration steps within each control volume had a negligible (less than 1%) effect on cumulative particle deposition values. Due to relatively small particle response times, double precision calculations have been employed. It was found that approximately 20,000 particle trajectories were required to produce convergent cumulative deposition values based on a 1% relative error criterion. As such, 20,000 particles have been initialized in all deposition cases considered. Results 9.4.5.1 Validation Studies Validations of velocity field values for the structured hexahedral mesh scheme applied to a bifurcation geometry have been reported in a previous study. Briefly, a single bifurcation model was considered with a characteristic Reynolds number of 518 and results were compared to the empirical velocity field data of [Zhao and Lieber]432. For steady inhalation flow, the velocity field results of [Longest and Vinchurkar] indicate good quantitative agreement with the empirical data of [Zhao and Lieber]. 9.4.5.2 Grid Convergence To evaluate grid convergence for each mesh style considered, low, mid and high-resolution comparisons between coarse and fine grids have been considered for the planar geometry. Results of this comparison in the form of grid convergence values and required simulation times are reported in Table 9.7 (a-d) and are discussed below. The reported grid convergence results are for the planar bifurcation model (Error! Reference source not found. a). Similar grid convergence results were o bserved for the out-of-plane configuration (Error! Reference source not found. b). The number of grid cells required is based on the presence of one symmetry plane, i.e., one-half of the geometry is meshed. As described, grid convergence has been based on comparisons between coarse and fine grid solutions at 1000 points concentrated in the region of the bifurcation. A layer of near-wall comparison points was positioned to be less than 5% of the internal radius away from the wall. Selections of other sets of 1000 points as well as doubling the number of points considered had a negligible (i.e., less than 1%) impact on the grid convergence values reported. For the structured hexahedral mesh, successive grid refinements resulted in an effective reduction of εrms values (Table 9.7a). For the high resolution case, an εrms of 1.99% was obtained. In comparison to other relative error estimates, this value is relatively high. However, the selection of 1000 points with many locations near the wall and in low velocity positions produces a very rigorous 432

Zhao Y, Lieber BB. Steady inspiratory flow in a model symmetrical bifurcation. J Biomech Eng 1994.

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condition for testing grid convergence. Moreover, errors on the order of 1% are expected to arise from the linear interpolation algorithm used to calculate values at the positions of interest for comparisons of the coarse and fine grid solutions. Therefore, achieving εrms values below 1% may not be possible with the rigorous grid convergence method employed. In this study, values of εrms on the order of approximately 1% are considered to represent a well converged solution. Accounting for the grid reduction factor used in the high resolution case results in a GCI value of 4.27% for the structured hexahedral mesh with 214 K control volumes. Grid convergence estimates for the unstructured hexahedral mesh are reported in Table 9.7 b. These results are highly similar to the grid convergence values observed for the base case. That is, an εrms value of 1.95% is achieved for the high resolution case. However, the number of grid cells required to achieve this level of grid convergence was increased from 214K for the structured hexahedral mesh to 318 K for the unstructured hexahedral mesh. This increase in cell number resulted in a 10% increase in solution time. Grid convergence index values on the order of 10%, as observed for the medium-level resolution, are shown to result in visible differences between velocity profiles. For the high resolution case, which is characterized by a GCI of 4.32%, differences in the velocity profiles are much less discernable. For the prismatic mesh configuration, εrms and GCI values are similar to those observed for the hexahedral style meshes (Table 9.7 c). However, to achieve this level of grid convergence, grid resolution was increased by approximately 30–40% for each case considered. This increase in

(a) Structure (Hexa)

(b) Un-Structured (Hexa)

(c) Un-Structured (Prism)

(d) Un-Structured (Hybrid)

Table 9.7

Grid Convergence – (Courtesy of Samir Vinchurkar & Worth Longest)

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grid density produces an associated increase in simulation time of approximately 20%. Furthermore, it is observed that the medium-level resolution case of the unstructured prism mesh results in a GCI value of approximately 6.6%, which is consistent with the high resolution prismatic case and significantly lower than with the medium resolution hexahedral meshes. Grid convergence values for the hybrid meshes are significantly higher than values reported for the their configurations (Table 9.7 d). The minimum GCI value for the hybrid style was 17.7% and occurred medium-level grid density. For the high-level resolution condition, the GCI value increased to 21.3%. Further increases in grid density resulted in a higher GCI value. This increase may be a result of round-off errors arising from an over-resolved grid. Furthermore, this level of grid convergence is consistent with GCI values observed for purely tetrahedral meshes with and without flow adaption. As a result, the hybrid mesh style results in GCI values that are significantly higher than observed for the other meshes considered in this study and appears to provide little advantage to purely tetrahedral style meshes. The higher GCI values of the hybrid configuration may largely be a result of mesh elements not aligning with the direction of predominate flow. 9.4.5.3 Velocity Fields Velocity vectors, contours of velocity magnitude and streamlines of secondary motion are resented in Figure 9.11 for the high resolution cases of the four mesh styles considered in the planar bifurcation model. Midplane velocity fields appear highly similar among the hexahedral and prismatic meshes (Figure 9.11 c). However the hybrid mesh results in a significant reduction in midplane velocity gradients, which may arise from artificial or numerical dissipation (Figure 9.11 d). Similarly, secondary motions viewed at cross-sectional slice locations appear similar among the first three mesh styles considered (Figure 9.11 c). A single vortex is observed for the upper half of the geometry at Slice 1. The second carinal ridge produces a pair of counter rotating vortices for the inner branch of G5, as observed in Slice 2 (Figure 9.11c). However, due to the highly dissipative conditions of the hybrid mesh, only one fully formed vortex is observed in each of the three crosssectional planes considered (Figure 9.11 d). In summary, midplane velocity vectors appear relatively consistent among the four meshes considered, with some variations observed for the hybrid configuration. Secondary velocity profiles appear similar between the two hexahedral mesh styles. However, secondary velocity profiles are significantly different for the non-hexahedral meshes with the largest variations occurring for the hybrid configuration. In order to better evaluate differences among the solutions of the meshes considered, midplane velocity profiles have been plotted at Slices 1–3 for high resolution conditions in the planar model. At each location, velocity profiles for the hexahedral and prismatic meshes are similar. However, minor differences among the first three solutions are discernable. This observation highlights the fact that a high level of grid convergence does not ensure an exact match among solutions of different mesh types. In contrast to the hexahedral and prismatic solutions, the hybrid configuration results in significantly different velocity profiles. Velocity values for the hybrid solution again appear to be influenced by a high degree of dissipation. Considering Slice 3, differences between the first three solutions and the hybrid configuration vary between approximately 30% to one order of magnitude. 9.4.5.4 Particle Deposition Deposition locations for the four mesh styles considered and the planar geometry with 10 μm particles are shown in Figure 9.12. The 10 μm aerosols deposit primarily by impaction. Qualitatively, the observed deposition locations are very similar between the structured and unstructured hexahedral meshes. Furthermore, the hexahedral mesh styles exhibit very distinct divisions between regions of deposition and areas devoid of particle–wall interactions. In contrast, particle deposition locations for the prismatic and hybrid meshes appear more diffuse. This effect

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may be the result of fewer mesh elements aligned with the flow, especially for the hybrid configuration. Nevertheless, each of the mesh styles considered emphasizes local accumulations of particles, referred to as hotspots, occurring just upstream of the bifurcation points and continuing downstream for approximately one-half the branch lengths. For 10 μm particles, the structured hexahedral, unstructured hexahedral and prismatic high resolution meshes all match the experimental data of [Oldham et al]. For these three solutions, variations from the cumulative particle deposition experimental data are within 2–3%. Furthermore, these solutions result in a final deposition fraction that is within approximately 1% of the experimentally reported value of 81%. Differences in cumulative deposition values among the solutions for the hexahedral and prismatic meshes vary by less than 1%. In contrast, cumulative

Figure 9.11 Velocity Vectors (a) Structured Hexahedral Mesh with 214 K C.V. (b) Unstructured Hexahedral Mesh with 318 K, C. V. (c) Prismatic Mesh with 510K C. V, (d) Hybrid Mesh with 608 K C. V. – (Courtesy of Samir Vinchurkar & Worth Longest)

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deposition results for the high-resolution hybrid mesh and 10 μm particles are significantly lower than the experimental data. The hybrid mesh considered is observed to under-predict cumulative deposition by approximately 20%. (see [Samir Vinchurkar & Worth Longest]433). As particle size decreases, larger differences are observed among the cumulative deposition predictions for the mesh styles considered. For 5 μm aerosols, the structured and unstructured hexahedral meshes are in close agreement with a final deposition fraction between 5% and 6%. In contrast, the prismatic mesh predicts a cumulative deposition of 11%, which is approximately double the hexahedral mesh estimates. Results for the hybrid mesh and 5 μm particles are even higher, with a total deposition fraction of 12%. Considering 3 μm particles, close agreement is observed between the hexahedral mesh predictions with a total deposition fraction of 0.3%. In contrast, the prismatic and hybrid configurations predict a deposition rate of approximately 1.8%. A similar trend is observed for 1 μm aerosols. Again, results for the structured and unstructured hexahedral configurations are in close agreement with a total deposition fraction ranging between 0.12% and 0.17%. However, predictions of the prismatic and hybrid meshes are significantly higher by a factor

Figure 9.12 Deposition Locations for 10 lm Particles in the Planar Geometry for the (a) Structured Hexahedral Mesh, (b) Unstructured Hexahedral Mesh, (c) Prismatic Mesh, and (d) Hybrid Mesh – (Courtesy of Samir Vinchurkar & Worth Longest)

Samir Vinchurkar, P. Worth Longest, “Evaluation of hexahedral, prismatic and hybrid mesh styles for simulating respiratory aerosol dynamics”, Computers & Fluids, 2008. 433

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of approximately five. In general, cumulative deposition results are consistent between the structured and unstructured hexahedral meshes for the planar geometry. Results for the prismatic and hybrid meshes differ from the hexahedral results by values ranging from 20% (10 μm) to a factor of five (1 μm). Deposition predictions of the prismatic and hybrid meshes are also generally higher than for the hexahedral models. Differences in deposition results between the hexahedral and prism/hybrid meshes appears to increase with decreasing particle size. For all particle sizes considered in respiratory generations G3–G5, impaction is the primary deposition mechanism. However, the smaller particles considered have less inertia and are influenced to a greater extent by the secondary velocity patterns. Significant differences in secondary velocity profiles were observed between the hexahedral and other mesh styles considered in Figure 9.11. Therefore, it is concluded that differences in secondary motion patterns associated with mesh style are partially responsible for increased differences in deposition patterns as particle size is reduced. Furthermore, increase in secondary motion associated with out of plane bifurcations may induce additional discrepancies among the models considered. Cumulative deposition results for the out-ofplane geometry and particle sizes of 3 and 10 μm are shown in434. As with the planar geometry for 10 μm particles, close agreement is observed between the hexahedral and prismatic mesh configurations with a total deposition rate of approximately 90%. The hybrid mesh results in an 85% deposition value, which is in relatively close agreement with the other mesh styles considered. However, significant differences in model predictions are again observed as the particle size is decreased. For 3 μm aerosols, results for the structured and unstructured hexahedral meshes appear to be in close agreement with a total deposition rate of approximately 1.8%. Deposition results for the prismatic and hybrid meshes are approximately six times higher than the other model predictions with a total deposition fraction of 11%. Discussion In this study, the effects of mesh style have been evaluated with respect to grid convergence, velocity fields and particle deposition values in a double bifurcation model of the respiratory tract. Mesh styles considered include structured hexahedral, unstructured hexahedral, prismatic and hybrid configurations. Particles ranging from 1 to 10 μm have been evaluated in planar and out-of-plane geometries. Deposition results for 10 μm particles in the planar geometry were found to be in close agreement with the experimental deposition data of [Oldham et al.] on a highly localized basis. In general, grid convergence, velocity fields, and local particle deposition values were consistent between the structured and unstructured hexahedral meshes. Both hexahedral meshes considered resulted in GCI values of approximately 5% and nearly identical midplane and secondary velocity patterns. Furthermore, local particle deposition profiles were largely similar for the hexahedral meshes across the range of particle sizes evaluated. Considering the prismatic mesh, GCI values were comparable to the hexahedral configuration with only a moderate increase in control volume number. Prismatic velocity fields were consistent with the hexahedral results, with some minor variations in the secondary velocity profiles. However, the prismatic mesh resulted in significant differences in local deposition profiles for particles less than 10 μm. The hybrid mesh resulted in a GCI value that was significantly higher than observed for the other meshes. This increase in GCI occurred despite a significant increase in the number of cells in the hybrid mesh. The velocity field for the hybrid configuration differed from the hexahedral and prismatic solutions by up to an order of magnitude at some locations with significant differences in the secondary vortex patterns. Moreover, deposition results for the hybrid mesh differed from the hexahedral results by values ranging from 20% (10 μm) Samir Vinchurkar, P. Worth Longest, “Evaluation of hexahedral, prismatic and hybrid mesh styles for simulating respiratory aerosol dynamics”, Computers & Fluids, 2008. 434

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to a factor of five (1μm). For the out-of-plane bifurcating geometry, local deposition results were generally consistent for 10 μm aerosols, but differed significantly for 3 μm particles among the mesh styles considered. This study highlights the effects of mesh style on grid convergence and related solution variables for an internal biofluid flow field. For any CFD problem, the required quality of the solution is often weighted against the time and resources available for mesh development. Structured hexahedral meshes are often thought to provide the highest quality solution, but the associated mesh construction time may be prohibitively expensive. In this study, structured hexahedral and unstructured hexahedral mesh schemes have been shown to provide highly comparable grid convergence values, velocity fields and particle deposition profiles. Moreover, both of these mesh styles predicted deposition results in very close agreement with experimental data for 10 μm aerosols in a planar geometry. As illustrated in Figure 9.10 construction of the unstructured hexahedral mesh requires a less complex blocking schemes than for the structured hexahedral configuration. For example, construction of the structured hexahedral mesh requires the creation of 178 blocks in comparison to 80 blocks for the unstructured hexahedral mesh. Therefore, the unstructured hexahedral mesh offers a significant savings in construction time without an appreciable loss in solution performance. Compared with the purely tetrahedral meshes considered in [Longest and Vinchurkar], the hybrid mesh employed in this study showed no improvement in performance. Construction of the hybrid mesh did require subdivision of the PRB surface geometry into rectangular faces. In contrast, construction of purely tetrahedral meshes does no require subdividing the surface into rectangular faces. As a result, purely tetrahedral and flow adaptive tetrahedral meshes may be advantageous in comparison to the hybrid mesh considered in this study. Furthermore, the use of tetrahedral meshes may be preferred when rapid approximate solutions are the top priority. This scenario may arise for patient-specific modeling in the clinical setting. That is, approximate solutions with rapidly generated tetrahedral meshes may be necessary in order to make true patient-specific modeling a reality in the clinical setting . 9.4.6.1 Advantages of Hexahedral Structured Mesh In this study, hexahedral and prismatic meshes were found to provide adequate grid convergence and similar velocity fields. For particle deposition, hexahedral mesh configurations appear to provide the best solution. The observed better performance of the hexahedral and prismatic meshes in comparison to the hybrid mesh may occur for two reasons: First, both hexahedral and prismatic meshes can be aligned with the predominate direction of flow. This alignment is reported to reduce numerical diffusion errors. Furthermore, discretization errors partially cancel on opposite hexahedral faces. In contrast, mainly tetrahedral meshes cannot be aligned with the direction of predominate flow, thereby increasing the potential for numerical diffusion. Therefore, numerical diffusion errors associated with randomly oriented tetrahedral faces are one likely cause of the higher grid convergence values observed for these meshes. The occurrence of these errors is enhanced in the unidirectional flow system considered. The second possible factor responsible for the improved performance of the hexahedral solutions is the use of higher order elements. The hexahedral elements implemented provide more nodes per face for improved predictions of flux values and particle tracking. Some commercial CFD packages provide an increased number of nodes per face to account for this problem. However, the effect of increasing the number of nodes per face has not been quantified for internal biofluid flows. Furthermore, the effects of nodes per face on solution performance is expected to be a secondary factor in comparison to aligning the grid with the predominate direction of flow in the long and thin conduits of interest. Limitations of the current study include calculation of the GCI parameter at linearly interpolated points, the evaluation of a single software package, and the construction of only one style of hybrid

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mesh. The grid convergence parameter was evaluated at 1000 representative points throughout the flow field. These points include near-wall locations where minor variations in flow field velocities can result in very large relative errors. Modifying the number and location of these randomly selected points did not appreciably change the GCI value provided at least 1000 points were included. However, interpolation errors are present in determining values at comparison points. These errors are estimated to be on the order of approximately 1%. Nevertheless, the grid convergence algorithm employed provided an effective strategy for evaluating relative performance among the mesh styles considered that includes low velocity and near-wall regions. In this study, only one commercial software package was evaluated. Other software may improve the solution quality of the hybrid configuration. Moreover, many other hybrid mesh styles are possible. Nevertheless, evaluation of a representative state-of-the-art commercial software provides a valuable basis of comparison for various styles of meshes. Furthermore, this study highlights the advantages of aligning mesh elements with the predominate direction of flow, which is independent of the computational package considered. Conclusion In deduction, structured and unstructured hexahedral meshes have been shown to provide acceptable grid convergence values, comparable velocity fields and good agreement with experimental 10 μm particle deposition data in a branching respiratory geometry. Generation of the unstructured hexahedral mesh provided a significant time savings in pre-processing with an associated minimal increase in computational run time. In contrast, a hybrid mesh configuration of tetrahedral cells surrounded by multiple layers of near-wall hexahedral elements resulted in significantly higher grid convergence values and different velocity and particle deposition results. These findings emphasize the importance of aligning control volume gridlines with the predominate direction of flow and using higher order elements in biofluid applications with long and thin conduits. Future work is needed to better assess modified flux interpolation schemes, other hybrid configurations and the use of polyhedral elements. For further discussion, please refer to [Samir Vinchurkar, P. Worth Longest]435.

Case Study 5 - Comparison Between Structured Hexahedral and Hybrid Tetrahedral Meshes Generated by Commercial Software for CFD Hydraulic Turbine Analysis436 Reducing meshing time and improving hybrid tetrahedral meshes are desired goals in hydraulic turbine analysis. This paper compares two different meshing methodologies in an industrial application of Francis hydraulic turbines. The first meshing methodology is a structured hexahedral mesh designed by ANSYS ICEM CFD 13.0©437 and the second is a hybrid tetrahedral mesh developed by Pointwise 17.0 R1©438. This software promises to reduce meshing time in comparison to hexahedral structured mesh as well as greatly improve hybrid tetrahedral meshing. Two different spiral case geometries of the same hydraulic turbine are used for the comparison. The features of these geometries are explained, especially in terms of understanding the challenges posed by their meshing. The meshing methodologies and the advantages and disadvantages in the application of these meshes to these geometries are then explained in detail. The numerical method Samir Vinchurkar, P. Worth Longest, “Evaluation of hexahedral, prismatic and hybrid mesh styles for simulating respiratory aerosol dynamics”, Computers & Fluids, 2008. 436 Philippe Martineau Rousseau, Azzeddine Soulaïmani and Michel Sabourin, “Comparison between structured hexahedral and hybrid tetrahedral meshes generated by commercial software for CFD hydraulic turbine analysis”, Conference Paper, May 2013. 437 ANSYS ICEM CFD 13.0. Available from: http://www.ansys.com/Products/Other+Products/ANSYS+ICEM. 438 Pointwise 17.0 R1. Available from: http://www.pointwise.com/. 435

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for the flow calculation and the boundary conditions used to obtain the results are shown. The total head loss and the meridian velocity at the symmetrical plane are used to show similarities between the two meshing methodologies. An investigation of the small differences between the results is made, utilizing the velocity and total pressure contours. These analyses indicate that these two meshing methodologies achieve equivalent results for both spiral case geometries. Problem Description Structured hexahedral meshes are usually used to produce more accurate predictions of head losses for internal flows or of the drag force for external flows. The quality of their elements and their improved control over their distribution in the computational domain are the main factors that make them more appealing than tetrahedral meshes. However, designing a hexahedral mesh with a good level of quality for complex geometry requires a considerable time investment, very often from a few days to a few weeks. In addition, it is sometimes impossible to obtain a mesh of acceptable quality for very complex geometries. Unstructured tetrahedral meshes are preferred for their ability to quickly mesh complex geometries. However, the addition of structured elements in boundary layers is necessary to improve energy loss prediction. As a result, hybrid tetrahedral meshes are widely used in aerospace and hydraulic turbomachines. Please be advised that the hybrid tetrahedral mesh, often has problems in the transition between the boundary layer structured elements and the neighboring unstructured ones. The too-rapid increase of volume elements, as illustrated in Figure 9.13 (a), is often the source of inaccuracies and convergence problems. Other problems inherent to tetrahedral elements are the inappropriate spatial discretization of trailing and leading edges, which are also the cause of inaccuracy in the determination of the energy loss. Figure 9.13 (b), displays the correct transition between prismatic and volume elements in a boundary area mesh.

(a) Problematic Transition

Figure 9.13

(b) Correct Transition

Boundary Layer Transition Between Prismatic and Volume Elements – (Courtesy of Rousseau et al.)

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Geometry A turbine spiral case is the component before the runner of a Francis hydraulic turbine. More specifically, it begins after the penstock and ends at the runner’s entrance, and it also includes the stay vanes and the wicket gates. Figure 9.14 shows a half domain used for the calculation of a spiral case. The primary function of the spiral casing is to rotate the flow and distribute it equally to the runner [4]. The stay vanes and mostly the wicket gates induce a direction to the flow at the entrance of the runner. Ideally these functions are carried out with minimum head loss and Figure 9.14 Example of a hydraulic turbine evenly within each stay vane and wicket gate Figure 9.15 spiral Geometry of thedomain) Stay Vanes and case (half channel. For example, the effect of Wicket Gates, Left: Geometry A, Right: – (Courtesy of RousseauGeometry et al.) B – recirculation zones at the stay vanes could (Courtesy of Rousseau et al.) affect the distribution and direction of the flow at the runner. The behavior of the runner as well as of the draft tube could thus be affected. In this paper, we consider two spiral case geometries of the same hydraulic turbine and these are scaled at small model dimensions. Each spiral case has 24 stay vanes and wicket gates. Five different models of stay vanes are utilized in each spiral case. These models are characterized by their different cord lengths, which decrease from the entrance to the end of the spiral case. The wicket gates are identical within the same spiral case. The difference between the two spiral cases lies with the stay vanes and the wicket gates. The stay vane leading edge incidence angle of the geometry B is better aligned with the flow and rounded. The trailing edges of the stay vanes and wicket gates are tapered. Figure 9.15 shows the differences between the two geometries with the same model of stay vanes and wicket gates. These changes improve the flow in the geometry B by reducing the separation on the upper surface of the stay vanes. This greatly eliminates the recirculation zones. Replacement of the chamfer by a rounded leading edge on the stay vanes and refinement of the trailing edge of the wicket gates also decrease wakes in the flow. The flow is generally more uniform in the geometry B. The total head loss through the spiral case is also greatly reduced. The complexity of the flow and the different models of stay vane in the spiral case prevent any periodic simplification of the computational domain. However, it is simplified symmetrically in the horizontal plane. The absence of the runner after the exit of the spiral case allows this simplification. The computation domain of the geometry A is shown in Figure 9.14. The exit of the spiral case is far from the trailing edge of the wicket gates to reduce the effect of the outflow condition on the flow. Mesh Description Some statistics of the meshes used in CFD are shown in Table 9.8. Three different mesh densities are used for the geometry A; coarse (C), medium (M) and fine (F). The geometry B used two meshes; coarse (C) and medium (M). The increase in the density of elements on the surface profiles and walls of the spiral case is the main difference between each mesh. The thickness of the first element on the wall is the same for all meshes. It is selected to achieve a Y+ of less than 5, and averages close to 2. This value is a compromise with respect to the size of the mesh. In contrast to external flows, the internal flow contains several solid walls. Obtaining a Y+ close to one would be ideal, but it was impossible to achieve with our available computing resources.

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Geometry A Table 9.8

Geometry B

Mesh Densities for Structured Hexahedral and Hybrid Un-Structural Tetrahedral – (Courtesy of Rousseau et al.)

9.5.3.1 Structured Hexahedral Meshes Structured hexahedral meshes are designed with ICEM CFD 13.0 © and are of a multiblock type. Figure 9.16 show the symmetry surface of the medium hexahedral mesh at different resolutions. In this case, the structured hexahedral mesh has the advantage of an adequate definition of the leading and trailing edges of the hydraulic profiles. Elements in the wake are also of an appropriate density and positioned correctly. However, the use of structured hexahedral mesh in a spiral case brings several drawbacks. For example, the junction of the blocking of each channel of the spiral case leads to a needlessly high element density at the domain exit, see the [Rousseau et al.]439. This junction in the blocking also leads to poor-quality elements on the upper surface of the stay vanes, particularly near the leading edge. It is the same at the leading and trailing edges of the wicket gates. The time required to construct a structured hexahedral mesh for a spiral case is another drawback. The large number of operations related to the multiple blocks is the main cause440. Figure 9.16 Structured Hexahedral Mesh of the Geometry A on the Symmetrical Surface and Close Up – (Courtesy of Rousseau et al.)

Philippe Martineau Rousseau, Azzeddine Soulaïmani and Michel Sabourin, “Comparison between structured hexahedral and hybrid tetrahedral meshes generated by commercial software for CFD hydraulic turbine analysis”, Conference Paper, May 2013. 440 See Previous. 439

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9.5.3.2 Hybrid Tetrahedral Mesh Unlike structured hexahedral mesh, the hybrid tetrahedral design does not require complex blocking. The construction of the mesh with Pointwise 17.0 R1© begins with the design of a surface mesh. The volume mesh is then performed by a Pointwise tool called T-Rex, with smooth transition between prismatic and volume elements as shown in Figure 9.13 (b). These tools allow a good-quality volume mesh to be created in a shorter timeframe than with the hexahedral structured mesh. The mesh density is generally adequate in every place of the spiral case. In addition, unlike the structured hexahedral topology, the wall elements are of a higher quality because they are more orthogonal. Figure 9.17 (left) shows an example of the hybrid tetrahedral mesh for the geometry A. A drawback of using hybrid tetrahedral mesh is the difficulty to obtain a density of elements equivalent to that of the structured hexahedral in the wake of hydraulic profiles, as illustrated in Figure 9.17 (right). In fact, the size of the elements increases too quickly. This difference may lead to an overestimation of the dissipation in the wake.

Figure 9.17 Hybrid Tetrahedral Medium Mesh on the Symmetric Surface of the Geometry A (left) & Mesh in the wake of a Hydraulic Profile (wicket gates trailing edge)(right) – (Courtesy of Rousseau et al.)

CFD Solution Strategy and Boundary Conditions The flow is computed for a Reynolds number of , based on the runner diameter (350 mm). It is modeled by the Reynolds Averaged Navier-Stokes equations (RANS) and the standard SST turbulence. The presence of a separation and recirculation zone in the geometry A justifies the use of this turbulence model441-442. The second-order advection scheme of ANSYS CFX (blend factor = 1) is used to limit the numerical diffusion. The convergence is obtained when the RMS residuals of the momentum and mass conservation equations are less than 10-5. It is also verified that the total head loss in the spiral case has been stabilized. However, the presence of large recirculation zones prevents this level of convergence to be achieved for all calculations, but the total head loss is always stabilized. This implies that the flow is locally weakly unsteady. The lack of information about the flow boundary conditions used in the physical setup lead us to assume the following conditions: the flow rate corresponding to a maximum opening of the wicket gates is imposed at the entrance of the spiral case by a velocity profile. This profile corresponds to a fully developed turbulent flow in a circular duct as described by the Power Law443. A zero static pressure is imposed at the spiral case exit. However, these conditions can only be valid in a comparative analysis situation. Shur, M., et al., Comparative Numerical Testing of One- and Two-Equation Turbulence Models for Flows with Separation and Reattachment, 33rd Aerospace Sciences Meeting and Exhibit, American Institute of Aeronautics and Astronautics: Reno, NV, 1995. 442 Bardina, J.E., P.G. Huang, and T.J. Coakley, Turbulence Modeling Validation, Testing, and Development, NASA. 443 R. Munson, B., et al., Fundamentals of Fluid Mechanics. Sixth Edition ed2009, Hoboken, NJ: Wiley. 724. 441

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Results Analysis of the total head loss as a function of the radius provides details about the loss through the spiral casing components, which in turn allows identification of the component that causes the largest total head loss. This information is used to amend the problematic component, for example, the stay vanes in the geometry A. Ultimately, the analysis helps in calculating the hydraulic efficiency of the turbine. The meridian velocity on the symmetrical plane is also used to compare the two meshes. It provides qualitative information on the flow. For example, it shows the effect of recirculation zones or of an obtuse trailing edge. It also indicates if the hybrid tetrahedral mesh overestimates the dissipation of the wake. The meridian velocity is measured at 10 % upstream of the inlet radius of the stay vanes, between the end of the stay vanes and the beginning the guide vanes, and at the average radius of the runner inlet. The difference of the total cumulative loss between the two types of mesh is approximately 10% and occurs predominantly near the trailing edge of the stay vanes and upstream of the leading edge of the guide vanes. Figure 9.18 confirms that the total head loss difference originates at the end of the stay vanes. In fact, the hybrid tetrahedral mesh models a larger recirculation zone and thus a larger wake. The better quality of the prism elements of the tetrahedral mesh on the wall could be the cause for that larger wake. Furthermore, Figure 9.18 shows an effect of the second order advection scheme (blend factor = 1) used in the geometry A by the non-physical total pressure augmentation (red contour plot). This second order scheme could lead to local instabilities in cases of sudden flow direction change or coarse meshes. The adaptive CFX advection scheme (high resolution) should eliminates these instabilities but will induce more numerical diffusion in the presence of large recirculation zones.

Figure 9.18 Relative Total Head Loss on the Meridian Plane for the Geometry A with fine mesh, left: Structured Hexahedral, right: Hybrid Tetrahedral – (Courtesy of Rousseau et al.)

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Figure 9.19 show the meridian velocity profiles for the geometry A with medium and fine meshes. For both grids, the meridian velocity is almost identical at the entrance of the stay vanes. The slight difference is due to the azimuthal change in the distribution of the flow. It should be noted that the sudden jump in the meridian velocity corresponds to the end of the spiral case at an azimuthal position of 40°. The wave pattern in the velocity profile shows the influence of the leading edge of the stay vanes. The velocity profile in the gap between the stay vanes and the wicket gates is very similar for all meshes. However, there remains a slight difference caused by the largest recirculation zone in the hybrid tetrahedral mesh. The velocity profile at the entrance of the runner differs by its faster dissipation of the wake and a gap in the velocity profile. The former tends to carry out a smoothing of the velocity profile, and the latter results from the difference between the recirculation zone in the hybrid tetrahedral mesh. The velocity profile at the entrance of the runner differs by its faster dissipation of the wake and a gap in the velocity profile. The former tends to carry out a smoothing of the velocity profile, and the latter results from the difference between the recirculation zones of the two meshes. As noted on the velocity profile at the entrance of the stay vanes, the recirculation zone differences slightly change the flow distribution in the spiral case. These differences between the two meshes are slightly more pronounced with increased refinement. In fact, the recirculation zone is larger with hybrid mesh. In contrast to the geometry A, it appears that the evolution of the total head loss is very similar for both types of meshes. Only the coarse hybrid tetrahedral mesh differs in the total head loss to the end of the stay vanes and wicket gates. The too-rapid growth of the tetrahedral mesh at the trailing edge explains this difference.

Figure 9.19

Meridian Velocity Near a Stay Vane with fine mesh for Geometry A, left: Structured Hexahedral, right: Hybrid Tetrahedral – (Courtesy of Rousseau et al.)

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For the geometry, only medium meshes are chosen due to the strong similarity of their assessment of the total head loss. As expected, the meridian velocity at the entrance of the stay vanes is almost identical for the two meshes. (see the [Roussea et al.]444). Similarly, the velocity is almost identical between the stay vanes and the wicket gates. The meridian velocity at the runner entrance shows that the hybrid tetrahedral mesh dissipates faster the wake. In fact, the extreme values caused by the wicket gate’s wake are dispelled by this mesh. Figure 9.20 shows the overall effect of the dissipation of the wake caused by the too-rapid growth of the tetrahedral mesh. However, a strong similarity of the flow is observed between the two mesh types.

Figure 9.20

Meridian Velocity on the Meridian Plane for the Geometry B – (Courtesy of Rousseau et al.)

Conclusion The comparison between the structured hexahedral and hybrid tetrahedral meshes in the complex geometry of a hydraulic turbine spiral case gives an advantage to the latter. In fact, Pointwise software eliminates many defects inherent to hybrid tetrahedral mesh, such as inadequate definition of hydraulic profiles and poor transition between prismatic and volume elements. This mesh also leads to higher quality elements near the walls. Furthermore, a significant savings in turnaround time is obtained for the mesh construction compared to the hexahedral mesh. Typically, the mesh design time is reduced between five and ten times with the construction of a hybrid tetrahedral mesh. The results show a great similarity of the flow for the two meshes in the two geometries. However, the flow in the geometry A differs in the recirculation zones in the upper surface of the stay vanes. The Philippe Martineau Rousseau, Azzeddine Soulaïmani and Michel Sabourin, “Comparison between structured hexahedral and hybrid tetrahedral meshes generated by commercial software for CFD hydraulic turbine analysis”, Conference Paper, May 2013. 444

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hybrid tetrahedral mesh models a larger recirculation zone than that of the structured hexahedral. The higher quality of the prismatic elements on the wall could be one cause. This difference modifies the evolution of the total head loss and slightly alters the meridian velocity profile. In addition to these differences, there is also a slightly faster dissipation of the wake downstream of hydraulic profiles in the hybrid tetrahedral mesh. The too-rapid growth of tetrahedral elements is the main cause. However, this disadvantage could be reduced with a finer mesh (and associated computational resources). In the absence of detailed experimental results about the flow in the spiral case, it is not possible to conclude on the accuracy of each mesh. In regard to the geometry B, the flow and the evolution of losses are virtually identical. As in the geometry A, there is only a slightly larger dissipation of the wakes due to the rapid expansion of the size of tetrahedral mesh elements at the trailing edge. Finally, the application of two types of mesh in both geometries shows similar results in terms of hydraulic performance. However, it is interesting to note that the evolution of the total head loss function of the radius in the spiral case shows that the hexahedral mesh requires fewer nodes than the tetrahedral hybrid mesh to achieve mesh independence. This can be an important factor when the available computing power is limited or the number of licenses for commercial software becomes an issue. The application of the hybrid tetrahedral mesh is currently being used for the calculation of the entire flooded parts of a Francis hydraulic turbine. That study should allow demonstrating the validity of this meshing method compared with experimental results.

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10 Mesh Sensitivity and Mesh Independence Study Although mesh sensitivity, in our opinion, could be grouped as one of mesh quality criteria, it is still debatable a, so we leave it as such. But more importantly what is the difference between mesh sensitivity and mesh independence? Very little. Although some argue that grid sensitivity is a real (measured quantity) while grid independence is merely a mist and cannot be truly achieved. So it dependence who is your audience. Here we homage first mesh sensitivity than mesh independence study.

Different Types of Mesh Sensitivity Several methods concerning the derivation of mesh sensitivity equations are currently available. Among the most frequently mentioned are : • • • • •

Direct (Analytical) Differentiation (DD), Adjoint Variable (AV), Symbolic Differentiation (SD), Automatic Differentiation (AD), ( e.g. Odyssée or ADIFOR) Finite Difference (FD), (Brute Force)

Each technique has its own unique characteristics. For example, the Direct Differentiation, used here, has the advantage of being exact, due to direct differentiation of governing equations with respect to design parameters, but limited in scope. By far, the most used sensitivity analysis, is Adjoint Variables techniques, especially for aerodynamic optimization. Due to apparent popularity, we consider these in more details. Symbolic Differentiation Manipulates mathematical expressions in the code. If you ever used Matlab or Mathematica, then you probably used it. For every math expression they know the derivative and use various rules (product rule, chain rule) to calculate the resulting derivative. Then they simplify the end expression to obtain the resulting expression. Automatic Differentiation Manipulates blocks of computer each element of a program (when you define any operation in code, you need to register a gradient for this operation). It also uses chain rule to break complex expressions into simpler ones. 10.1.2.1 Symbolic vs Automatic Differentiation You might think that Automatic differentiation is the same as Symbolic differentiation (in one place they operate on math expression, in another on computer programs). And yes, they are sometimes very similar. But for control flow statements (`if, while, loops) the results can be very different: symbolic differentiation leads to inefficient code (unless carefully done) and faces the difficulty of converting a computer program into a single expression programs445. Finite Differencing This method is easy implement, but cost insensitive. There are a series of issues with this approach. Accuracy is the main drawback of the method especially for non-linear problems such as those of aerodynamic nature. Cost is also something that should not be underestimated. For instance, considering the case where the grid sensitivity needs to be computed, the number of operations that 445

Stackoverflow blog.

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need to be performed is 𝑁𝐷𝑉×𝑚, where m is the number of step size tested which is unknown a priori and is specific for each design variable (DV). It is obvious that the cost quickly becomes unmanageable446.

Mesh Sensitivity via Direct Differentiation (DD) The general equations can be written as Where R is the residual vector, X mesh vector, and P is the meshing parameters. This results in a large system of linear equations in delta form at each time step as for a steady-state solution (i.e., t → ∞) reduces to

𝐑( 𝐗 (𝐏), 𝐏) = 0

Eq. 10.1 Where the explicit dependency of R on grid and vector of parameters P is evident. The parameters P control the grid X. Using chain rule of differentiation

∂𝐑 ∂𝐗 ∂𝐑 [ ][ ] + [ ] = 0 ∂𝐗 ∂𝐏 ∂𝐏

Eq. 10.2 Further simplification could include the vector of grid sensitivity which is

 X   X   XB   =    P   XB   P  Eq. 10.3 Where XB denotes the boundary nodes447. Surface Modeling Using NURBS Among many ideas proposed for generating any arbitrary surface, the approximate techniques of using spline functions are gaining a wide range of popularity. The most commonly used approximate representation is the Non-uniform Rational B-Spline (NURBS) function. They provide a powerful geometric tool for representing both analytic shapes (conics, quadrics, surfaces of revolution, etc.) and free-form surfaces448; or occasionally called Free From Deformations (FFD). The surface is influenced by a set of control points and weights to where unlike interpolating schemes the control points might not be at the surface itself. By changing the control points and corresponding weights, the designer can influence the surface with a great degree of flexibility without compromising the accuracy of the design. The relation for a NURBS curve is n

X (r) =  R i,p (r) Di i =0

i = 0,........., n

R i,p (r) =

N i,p (r) ωi n

N i =0

i, p

(r) ωi

Eq. 10.4 where X (r) is the vector surface coordinate in the r-direction, Di are the control points (forming a 446 Gabriele Luigi Mura,

“Mesh Sensitivity Investigation in the Discrete Adjoint Framework”, Thesis submitted to University of Sheffield in partial fulfilment of the requirement for the degree of Doctor of Philosophy, 2017. 447 Sadrehaghighi, I., Smith, R.E., Tiwari, S., N., “Grid Sensitivity and Aerodynamic Optimization Of Generic Airfoils”, Journal of Aircraft, Volume 32, No. 6, Pages 1234-1239. 448 Tiller, W., “Rational B-Splines for Curve and Surface Representation," Computer Graphics and Applications, Volume 3, N0. 10, September 1983.

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control polygon), ωi are weights, Ni,p (r) are the p-th degree B-Spline basis function, and Ri,p(r) are known as the Rational basis functions set n

∑ R i,p (r) = 1 , R i,p (r) ≥ 0 i=1

Eq. 10.5 Figure 10.2 represents a six control point representation of a generic airfoil. The points at the leading and trailing edges are fixed. Two control points at the 0% chord are used to affect the bluntness of the section. Similar procedure can be applied to other airfoil geometries such as NACA four or five digit series. Another example Figure 10.1 shows two airfoils NACA0012 and RAE2822 parameterized using B-Spline curve of order 4 with control points. The choice for number of control points and their locations are best determined using an inverse B-Spline interpolation of the initial data. The algorithm yields a system of linear equations with a positive and banded coefficient matrix. Therefore, it can be solved safely using techniques such as Figure 10.2 Six Control Point Representation of a Gaussian elimination without pivoting. Generic Airfoil The procedure can be easily extended to cross-sectional configurations, when critical cross-sections are denoted by several circular conic sections, and the intermediate surfaces have been generated using linear interpolation. Increasing the weights would deform the circular segments to other conic segments (elliptic, parabolic, etc.) as desired for different flight regions. In this manner, the number of design parameters can be kept to a minimum, which is an important factor in reducing costs. The practice is effortlessly applicable to 3D for example like the common wing & fuselage as designated in Figure 10.3 [Kenway et al.]449. The choice for number of control points and their locations are best determined using an inverse B-Spline interpolation of the initial data. The algorithm yields a system of linear equations with a positive and banded coefficient matrix.

Figure 10.1

B-Spline Approximation of NACA0012 (left) and RAE2822 (right) Airfoils

Gaetan K.W. Kenway, Joaquim R. R. A. Martins, and Graeme J. Kennedy, “Aero structural optimization of the Common Research Model configuration”, American Institute of Aeronautics and Astronautics. 449

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Therefore, it can be solved safely using techniques such as Gaussian elimination without pivoting. The procedure can be easily extended to cross-sectional configurations, when critical cross-sections are denoted by several circular conic sections, and the intermediate surfaces have been generated using linear interpolation. Increasing the weights would deform the circular segments to other conic segments (elliptic, parabolic, etc.) as desired for different flight regions. In this manner, the number of design parameters can be kept to a minimum, which is an important factor in reducing costs. An efficient gradient-based algorithm for aerodynamic shape optimization is presented by [Hicken and Zingg]450 where to integrate geometry parameterization and mesh movement. The generalized Bspline volumes are used to parameterize both the surface and volume mesh. Volume mesh of B-spline control points mimics a coarse mesh where a linear elasticity mesh-movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically. Using this approach, mesh-movement time is reduced by two to three orders of magnitude relative to a node-based movement.

Figure 10.3

Free Form Deformation (FFD) for Volume Grid with Control Points (Courtesy of Kenway et al.)

10.2.1.1 Case Study - 2D Study of Airfoil Grid Sensitivity via Direct Differentiation (DD)451 The structured grid sensitivity of a generic airfoil with respect to design parameters using the NURBS parameterization is discussed. The geometry, as shown in Figure 10.2, has six pre-specified control points. The control points are numbered counter-clockwise, starting and ending with control points (0 and 5), assigned to the tail of the airfoil. A total of 18 design parameters (i.e., three design parameters per control point) available for optimization purpose. Depending on desired accuracy and degree of freedom for optimization, the number of design parameters could be reduced for each particular problem. For the present case, such reduction is achieved by considering fixed weights and chord-length. Out of the remaining four control points with two degrees of freedom for each, Jason E. Hickenand, David W. Zingg, “Aerodynamic Optimization Algorithm with Integrated Geometry Parameterization and Mesh Movement”, AIAA Journal Vol. 48, No. 2, February 2010. 451 Sadrehaghighi, I., Smith, R.E., Tiwari, S., N., “Grid Sensitivity and Aerodynamic Optimization Of Generic Airfoils”, Journal of Aircraft, Volume 32, No. 6, Pages 1234-1239. 450

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control points 1 and 5 have been chosen as a case study. The number of design parameters is now reduced to four with XD = {X1, Y1, X5, Y5}T, with initial values specified in Figure 10.2. Non-zero contribution to the surface grid sensitivity coefficients of these control points are the basis functions R1,3(r) and R5,3(r). The sensitivity gradients are restricted only to the region influenced by the elected control point. This locality feature of the NURBS parameterization makes it a desirable tool for complex design and optimization when only a local perturbation of the geometry is warranted. Similar results can be obtained for design control point 5 where the sensitivity gradients are restricted to the lower portion of domain. Figure 10.4 shows C-type dual blocks structured grid and its sensitivity with respect to NURBS input for different design control points.

Adjoint Variable Sensitivity Analysis (AV) Following closely the development in [Luo and Liu]452, in the discipline of aerodynamics, a performance function I is usually dependent on flow solutions w and aerodynamic shape, which is essentially dependent on the geometric parameters x = {x1; · · · ;xN}, and subsequently the grid. The sensitivities of performance function to geometric parameters can be given as

Figure 10.4

Sample Grid and Grid Sensitivity

δ𝐈 ∂𝐈 δ𝐰 ∂𝐈 = + δxi ∂𝐰 δxi ∂xi

Eq. 10.6 In the meantime, δw implicitly depends on aerodynamic shape change through the governing flow equations R(w , x) = 0. A similar form as Eq. 10.6 for R can be given as

452 Jiaqi Luo, Feng Liu, “Performance Impact Of Manufacturing Tolerances for a Turbine

Blade Using Second Order Sensitivities”, Proceedings of ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition, GT2018, June 11-15, 2018, Oslo, Norway.

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δ𝐑 ∂𝐑 δ𝐰 ∂𝐑 = + =0 δxi ∂𝐰 δxi ∂xi

Eq. 10.7 By introducing a series of co-state variables, Ψ and subtracting the product of ΨT and Eq. 10.6 from Eq. 10.7, we can get

δ𝐈 ∂𝐈 ∂𝐑 δ𝐰 ∂𝐈 ∂𝐑 ={ − ψT + { − ψT } } δxi ⏟∂𝐰 ∂𝐰 δxi ∂𝐱 𝐢 ∂xi 0

Eq. 10.8 The crucial issue of the adjoint method is to eliminate the effects of δw on δI to avoid the calculation of δw due to the change of aerodynamic shape. It can be achieved if the adjoint operator Ψ satisfies the adjoint equations, therefore the first term in Eq. 10.8 is set to zero. Then the sensitivities can be determined by

δ𝐈 ∂𝐈 ∂𝐑 = − ψT δxi ∂xi ∂xi

Eq. 10.9 Once the flow solutions and adjoint solutions are obtained by solving the governing flow equations and the adjoint equations, respectively, the complete sensitivities can be calculated by deforming the aerodynamic shape and thus the grid for each geometric parameter. Considering that the evaluation of an aerodynamic objective function involves perturbing the grid, it is to be expected that the sensitivities of the objective function to the design variables will in some way involve sensitivities of the grid perturbation algorithm. When evaluating the gradient using the discrete adjoint method, these mesh sensitivities are implicitly included in the terms ∂I/∂xi and ∂R/∂xi .453 Table show the pro

Method

Pro

Finite Difference

Easy To Implementation

Analytical

No Convergence Issue Low Memory and CPU Requirements

Adjoint Variables

Independent from the 𝑁𝐷𝑉

Automatic Differentiation

Easy to Implement - Largely Automatic

Table 10.1

Cons Prone to Cancellation and Round-off errorCost scales linearly with the 𝑁𝐷𝑉 - Poor accuracy and Consistency The Pre-Processing cost is Proportional to the 𝑁𝐷𝑉 Needs the Solution of a Large Linear System Increase in Memory When Used in Reverse Mode

Pros & Cons of Different Grid Sensitivity Method (NDV = Number of Design Variable)

Chad Oldfield, “An Adjoint Method Augmented with Grid Sensitivities for Aerodynamic Optimization”, A thesis submitted in conformity with the requirements for the degree of M.A.Sc. Graduate Department of Aerospace Engineering University of Toronto, 2006. 453

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and cons of different mesh sensitivity routine as envisioned by [Gabriele Luigi Mura]454.

Mesh Independence Study To perform a Mesh Independent Study, is fairly straight forward as seen in a compressible flow over a forwarding step size example (see Figure 10.5): 1. Run the initial simulation on your initial mesh and ensure convergence of residual error to 10-4, monitor points are steady, and imbalances below 1%. If not refine the mesh and repeat. 2. Once you have met the convergence criteria above for your first simulation, refine the mesh globally so that you have finer cells throughout the domain. Generally we would aim for around 1.5 times the initial mesh size. Run the simulation and ensure that the residual error drops below 10-4, that the monitor points are steady, and that the imbalances are below 1%. At this point you need to compare the monitor point values from Step 2 against the values from Step 1. If they are the same (within your own allowable tolerance), then the mesh at Step 1 was accurate enough to capture the result. If the value at Step 2 is not within acceptable values of the Step 1 result, then this means that your solution is changing because of your mesh resolution, and hence the solution is not yet independent of the mesh. In this case you will need to move to Step 3.

Figure 10.5

Effects of Mesh Density on Solution Domain

Gabriele Luigi Mura, “ Mesh Sensitivity Investigation in the Discrete Adjoint Framework”, Thesis is submitted to University of Sheffield in partial fulfilment of the requirement for the degree of Doctor of Philosophy, 2017. 454

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3. Because your solution is changing with the refinement of mesh, you have not yet achieved a mesh independent solution. You need to refine the mesh more, and repeat the process until you have a solution that is independent of the mesh. You should then always use the smallest mesh that gives you this mesh independent solution (to reduce your simulation run time). Briefly, according to [Ssheshan Pugazhendh], mesh or grid independent solution is a solution that does not vary significantly even when you refine your mesh further. The answer comes through the question that emphasizes the independence of numerical solution from grid structure, also called mesh. In every computational analysis, mesh independence studies, also expressed as mesh convergence, ought to be conducted to sustain credible results. Otherwise, the results that obtained would be considered as skeptical. (see Figure 10.6) Figure 10.6

Mesh Independence

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11

Mesh Quality Background

Make no mistake about it, mesh quality can have a large influence upon the accuracy (and efficiency) of a simulations based on the solution of partial differential equations (PDE)'s. Most argue that your CFD solution is as good as mesh it has. Many factors go into the influence of mesh on accuracy including the type of physics being simulated, details of the solution to the particular simulation, the method of discretization, and geometric mesh properties having to do with spacing, curvature, angles, smoothness, etc,455. The general consensus is that a good quadrilateral mesh would be formed by two families of orthogonal, or at least nearly orthogonal, curves with a smooth gradation between a coarse mesh in the far field and a fine mesh near the boundary. The following provisional definition is accepted as Mesh Quality concerns the characteristics of a mesh that permit a particular numerical PDE simulation to be efficiently performed, with fidelity to the underlying physics, and with the accuracy required for the problem. This description hints at several issues. First, mesh quality depends on the particular calculation which is undertaken and thus changes if a different calculation is performed. Second, a mesh should do no harm, i.e., it should not create difficulties for the simulation. As mesh generation methods evolved to handle complex three dimensional configurations, and the choice of element type broadened to include not just hexahedra but also tetrahedral and prisms, visual inspection of a mesh became much more difficult. The task was aided considerably by the advent of computer workstations with a powerful graphics capability and the development of good graphics software to view CFD solutions. Today, of course, it is often possible to undertake a CFD simulation and view the results on a laptop computer. Despite these developments in computer graphics and visualization software it is almost impossible to check a mesh with several million points around a complete aircraft and decide whether the quality and distribution of the mesh elements is acceptable. Even if this were a feasible option, visual inspection of large meshes is extremely time consuming and is clearly unacceptable in a design environment where a rapid turnaround is essential and numerous design variations must be evaluated in a timely manner.

Mesh Quality Metric There is a move towards quantifying the mesh in terms of criteria that can measure the element quality and the gradation in mesh element size in a precise way. At the very least, analyzing the mesh in this way allows one to identify the hot spots and thus decide where a careful visual inspection maybe needed (eye pleasing ?). As this approach develops and gains in sophistication one can envision a time when visual assessment is replaced by a different aesthetic, one based entirely on mathematical criteria. The question of whether a mesh is sufficiently fine to achieve a solution that has a required level of accuracy depends to a large extent on the discretization of the flow equations. If the discretization has a formal order of accuracy O(h) where h is the local mesh width (i.e. the linear extent of a mesh element) one would generally expect that the solution error on a good quality mesh should also scale in the same way. By comparing computed solutions on a sequence of three progressively larger meshes it is possible to check this assumption by exploiting a generalized form of Richardson Extrapolation. Deciding whether mesh convergence has been achieved for a flow field computation over a given configuration and, if not, how fine a mesh one actually needs to achieve specified accuracy is critically important (i.e., discretization error will be dealt in detail). In general,

Patrick M. Knupp, “Remarks on Mesh Quality”, 45th AIAA Aerospace Sciences Meeting and Exhibit, 7-10 January, 2007, Reno, NV. 455

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• • • •

Rate of convergence Solution accuracy Grid Independence result CPU time required

Now these days most of grid generation routines have sophisticated software of grid quality which shows the results graphically. Important metrics such as Volume, Orthogonality, Skewness, Stretching, Centroids, etc., are available on most grids generation software. Figure 11.1 shows the mesh quality (Volume, AR, and Stretching) for benchmark test case Turk/Hron.

(a) Volume

(b) Aspect Ratio

(c) Stretching

Figure 11.1

Predicted Mesh Quality (Volume, Aspect Ratio, and Stretch)

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Mesh Quality from User’s Perspective The importance of a priori indicators of mesh quality is exemplified by NASA’s Stephen Alter456, who defined and demonstrated the utility of his grid quality (GQ) metric that combines both orthogonality and stretching into a single number. Driven by the desire to ensure the accuracy of supersonic flow solutions over blunt bodies computed using a Thin Layer Navier-Stokes (TLNS) solver, he has established criteria for the GQ metric that give him confidence prior to starting a CFD solution. Two aspects of GQ are notable. First, this metric’s reliance on orthogonality is closely coupled to the numerics of the solver where TLNS assumptions break down when the grid lacks orthogonality. Second, use of a global metric aids decision making, or as Thornburg457 wrote, “A local error estimate is of little use.” GQ represents domain expertise where the use of specific criteria within a specific application domain458. Mesh Quality from Researcher’s Perspective Dannenhoffer459 reported on an extensive benchmark study that involved parametric variation of a structured grid’s quality for a 5 degree double-wedge airfoil in Mach 2 inviscid flow at 3 degrees angle of attack. Variations of the mesh included resolution, aspect ratio, clustering, skew, taper, and wiggle (using the Verdict definitions). Dannenhoffer’s main conclusion was very interesting: there was little (if any) correlation between the grid metrics and solution accuracy. This may have been exacerbated by the fact that he found it difficult to change one metric without influencing another (e.g. adding wiggle to the mesh also affected skew) or it may have been due to the specific flow conditions. Dannenhoffer also introduced the concept of grid validity (as opposed to grid quality), which is intended to measure whether the grid conforms to the configuration being modeled (which in practice it sometimes does not). He proposed three types of validity checks: 1. Type 1 checks whether cells have positive volumes and faces that do not intersect each other. 2. Type 2 checks whether interior cell faces match uniquely with one other interior face and whether boundary cell faces lie on the geometry model of the object being meshed. 3. Type 3 checks whether each surface of the geometry model is completely covered by boundary cell faces, whether each hard edge of the geometry is covered by edges of boundary cell faces, and whether the sum of the boundary faces areas matches the actual geometry surface area. [Christopher Roy]460 from Virginia Tech showed a counter-intuitive example (at least from the standpoint of a priori metrics) that the solution of 2D Burger’s equation on an adapted mesh (with cells of widely varying skew, aspect ratio, and other metrics) has much less Discretization Error (DE) than the solution on a mesh of perfect squares as seen in Figure 11.2461. From this example alone, it is clear that metrics based solely on cell geometry are not good indicators of mesh quality as it pertains to solution accuracy. Mesh Quality from Solver’s Perspective The common thread among all participating CFD solvers was that convergence and stability are more directly affected by mesh quality than solution accuracy. Stephen Alter, “A Structured-Grid Quality Measure”, NASA Langley. Thornburg, Hugh J., “Overview of the PETTT Workshop on Mesh Quality/Resolution, Practice, Current Research, and Future Directions”, AIAA paper no. 2012-0606, Jan. 2012. 458 Another Fine Mesh, Pointwise blog, posted on July 5, 2012 by John Chawner. 459 John Dannehoffer , “On Grid Quality and Validity”, Syracuse University. 460 Christopher Roy, “Discretization Error”, Virginia Tech. 461 A simple demonstration of how a poor mesh from a cell geometry perspective (right) results in lower discretization error than one with “perfect” cells (left). 456 457

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Figure 11.2

A simple Demonstration of How a Poor Mesh from a Cell Geometry Perspective

11.2.3.1 CFD++ Metacomp Technologies’ Vinit Gupta462 cited cell skewness and cell size variation as two quality issues to be aware of for structured grids. In particular, grid refinement across block boundaries in the far field where gradients are low has a strong, negative impact on convergence. For unstructured and hybrid meshes, anisotropic tets in the boundary layer and the transition from prisms to tets outside the boundary layer also can be problematic. Gupta also pointed out two problems associated with metric computations. Cell volume computations that rely on a decomposition of a cell into tets are not unique and depend on the manner of decomposition. Therefore, volume (or any measure that relies on volume) reported by one program may differ from that reported by another. Similarly, face normal computations for anything but a triangle are not unique and also may differ from program to program. (This is a scenario can be often encountered when there is a disagreement with a solver vendor over a cell’s volume that turns out to be the result of different computation methods.) 11.2.3.2 Fluent and CFX ANSYS’ Konstantine Kourbatski463 showed how cell shapes that differ from perfect (dot product of face normal vector with vector connecting adjacent cell centers) make the system of equations stiffer slowing convergence. He then introduced metrics, Orthogonal Quality and two skewness definitions, with rules of thumb for the Fluent solver. It was interesting to note that the orthogonality measure ranges from 0 (bad) to 1 (good) whereas the skewness metric is directly opposite: 0 is good and 1 is bad. Another example of a metric criterion was that aspect ratios should be kept to less than 5 in the bulk flow. Kourbatski also provided guidelines for the CFX solver. He also pointed out that resolution of critical flow features (e.g. shear layers, shock waves) is vital to an accurate solution and that bad cells in benign flow regions usually do not have a significant effect on the solution. 11.2.3.3 Kestrel Kestrel, the CFD solver from the CREATE-AV program, was represented by David McDaniel464 from the University of Alabama at Birmingham. At the start, he made two important statements. First, their Vinit Gupta, “CFD++ Perspective on Mesh Quality”, Metacomp Technologies. Konstantine Kourbataski, “Assessment of Mesh Quality in ANSYS CFD”, ANSYS. 464 David McDaniel, “Kestrel/CREATE-AV Perspective on Mesh Quality”, University of Alabama at Birmingham. 462 463

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goal is to “do well with the mesh given to us.” (This is similar to Pointwise’s approach to dealing with CAD geometry – do the absolute best with the geometry provided.) Second, he notes that mixedelement unstructured meshes (their primary type) are terrible according to traditional mesh metrics, despite being known to yield accurate results. This same observation is true for adaptive meshes and meshes distorted by the relative motion of bodies within a mesh (e.g. flaps deflecting, stores dropping). More significantly, McDaniel notes a “scary” interdependence between solver discretization and mesh geometry by recalling Mavriplis’ paper on the drag prediction workshop465 in which two extremely similar meshes yielded vastly different results with multiple solvers. To address mesh quality, Kestrel’s developers have implemented non-dimensional quality metrics that are both local and global and that are consistent in the sense that 0 always means bad and 1 always means good. The metrics important to Kestrel are an area-weighted measure of quad face planarity, an interesting measure of flow alignment with the nearest solid boundary, a least squares gradient that accounts for the orientation and proximity of neighbor cell centroids, smoothness, spacing and isotropy. Differing from Dannenhoffer’s result, McDaniel showed a correlation of mesh quality with solution accuracy with the caution that a well resolved mesh can have poor quality and still produce a good answer. (In other words, more points always is better; see Figure 11.3).

Figure 11.3

Using Kestrel one can Show a Correlation Between Mesh and Solution Quality

11.2.3.4 STAR-CCM+ Alan Mueller’s466 presentation on CD-adapco’s STAR-CCM+ solver began by pointing out that mesh quality begins with CAD geometry quality and manifests as either a low quality surface mesh or an inaccurate representation of the true shape. This echoes Dannenhoffer’s grid validity idea. After introducing a list of their quality metrics, Mueller makes the following statement, “Results on less than perfect meshes are essentially the same (drag and lift) as on meshes where considerable resources were spent to eliminate the poor cells in the mesh.” Here we note that the objective Mavriplis, Dimitri J., “Grid Quality and Resolution Issues from the Drag Prediction Workshop Series”, AIAA paper 2008-930, Jan. 2008. 466 Alan Mueller, “A CD-Adapco Perspective on Mesh Quality”,CD-Adapco. 465

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functions are integrated quantities (drag and lift,) instead of distributed data like pressure profiles. After all, integrated quantities are the type of engineering data we want to get from CFD. This insensitivity of accuracy to mesh quality supports Mueller’s position that poor cell quality is a stability issue. Accordingly, the approach with STAR-CCM+ is to be conservative opt for robustness over accuracy. Specifically, they are looking for metrics that will result in division by zero in the solver. Skewness as it effects diffusion flux and linearization is one such example. 11.2.3.5 Deducing Results 1. CFD solver developers believe mesh quality affects convergence much more than accuracy. Therefore, the solution error due to poor or incomplete convergence cannot be ignored. 2. One researcher was able to show a complete lack of correlation between mesh quality and solution accuracy. It would be valuable to reproduce this result for other solvers and flow conditions. 3. Use as many grid points as possible (Dannenhoffer, McDaniel). In many cases, resolution trumps quality. However, the practical matter of minimizing compute time by using the minimum number of points (what Thornburg called an optimum mesh) means that quality still will be important. 4. A priori metrics are valuable to users as an effective confidence check prior to running the solver. It is important that these metrics account for cell geometry but also the solver’s numerical algorithm. The implication is that metrics are solver-dependent. A further implication is that Dannehoffer’s grid validity checks be implemented. 5. There are numerous quality metrics that can be computed, but they are often computed inconsistently from program to program. Development of a common vocabulary for metrics would aid portability. 6. Interpreting metrics can be difficult because their actual numerical values are non-intuitive and stymie development of domain expertise. A metric vocabulary should account for desired range of result numerical values and the meaning of “bad” and “good.” Some Geometric Properties 11.2.4.1 Aspect ratio Prime example would be the Aspect Ratio for a simplified geometry is shown in Figure 11.1 (b) and defined for tetrahedral cells as the ratio between the maximum edge length l and the minimum cell height h as

ARi =

Max (li ) Min (h i )

Eq. 11.1 Similiarly, for hex and polyhedral cells. 11.2.4.2 Orthogonality The concept of mesh orthogonality relates to how close the angles between adjacent element faces or adjacent element edges are to some optimal angle (for example, 90º for quadrilateral faced elements and 60º for triangular faces elements). The most relevant measure of mesh orthogonality, as defined by the

Figure 11.4

Concept of Orthogonality in Cells

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CFX-Solver is illustrated in Figure 11.4. It involves the angle between the vector that joins two mesh (or control volume) nodes (s) and the normal vector for each integration point surface (n) associated with that edge. Significant orthogonality and non-orthogonality are illustrated at ip1 and ip2, respectively. 11.2.4.3 Skewness Skew of triangular elements is calculated by finding the minimum angle between the vector from each node to the opposing mid-side, and the vector between the two adjacent mid-sides at each node of the element as shown in Figure 11.5 (a). The minimum angle found is subtracted from ninety degrees and reported as the element‘s skew. Skew in quads is calculated by finding the minimum angle between (b) Warpage calculation of two lines joining opposite mid-sides (a) Skewness in Triangle a quadrilateral element of the element. Ninety degrees minus the minimum angle found is Figure 11.5 Skewness and Warpage reported as skew of the element. Maximum of 60-70 skewed elements are accepted in most of the solver beyond this limit solver can complain about the skewness of the grid. 11.2.4.4 Warpage This is the amount by which an element (or in the case of solid elements, an element face) deviates from being planar. Since three points define a plane, this check only applies to quads. The quad is divided into two trias along its diagonal, and the angle between the trias normal is measured as shown in Figure 11.5 (b). The maximum angle found between the planes is the warpage of the element. Warpage in three-dimensional elements is performed in the same fashion on all faces of the element. Warpage of up to five degrees is generally acceptable. 11.2.4.5 Jacobian This measures the deviation of an element from its ideal or "perfect" shape, such as a triangle‘s deviation from equilateral. The Jacobian value ranges from -1.0 to 1.0, where 1.0 represents a perfectly shaped element. As the element becomes more distorted, the Jacobian value approaches zero. A Jacobian value of less than zero represents a concave element, which most analysis codes do not allow. So it is a good practice to keep the Jacobian of the grid greater than zero467. 11.2.4.6 Tetrahedral Volume Since the tetrahedral is the most elementary of volumes, we start with that. Calculates the volumes of a tetrahedron and a parallelepiped which encompasses it with 6 volumes, given four vertices in right hand side order (see Figure 11.6): 467

HyperMesh9.0 Manual, Altair Inc.

Figure 11.6

Tetrahedral Volume

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VP = AD.(AB  AC) = (x 4 − x1 )[(y 2 − y1 ) − (z 2 − z1 )(y3 − y1 )] + (y 4 − y1 )[(z 2 − z1 ) − (x 2 − x1 )(z 3 − z1 )] + (z 4 − z1 )[(x 2 − x1 ) − (y 2 − y1 )(x 3 − x1 )] Vtet = Eq. 11.2

VP 6

11.2.4.7 Polygonal Face Area and Centroid A polyhedral cell consists of a number of polygonal faces Figure 11.7 Triangulation of a polygon forming a closed volume. The area vector and the centroid location of each face needs to be computed. This can be achieved via triangulation of the polygon around a given point f as shown in Figure 11.7468. A convenient starting location for the point f is the midpoint (simple average of the nodes of the polygon).

1 rf = Nf

Nf

r i =1

i

Eq. 11.3 where Nf denotes number of face nodes and r is the position vector. The area of each of the triangular patches are added to get the area of the polygon face

A tri =

(ri − rf )  (ri +1 − rf ) 2

for i = 1, N f

Nf

A f =  A tri i =1

Eq. 11.4 And rN+1 = r1. Centroid of the face is computed in a similar fashion as:

1 rf = Af

Nf

A tri (rn + rn +1 + rf )

i =1

3



Eq. 11.5 Note that the face centroid rf was initially taken as simply the midpoint of the nodes but it is updated at the end of the process. In the case of a planar polygon, this updated location reflects the true centroid of the polygon. However, while not desirable, polygon nodes may be highly no coplanar in practice. This introduces ambiguity to the centroid location as no unique definition exists based solely on the knowledge of the node coordinates. In this case, simply iterating over Eq. 11.4 & Eq. 11.5 until convergence provides a reasonable answer. The triangulated polygonal face, even if noncoplanar, is still attached to each of the vertices defining it as opposed to an approach where one might fit a planar surface to the vertices. This ensures that, once all the faces of a cell is processed, a water-tight control volume is achieved. We note once again that regardless of the aforementioned

Emre Sozer, Christoph Brehm and Cetin C. Kiris,”Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers”, American Institute of Aeronautics and Astronautics. 468

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ambiguity for non-coplanar polygons, consistency can be retained if the cells sharing a face use the same face centroid and area for their reconstruction and flux integration469. 11.2.4.8 Polyhedral Volume and Centroid The volume and the centroid location of a polyhedral cell can be computed via tetrahedralization, basically by extending the logic presented in the previous section to 3D. Figure 11.8 shows a single face of a polyhedral cell and the corresponding tetrahedralization around a midpoint m.

1 rm = Nc

Nc

1 Vtet = A tri .(rm − rf ) 3

r i =1

i

Eq. 11.6 Where Atri and rf for a given face f is obtained previously. This usage of face triangulation around the previously calculated centroid ensures that a consistent volume is obtained. The integrated volume and the centroid of the polyhedral cell is then calculated via summation of the contributions from Nf

Ni

V =  Vtet f =1 i =1

,

rc =

1 Nf Ni  (rf,i + rf,i+1 + rf + rm )Vtet 4V f =1 i =1

Eq. 11.7 Where Nf is the number of faces, Ni is the number of face nodes470.

Best Practice for Mesh Generation There is no definite guild lines for meshing per say. It really depends to whom you are talking and application in hand. Therefore, each disciplines have its guide lines. In general, the mesh should exhibit some minimal grid quality as defined by measures of orthogonality (specially at boundaries), relative grid spacing (15% to 20% stretching is considered a maximum value), grid Figure 11.8 Tetrahedralization of a skewness, aspect ratio etc471. Also the maximum Polyhedral (showing a single face) spacing should be according to the desired resolution of the physical phenomena. Optimum quality measures for the surface as well as volume grid have been described. To resolve the boundary layer it is required to cluster the grid in the direction normal to the surface with the spacing of the first grid point off the wall to be well within the laminar sub-layer of the boundary layer. Hexahedra or prisms elements are employed to discretize boundary layers to preserve the accuracy in the wall normal direction for highly stretched viscous grid. In case of turbulent flows, it is recommended that the first point off the wall should exhibit a wall normal dimensionless distance (y+) value of less than 1.0.

Emre Sozer, Christoph Brehm and Cetin C. Kiris,”Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers”, American Institute of Aeronautics and Astronautics. 470 Emre Sozer, Christoph Brehm and Cetin C. Kiris,”Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers”, American Institute of Aeronautics and Astronautics. 471 Abhishek Khare, Ashish Singh, and Kishor Nokam, “Best Practices in Grid Generation for CFD Applications Using HyperMesh”, Member of Technical Staff Computational Research Lab. 469

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Geometry Modeling and Geometry Cleanup Object about which flow has to be simulated requires modeling. This generally involves modeling the object geometry with some CAD software package. For doing this approximations and simplifications of the geometry are required to allow an analysis with reasonable effort. Unfortunately, CAD models are developed primarily for manufacturing purposes, and typically include details irrelevant for CFD simulations or omitting key components, such as boundaries to close a domain. While it appears that CAD designers are beginning to take better account of the needs of CAE engineers in their work, it is unlikely that CAD models will be delivered CFD ready for some time yet to come. There can be errors in CAD data in the form of gaps, overlaps, non-physical protrusions. So we need a lot of cleanup of the imported CAD geometry. Geometry cleanup is a time consuming step and it requires some intelligence to decide which feature of geometry has to remove and which feature to retain. Usual practice is to retain the details that matter for simulation and ensure water tight geometry. Computational Domain After importing geometry into grid generator, decisions has to be made about the extent of the finite flow domain which is called computational domain in which the flow has to be simulated. Shape and size of flow computational domain is depends on the geometry and the physics of flow. Since this modeled geometry along with the flow domain are used as an input for the grid generation hence the modeling should takes into account the structure and topology of the grid generation. For external flows, decision of computational domain is based on to replicate the actual physics. Many times this domain is decided by wind tunnel dimensions and the blockage ratio. A ratio of model frontal area to wind tunnel cross sectional area (Blockage ratio) should be less than 5%. In internal flows the flow path recognition is one of the major works for internal flows, specifically for conjugate heat transfer analysis. From the model search the interface between solid and fluid. Choice of Grid The choice of whether to use a structured or an unstructured mesh is problem specific which is discussed in detail before. Some advantages of structured meshes that hold generally over most applications are simplicity, availability of code, and suitability for multigrid and finite difference methods. On the other hand, unstructured meshes conform to the domain more easily and allow element sizes to vary more dramatically. Structured meshes are currently more popular, but unstructured are catching up472. Here some guidelines are listed based on three parameters for choosing between structured and unstructured grid: 1. Complex geometry: Unstructured grid generation is usually much faster than structured one. However, if the geometry is only slightly modified from a previously existing geometry with a structured grid, then structured grid generation can occur very rapidly. For a particular problem structured grid can take say a man weeks to one man month. On the other hand unstructured grid will take a man hour to a few days. 2. Accuracy: For simpler problem such as airfoil (single element) or an isolated wing, structured grids are generally more accurate per unknown than unstructured. However, for more complex flows, the adaptively facilitated by an unstructured grid may allow more accurate solutions. 3. Convergence: Structured grid calculations usually take less time than an unstructured grid calculation because, to date, the existing algorithms are more efficient.

Abhishek Khare, Ashish Singh, and Kishor Nokam, “Best Practices in Grid Generation for CFD Applications Using HyperMesh”, Member of Technical Staff Computational Research Lab. 472

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Surface Meshing Once the water tight geometry is ready, it is time to create surface mesh on the model surfaces to ensure good quality surface and volume mesh. Volume mesh largely depends on the quality of surface mesh e.g. min/max angle, skewness etc. High surface deviation areas should be meshed by dense grid with smooth transition to low surface deviation areas. According to surface mesh requirement, surfaces can be subdivided into parts. All sharp edges and turnings should be assured by clustering for good quality volume mesh generation. For external aerodynamics like aero and auto domains, a hybrid mesh is fast and cost effective. Volume Meshing Once the surface grid is ready, to generate the volume grid, it is good practice to check whether it is forming a closed volume or not. If the surface grid forms a closed volume, volume grid generation can be started. For structured grid we need to do mapping of the corresponding surfaces as per the topology of the domain. In case of the unstructured grid we need to decide some parameters like boundary layer thickness, element growth ratio, Y+ values, which are required for volume grid generation. Most important parameter is the first point distance from the wall. Placement of the first point near to Figure 11.9 General estimation of surface mesh element size the wall depends on the grid resolution required. This is discussed in more detail in next section which is boundary layer grid generation. The second crucial parameter is the stretching ratio (SR). The value of the SR should be taken in such a way, so that the size of the elements varies smoothly in the domain. The recommended value of the SR is around 1.1 to 1.3. Figure 11.9 shows the general estimation of surface mesh element size based on free stream velocity and Y+ value473. This estimation works as well for automobile aerodynamics. Boundary Layer Meshing Successful computations of turbulent flows require some boundary layer consideration during the mesh generation. Since turbulence (through the spatially-varying effective viscosity) plays a dominant role in the transport of mean momentum and other scalars for the majority of complex turbulent flows, one must ascertain that turbulence quantities are properly resolved, if high accuracy is required. Due to the strong interaction of the mean flow and turbulence, the numerical results for turbulent flows tend to be more susceptible to grid dependency than those for laminar flows. To resolve the boundary layer it is required to cluster the grid in the direction normal to the surface with the spacing of the first grid point off the wall to be well within the laminar sub-layer of the boundary layer. Hexahedral or prism elements are employed to discretize boundary layers to preserve the accuracy in the wall normal direction for highly stretched viscous grid. In case of turbulent flows, it Macro Lanfrit, “Best Practices Guidelines for Handling Automotive External Aerodynamics with Fluent“, Fluent Deutschland GmbH, Birkenweg 14a, 64295 Darmstadt/Germany, 2005. 473

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is recommended that the first point off the wall should exhibit a wall normal dimensionless distance (y+) value of less than 1.0. This avoids the use of wall functions which generally over predicts the viscous drag in comparison to solve to wall approach. In complex geometries, particularly at high Reynolds numbers, the condition y+ < 1 can be rather stringent, requiring an excessive total number of grid points. In this case, the user should consider the use of wall functions, which allow much larger values for y+, at the expense of additional modeling assumptions introduced in the wall-fluxes and wall adjacent centroids. In case of LES simulation there are no computational restrictions on the near-wall mesh spacing. However, for best results, it might be necessary to use very fine near-wall mesh spacing (on the order of Y+)474. Guidelines for Aerodynamics in General Accurate Cl and Cd predictions are the major focusing areas, while dealing with aero aerodynamics. Decision of element size based on local chord length should be the strategy for surface gridding here. A size of 0.1% of local chord length at leading and trailing edge is good enough to resolve the wing flow physics. 2% of reference chord length produces good quality surface mesh near fuselage nose and after body. Approximately 5% of local chord length is fair enough to resolve flow phenomenon along span wise direction. Since Prediction of viscous drag is crucial here, boundary layer resolution plays a vital role in the prediction of Cd values. Y+ is the governing factor for boundary layer meshing. It is preferred that the value of Y+ must be less than 1 to avoid any kind of wall functions in simulation. Start from a coarse mesh to fine, Y+ can be 1, 0.67, 0.44, and 0.3. This may generate from 5 million to 100 million cells depending on other volume meshing parameters. One should be very careful while stretching cells from boundary layer to outer far field domain. Within the boundary layer, it is a good practice to keep first two cells with constant normal spacing. In general the limit of cell volume growth should be less than 1.25. Outside the viscous layer this can be 3 times for unstructured and 1.5 times for structured mesh. Guidelines for Auto Aerodynamics Surface meshing for ground vehicle aerodynamics should resolve both, boundary layer (less extent) and flow separation regimes (greater extent). In auto aerodynamics sector, the geometries and are quite complex and even if it is simpler, then also the flow physics is complex like modeling of under hood aerodynamics, rotating wheel aerodynamics, rear end aerodynamics etc. If we talk about Formula 1 race car, then it is complex from geometry as well as aerodynamics point of view. In ground vehicle aerodynamics (Mach < 0.3), pressure drag dominates over viscous drag (roughly pressure drag is more than 90% of total drag), so the mesh generation effort should be focused on capturing highly separated flows. Generally for surface gridding, element size variation less than 3% of vehicle reference length is acceptable with clustered mesh at corners and geometric turning. More to say, if it is less than or equal to 1%, the quality of mesh is of high standard. These numbers are independent of surface grid topology. Near the stagnant point and separation areas, the clustering must insure a ratio less than 1.5 between maximum and minimum element size. The user should first decide the degree of resolution needed for the simulation. The standard practice is to calculate an average surface element size, by means of viscous layer resolution criteria (y+ values) for near wall modeling. It can also be limit by aspect ratio at the wall. Aspect ratio values may vary from order of 2 to order of 4 depending upon the curvature in the geometry. Based on Ref 475[6] and our experience, choosing a coarse surface mesh will lead to an initial mesh of approx. 2-5 million cells. A medium resolution, which currently is part of the standard approach, will lead to meshes that consist of approx. 5-10 million cells, while a fine resolution will correspond to meshes beyond 10 HyperMesh9.0 Manual, Altair Inc. Macro Lanfrit, 2005, “Best Practices Guidelines for Handling Automotive External Aerodynamics with Fluent“, Fluent Deutschland GmbH, Birkenweg 14a, 64295 Darmstadt/Germany. 474 475

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million cells. The overall mesh largely depends on the complexity of model and volume mesh parameters. Improvement of Grid Quality Since the quality of the grid strongly influence the accuracy of the solution. It is required that the resulting grid should have the elements which are as regular as possible. Also the variation of the elements size should be as smooth as possible. After the grid is generated two procedures can be applied to improve the quality of the unstructured grid, namely Edge Swapping and Grid Smoothing. These procedures do not change the total numbers of the elements inside the grid and most of the commercial grid generators are having these for the grid quality476-477.

Blazek, J, “Computational Fluid Dynamics: Principles and Applications”, Elsevier Science Publication, Oxford, UK, 2005. 477 Sven Perzon, “On blockage effects in wind tunnel - A CFD Study”, SAE - 2001-01-0705, 2001. 476

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12 Appendix A A.1

Computer Code for a Transfinite Interpolation

This subroutine is based on a transfinite interpolation with a Lagrangian blending function. The following section describes the subroutine arguments. Nomenclature F

:

IL, JL :

Grid position (x, y, or z) Number of grid points in i and j-directions, respectively.

II(i), JJ(j) : This array stores the locations of known grid lines in i- and j-directions, respectively (1 for known grid lines). IS, IE, JS, JE : Starting and ending of region (computational) of interest. IMAX, JMAX : Array dimension Example: Consider surface IV in Figure 12.1. In this case, five grid lines are known: two lines at GH, two lines at GJLN, and one line at NO. The size of the grid is 95 in the I-direction and 50 in the Jdirection. Point G is at (70, 15) and point O is at (95, 25).

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268

269

270

Figure 12.1

Symmetry plane (XY)