Mesh Generation in CFD

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CFD Open Series Revision 1.86.1

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

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Introduction ................................................................................................................................ 20

1.1 The Black Box Dilemma .................................................................................................................................... 20 1.1.1 Trust the Mesh Generated by the Software, or Take a Proactive Approach? .................. 20 1.1.2 Not All the Meshes Created Equal ..................................................................................... 20 1.1.3 The Mesh Types ................................................................................................................. 21 1.1.4 Regional Meshing .............................................................................................................. 22 1.1.5 Simulation Cost .................................................................................................................. 22 1.1.6 Physics vs. Mesh ................................................................................................................ 23 1.1.7 Meshing Generalities ......................................................................................................... 23

Computer-Aided Design (CAD) ............................................................................................. 25

2.1 Software and Technology ................................................................................................................................ 25 2.1.1 Commercially Available CAD Systems: .............................................................................. 26 2.1.2 Freeware and Open Source ............................................................................................... 27 2.2 Solid (Geometry) Modeling ............................................................................................................................. 27 2.2.1 Principal Characteristics of a Solid Modeling Software ..................................................... 27 2.2.2 Feature-Based Modeling ................................................................................................... 27 2.2.3 Constraint-Based Modeling ............................................................................................... 27 2.2.4 Parametric Modeling ......................................................................................................... 28 2.2.5 History-Based Modeling .................................................................................................... 28 2.2.6 Associative Modeling ......................................................................................................... 28 2.3 Constructive Solid Geometry (CSG) Representation of Solids.......................................................... 28 2.3.1 Basic Primitives .................................................................................................................. 28 2.3.2 Regularized Boolean Operators ......................................................................................... 29 2.4 The CSG Tree ......................................................................................................................................................... 29 2.5 Geometry Related Issues For Mesh Generation ..................................................................................... 29 2.5.1 Understanding the Analysis Requirements ....................................................................... 30 2.5.2 Disfeaturing ....................................................................................................................... 31 2.5.3 “Dirty” Geometry ............................................................................................................... 32

Structured Mesh Generation ................................................................................................. 33

3.1 Classification of Mesh Generation Techniques ....................................................................................... 33 3.2 Grid Topology ....................................................................................................................................................... 35 3.3 Conformal Mapping (The Sponge Analogy) ............................................................................................. 35 3.4 Domain Decomposition with Multi-Blocking (Multi-Sponge Analogy) ........................................ 37 3.5 Further Remarks on Block Topology For Structured Multi-Block Meshing ............................... 39 3.5.1 Background and Literature Survey .................................................................................... 39 3.5.2 Hybrid Blocking .................................................................................................................. 41 3.5.3 Hierarchical Geometry Handling ....................................................................................... 43 3.5.4 Body Force Modeling ......................................................................................................... 44 3.5.5 Results ............................................................................................................................... 44 3.5.5.1 NASA CRM Wing-Body-Tail ...................................................................................... 44 3.5.5.2 Jet-Wing-Flap ........................................................................................................... 45 3.5.5.3 Engine-Wing-Flap .................................................................................................... 47 3.6 Field (Domain) Discretization Process (Mesh Generation) ............................................................... 48

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3.7 Structured Grid Generation ............................................................................................................................ 49 3.7.1 Complex Variables ............................................................................................................. 49 3.7.2 Algebraic Methods -Transfinite Interpolation (TFI) ........................................................... 49 3.7.2.1 Blending Function .................................................................................................... 49 3.7.2.1.1 Case Study- Rapid Meshing System For Turbomachinery .................................. 52 3.7.3 PDE Smoother .................................................................................................................... 53 3.7.3.1 Elliptic Schemes ....................................................................................................... 54 3.7.3.1.1 Case Study – Orthogonal Elliptic Mesh Smoother............................................. 55 3.7.3.1.2 Orthogonality Adjustment Algorithm ................................................................ 55 3.7.3.1.3 Stretching Functions ........................................................................................... 56 3.7.3.1.4 Extension to 3D .................................................................................................. 56 3.7.3.1.5 Mesh Quality Analysis ........................................................................................ 57 3.7.3.2 Hyperbolic Schemes ................................................................................................ 57 3.7.3.3 Parabolic Schemes ................................................................................................... 58 3.7.4 Variational Method............................................................................................................ 58 1.8 Structured Adaptive Grid.................................................................................................................................. 59 3.7.5 Case Study – 2D Euler Flow Over an NACA Airfoil ............................................................. 60

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Un-Structured Mesh Generation .......................................................................................... 63 4.1 Advancing Front Method ................................................................................................................................. 63 4.1.1 Advancing Front Triangular Mesh Generator .................................................................... 64 4.2 Advancing Front Quadrilateral Meshing Using Triangle Transformations ................................ 66 4.2.1 Outline of Quad-Morphing Algorithm ............................................................................... 66 4.2.1.1 Initial Triangle Mesh ................................................................................................ 66 4.2.1.2 Front Definition ....................................................................................................... 66 4.2.1.3 Front Edge Classification ......................................................................................... 66 4.2.1.4 Front Edge Processing ............................................................................................. 66 4.2.1.5 Topological Clean-up and Final Smoothing Process ................................................ 68 4.2.1.6 Example Problems ................................................................................................... 68 4.2.1.7 Conclusion ............................................................................................................... 71 4.3 Delaney Triangulation Method ...................................................................................................................... 71 4.3.1 Properties of Delaunay Triangulation................................................................................ 71 4.3.1.1 Delaunay Lemma ..................................................................................................... 72 4.3.1.2 Compactness ........................................................................................................... 72 4.3.2 Algorithms ......................................................................................................................... 72 4.3.3 Advantages ........................................................................................................................ 74 4.3.4 Delaunay Adaptive Refinement ......................................................................................... 75 4.3.5 Voronoi Diagrams .............................................................................................................. 76 4.3.6 Restricted Delaunay Triangulation .................................................................................... 76 4.4 Anisotropic Mesh Generation ........................................................................................................................ 77 4.4.1 Case Study - Anisotropic Mesh Generation via Discretized Riemannian Delaunay Triangulations ................................................................................................................................... 78 4.4.1.1 Anisotropic Delaunay Triangulations ...................................................................... 80 4.4.1.1.1 Locally Uniform Anisotropic Meshes.................................................................. 80 4.4.1.1.2 Metric Tensor ..................................................................................................... 80 4.4.1.1.3 Distortion ............................................................................................................ 81 4.4.1.1.4 Locally Uniform Anisotropic Meshes.................................................................. 82 4.4.1.1.5 The Star Set ........................................................................................................ 83

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4.4.1.1.6 Stars and Inconsistencies ................................................................................... 83 4.4.1.2 Refinement Algorithm ............................................................................................. 84 4.4.1.3 Discussion on the Parameters ................................................................................. 84 4.4.1.3.1 Parameter φ0 ..................................................................................................... 84 4.4.1.3.2 Parameters r0 and ρ0 .......................................................................................... 85 4.4.1.3.3 Parameters β and δ ............................................................................................ 86 4.4.1.3.4 Parameters σ0 ..................................................................................................... 87 4.4.1.4 Results and Limitations ........................................................................................... 87 4.4.1.4.1 Uniform Metric Fields......................................................................................... 87 4.4.1.4.2 Shock-Based Metric Fields on Planar Domains .................................................. 87 4.4.1.4.3 Starred ................................................................................................................ 87 4.4.1.4.4 Hyperbolic .......................................................................................................... 88 4.4.1.4.5 Swirl .................................................................................................................... 89 4.4.1.4.6 Curvature-Based Metrics Fields on Surfaces ...................................................... 89 4.4.1.4.7 Optimization ....................................................................................................... 89 4.4.1.5 Discrete Riemannian Voronoi Diagrams ................................................................. 90 4.4.1.5.1 Advantages Over Isotropic Canvasses ................................................................ 91 4.4.1.5.2 Straight Riemannian Delaunay Triangulation..................................................... 91 4.4.1.5.3 Curved Riemannian Delaunay Triangulation ...................................................... 92 4.4.1.6 Conclusion ............................................................................................................... 94 4.5 Octree Decomposition ....................................................................................................................................... 94 4.6 Unstructured Hexahedral Meshes................................................................................................................ 96 4.6.1 Conversion of Triangular to Quadrilateral Meshes (2D) ................................................... 97 4.7 Overset Grids ........................................................................................................................................................ 98 4.8 Cartesian Grids ..................................................................................................................................................... 99 4.8.1 Background and Cartesian Grid Origins............................................................................. 99 4.8.2 Cartesian Grids Schemes ................................................................................................. 101 4.8.2.1 Adaptive Mesh Refinement ................................................................................... 102 4.8.2.2 Immersed Boundary Methods............................................................................... 104 4.8.2.3 Volume of Fluid Methods ...................................................................................... 105 4.8.2.4 Reconstruction Schemes ....................................................................................... 105 4.8.2.5 Cut Cell Based Methods ........................................................................................ 106 4.8.2.6 Chimera Grid Schemes .......................................................................................... 107 4.8.2.7 Hybrid Grid Schemes ............................................................................................. 107 4.8.2.7.1 Composite Grid Approach ................................................................................ 108 4.8.3 Discussion ........................................................................................................................ 110 4.9 Trimmed (SAMM) Cells ................................................................................................................................. 111 4.10 Polyhedral Cells .......................................................................................................................................... 111 4.10.1 Cell Decomposition .......................................................................................................... 111 4.10.2 Mesh Duality .................................................................................................................... 112 4.10.3 Methodology ................................................................................................................... 113 4.11 Treatment of Boundary Layer .............................................................................................................. 114 4.12 Domain Mesh Stretching in Unstructured Environment ........................................................... 114 4.13 Spatial (Field) Discretization ................................................................................................................ 116 4.14 Considerations for the Navier-Stokes Equation ............................................................................ 117 4.15 Unstructured Quadrilateral Mesh Generation ............................................................................... 119 4.15.1 Geometry Representation ............................................................................................... 119 4.15.2 Local Mesh Generation Algorithm ................................................................................... 120

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4.16

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Connectivity Information and Data Structure................................................................................ 121

Hybrid Meshes ........................................................................................................................ 123 5.1 Accuracy Consideration................................................................................................................................. 123 5.1.1 Comparing Mesh Type for Viscous Accuracy................................................................... 124 5.1.2 Effect of Prismatic Extrusion Sub-Layer in Viscous Layer ................................................ 124 5.2 Meshing Tools in CD-Adapco® .................................................................................................................... 125 5.2.1 A Novel Methodology for Extrusion Layer Meshing ........................................................ 126 5.3 Mesh Refinement.............................................................................................................................................. 128 5.3.1 R-refinement.................................................................................................................... 129 5.3.2 H-refinement ................................................................................................................... 129 5.3.2.1 Isotropic vs. Anisotropic Meshing ......................................................................... 130 5.3.3 P-refinement.................................................................................................................... 131 5.4 Mesh Modification Operators...................................................................................................................... 131 5.4.1 Coarsening Triangulation Regions ................................................................................... 131 5.4.1.1 Case Study - Numerical Testing for Engine Nacelle ............................................... 132 5.4.1.2 Coarsening With/Without Local Re-Triangulation ............................................... 133 5.4.2 Refinement of Triangulation Region ............................................................................... 134 5.4.2.1 Local Re-Triangulation ........................................................................................... 134 5.4.3 Refinement of Hexahedral Region (Near Wall) ............................................................... 136 5.4.3.1 Improvement to Near-Field Grid Generation Procedure (Hexahedral) ................ 137 5.4.4 Discretization Improvement Through Chimera Technique for Sharp Corners ................ 139 5.5 Case Study 1 - Hybrid Unstructured Meshes for Common Research Model (CRM & JSM) via ANSA®................................................................................................................................................................................ 139 5.5.1 Geometry and Mesh Generation Background................................................................. 140 5.5.2 Geometry Handling.......................................................................................................... 141 5.5.2.1 The CRM Model .................................................................................................... 141 5.5.2.2 The JSM Model ...................................................................................................... 142 5.5.3 Surface Meshing .............................................................................................................. 142 5.5.4 Volume Meshing .............................................................................................................. 146 5.5.4.1 Extrusion Layers Generation ................................................................................. 146 5.5.4.2 Tetra Meshing........................................................................................................ 148 5.5.5 Sample CFD Results ......................................................................................................... 148 5.5.5.1 CRM ....................................................................................................................... 148 5.5.5.2 JSM ........................................................................................................................ 149 5.6 Case Study 2 - A 3D Hybrid Grid Generation Technique and a Multigrid/Parallel Algorithm Based on Anisotropic Agglomeration Approach ............................................................................................. 150 5.6.1 Statement of the Problem ............................................................................................... 150 5.6.2 Introduction, Background and Contributors ................................................................... 150 5.6.3 Hybrid Grid Generation Technique based on Anisotropic Agglomeration Approach ..... 152 5.6.3.1 Prism Grid Generation Method Based on Anisotropic Agglomeration Approach 153 5.6.3.1.1 Volume Agglomeration .................................................................................... 153 5.6.3.1.2 Interface Agglomeration .................................................................................. 153 5.6.4 Multigrid/Parallel Algorithm............................................................................................ 154 5.6.5 Multi-Level Coarser Grid Generation Based on Anisotropic Agglomeration Approach .. 155 5.6.6 Applications and Discussions ........................................................................................... 156 5.6.6.1 Subsonic Turbulence Flow over 2D 30P30N Airfoil ............................................... 156 5.6.6.2 Transonic Turbulence Flow over ONERA M6 Wing ............................................... 157

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5.6.6.3 Transonic Turbulence Flow over DLR-F6 Wing-Body Configuration ..................... 157 5.6.7 Concluding Remarks ........................................................................................................ 162 5.7 Recent Advances in Hybrid Mesh Generation and Literature Survey ........................................ 162 5.7.1 Parallel Consideration...................................................................................................... 162 5.7.2 Local Remeshing .............................................................................................................. 162 5.7.3 Background Mesh ............................................................................................................ 163 5.7.4 Boundary Viscous Meshes & Sharp Corners ................................................................... 164 5.7.5 Procedures for Mesh Generation .................................................................................... 164 5.7.6 Dynamic Mesh ................................................................................................................. 166 5.7.7 Adaptation ....................................................................................................................... 167 5.7.8 Special Issues ................................................................................................................... 168 5.7.8.1 Centaur® ................................................................................................................. 168 5.7.8.2 Pointwise® .............................................................................................................. 169 5.8 Listing of Available Meshing Software .................................................................................................... 170 5.9 Meshing Challenges for Applied Aerodynamics .................................................................................. 170 5.9.1 Experience ....................................................................................................................... 170 5.9.2 Observations .................................................................................................................... 171

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Adaptive Mesh (Unstructured) ......................................................................................... 172

6.1 Adaptive Meshing by Subdivision ............................................................................................................. 172 6.2 Adaptive Mesh Refinement (AMR) ........................................................................................................... 173 6.2.1 Generalities...................................................................................................................... 174 6.2.2 Cell Division for a Geometry ............................................................................................ 175 6.2.2.1 Division Criteria ..................................................................................................... 176 6.2.3 Uniform AMR ................................................................................................................... 176 6.2.3.1 Transient Inviscid Flow .......................................................................................... 177 6.2.4 Case Study 1 - An Adaptive Hybrid Mesh Generation Method for Complex Geometries 177 6.2.4.1 Mesh Stitching ....................................................................................................... 178 6.2.4.1.1 Removal of Background Mesh Elements .......................................................... 178 6.2.4.2 Triangulation ......................................................................................................... 178 6.2.4.3 Test Cases .............................................................................................................. 178 6.2.4.3.1 30P30N Multi-Element Airfoil .......................................................................... 178 6.2.4.3.2 2D Fuel Cell Slice............................................................................................... 179 6.2.5 Case Study 2 – Unstructured Mesh Adaptation for 2D Airfoil......................................... 179 6.2.5.1 Adaption Control Mechanism ............................................................................... 181 6.2.6 Case Study 3 – Parallel Implementation of Unstructured Mesh Refinement of Duct Flow 181 6.2.7 Case Study 4 – Generic Transonic Store Release............................................................. 182 6.2.8 Case Study 5 - Adaptive Hybrid Mesh Refinement for Multiphysics Applications .......... 184 6.2.8.1 Adaptive Hybrid Mesh Optimization ..................................................................... 184 6.2.8.2 Hybrid Adaptive Meshing ...................................................................................... 186 6.2.8.3 Meshing and Load Balancing. ................................................................................ 188 6.2.8.4 Conclusions............................................................................................................ 189 6.3 Strategies for Driving Mesh Adaptation in CFD ................................................................................... 189 6.3.1.1 Feature-Based Adaption ........................................................................................ 189 6.3.1.2 Discretization Error and Recovery-Based Adaption .............................................. 191 6.3.1.3 Adjoint-Based Adaption ........................................................................................ 191

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6.3.1.4 Truncation Error-Based Adaption.......................................................................... 191 6.3.2 Current Approach for Performing Mesh Adaptation ...................................................... 191 6.3.3 Case Study - Mesh Adaption Results for 1D Burgers Equation (Re = 32) ........................ 193 6.4 A Solution-Based Adaptive Redistribution Method for Unstructured Meshes....................... 194 6.4.1 Introduction & Literature Survey .................................................................................... 194 6.4.2 Feature Detection ............................................................................................................ 195 6.4.3 Extraction of Solution Feature Surfaces .......................................................................... 196 6.4.3.1 Case Study 1 - NACA0012 Wing-Section ............................................................... 197 6.4.3.2 Case Study 2 - Capsule Model ............................................................................... 199

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Dynamic Meshing ................................................................................................................... 202

7.1 Type of Mesh Motion ...................................................................................................................................... 202 7.2 Mesh Deformation ........................................................................................................................................... 203 7.3 Finite Volume in Dynamic Mesh ................................................................................................................ 203 7.4 Dynamic Mesh Techniques .......................................................................................................................... 204 7.4.1 Laplacian Mesh Morphing ............................................................................................... 204 7.4.2 Pseudo-Solid Equation ..................................................................................................... 205 7.4.2.1 Case Study – Motion of a Cylinder ........................................................................ 205 7.4.3 Biharmonic Equation ....................................................................................................... 206 7.4.4 Radial Basis Function ....................................................................................................... 207 7.4.4.1 Case Study - Application to Outlet Guide Vane Geometry Matching ................... 208 7.4.5 Generalized Grid Interface .............................................................................................. 210 7.4.6 Overset Methods ............................................................................................................. 212 7.4.7 Delaunay Method ............................................................................................................ 212 7.4.7.1 Case Study - Airfoil Rotation .................................................................................. 213 7.4.8 Spring Analogy ................................................................................................................. 214 7.4.9 Six Degrees of Ferndom (6 DOF)...................................................................................... 214 7.4.9.1 Transitional Deformation ...................................................................................... 214 7.4.9.2 Rotational Deformation......................................................................................... 214 7.5 Dynamically Adaptive Mesh Refinement (DAMR) ............................................................................. 215 7.5.1 Case Study - Dynamically Adaptive Mesh Refinement FDTD: A Stable and Efficient Technique for Time-Domain Simulations ....................................................................................... 216 7.5.1.1 Numerical Results .................................................................................................. 217

Assessment of Mesh Types ................................................................................................. 219 8.1 Structured vs. Unstructured ........................................................................................................................ 219 8.1.1 Time and Memory ........................................................................................................... 219 8.1.2 Resolution ........................................................................................................................ 219 8.1.3 Alignment ........................................................................................................................ 219 8.1.4 Definable Normal............................................................................................................. 220 Effect of Cell Topology in Truncation Error ..................................................................... 220 8.1.5 Polyhedral vs. Tetrahedral ............................................................................................... 220 8.1.5.1 Boundary Prismatic Cells ....................................................................................... 221 8.2 Accuracy Assessment of Gradient Calculation Methods ................................................................. 223 8.2.1 Geometric Properties ...................................................................................................... 223 8.2.2 Literature Survey ............................................................................................................. 223 8.2.3 Gradient Calculation ........................................................................................................ 224

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8.2.3.1 Green-Gauss Gradient Method ............................................................................. 224 8.2.3.2 GG-Simple Face Averaging .................................................................................... 225 8.2.3.3 GG-Inverse Distance Weighted (IDW) Face Interpolation .................................... 225 8.2.4 Visual Inspection .............................................................................................................. 227 8.2.5 Results Based on L2 Norm ................................................................................................ 228 8.2.6 Concluding Remarks ........................................................................................................ 229

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Case Studies Involving Comparisons of Structured vs. Unstructured Meshes . 230 9.1 Case Study 1 – Flow through Pipe with 90 degree Bend ................................................................. 230 9.2 Case Study 2 - Comparison of Hexa-Structured and Hybrid-Unstructured Meshing Approaches for Numerical Prediction of the Flow Around Marine Propellers .................................. 231 9.2.1 Introduction & Contributions .......................................................................................... 232 9.2.2 Propeller Models ............................................................................................................. 233 9.2.3 Numerical Method........................................................................................................... 233 9.2.4 Meshing ........................................................................................................................... 234 9.2.5 Results ............................................................................................................................. 235 9.2.5.1 Propeller A ............................................................................................................. 236 9.2.5.2 Propeller P5168 ..................................................................................................... 237 9.2.6 Conclusions ...................................................................................................................... 238 9.3 Case Study 3 – Structure & Unstructured Hybrid Meshing and its effect on Quality of Solution on Turbine Blade ........................................................................................................................................ 239 9.3.1 Applications ..................................................................................................................... 239 9.3.2 Results ............................................................................................................................. 239 9.4 Case Study 4 - Evaluation of Structured vs. Unstructured Meshes for Simulating Respiratory Aerosol Dynamics ............................................................................................................................... 240 9.4.1 Bifurcation Model, Boundary Conditions, and Contributions ......................................... 240 9.4.2 Mesh Types ...................................................................................................................... 241 9.4.2.1 Structured .............................................................................................................. 242 9.4.2.2 Unstructured ......................................................................................................... 242 9.4.3 Governing Equations ....................................................................................................... 243 9.4.4 Numeric Method ............................................................................................................. 244 9.4.5 Results ............................................................................................................................. 246 9.4.5.1 Validation Studies .................................................................................................. 246 9.4.5.2 Grid Convergence .................................................................................................. 246 9.4.5.3 Velocity Fields ........................................................................................................ 248 9.4.5.4 Particle Deposition ................................................................................................ 248 9.4.6 Discussion ........................................................................................................................ 251 9.4.6.1 Advantages of Hexahedral Structured Mesh ........................................................ 252 9.4.7 Conclusion ....................................................................................................................... 253 9.5 Case Study 5 - Comparison Between Structured Hexahedral and Hybrid Tetrahedral Meshes Generated by Commercial Software for CFD Hydraulic Turbine Analysis........................... 253 9.5.1 Problem Description ........................................................................................................ 254 9.5.2 Geometry ......................................................................................................................... 255 9.5.3 Mesh Description............................................................................................................. 255 9.5.3.1 Structured Hexahedral Meshes ............................................................................. 256 9.5.3.2 Hybrid Tetrahedral Mesh ...................................................................................... 257 9.5.4 CFD Solution Strategy and Boundary Conditions ............................................................ 257 9.5.5 Results ............................................................................................................................. 258

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9.5.6

Conclusion ....................................................................................................................... 260

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Mesh Sensitivity and Mesh Independence Study ........................................................ 262

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Mesh Quality ............................................................................................................................ 270

10.1 Different Types of Mesh Sensitivity.................................................................................................... 262 10.1.1 Symbolic Differentiation .................................................................................................. 262 10.1.2 Automatic Differentiation ............................................................................................... 262 10.1.2.1 Symbolic vs Automatic Differentiation.................................................................. 262 10.1.3 Finite Differencing ........................................................................................................... 262 10.2 Mesh Sensitivity via Direct Differentiation (DD) .......................................................................... 263 10.2.1 Surface Modeling Using NURBS....................................................................................... 263 10.2.1.1 Case Study - 2D Study of Airfoil Grid Sensitivity via Direct Differentiation (DD) .. 265 10.3 Adjoint Variable Sensitivity Analysis (AV) ...................................................................................... 266 10.4 Mesh Independence Study ..................................................................................................................... 268

11.1 Background .................................................................................................................................................. 270 11.2 Mesh Quality Metric .................................................................................................................................. 270 11.2.1 Mesh Quality from User’s Perspective ............................................................................ 272 11.2.2 Mesh Quality from Researcher’s Perspective ................................................................. 272 11.2.3 Mesh Quality from Solver’s Perspective.......................................................................... 272 11.2.3.1 CFD++..................................................................................................................... 273 11.2.3.2 Fluent and CFX ....................................................................................................... 273 11.2.3.3 Kestrel .................................................................................................................... 273 11.2.3.4 STAR-CCM+ ............................................................................................................ 274 11.2.3.5 Deducing Results ................................................................................................... 275 11.2.4 Some Geometric Properties ............................................................................................ 275 11.2.4.1 Aspect ratio ........................................................................................................... 275 11.2.4.2 Orthogonality ........................................................................................................ 275 11.2.4.3 Skewness ............................................................................................................... 276 11.2.4.4 Warpage ................................................................................................................ 276 11.2.4.5 Jacobian ................................................................................................................. 276 11.2.4.6 Tetrahedral Volume............................................................................................... 276 11.2.4.7 Polygonal Face Area and Centroid ........................................................................ 277 11.2.4.8 Polyhedral Volume and Centroid .......................................................................... 278 11.3 Best Practice for Mesh Generation ..................................................................................................... 278 11.3.1 Geometry Modeling and Geometry Cleanup .................................................................. 279 11.3.2 Computational Domain ................................................................................................... 279 11.3.3 Choice of Grid .................................................................................................................. 279 11.3.4 Surface Meshing .............................................................................................................. 280 11.3.5 Volume Meshing .............................................................................................................. 280 11.3.6 Boundary Layer Meshing ................................................................................................. 280 11.3.7 Guidelines for Aerodynamics in General ......................................................................... 281 11.3.8 Guidelines for Automotive Aerodynamics ...................................................................... 281 11.3.8.1 Case Study - Best Practice & Guidelines for Handling Automotive External Mesh Generation with FLUENT ........................................................................................................... 282 11.3.8.1.1 Meshing Strategies ........................................................................................... 282 11.3.8.1.2 Strategy A (Adaption) ....................................................................................... 282

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11.3.8.1.3 Strategy B (Boxes) ............................................................................................ 282 11.3.8.1.4 Strategy C (Controls) ........................................................................................ 283 11.3.8.1.5 Surface Meshing ............................................................................................... 283 11.3.8.1.6 Volume Meshing............................................................................................... 284 11.3.8.1.7 Prismatic Layers ................................................................................................ 284 11.3.8.1.8 Hybrid Mesh Transition .................................................................................... 285 11.3.8.1.9 Tetrahedral Mesh ............................................................................................. 285 11.3.8.1.10 Hex-core Mesh ............................................................................................... 286 11.3.8.1.11 Buffer Layers ................................................................................................... 286 11.3.8.1.12 Local Refinement ............................................................................................ 286 11.3.8.1.13 Checking Quality ............................................................................................. 287 11.3.9 Improvement of Grid Quality .......................................................................................... 287

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Appendix A ............................................................................................................................... 288

A.1

Computer Code for a Transfinite Interpolation ................................................................................... 288

List of Tables Table 3.1 NASA CRM Free-Stream Conditions ................................................................................................ 44 Table 4.1 Nomenclature ........................................................................................................................................... 79 Table 4.2 Comparison of the number of vertices and quality of the mesh for different values of δ - (Courtesy of [Labbe]) .................................................................................................................................................. 86 Table 5.1 Near-Field Grid Details....................................................................................................................... 138 Table 5.2 Abbreviations......................................................................................................................................... 140 Table 5.3 Currently Available Grid Generation Software ........................................................................ 169 Table 9.1 Dimensions of Domains – (Courtesy of Morgut & Nobile) .................................................. 234 Table 9.2 Grids for Propeller A– (Courtesy of Morgut & Nobile) ......................................................... 234 Table 9.3 Grids for Propeller P5168 – (Courtesy of Morgut & Nobile) .............................................. 234 Table 9.4 Results of Propeller A– (Courtesy of Morgut & Nobile) ....................................................... 236 Table 9.5 Experimental setup of Propeller P5168 ..................................................................................... 237 Table 9.6 Relative Percentage Differences of Computed Values Between Finer and Coarser Mesh for propeller P5168 – (Courtesy of Morgut & Nobile) ...................................................................... 237 Table 9.7 Grid Convergence – (Courtesy of Samir Vinchurkar & Worth Longest) ........................ 247 Table 9.8 Mesh Densities for Structured Hexahedral and Hybrid Un-Structural Tetrahedral – (Courtesy of Rousseau et al.) ................................................................................................................................... 256 Table 10.1 Pros & Cons of Different Grid Sensitivity Method (NDV = Number of Design Variable) ........................................................................................................................................................................... 267

List of Figures Figure 1.1 Meshes Created using ANSYS Mosaic-Enabled Poly-Hex Core Meshing - Courtesy of Sheffield Hallam University ......................................................................................................................................... 21 Figure 1.2 Methodology of General Grid Generation .................................................................................... 23 Figure 2.1 Anatomy of commercial CAD Systems .......................................................................................... 26 Figure 2.2 Fighter Airplane F-16 calculation ................................................................................................... 26 Figure 2.3 Example of a CSG Tree ......................................................................................................................... 29 Figure 2.4 Geometry Import and Preparation................................................................................................. 30 Figure 2.5 Different Analysis Require Different Geometric Representations .................................... 31 Figure 2.6 Small Feature (Left) vs Removed (Right) .................................................................................... 31 Figure 3.1 Classification of Grid Generation Algorithms (Courtesy of Steven Owen) .................... 34

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Figure 3.2 Domain Topology (O-Type, C-Type, and H-Type; from left to right) ............................... 35 Figure 3.3 Sponge Analogy ...................................................................................................................................... 35 Figure 3.4 Dual Block Grid Topology for a Generic Wing-Fuselage Configuration .......................... 36 Figure 3.5 Topology and Grid on a Multi-Block Wings via GridPro® ..................................................... 37 Figure 3.6 Multi Block Representation for C-H Mesh Around a Wing ................................................... 37 Figure 3.7 Multi-Block Gridding of Turbine Blade - (Courtesy of GridPro) ....................................... 37 Figure 3.8 Schwarz Concept of Iterating Between Domains ..................................................................... 38 Figure 3.9 Domain Decomposition for M6 Wing using TIL Scripts (Courtesy of GridPro) ........... 38 Figure 3.10 Jet-Wing-Flap: Medial Axis Transform (Compression Shock) and Expansion Features Close to the Geometry – Courtesy of [Ali et al]................................................................................. 41 Figure 3.11 Two dimensional jet-wing-flap geometry: (a) the distance field; (b) distance field wrap and (c) corresponding medial axis (d) hybrid blocking around 2D geometry – Courtesy of [Ali et al] .............................................................................................................................................................................. 42 Figure 3.12 Hierarchical Geometry Handling Strategy................................................................................ 43 Figure 3.13 NASA CRM Wing-Body-Tail (c) Hybrid Blocking (d) Mesh Cut Section – Courtesy of [Ali et al.] ........................................................................................................................................................................ 45 Figure 3.14 Jet-Wing-Flap (a) CAD, (b) CAD and the Medial Axis Cut Section, (c) Inner Hybrid Blocking – Courtesy of [Ali et al] ............................................................................................................................... 46 Figure 3.15 Jet-Wing-Flap with Modified Inner Blocking (To Accommodate Shear Layers) – Courtesy of [Ali et al]...................................................................................................................................................... 46 Figure 3.16 Engine-Wing-Flap (a) Schematics Showing Geometric Zones and Domains with Different Block Topologies (b) Cut Section at z = 0 Plane – Courtesy of [Ali et al] ............................. 47 Figure 3.17 Examples of Structured grids for Turbine Blade ................................................................... 48 Figure 3.18 Example of Unstructured Tetrahedral Grids ........................................................................... 48 Figure 3.19 Exponential Distribution Functions ........................................................................................... 50 Figure 3.20 Grid for Dual-Block Generic Wing-Fuselage Geometry....................................................... 51 Figure 3.21 Hyperbolic Tangent Distribution Functions ............................................................................ 51 Figure 3.22 Single Passage PADRAM Mesh for the Swept Back OGV..................................................... 52 Figure 3.23 Detail of the Splitter C-Mesh, Hub Mesh and the Engine-Core Exit Mesh ................... 53 Figure 3.24 Typical Elliptic Grid for an Airfoil with Orthogonality Enforced on the Boundary . 55 Figure 3.25 Orthogonality Adjustments – (Courtesy of Chaitanya Varier) ......................................... 56 Figure 3.26 Euler Solution on a HSCT Wing-Fuselage ................................................................................. 58 Figure 3.27 Folded Grid by Transfinite Interpolation - Smooth Grid by Winslow Functional.... 59 Figure 3.28 1D Weight Function for High Gradient and Curvature........................................................ 60 Figure 3.29 Mesh and Mach Contours for Transonic Flow ........................................................................ 61 Figure 3.30 Grid Adaption and Mack Contours for Supersonic Airfoil ................................................. 62 Figure 4.1 Closing stage of a Moving Front Method ...................................................................................... 63 Figure 4.2 Mesh parameters ................................................................................................................................... 64 Figure 4.3 Surface Mesh of SGI Logo ................................................................................................................... 65 Figure 4.4 States of a front edge – (Courtesy of Owen et al.) .................................................................... 66 Figure 4.5 Steps demonstrating process of generating a quadrilateral from Front NA-NB (Courtesy of Owen et al.) .............................................................................................................................................. 67 Figure 4.6 Progression of Q-Morph- (Courtesy of Owen et al.) ................................................................ 68 Figure 4.7 Comparison of Q-Morph with Lee’s Algorithm Illustrating Element Boundary.......... 69 Figure 4.8 Results of Q-Morph Compared with Lee’s (1994) Advancing Front Indirect............... 69 Figure 4.9 Large Transition Mesh for CFD Application - (Courtesy of Owen et al.) ........................ 70 Figure 4.10 Success and failure of the in sphere test of abcd with e. .................................................... 72 Figure 4.11 Relationship Between Delaunay Triangles and the Voronoi Diagram ......................... 73 Figure 4.12 Two-Three Tetrahedral swap ........................................................................................................ 73 Figure 4.13 Robust and Fast way to Detect if point D lies in the Circumcircle of A, B, C ............... 74

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Figure 4.14 Delaunay Triangulation (white) and Voronoi Diagram (blue) – Courtesy of [Labbe]) ............................................................................................................................................................................... 74 Figure 4.15 2D Delaunay triangulation of a set of vertices (black) restricted to a curve (blue) 77 Figure 4.16 Representation of a 3D Metric with Eigenvalues λ1, λ2 and λ3 as an Ellipsoid – (Courtesy of [Labbe]) ..................................................................................................................................................... 81 Figure 4.17 An anisotropic uniform Delaunay triangulation (orange) and the corresponding stretched ............................................................................................................................................................................. 83 Figure 4.18 Two stars Sp and Sq forming an inconsistent configuration - (Courtesy of [Labbe]) ................................................................................................................................................................................................. 84 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]) ....................................................................................................................... 85 Figure 4.20 A square of side 10 and centered on the origin, endowed with the “Starred” metric field ........................................................................................................................................................................................ 88 Figure 4.21 Anisotropic Triangulation of a Rectangle Endowed with the Hyperbolic Shock Metric Field - (Courtesy of [Labbe]) ........................................................................................................................ 88 Figure 4.22 A square of side 6 and centered on the origin, endowed with the “Swirl” metric field - (Courtesy of Labbé et al.) .............................................................................................................................. 89 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.) ....................................................................................................................................................................... 90 Figure 4.24 Isotropic and Anisotropic Canvas Sampling - (Courtesy of [Labbe]) ............................ 91 Figure 4.25 Unit Sphere Endowed with the Hyperbolic Metric field - (Courtesy of [Labbe]) ..... 92 Figure 4.26 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]) ............................................................................... 92 Figure 4.27 Discrete Riemannian Voronoi Diagram (top) and Curved Riemannian Delaunay Triangulation (bottom) Endowed with the Hyperbolic Shock Metric Field - (Courtesy of [Labbe]) ................................................................................................................................................................................................. 93 Figure 4.28 Converging of an Octree Decomposition Around an Airfoil .............................................. 94 Figure 4.29 A close-up view of Nasty Cheese a well-known test-case featuring 30◦ Dihedral angles – (Courtesy’s of [Mar´echal]) ........................................................................................................................ 95 Figure 4.30 Hierarchy of Meshing Methodologies......................................................................................... 96 Figure 4.31 Quadrilateral Mesh Generation..................................................................................................... 97 Figure 4.32 Overset Mesh Combination............................................................................................................. 98 Figure 4.33 Two Counter-Rotating Objects Embedded in Two Overset Regions with Background Mesh – (Courtesy of Siemens) .......................................................................................................... 99 Figure 4.34 Example of Cartesian Grid Near Curved Surface – (Courtesy of NASA Ames) .......... 99 Figure 4.35 Solid Surface Over-Layer Cartesian Cell and Resulting Cut and Split Cell – (Courtesy of NASA Ames) ............................................................................................................................................................... 100 Figure 4.36 Example of Merge Cell Creation – (Courtesy of NASA Ames) ........................................ 101 Figure 4.37 Semi-Automatic Cartesian Mesh Generation with CFOW – Courtesy of Kawasaki .............................................................................................................................................................................................. 101 Figure 4.38 Boundary-Fitted Layer Grid for Multi-Element Airfoil .................................................... 102 Figure 4.39 Example Adaptive Grid for Supersonic Wedge Flow – (Courtesy of NASA Ames) 102 Figure 4.40 Schematic image of Adaptive Mesh Refinement – (Courtesy of Hiroshi Abe) ........ 103 Figure 4.41 Pressure Contours in 2D Backward Step ............................................................................... 104 Figure 4.42 Example Chimera Grid Near Curved Surface (Courtesy of NASA Ames) ................. 106 Figure 4.43 Example Hybrid Grid Near Curved Surface – (Courtesy of NASA Ames).................. 108 Figure 4.44 Basic Superposition Example – (Courtesy of Kalinin, Mazo and Isaev) .................... 109

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Figure 4.45 Example of Cartesian Grid on a Generic Airplane – (Source: Richard Smith 1996) .............................................................................................................................................................................................. 110 Figure 4.46 Meshing Types in SAMM ............................................................................................................... 111 Figure 4.47 Typical Polyhedral Cell and their Decomposition .............................................................. 112 Figure 4.48 Polyhedral meshing using Delaunay triangulation............................................................ 112 Figure 4.49 Dual surface Triangulation resulting in Polyhedron ......................................................... 113 Figure 4.50 Boundary Layer Prisms Generated on a Cascade of a 2D Triangulation and Dual Polyhedron ...................................................................................................................................................................... 115 Figure 4.51 Concept of cascading for boundary layer in 3D................................................................... 116 Figure 4.52 Dual Mesh for Mixed Triangular-Quadrilateral Unstructured Mesh ................................ 117 Figure 4.53 Conventional configuration geometry (a), final structural mesh (Courtesy of Hwang & Martins) ........................................................................................................................................................ 119 Figure 4.54 The Six Steps of the Unstructured Quad Meshing Algorithm ........................................ 120 Figure 5.1 Hybrid Grid and Steady State Solution ...................................................................................... 123 Figure 5.2 Comparison of different mesh types for RANS Computations......................................... 124 Figure 5.3 Constructions of Hybrid mesh ...................................................................................................... 125 Figure 5.4 Predominantly polyhedral meshing ........................................................................................... 125 Figure 5.5 Combined Volume and Extrusion Layer Meshes ................................................................... 126 Figure 5.6 Polyhedral cells for HL CRM – Courtesy of Siemens ............................................................ 126 Figure 5.7 Meshing Tools in CD-Adapco ......................................................................................................... 127 Figure 5.8 Meshes Generated by a) Proposed Algorithm and b) Leading Commercial Vendor .............................................................................................................................................................................................. 128 Figure 5.9 Adaptive Mesh Refinement Types ............................................................................................... 129 Figure 5.10 An H-refinement mesh about a Shuttle-like body (left) and Computed CP (right) 130 Figure 5.11 Isotropic vs. Anisotropic Meshing............................................................................................. 130 Figure 5.12 Coarsening by Edge Collapsing – Courtesy of [Cavallo]................................................... 132 Figure 5.13 Hierarchy of Successively Coarser Meshes Obtained by Uniform ............................... 133 Figure 5.14 Coarsening ratio for coarsening with and without local Retriangulation. ............... 134 Figure 5.15 3 to 2 and 2 to 3 Swap ................................................................................................................... 135 Figure 5.16 Comparison of SolarChimera5 and Solar Grid at x =1454 inch plane; Viscous Wall Surface in Dark .............................................................................................................................................................. 138 Figure 5.17 Comparison of Coarse, Medium and Fine Grids: lateral view on fore-body with Symmetry......................................................................................................................................................................... 138 Figure 5.18 local dissipation error of drag coefficient on field cut-plane at x=1400 inch; isometric/downstream view ................................................................................................................................... 139 Figure 5.19 JSM Model with Engine Nacelle.................................................................................................. 140 Figure 5.20 Computational Domain of the HL-CRM Gapped Flaps Model ........................................ 141 Figure 5.21 Computational Domain and Separation of Zones of the JSM Model with Engine Nacelle ............................................................................................................................................................................... 142 Figure 5.22 Three Locations of Problematic Areas of the JSM Geometry for the Generation of Boundary Layers ........................................................................................................................................................... 143 Figure 5.23 Batch Mesh setup for the JSM Model with Size Boxes for Local Mesh Control....... 143 Figure 5.24 Resulting Layers for Isotropic Surface Mesh (Top) and Anisotropic (Bottom) ..... 144 Figure 5.25 Close ups of Coarse CRM Gapped Flap Model with Comparison of Tridiagonal Dominant (Top) vs. Quad Dominant (Bottom) Surface Mesh .................................................................... 145 Figure 5.26 Volume Mesh of the JSM................................................................................................................ 147 Figure 5.27 CL and CD for CRM Geometry at 8 degree AoA using OpenFOAM and STAR-CCM+ .............................................................................................................................................................................................. 148 Figure 5.28 Lift and Drag Coefficients for the JSM Geometry using OpenFOAM and STAR-CCM+ .............................................................................................................................................................................................. 149

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Figure 5.29 Interface Agglomeration Procedure Wing – Courtesy of [Laiping et al.] ................. 154 Figure 5.30 Initial Hybrid Grids and Coarsen Grids Wing – Courtesy of [Laiping et al.] ........... 155 Figure 5.31 Initial Hybrid Grids and Coarsening Grids over 30P30N Airfoil Wing – Courtesy of [Laiping et al.] ................................................................................................................................................................ 156 Figure 5.32 CP Distribution on Solid Wall Wing Courtesy of [Laiping et al.].................................. 157 Figure 5.33 Initial Hybrid Grids and Coarsening Grids over ONERA M6 Wing – Courtesy of [Laiping et al.] ................................................................................................................................................................ 157 Figure 5.34 Close-up Views of Hybrid Grids After Agglomeration Wing – Courtesy of [Laiping et al.]................................................................................................................................................................................... 158 Figure 5.35 Aerodynamic Force Coefficients for Different Angles of Attack (M∞ = 0.75) Wing – Courtesy of [Laiping et al.] ....................................................................................................................................... 159 Figure 5.36 Hybrid Grids over DLR-F6-WBNP Configuration Wing – Courtesy of [Laiping et al.] .............................................................................................................................................................................................. 160 Figure 5.37 CP Distributions at Three Typical Sections (M = 0.75, α = 1.0 deg) Wing – Courtesy of [Laiping et al.]........................................................................................................................................................... 161 Figure 5.38 Local Remeshing .............................................................................................................................. 163 Figure 5.39 Hybrid Mesh on a Wing-Body-Pylon-Nacelle Configuration – Courtesy of Centrum® .............................................................................................................................................................................................. 165 Figure 5.40 Meshing Aircraft Landing & Takeoff – Courtesy of Centaur© ........................................ 169 Figure 5.41 HiLiftPW-3 Experience .................................................................................................................. 171 Figure 6.1 Example Adaptive Grid for Supersonic Wedge Flow ........................................................... 174 Figure 6.2 Schematic image of Adaptive Mesh Refinement .................................................................... 174 Figure 6.3 Octree Data Structure of Adaptive Cartesian Grid Method ............................................... 175 Figure 6.4 Schematic 2D view of angular variation of normal .............................................................. 176 Figure 6.5 Pressure Contours in 2D Backward Step .................................................................................. 176 Figure 6.6 Selected Initial Meshes for the Transient Adaptive Procedure (Meshes 3, 20, 27 and 29) ....................................................................................................................................................................................... 177 Figure 6.7 30P30N Multi-Element Airfoil & close up of slat................................................................... 179 Figure 6.8 2D Fuel Cell Slice & Zoomed........................................................................................................... 179 Figure 6.9 Grid Adaption using Supersonic Flow for an Airfoil (bow shock).................................. 180 Figure 6.10 NACA 0012 Transonic test case: M∞ = 0.8, α=1.25 ............................................................. 180 Figure 6.11 Two-Pass Approach for Parallel Coarsening and Refinement. ........................................... 182 Figure 6.12 Store position, orientation, and surface pressures at selected points in trajectory ........ 182 Figure 6.13 Adapted Mesh Partitioning During Store Dispense ................................................................ 183 Figure 6.14 Inter-Processor Partitioning Based on Laplace Coefficients ......................................... 184 Figure 6.15 Hybrid Icosahedra Surface Mesh (left) and Multi-Material Hybrid Volume Mesh (right) – (Courtesy of Khamayseh and Almeida)............................................................................................. 185 Figure 6.16 HTTR Multi-Material Geometry, Initial Coarse Mesh (left), Refined Mesh (right) ) – (Courtesy of Khamayseh and Almeida) ............................................................................................................... 186 Figure 6.17 Orography field (left), r-adaptivity (center) and h-adaptivity (right) for climate modeling ) – (Courtesy of Khamayseh and Almeida) .................................................................................... 186 Figure 6.18 Coupled orography field transfer with h-adaptivity. Planar orography field (top), .............................................................................................................................................................................................. 187 Figure 6.19 Meshing and Partitioning of Centrifugal Contactor ) – (Courtesy of Khamayseh and Almeida) ........................................................................................................................................................................... 188 Figure 6.20 Discretization Error in the Drag Coefficient for Transonic Flow over an Airfoil (Reproduced from Dwight) ...................................................................................................................................... 190 Figure 6.21 Steady-State Burgers Equation for Reynolds Number 32............................................... 193 Figure 6.22 Adaption Schemes Applied to Burgers Equation Left) numerical solutions and right) local nodal spacing Δx. ................................................................................................................................... 194

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Figure 6.23 NACA0012 wing-Section Adaption........................................................................................... 198 Figure 6.24 Extraction of a Flow Feature & Redistributed Volume Meshes ...................................... 199 Figure 6.25 Hybrid Meshes for the NACA0012 Wing-Section and Cp Distribution (-1.0 to 1.0) .............................................................................................................................................................................................. 200 Figure 6.26 Adaptive Remeshing of Capsule................................................................................................. 200 Figure 7.1 Mesh Deformation Problem ........................................................................................................... 203 Figure 7.2 Cylinder Motion in 2D....................................................................................................................... 206 Figure 7.3 Mesh Deformation via Bi-Harmonic Equations ..................................................................... 207 Figure 7.4 Mesh Deformation via Laplace & RBF Methods ..................................................................... 208 Figure 7.5 Outlet Guide Vane (OGV) Boundary Surfaces Defined for a Single Passage............... 209 Figure 7.6 Typical Angular Variation Between the Computed Distance Field Vector and the Surface Normal (OGV not shown to scale) ......................................................................................................... 210 Figure 7.7 GGI interface ......................................................................................................................................... 211 Figure 7.8 Overset Method ................................................................................................................................... 212 Figure 7.9 Delaunay Method of Dynamic mesh ........................................................................................... 213 Figure 7.10 Mesh before and after the translational deformations .................................................... 214 Figure 7.11 Mesh Before and After the x-axis Rotational Deformation............................................. 215 Figure 7.12 Spiral Inductor Geometry where P1 and P2 denote port 1 and Part 2...................... 216 Figure 7.13 Vertical field evolution and associated mesh refinement in the microstrip spiral inductor, simulated by a two-level dynamic AMR-FDTD ............................................................................. 217 Figure 8.1 Backward facing step in a duct using Polyhedral, Hexahedral and Tetrahedral cells .............................................................................................................................................................................................. 219 Figure 8.2 Effect of truncation error on Hex and Tet cells ...................................................................... 220 Figure 8.3 Average Bees Being Smarter than CFD Engineer? (Courtesy of Stephen Ferguson) .............................................................................................................................................................................................. 220 Figure 8.4 Polyhedral cells vs Tetrahedral cells .......................................................................................... 221 Figure 8.5 Boundary prims cells for tetrahedral (left) and polyhedral (right) cells – (Courtesy of CD-Adapco) ................................................................................................................................................................ 222 Figure 8.6 GG simple face averaging ................................................................................................................ 224 Figure 8.7 GG Inverse Distance Weighted (IDW) Face Interpolation ................................................. 225 Figure 8.8 Methodologies for various Gradient Order of Accuracy..................................................... 227 Figure 8.9 Global Error Norms for x-Direction Gradient for Various Gradient Methods ........... 228 Figure 9.1 Comparison of Hex (16 K Cells) and Tet (440 K Cells) for a Pipe with 90 Degree Bend ................................................................................................................................................................................... 230 Figure 9.2 Results of Hex vs Tet Meshes as well as Hybrid Mesh in a Pipe with 90 Degree Bend .............................................................................................................................................................................................. 231 Figure 9.3 Design of Propellers, (left) Propeller P5168, .......................................................................... 233 Figure 9.4 Computational Domain– (Courtesy of Morgut & Nobile) .................................................. 233 Figure 9.5 Meshing for Propeller P5168– (Courtesy of Morgut & Nobile)....................................... 235 Figure 9.6 KT , KQ and η Curves of Propeller A – (Courtesy of Morgut & Nobile) ........................... 236 Figure 9.7 KT and KQ curves of Propeller P5168 – (Courtesy of Morgut & Nobile)....................... 237 Figure 9.8 Flow Around Turbine Blade – (Courtsy of Sasaki et al.) .................................................... 239 Figure 9.9 Geometric Blocking Used (a) Structured Hexahedral (178 Blocks) and (b) Unstructured Hexahedral (80 Blocks) – (Courtesy of Samir Vinchurkar & Worth Longest)........ 241 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) 242 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) ........................................................................ 249

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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) .......................................................... 250 Figure 9.13 Boundary Layer Transition Between Prismatic and Volume Elements – (Courtesy of Rousseau et al.)......................................................................................................................................................... 254 Figure 9.14 Example of a hydraulic turbine spiral case (half domain) .............................................. 255 Figure 9.15 Geometry of the Stay Vanes and Wicket Gates, Left: Geometry A, Right: Geometry B – (Courtesy of Rousseau et al.) ................................................................................................................................ 255 Figure 9.16 Structured Hexahedral Mesh of the Geometry A on the Symmetrical Surface and Close Up – (Courtesy of Rousseau et al.) ............................................................................................................. 256 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.).............................................................................................................................................................. 257 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.) ................ 258 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.) ................................................ 259 Figure 9.20 Meridian Velocity on the Meridian Plane for the Geometry B – (Courtesy of Rousseau et al.).............................................................................................................................................................. 260 Figure 10.1 B-Spline Approximation of NACA0012 (left) and RAE2822 (right) Airfoils ........... 264 Figure 10.2 Six Control Point Representation of a Generic Airfoil ...................................................... 264 Figure 10.3 Free Form Deformation (FFD) for Volume Grid with Control Points (Courtesy of Kenway et al.) ................................................................................................................................................................. 265 Figure 10.4 Sample Grid and Grid Sensitivity............................................................................................... 266 Figure 10.5 Effects of Mesh Density on Solution Domain ........................................................................ 268 Figure 10.6 Mesh Independence ........................................................................................................................ 269 Figure 11.1 Predicted Mesh Quality (Volume, Aspect Ratio, and Stretch) ....................................... 271 Figure 11.2 A simple Demonstration of How a Poor Mesh from a Cell Geometry Perspective 273 Figure 11.3 Using Kestrel one can Show a Correlation Between Mesh and Solution Quality .. 274 Figure 11.4 Concept of Orthogonality in Cells .............................................................................................. 275 Figure 11.5 Skewness and Warpage................................................................................................................. 276 Figure 11.6 Tetrahedral Volume ........................................................................................................................ 276 Figure 11.7 Triangulation of a polygon ........................................................................................................... 277 Figure 11.8 Tetrahedralization of a Polyhedral (showing a single face) .......................................... 278 Figure 11.9 General Estimation of Surface Mesh Element Size ............................................................. 280 Figure 11.10 Mesh Resolution for Sideview Mirror................................................................................... 283 Figure 11.11 Prism Layer Growth ..................................................................................................................... 284 Figure 11.12 Handling Prism Sides using Non-conformal Interfaces................................................. 285 Figure 11.13 Impact of Local Refinement on Tetrahedral Mesh .......................................................... 286 Figure 12.1 Symmetry plane (XY) ..................................................................................................................... 289

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.

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➢ 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. ➢ 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. 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.

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➢ 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. ➢ 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.

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➢ 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. ➢ Zaib Ali, James Tyacke, Paul G. Tucker, Shahrokh Shahparb, “Block topology generation for structured multi-block meshing with hierarchical geometry handling”, 25th International Meshing Roundtable (IMR25), 2016. ➢ Jeffrey Slotnick, “Meshing Challenges for Applied Aerodynamics”, Boeing, January 2019. ➢ Shahrokh Shahpar and Leigh Lapworth, “PADRAM: Parametric Design And Rapid Meshing System For Turbomachinery Optimization”, Proceedings of ASME Turbo Expo 2003, USA. ➢ Marco Lanfrit, “Best practice guidelines for handling Automotive External Aerodynamics with FLUENT”, Fluent Deutschland GmbH, Birkenweg 14a, 64295 Darmstadt/Germany.

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1 Introduction 1.1 The Black Box Dilemma1 1.1.1 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. 1.1.2 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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1.1.3 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. 1.1.4 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.” 1.1.5 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.” 1.1.6 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. 1.1.7 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.

2.1 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.]. 2.1.1

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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➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ ➢ 2.1.2 ➢ ➢ ➢ ➢ ➢ ➢ ➢

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

2.2 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. 2.2.1

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: 2.2.2

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. 2.2.3

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. 2.2.4

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. 2.2.5

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. 2.2.6

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.

2.3 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. 2.3.1

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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2.3.2

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.

2.4 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.

2.5 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. In summary, the geometry import can be devised in three steps, following [Yasuda]10 for HL CRM mesh development, where CFLOW compromises a grid generator and flow solver.

Figure 2.4

2.5.1

Geometry Import and Preparation

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 2.5. 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.

Hidemasa Yasuda, Taku Nagata, Atsushi Tajima, Akio Ochi, “KHI Contribution to GMGW-1”, Kawasaki, 1st Geometry and Mesh Generation Workshop Denver, CO June 3-4, 2017. 10

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Figure 2.5

2.5.2

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 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

Figure 2.6

Small Feature (Left) vs Removed (Right)

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difficulties associated with mesh generation. Artifact features can be created from any modeling system and cannot be suppressed in the original modeling system. Figure 2.6 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. 2.5.3

“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 the geometry to represent it as a valid model in the non-native system11-12-13. 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. 13 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. 11 12

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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 equations14. 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 community in mesh related problems15. 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.16.

3.1 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 blocks17. 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 phenomenon18. 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 reference19 (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 Richardson LF. Weather prediction by numerical process. Cambridge: Cambridge University Press; 1921. Edelsbrunner H. “Geometry and topology for mesh generation”, Cambridge: Cambridge university, 2001. 16 Baker, T., “Mesh generation: Art or science?” MAE Department, Princeton University, Princeton, NJ. 17 Steven J. Owen, “A Survey of Unstructured Mesh Generation Technology”, Carnegie Mellon University, PA. 18 Introduction: An Initial Guide to CFD and to this Volume; page 1, 2007. 19 Steven Owen: Introduction to unstructured mesh generation, 2005. 14 15

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➢ 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

Figure 3.1

Classification of Grid Generation Algorithms (Courtesy of Steven Owen)

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3.2 Grid Topology Before we pay attention to the individual cell topology, we consider domain topology which are compared for the 2D case, namely H, C, and O topologies. (See Figure 3.2). Meshes with H-H and CH topology were constructed for 3D comparison; however due to the incompatibility of the C-H structure on a sharp wing tip or trailing edge with the current solver, no C-H studies are included. Most of the studies were under lifting inviscid flow conditions. Multiple studies were conducted under turbulent conditions but only one is included. Overall, when it comes to topology, the H mesh scores first place followed by the C mesh and the O mesh comes last. When it comes to mesh parameters, the studies show that with carefully chosen mesh spacing around the leading edge, good orthogonality and skewness factors, smooth spacing variation, and a reasonable number of nodes, excellent CFD results can be obtained from the mesh in terms of accuracy of computed functional, determined convergence order and adjoin error estimation. Now with regard to Topology of individual cells where three types are considered; Hexahedral, Tetrahedral, and Polyhedral. In essence, topology is a structure of blocks that acts as a framework for placing mesh elements. Blocks are laid out without gaps with shared edges and corners and contain same number of elements along each side. Number of blocks will dictate the skewness of the grid elements.

Figure 3.2

3.3

Domain Topology (O-Type, C-Type, and H-Type; from left to right)

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 Figure 3.3 Sponge Analogy rectangle logical form of grid lattice is still

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preserved20. Figure 3.3 spectacles a simply connected (as oppose to 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 (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.

Figure 3.4

Dual Block Grid Topology for a Generic Wing-Fuselage Configuration

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. 20 21

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3.4 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 3.6 illustrates this idea by showing a schematic of a three block

Figure 3.6

Multi Block Representation for C-H Mesh Around a Wing

Figure 3.7 Multi-Block Gridding of Turbine Blade - (Courtesy of GridPro)

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

(a) M6 Wing

Figure 3.5

(b) Reference H

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

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easily meshed either by solving a partial differential equation or, alternatively, by an algebraic 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 Iterate shown on Figure 3.4. An example showing a multi block conformal mapping Figure 3.8 Schwarz Concept of Iterating Between Domains for a M6 and Reference-H wings is illustrated in Figure 3.5 (a) and Figure 3.5 (b). Another example of multi-block structure gridding for a Turbine Blade is giving by Figure 3.7. GridPro© has developed a Topology Input Language (TIL) which can be used for similar geometries with minimal effort25. The domain decomposition 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 of the subdomain is qualitatively or quantitatively “easier” than 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 concept26. 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 3.8. 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 3.9. Other venders have their own scripts or input data depending. Another example is Poitwise® which uses Glyph or newer Glyph2 as a

Figure 3.9

Domain Decomposition for M6 Wing using TIL Scripts (Courtesy of GridPro)

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. 26 David E. Keyes, “Domain Decomposition Methods for Partial Differential Equations”, Department of Applied Physics & Applied Mathematics Columbia University. 24

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scripting for the geometry. 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 hard to come by in automated unstructured grid generates.

3.5 Further Remarks on Block Topology For Structured Multi-Block Meshing 3.5.1 Background and Literature Survey Multi-block structured mesh generation is among the most widely used meshing techniques in flow simulations. There is essentially a fly by process and there in no specific rule, just only the users ingenuity. But to be consistent, we include the development by [Ali et al.]27 which offers some pointers in the topic. This is basically a two-stage process. In the first stage, a suitable blocking topology is generated which divides the complex domain into simple sub-domains. The resulting blocks are subsequently meshed. This structured blocking offers an efficient meshing strategy for topologically simple configurations and standard templates exist for partitioning of such domains. For example, the H-O-H type blocking is commonly used to mesh the turbine blade passage. However, the modern day design challenges demand the computational analysis of more realistic geometries. Aeroengine domains, for example, involving the fan, outlet guide vanes, gearbox shaft and nozzle coupled with wing, flap and pylon are now being used for flow simulations. Meshing such multiply linked and more diverse geometries requires significant user intervention, or writing of templates as part of a library. Thus, an automatic or semiautomatic blocking strategy can be beneficial to reduce the CFD design cycle time and could be a better alternative to the unstructured or hybrid meshing methods. Fully automatic 3D block topology generation is a complex problem and currently there is no ideal block topology algorithm with all the desired features for structured mesh generation. However various automatic blocking approaches have been proposed with varying levels of automation and geometric complexity handling. This include approaches based on medial axis28-29, paving/plastering30-31 and more recently methods based on cross/frame field The medial axis transform (MAT) based algorithms for the domain decomposition have been presented in. Here the medial axis is generated using the Voronoi based method. A subdivision is created resulting in one block for each medial vertex, medial edge and medial face. A midpoint subdivision is then used for meshing the blocks. An alternative has been presented by [Rigby]32 called the ‘TopMaker’ approach, which makes use of medial vertices and parts of medial axis to block the domain. Medial vertices are defined as the points which are equidistant from three locations form the domain boundary. Consequently, six types of medial edges and appropriate rules are defined for creating the blocks. Further enhancements have been included to produce a good quality mesh however this technique has yet to be extended for 3D.

Zaib Ali, James Tyacke, Paul G. Tucker, Shahrokh Shahparb, “Block topology generation for structured multiblock meshing with hierarchical geometry handling”, 25th International Meshing Roundtable (IMR25), 2016. 28 T. Tam, C. Armstrong, 2D finite element mesh generation by medial axis subdivision, Adv. Eng. Software (1991). 29 D. Sheehy, C. Armstrong, D. Robinson, Computing the medial surface of a solid from a domain Delaunay triangulation, ACM Symposium on Solid Modeling Foundations and Applications, 1995. 30 T. D. Blacker, R. J. Myers, Seams and wedges in Plastering: A 3D hexahedral mesh generation algorithm, Engineering with Computers 2 (1993). 31 M. L. Staten, S. J. Owen, T. D. Blacker, Unconstrained paving and plastering: A new idea for all hexahedral mesh generation, Proceedings of 14th International Meshing Roundtable, Sandia National Lab, 2005. 32 D. Rigby, “A technique for automatic multi-block topology generation using the medial axis”, NASA/CR FEDSM2003-45527 (2004). 27

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Distance field based approaches are also widely used for the medial axis approximation and domain decomposition33-34. A hybrid approach called differential MAT or d-MAT approach is presented in [Xia and Tucker]35-36. Here, the hyperbolic-natured eikonal, level set equation is used to calculate the distance field37. Medial axis point clouds are then extracted from the Laplacian or Hessian determinant of the distance field. A thinning algorithm is then used for thinning the point clouds into curves and surfaces. For illustration of the model, please refer to [Ali et al.]38. Recent advancements in mesh generation are the methods based on the cross-fields (frame fields in 3D). A cross field is defined by assigning a set of four unit vectors to points at the discrete locations. These unit vectors form a regular cross on the tangent plane. Thus the size and the orientation of the quadrilateral cells can be specified by the cross field. A number of approaches have been put forward for 2D and 3D cross field based domain decomposition and mesh generation. To generate the block topology, the partitioning created by connecting the cross-field streamlines to the singularities can be used. The resulting blocks of the cross field can then be mapped to a grid. The cross field approach towards domain decomposition and mesh generation is novel and efficient but quite complex and expensive. [LayTracks3D]39, is a hybrid hexahedral meshing method combining medial axis based decomposition and the advancing front method. This technique produces good quality hexahedral meshes but degenerate cells can be formed around the sharp concave features. [Malcevic]40 presents another automated blocking strategy based on a Cartesian fitting method. While preserving the topology definition, a forward geometry simplification is performed followed by fitting the model into a Cartesian framework. The next step is blocking the domain after which the blocked model is mapped back on to the original geometry. Further operations such as removing singularities by J-grid wrapping are performed to enhance the mesh quality. This technique has been applied for meshing the end-wall cavities found in turbomachinery. This technique is very simple and but has only been demonstrated for 2D cases so far. The method sometimes produces some unnecessary mesh clustering across the block interfaces. An assessment of various automatic block topology generation techniques surveyed above has been performed in41-42. The comparison has been carried out using an adjoint based error analysis of the meshes generated by these block topologies. It is found that, in general, the medial axis based approaches provide optimal blocking and yields better accuracy in computing the functional of interest. Mostly, domains having internal flows were used for this assessment. However, the medial axis based methods may not always yield an optimal block topology when dealing with complex 3D geometries and external flows. To P.-E. Danielsson, “Euclidean distance mapping, Computer Graphics and image processing”, (1980). J. Vleugels, M. Overmars, Approximating generalized Voronoi diagrams in any dimension (1995). Technical report UU-CS-1995-14 Utretcht University. 35 H. Xia, P. Tucker, Finite volume distance field and its application to medial axis transforms, International Journal for Numerical Methods in Engineering 82(1) (2010). 36 H. Xia, P. Tucker, Fast equal and biased distance fields for medial axis transform with meshing in mind, Applied Mathematical Modelling (2011). 37 P. Tucker, Differential equation-based wall distance computation for DES and RANS, Journal of Computational Physics 190(1) (2003). 38 Zaib Ali, James Tyacke, Paul G. Tucker, Shahrokh Shahparb, “Block topology generation for structured multiblock meshing with hierarchical geometry handling”, 25th International Meshing Roundtable (IMR25), 2016. 39 W. R. Quadros, LayTracks3D: a new approach to meshing general solids using medial axis transform, Procedia Engineering 82 (2014). 40 I. Malcevic, Automated blocking for structured CFD gridding with an application to turbomachinery secondary flows, 20th AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii, 2011. 41 Z. Ali, P. G. Tucker, Multiblock structured mesh generation for turbomachinery flows, Proceedings of the 22nd International Meshing Roundtable, Springer, 2014. 42 Z. Ali, Optimal block topology generation for CFD meshing, Ph.D. thesis, Department of Engineering, University of Cambridge, UK, 2015. 33 34

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overcome this limitation, a hybrid blocking technique is illustrated which makes use of the distance field iso-surface in addition to the medial axis transform. This is demonstrated using a wing-bodytail and a jet-wing-flap configurations. In addition to that, to reduce the meshing effort, a hierarchical geometry handling approach is also defined and applied to an engine-wing-flap configuration. These approaches are described next. 3.5.2

Hybrid Blocking

Consider an aero-engine jet-wing-flap (JWF) domain with a far-field as shown in the Figure 3.10. The medial axis close to the JWF geometry is shown in the enlarged view of 2D slice of domain in the Figure 3.10. Here the distance field and the corresponding medial axis is computed with respect to the geometry and the cylindrical far-field thus avoiding effect of the inlet and exit boundaries. The solid lines represent the shock features i.e. the medial axis and the dashed lines show the expansion features. Following this medial axis branches and even connecting the hanging and expansion features, a poor quality blocking would be achieved. This is also because, to generate small branches of the medial axis between the internal aeroengine geometry parts would have to be combined with very large branches between the aeroengine and the far-field.

Figure 3.10

Jet-Wing-Flap: Medial Axis Transform (Compression Shock) and Expansion Features Close to the Geometry – Courtesy of [Ali et al]

To overcome this limitation, the distance field function d can be used. An iso surface (contour in 2D) of d is wrapped around the geometry to facilitate the MAT based block topology generation. The wall distance computation is an intermediate step in the distance field based medial axis approximation and hence is available for use without any extra cost. Thus using an iso surface of the distance field instead of the far field for the medial axis computation can significantly improve the medial axis based blocking. The hybrid blocking procedure is described below with the help of a simple 2D JWF geometry. The extension of this methodology to the 3D cases also follows the same procedure as demonstrated later.

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1. The distance field is computed around the domain of interest as shown in the Figure 3.11 (a). An exact equation for the wall distance d is the hyperbolic eikonal equation which models the front propagation from the surface at unit velocity.

Eq. 3.1

|∇d| = 1 + Γ ∇2 d where Γ −→ 0 yielding viscosity solutions. The first arrival time of the front for a unit velocity is equal to the wall distance. The eikonal equation is solved using a fast marching method43. A suitable iso surface is then extracted from d. This iso surface selection is currently arbitrary but it can be linked to a criteria. For example, the width of the shear layer regions in the jet wake can dictate this selection upstream or it could be based upon the dimensionless wall distance y+ value. The iso surface acts like a virtual geometry or a wrap around the real domain (see Figure 3.11 (b)).

2. The next step is approximation of the medial axis between the geometry and the distance field wrap. The Voronoi diagram based algorithm of [Dey and Zhao]44 is used here for the medial axis approximation. This algorithm provides a more stable and continuous medial axis for complex 3D domains than the voxel thinning approach. The input to this program is the point cloud data of the geometry and the distance field iso surface. It makes use of the observation that certain Voronoi facets are positioned close to the medial axis if their dual Delaunay edges tilt away from the surface or are very long. Hence, the angle condition and the ratio condition are defined to filter such tilted and long Delaunay edges and the medial axis is approximated. Let VP be the Voronoi diagram for a dense point set P from a smooth compact surface S ⊂ R3. This Vornoi diagram is a cell complex comprising of Voronoi cells Vp p ∈ P and their facets, edges and vertices. Also, for each point x ∈ S ,

Eq. 3.2

Vp = x ∈ ℝ3 ‖p − x‖ ≤ ‖q − x ‖ ,

∀q ≠ p

where p and q are any two points in P. Let DP be the Delaunay triangulation of P and dual to the Voronoi complex. The Delaunay triangles incident to point p which are dual to the Voronoi edges intersected by a tangent plane at p are used to construct the criteria. All the Delaunay

Figure 3.11 Two dimensional jet-wing-flap geometry: (a) the distance field; (b) distance field wrap and (c) corresponding medial axis (d) hybrid blocking around 2D geometry – Courtesy of [Ali et al]

H. Xia, P. Tucker, Finite volume distance field and its application to medial axis transforms, International Journal for Numerical Methods in Engineering 82(1) (2010). 44 T. K. Dey, W. Zhao, Approximate medial axis as a Voronoi subcomplex, Computer-Aided Design 36 (2004). 43

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edges that make relatively large angle with the planes of the triangles are filtered. If the angle between the vector tpq from p to q and the normal nptu to a triangle ptu is less than a threshold angle π/2 − θ for all the triangles, then that associated Delaunay edge is filtered i.e.

max ∠ 𝐧𝐩𝐭𝐮 , t pq < Eq. 3.3

π −θ 2

gives good results. The ratio condition is based on the comparison of the length of the Delaunay edges with the circum-radii of the triangles. Thus those Delaunay edges are filtered which satisfy the criteria:

min Eq. 3.4

‖p − q‖ >ρ r

Where ∥p − q∥ defines the length of a Delaunay edge and ρ is the circum-radius of a triangle ptu. A value of ρ = 8 is normally used for dense point clouds. The medial axis is generated as a continuous surface which can be imported into the mesh generator. The medial axis for the JWF slice is shown in the Figure 3.11 (c). 3. To complete the blocking process, additional rules as described in the Section 1 are manually used. Applying the rules, for example to the 2D JWF slice, the expansion features are connected to the nearest medial vertex or otherwise the medial axis as shown in the Figure 3.11 (d). 4. Once the critical parts of the domain have been blocked using the medial axis, the far-field region can be partitioned using simple Cartesian fitting or H-type blocks. This is shown, for example, in Figure 3.11 (d) with the green lines. This resulting domain decomposition is significantly better than the one obtained initially shown in the Figure 3.10. There can still be some regions where the block topology is unsatisfactory. Such areas must be manually altered. Hence, a semi-automatic blocking process arises. The mesh is then generated in the commercial program Pointwise. 3.5.3

Hierarchical Geometry Handling

Modern day aero-engine design challenges demand performing realistic and multi-scale CFD simulations of strongly coupled systems. This means that geometries are becoming highly complex and as a result more effort is consumed in the mesh generation and flow modeling process. This can have a strong impact on the duration of the design optimization cycle. To this end, a combination of the high and low fidelity (structured multi-block) meshing/modeling techniques can be employed using a hierarchy of the Figure 3.12 Hierarchical Geometry Handling Strategy methods shown in the Figure 3.12. At the bottom of this hierarchy is the smeared representation of the geometry through the use of the lower order methods such as

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immersed boundary method (IBM) and body force model (BFM)45-46. Next in the hierarchy is the real geometry resolved by the overset meshes which present a useful option for integrating various domains together without adding extra complexity. The information between the overlapping parts is exchanged through interpolation. At the top is the cost effective alternative to the fully resolved/meshed geometry, a combination of various high and low fidelity methods allows rapid addition and modeling of arbitrary geometry effects. This helps in reducing the design cycle time. The objective here is not to present any novel mesh generation method but to apply a zonal/hierarchical approach to handle complex domains to reduce the meshing effort. Such an approach can be efficiently used to rapidly explore the design space and can aid the development of high fidelity numerical tools. 3.5.4

Body Force Modeling

The engine-wing-flap geometry case to be presented later employs the hierarchical geometry handling approach and uses a body force method to model the effect of the fan and outlet guide vanes. This model is outlined next. Assuming an infinite number of blades, the aerodynamic effects of a blade row are modeled using an axisymmetric flow in each infinitesimal blade passage and the body forces are added as source terms. The parallel and normal components of these forces in cylindrical coordinates system are given as:

kp kp kp Vx Vrel , Fp,θ = Vθ Vrel , Fp,r = VV s s s r rel k n 𝑉𝜃 k n 𝑉𝑥 = f (Vx Vθ , α) , Fn,θ = f (Vx Vθ , α) s 𝑉𝑟𝑒𝑙 s 𝑉𝑟𝑒𝑙

Fp,x = Fn,x

Eq. 3.5 Here, Vx, Vθ and Vr are the axial, tangential and radial velocity components. Vrel is the magnitude of the fluid velocity relative to the blade. Kp and Kn are calibration constants. Also, α and s are the local blade metal angle and blade pitch respectively. The above equations can also be modified to produce local blockage terms. 3.5.5

Results

3.5.5.1 NASA CRM Wing-Body-Tail In this section, the hybrid blocking is applied to partition the domain around a 3D NASA Common Research Model (CRM) horizontal wing-body-tail configuration. This model represents a modern, transonic and commercial aircraft designed to cruise at M∞ = 0.85 and CL = 0.5. The geometric and aerodynamic details about the model as described. Further information M∞ 0.85 can be obtained from the development by [Ali et al.]47. The medial axis P 201326.9 Pa total around the wing and tail also provides a block topology similar to OTtotal 310.93 k type or C-type meshes. To assist the blocking, expansion features at the trailing edges of the wing and the tail are joined to the nearest medial Table 3.1 NASA CRM axis . After the blocking around the geometry is complete, the far-field Free-Stream Conditions domain partitioning is carried out. The region is partitioned to create a H-type mesh. The block topology around the model is shown in the T. Cao, P. Hield, P. G. Tucker, Hierarchical immersed boundary method with smeared geometry, 54th AIAA Aerospace Sciences Meeting, AIAA Science and Technology Forum and Exposition, 2016, pp. 2016–2130. 46 T. Cao, N. R. Vadlamani, P. G. Tucker, A. R. Smith, M. Slaby, C. T. J. Sheaf, Fan-intake interaction under high incidence, Proc. of ASME Turbo Expo, Seoul, South Korea, 2016. ASME Paper Number GT2016–56561. 47 Zaib Ali, James Tyacke, Paul G. Tucker, Shahrokh Shahparb, “Block topology generation for structured multiblock meshing with hierarchical geometry handling”, 25th International Meshing Roundtable (IMR25), 2016. 45

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Figure 3.13 (c). The volume and the surface mesh cuts are displayed in the Figure 3.13 (d). The NASA CRM configuration has been the test case for the 4th and 5th AIAA CFD drag prediction workshops48. The aim of the workshop is to assess the state-of-the-art in the CFD methods for aircraft aerodynamic analysis. Here, we use the same flow conditions as given in the workshop to compute the flow around the test case. The result of MAT based topology was to create 256 blocks in 2 minutes, and gridding in 6 minutes. The simulations are performed in HYDRA which is an unstructured, finite volume, edge-based and compressible flow solver using MUSCL based flux differencing.

Figure 3.13

NASA CRM Wing-Body-Tail (c) Hybrid Blocking (d) Mesh Cut Section – Courtesy of [Ali et al.]

The simulation is carried out at M∞ = 0.85 and CL = 0.5 with Re = 5 × 106 based on the reference chord length Cref = 7.00532 m. Table 3.1 describes the free-stream flow conditions. A coarse mesh of approximately 4 million cells is used. The first grid node from the wall is located at y+ ≈ 1. The Spalart Allmaras (SA) turbulence model is used for this simulation. The flow angle for this mesh to gain CL = 0.5 is α = 2.36 deg. 3.5.5.2 Jet-Wing-Flap In this section, the 3D jet-wing-flap case is presented. The geometry comprises of co-axial nozzle, pylon and a wing with a flap as shown in the Figure 3.14 (a). This realistic aero-engine geometry has been used for detailed computational aero-acoustics analysis,. The pylon adds complexity to the otherwise cylindrical nozzle topology along with the wing and the flap. Hence, blocking such a case demands significant user insight. After wrapping the distance field, the medial axis is approximated. This is shown in the Figure 3.14 (b). To simplify the blocking procedure, small medial axis branches are removed for this case. This is followed by the inner blocking aided by the rules which is shown in the Figure 3.14 (c). The far-field domain decomposition can be then be carried out at this stage. However, one of the aims for the aero-acoustic jet simulations is to properly resolve the shear layers in the far-field. This requires a good quality mesh aligned with the shear layer regions. The current J. Vassberg, E. N. Tinoco, M. Mani, B. Rider, T. Zickuhr, D. Levy, O. P. Brodersen, B. Eisfeld, S. Crippa, R. A. Wahls, et al., Summary of the Fourth AIAA CFD Drag Prediction Workshop (2010). AIAA Paper No. AIAA-2010. 48

46

block topology as shown in the Figure 3.14 (c) is non-optimal for properly resolving the shear layers produced by the bypass and the core flows. Hence a manual alteration of the blocking around the pylon and the core flow exhaust is performed.

Figure 3.14

Jet-Wing-Flap (a) CAD, (b) CAD and the Medial Axis Cut Section, (c) Inner Hybrid Blocking – Courtesy of [Ali et al]

The modified inner block topology with the far-field decomposition are shown in the Figure 3.15. A RANS simulation using the SST k − ω is carried out on the mesh generated by the hybrid blocking. The first grid node from the wall is located at y+ ≈ 1. The two cases presented above show how the hybrid blocking approach can be effectively used to decompose and mesh the complex geometries. The medial axis based domain decompositions also provide meshes having better flow alignment when compared to other partitioning methods e.g. Figure 3.15 Jet-Wing-Flap with Modified Inner Blocking (To Cartesian fitting and cross Accommodate Shear Layers) – Courtesy of [Ali et al] field based techniques. Hence, this technique further enhances the scope and applicability of these MAT based blocking methods. Also, the blocking templates could be generated using this approach which can speed up the mesh generation process and aid an inexperienced CFD user.

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3.5.5.3 Engine-Wing-Flap In the last section, a coaxial nozzle with pylon and wing geometry was presented which makes the rear or downstream part of the aero-engine. To carry out a more realistic simulation, the front engine part containing the axisymmetric intake, hub and splitter geometry is added to this rear part using the overset mesh at the interface. This procedure avoids re-blocking the domains to have a cell to cell match between the two zones. Also, a smeared fan geometry is used where the fan is modeled using the BFM (see the schematic in Figure 3.16 (Top)). The other downstream components are imprinted and treated again with the BFM but with localized sources. These internal geometry components include the downstream vanes, gearbox shaft, and the engine supporting A-frames. Thus using a hierarchical geometry handling approach, a complex domain can be readily meshed and

Figure 3.16

Engine-Wing-Flap (a) Schematics Showing Geometric Zones and Domains with Different Block Topologies (b) Cut Section at z = 0 Plane – Courtesy of [Ali et al]

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analyzed for in a design optimization cycle. A cut section of the mesh at z=0 plane is shown in the Figure 3.16 (Bottom). The total mesh size is 50 million. The internal geometry in the front part is modeled using a body force model which has been extended to include the local blockage and wakes modeling. This is done by adding local enhanced source terms to generate wake zones, which is similar to adding the source terms in the IBM for simulating geometry or boundary on a nonconformal Cartesian mesh.

3.6 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

Figure 3.17

Examples of Structured grids for Turbine Blade

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 outcome of a CFD simulation and its accuracy can 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

Figure 3.18 Example of Unstructured Tetrahedral Grids

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essential and cannot be omitted. Without a grid, there is no possibility to start a CFD simulation. Figure 3.17 shows examples of 2D and 3D structured grids, while Figure 3.18 displays an example of mainly tetrahedral unstructured grids.

3.7 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 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. 3.7.1 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 Churchill49, Moretti, and Davis. 3.7.2 Algebraic Methods -Transfinite Interpolation (TFI) 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: 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 N

Eq. 3.6

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. 3.7 3.7.2.1 Blending Function The interpolation function defined by Eq. 3.7 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. 3.8 shows a typical Exponential blending function as:

r(ξ) = Yi + [Yi − Yi−1 ] 49

ξ − Xi −1 Xi − Xi−1 ] Exp (Bi−1 ) − 1

Exp [(Bi−1 )

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

for

Xi−1 ≤ ξ ≤ Xi

50

where

0 ≤ X𝑖 , Y𝑖 ≤ 1

,

0 ≤ ξ , r(ξ) ≤1

,

i =2, , , , m

Eq. 3.8 The integer m represents number of control points with coordinates {Xi, Yi}. The quantity Bi-1, called the stretching parameter, is responsible for grid density. Specifying B1, values of Bi ≥ 2 are obtained by matching the slopes at control points. This, guaranteeing a smooth grid transition between each region, can be accomplished using Newton's iterative scheme which is quadratically convergent. The greater the B , the less discontinuity will propagate. Similarly, a blending function could be constructed for η direction. The spline-blended form gives the smoothest grid with continuous second derivatives50. An example of exponential stretching (Bi = -2) is given by Figure 3.19 .

Figure 3.19

Exponential Distribution Functions

The exponential function, while reasonable, is not the best choice when the variation in grid spacing is large. The truncation error associated with the metric coefficients is proportional to the rate of change of grid spacing. A large variation in grid spacing, such as the one resulting from exponential function, would increase the truncation error, hence, attributing to the solution inaccuracies. A suggested alternative to exponential function has been the usage of hyperbolic sine function given as

r(ξ) = Yi + [Yi − Yi−1 ] where

ξ − Xi Xi − Xi−1 ] sinh (Bi−1 )

sinh [(Bi−1 )

0 ≤ X𝑖 , Y𝑖 ≤ 1

,

for

0 ≤ ξ , r(ξ) ≤1

,

Xi−1 ≤ ξ ≤ Xi i =2, , , , m

Eq. 3.9 The hyperbolic sine function gives a more uniform distribution in the immediate vicinity of the boundary, resulting in less truncation error. This criteria makes the hyperbolic sine function an excellent candidate for boundary layer type flows. A more appropriate function for flows with both viscous and non-viscous effect would be the usage of a hyperbolic tangent function such as

50 Joe

F. Thompson, Z. U. A. Warsi, C. Wayne Mastin, “Numerical Grid Generation -Foundations and Applications”, North Holland, 1985.

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(B ) ξ − Xi tanh [ i−1 ( − 1)] 2 Xi − Xi−1 r(ξ) = Yi + [Yi − Yi−1 ] for Xi−1 ≤ ξ ≤ Xi Bi−1 tanh [ ] 2 w here 0 ≤ X𝑖 , Y𝑖 ≤ 1 , 0 ≤ ξ , r(ξ) ≤ 1 , i =2, , , , m

Eq. 3.10 The hyperbolic tangent, as revealed in Figure 3.21, gives more uniform distribution on the inside

Figure 3.21

Hyperbolic Tangent Distribution Functions

as well as on the outside of the boundary layer to capture the non-viscous effects of the solution. Such overall improvement, makes the hyperbolic tangent a prime candidate for grid point distribution in viscous flow analysis, as publicized in Figure 3.20 for a generic wing-fuselage. A numerical approximation can be used to compute the grid-point distribution on a boundary curve. This approach is widely used for complex configurations and care must be taken to insure monotonicity of the distribution. For example, the natural cubic spline is C2 continuous and offers great flexibility in grid spacing control. However, some oscillations can be inadvertently introduced into the control function. The problem can be avoided by Figure 3.20 Grid for Dual-Block Generic using a smoothing cubic spline technique and Wing-Fuselage Geometry specifying the amount of smoothing as well as control points. Another choice would be the usage of a lower order polynomial such as Monotonic Rational Quadratic Spline (MRQS) which is always monotonic and smooth. Other advantage of MRQS over cubic spline is that it is an explicit scheme and does not require any matrix inversion. 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 Error! Reference s ource not found.. A pioneering work in control point form of Algebraic Grid Generation using a

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univariate interpolations can be attributed to [Eiseman and Smith]51. 3.7.2.1.1 Case Study- Rapid Meshing System For Turbomachinery PADRAM (Parametric Design and Rapid Meshing System), grid generation starts by dividing the computational blocks into sub-blocks for the purpose of generation of the algebraic grid and the control functions. O-type grid is used for the blades and H-type grids near the periodic boundary, upstream and downstream blocks, C-type grids are used for semi-infinite boundaries such as the splitter, pylon and RDF (Radial Drive Fairing). Transfinite interpolation (TFI) is used to generate the initial grid based on a linear interpolation of the specified boundaries. TFI is very easy to program, computationally efficient and the grid spacing is under direct control. PADRAM uses the following double clustering functions [ Shahpar and Lapworth]52: η−α

y=

β + 1 1−α (2α + β) [ + 2α − β ] β−1 η−α

β + 1 1−α (2α + 1) {1 + [ ] } β−1

where 1< β