58 T. S. Geiger and D. M. Dilts. ...... ACM Trans. Math. ...... is a relatively small correction, but there is a noticeable change between iterations n = 1 and n = 2.
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CFD Open Series Revision 1.85.3 a
Multiphysics Flow Ideen Sadrehaghighi, Ph.D.
Multiphysics Flows and FSI • Mesh Deformation Strageties
Coupling Mutiphysics Strategies • Sub-Iteration and Coupling Strageties
Aeroelasticity • Linear and Non-linear Elasiticity
Aeroacoustics • Generalized Scattering
ANNAPOLIS, MD
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Contents 1
Introduction .................................................................................................................................. 9 1.1
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Multiphysics Flow .................................................................................................................................................. 9 Aero-Elasticity .................................................................................................................... 9 Static Aero-elasticity ................................................................................................... 10 Dynamic Aero-elasticity ............................................................................................. 10 Single Discretization Methods ........................................................................................ 10 Multiple Discretization Methods..................................................................................... 10 Aero-Acoutics ................................................................................................................... 11
Multiphysics Flow ..................................................................................................................... 12 2.1 2.2 2.4 2.3
2.4
2.5
What Constitutes Multiphysics?.................................................................................................................... 12 Simulation Strategies for Multiphysics Problems.................................................................................. 13 Methodologies ....................................................................................................................................................... 13 PDE-Based Multiphysics Applications........................................................................................................ 15 Coupling ............................................................................................................................ 15 Strong vs. Weak Coupling of Physical Model ............................................................ 15 Tight vs. Loose Coupling of Numerical Models ......................................................... 15 Error and Residual ........................................................................................................... 16 Explicit Methods............................................................................................................... 16 Implicit Methods .............................................................................................................. 16 Multiscale Model .............................................................................................................. 16 Macro-Scale ................................................................................................................. 16 Meso-Scale ................................................................................................................... 16 Micro-Scale .................................................................................................................. 17 Operator Splitting ............................................................................................................ 17 Schur Complement ........................................................................................................... 17 Semi-Implicit Methods..................................................................................................... 17 Sub-Cycling ....................................................................................................................... 17 Attributes of Coupling Process as Devised by Supervisor ................................................................. 17 Dual Time Step ................................................................................................................. 17 Iteration Number ............................................................................................................. 18 Completion ....................................................................................................................... 18 Initialization ..................................................................................................................... 18 Uncertainties .................................................................................................................... 19 Interface Coupling Schemes............................................................................................................................ 19 Fixed point ........................................................................................................................ 19 Implicit Predictor-Corrector ........................................................................................... 20 Quasi-Newton Algorithm................................................................................................. 20 Choices of Relaxation Parameter .................................................................................... 21 Constant ....................................................................................................................... 21 Aitken ........................................................................................................................... 21 Steepest Descent ......................................................................................................... 22 Simple Search .............................................................................................................. 22
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Fluid-Structure Interactions (FSI)....................................................................................... 24
3.1
Comparing the Eulerian and Lagrangian Methods................................................................................ 25 What Is the ALE Method? ................................................................................................ 26 Applying the ALE Method to a Micro Pump ................................................................... 26 3.2 Interaction Between Fluids and Structures (FSI) .................................................................................. 28 3.3 Moving Boundaries and Mesh Deformation ............................................................................................ 28 Mesh Deformation using a Thermo-Elastic Analogy ..................................................... 29 Mesh Deformation via Linear Elasticity ......................................................................... 30 Isotropic vs Anisotropic Stress ....................................................................................... 31 3.4 Coupling Schemes ............................................................................................................................................... 31 3.5 Monolithic Approach using ALE .................................................................................................................. 32 3.6 Design of the Class Structure in OpenFOAM ............................................................................................ 33 3.7 Partitioned Approach - Loose or Strong Coupling ................................................................................ 36 Conditions at the Fluid-Solid Interface .......................................................................... 37 3.8 Case Study 1 - Monolithic Approach for Turek/Horn Benchmark Case (2D) .......................... 37 3.9 Case Study 2 - Modeling a Sphere Falling on a Water Surface.......................................................... 39 Oscillating Motion of a Buoyant Sphere ......................................................................... 39 Setting Up the Model........................................................................................................ 40 Analyzing the Simulation Results ................................................................................... 41 Concluding Thoughts ....................................................................................................... 42 3.10 Case Study 3 - Modeling Fluid-Structure Interaction in a Heart Valve ......................................... 42 Background ...................................................................................................................... 42 Advancing Heart Valve Research via Simulation........................................................... 42 Modeling the Opening and Closing of a Heart Valve in COMSOL Multiphysics ........... 43 Simulation Results for Fluid-Structure Interaction in a Heart Valve ................. 43 Improving the Design of Medical Devices with FSI Modeling ...................................... 43
Structural Design of Aircrafts ............................................................................................... 45 4.1 4.2 4.3 4.4
4.5 4.6 4.7 4.8 4.9
Generalities and Load Factor ......................................................................................................................... 45 Basics........................................................................................................................................................................ 45 Composite Material in Aircraft Structure.................................................................................................. 46 Three Design Phases .......................................................................................................................................... 46 Conceptual Design............................................................................................................ 47 Preliminary Design .......................................................................................................... 47 Detail Design..................................................................................................................... 49 System Integration ............................................................................................................................................. 49 Design of Composite Structures .................................................................................................................... 49 Constraints and Allowable .............................................................................................................................. 51 Non-Destructive Testing .................................................................................................................................. 51 Ultrasonic Methods .......................................................................................................... 52 Optimization Process......................................................................................................................................... 53 Multi-Objective Optimization .......................................................................................... 53 Multilevel Programming.................................................................................................. 53 Pareto Optimality ............................................................................................................. 53 Minimizing Weighted Sums ............................................................................................ 54 Weight and Performance Optimization.......................................................................... 54 Integrated Cost Optimization .......................................................................................... 54 Combined Cost/Weight Optimization ............................................................................ 55
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Interdisciplinary Wing Design – Structural Aspects ..................................................... 56
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Aero-Elasticity ............................................................................................................................ 57
5.1
Background............................................................................................................................................................ 57 Flutter and Limit Cycle Oscillation (LCO) ...................................................................... 59 5.2 Aero-Elastic Problem ......................................................................................................................................... 60 5.3 Formulation of Structure Behavior.............................................................................................................. 61 5.4 Linear Aero-Elastic ............................................................................................................................................. 61 Case Study – A Comparison of Linear and Non-Linear Flutter Prediction Methods .. 62 Aerodynamic Formulation and Solution ................................................................... 62 Structural Modeling .................................................................................................... 63 Description of Linear Methods .................................................................................. 63 Dynamic Test Cases - AGARD 445.6 Weakened Wing ............................................. 63 5.5 Generic Nonlinear Aero-Elastic ..................................................................................................................... 65 4.5.1 Statically or Dynamically Linear....................................................................................... 65 4.5.1 Statically or Dynamically Non-linear ................................................................................ 66 Case Study - Linear vs. Non-Linear Aero-Elastic Analysis of High Aspect-Ratio Wings 66 Wing Model.................................................................................................................. 67 Results and Discussion ............................................................................................... 67 Conclusions.................................................................................................................. 69 5.6 Kinematical Description of the Continuum .............................................................................................. 70 Lagrangian Algorithm ...................................................................................................... 70 Eulerian Algorithm .......................................................................................................... 70 Interface Tracking of Arbitrary Lagrangian–Eulerian (ALE) ....................................... 71 Material Acceleration....................................................................................................... 71 Conservation Form of ALE Equations ............................................................................ 71 Alternative form of Arbitrary Lagrange-Eulerian (ALE) .............................................. 72 Euler form of Arbitrary Lagrange-Eulerian (ALE) ........................................................ 73 Geometric Conservation Law (GCL) ............................................................................... 74
Case Studies for Aero-Elastic ................................................................................................ 76 6.1 6.2 6.3
6.4
Case Study 1 – Linear & Non-Linear Aero-Elastic behaviors of a 2D Flat Panel Flutter ........ 76 Method of Solution ........................................................................................................... 76 Concluding Remarks ........................................................................................................ 77 Case Study 2 – 3D Application of Non-Linear Fan-Flutter Analysis ............................................... 77 Simulation Results ........................................................................................................... 78 Coupling and Sub-Iteration ............................................................................................. 80 Case Study 3 - Multipoint Hydro-Structural Optimization of a Hydrofoil ................................... 80 Fluid–Structure Coupling Algorithm .............................................................................. 81 Geometry Perturbation Algorithm ................................................................................. 81 Optimization Algorithm ................................................................................................... 82 Adjoint Gradient Computation ........................................................................................ 82 Design Optimization Results ........................................................................................... 84 Single-Point Hydro-Structural Optimization ................................................................. 84 Multipoint Hydro-Structural Optimization .................................................................... 85 Case Study 4 - Steady State Aero-elastic Analysis for a Common Research Model Wings ... 88
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6.5
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Aero-Structural Analysis Mode ....................................................................................... 89 Aerodynamic Solver......................................................................................................... 89 Structural Solver .............................................................................................................. 89 Load and Displacement Transfer .................................................................................... 90 Mesh-Deformation Algorithm ......................................................................................... 91 Aero-Structural Analysis Methods.................................................................................. 92 Nonlinear Block Gauss-Seidel Method (NLBGS)....................................................... 92 Coupled Newton-Krylov Method (CNK).................................................................... 93 Results............................................................................................................................... 94 Jig Shape and Concept of Wing Box Design .................................................................... 98 Wing Box Geometry ......................................................................................................... 98 Aerodynamic Loads ......................................................................................................... 99 Structural Design ............................................................................................................. 99 Inverse Design Procedure ............................................................................................. 100 Case Study 5 - Structural Aspects of F11 Wing Design ..................................................................... 102 Problem Definition......................................................................................................... 102 F11 Wing Structure ........................................................................................................ 102 Results for F11 Wing ..................................................................................................... 105 Next Task ........................................................................................................................ 106
Aero-Acoustics ........................................................................................................................ 109
7.1 7.2
7.3
Lighthill's Equation ......................................................................................................................................... 110 Classification of Aero-Acoustics ................................................................................................................. 111 Linear Problems of Propagation and Generalized Scattering .................................... 111 The Linearized Navier-Stokes Equations ................................................................ 111 Modeling Considerations.......................................................................................... 113 Nonlinear Problems of Flow Acoustics (CAA) ............................................................. 114 Case Study - Evaluation Flap Edge Noise using Sub-Domain of Unstructured Grids ........... 115 Unstructured vs Structured Mesh for Flap Edge ......................................................... 115 Method of Solution ......................................................................................................... 115 Sub Domain Unstructured Refine Grid ......................................................................... 117 Sponge Zone ................................................................................................................... 117 Results............................................................................................................................. 118 Conclusion ...................................................................................................................... 119
List of Tables Table 2.1 Under Relaxation Choices ........................................................................................... 22 Table 3.1 Comparing Eulerian vs. Lagrangian Approach ......................................................... 25 Table 3.2 Algorithm Gauss-Seidel Multiphysics Coupling ........................................................ 28 Table 3.3 Characteristics of Turek/Horn benchmark run case ............................................... 39 Table 5.1 Main characteristics of the 3 different wing aspect-ratio configurations .............. 67 Table 6.1 Property Values ........................................................................................................... 76 Table 6.2 Aero-Structural Scaling Results (Courtesy of 63) ...................................................... 94 Table 6.3 Key Parameters in CRM Configuration – (Courtesy of 63) ........................................ 94 Table 6.4 Mesh Sizes for the Three Levels Used in the Parallel Scalability Study – (Courtesy of 63) .................................................................................................................................................. 95 Table 6.5 Material Properties .................................................................................................... 100
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List of Figures Figure 1.1 Pressure Contours on Elastic Band ............................................................................ 9 Figure 1.2 Local and Deformed Coordinate of Elasticity ........................................................... 9 Figure 2.1 Supervisor interaction with different domain codes .............................................. 14 Figure 2.2 Classification of FSI Methods .................................................................................... 15 Figure 2.3 Interface Coupling Algorithms.................................................................................. 21 Figure 3.1 Coupling between CFD & CSD ................................................................................... 24 Figure 3.2 Micro pump Mechanism ............................................................................................ 26 Figure 3.3 Interaction for Micro pump Mechanism .................................................................. 27 Figure 3.4 Mesh Deformation Driven by Boundary Displacement .......................................... 30 Figure 3.5 Coupling Schemes for FSI .......................................................................................... 32 Figure 3.6 Class Structure for Fluid, Solid, and Fluid-Solid Interaction Models .................... 34 Figure 3.7 Class Structure for Mechanical Laws and Dynamic Meshes ................................... 35 Figure 3.8 Coupling Schemes ...................................................................................................... 36 Figure 3.9 Turek/Horn Bench-Mark Case ................................................................................. 38 Figure 3.10 Sphere Falling on a Water Surface (Courtesy of Kumar) ..................................... 39 Figure 3.11 Displacement of the sphere Vs. time (Courtesy of Kumar) ................................. 41 Figure 3.12 Schematic of a heart. Image by Wapcaplet ( Licensed under CC BY-SA 3.0, via Wikimedia Commons) .................................................................................................................... 42 Figure 3.13 FSI Model of a Heart Valve Opening (left) and Closing (right) ............................ 43 Figure 4.1 Typical Load of Airliner (Courtesy of Airbus) ......................................................... 47 Figure 4.2 Levels in Detailed (Courtesy of H. Assler, Airbus Deutschland GmbH) ................. 48 Figure 4.3 Supply Hierarchy in the Commercial Aerospace Industry .................................... 49 Figure 4.4 Specific Strength and Stiffness of Different Metals and Alloys (Courtesy of Kaufmann) ....................................................................................................................................... 50 Figure 4.5 Portions of Composite material in Airbus Aircrafts (Courtesy of Airbus Industries) ....................................................................................................................................... 50 Figure 4.6 Ultrasonic Test Setup in Through Transmission Mode (Courtesy of Kaufmann et al.) ..................................................................................................................................................... 52 Figure 4.7 Design solutions, the corresponding Pareto points and the Pareto frontier ........ 54 Figure 4.8 Trade-off Between Acquisition and Operating Costs .............................................. 55 Figure 5.1 Schematics field of Aero-Elasticity ........................................................................... 57 Figure 5.2 Scope of Static and Dynamic Aero-Elasticity ........................................................... 58 Figure 5.3 Linear and nonlinear aero-elastic response ............................................................ 59 Figure 5.4 Sequence of a typical Aero-Elastic Problem ............................................................ 60 Figure 5.5 Flutter Boundaries for AGARD 445.6 Wing ............................................................. 64 Figure 5.6 Generic Nonlinear Aero-Elastic Behavior ................................................................ 65 Figure 5.7 Basic wing model configuration ............................................................................... 67 Figure 5.8 Lift coefficient (top row), and tip twist angle (bottom row) variations with AoA and Aspect Ratio (AR) (Courtesy of [Afonso, et al.]35) ................................................................. 68 Figure 5.9 Flutter speed difference (difference between linear and non-linear) in function of vertical tip displacement difference (Courtesy of [Afonso, et al.]35) ........................................... 69 Figure 5.10 One-dimensional example of Lagrangian, Eulerian and ALE mesh and particle motion .............................................................................................................................................. 70 Figure 6.1 Comparison of the Linear and Nonlinear Aero-Elastic Response of the Panel at M =2.3 Linear Theory Predicts Non-Physical Exponential Growth................................................. 76
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Figure 6.2 Rotor 67 grids at walls .............................................................................................. 77 Figure 6.3 Steady State relative Mack Number contours Experimental (left) and Computation (right) ........................................................................................................................ 78 Figure 6.4 Time History of 3 Nodal Displacements .................................................................. 79 Figure 6.5 Lift to Drag Ratio of the Single-Point Optimized Hydrofoil Compared to the Baseline* ........................................................................................................................................... 84 Figure 6.6 Comparison between the multipoint hydro-structural optimization result (red/blue) and the single-point hydro-structural optimized (at CL=0.65) hydrofoil (left/red) – Courtesy of 53) .................................................................................................................................. 86 Figure 6.7 Comparison of sectional geometries for the NACA 0009 baseline, the multipoint optimized design, and the single-point optimized design at ten sections along the span – (Courtesy of 53) ................................................................................................................................ 87 Figure 6.8 Load-Displacement Transfer Operation .................................................................. 91 Figure 6.9 Common Research Model Surface Pressure Coefficient (Courtesy of Kenway et al.) ..................................................................................................................................................... 96 Figure 6.10 CSM and CFD Solution Comparison for Each Mesh Level (Courtesy of Gaetan et al.) ..................................................................................................................................................... 97 Figure 6.11 Boeing 777 (Left) and CRM (Right) – (Courtesy of 63) .......................................... 98 Figure 6.12 The wing-box structure is clamped at the Symmetry Plane and Partially Constrained at the Wing-Fuselage Junction .................................................................................. 99 Figure 6.13 Free form deformation (FFD) Volume with Control Points Modified by the Optimization are shown as Red Dots – (Courtesy of Gaetan) .................................................... 100 Figure 6.14 Progression of the jig Aerodynamic surface ...................................................... 101 Figure 6.15 Airfoils of the F11-Wing (not to true scale) –(Courtesy of Anhalt et al.) .......... 102 Figure 6.16 Structural Layout of the Airbus A340 Wing (Courtesy of Airbus Industries) .. 103 Figure 6.17 Structural Layout of the F11 Wing (Courtesy of Anhalt) ................................... 103 Figure 6.18 Example of an Automatically Generated Wing Structure ................................... 104 Figure 6.19 Finite Element Model of the F11 Wing ................................................................ 105 Figure 6.20 Deformation of the F11 Wing ............................................................................... 106 Figure 6.21 New Structural Concept for Laminate Structures ............................................... 107 Figure 7.1 Sources of Aerodynamic Noise Generation in Passenger Car .............................. 109 Figure 7.2 Process for running a local DES around the flap edge based on the full steady RANS solution ................................................................................................................................ 116 Figure 7.3 Unstructured Grid for Flap Edge ............................................................................ 116 Figure 7.4 Third octave far field spectra from flap edge for various grids. The emission angle is 86 degrees .................................................................................................................................. 117 Figure 7.5 Log RMS Surface Pressure illustrating the major sources of noise on the flap .. 118 Figure 7.6 Time averaged vortex flow structures around the flap edge ............................... 119
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1 Introduction 1.1 Multiphysics Flow Multiphysics is a computational discipline which treats simulations that involve multiple physical models or multiple simultaneous physical phenomena. For example, combining chemical kinetics and fluid mechanics or combining finite elements with molecular dynamics. Multiphysics typically involves solving Coupled Systems of partial differential equations. Many physical simulations involve coupled systems, such as electric and magnetic fields for electromagnetism, pressure and velocity for sound, or the real and the imaginary part of the quantum mechanical wave function. Another case is the mean field approximation for the electronic structure of atoms, where the electric field and the electron wave functions are coupled1. Figure 1.1 Pressure Contours on Elastic Band Two of the main Multiphysics phenomena are Aero elasticity (deal with Aerodynamics and Structure) and Aeroacoustics (Aerodynamics and sound propagation). Figure 1.1 displays an example of FSI using an elastic band under the pressure. Aero-Elasticity Aero elasticity is not solely concerned with aircraft, and the topic is extremely relevant for the design of structures such as bridges, Formula1 racing cars, wind turbines, turbomachinery blades, helicopters, etc. However, here we only concentrate on fixed wing aircraft, with the emphasis being on large commercial aircraft, but the underlying principles have relevance to other applications. It is usual to classify aero-elastic phenomena as being either static or dynamic (see Figure 1.2).
Figure 1.2
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Local and Deformed Coordinate of Elasticity
From Wikipedia, the free encyclopedia.
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Static Aero-elasticity Static aero-elasticity considers the non-oscillatory effects of aerodynamic forces acting on the flexible aircraft structure. The flexible nature of the wing will influence the in-flight wing shape and hence the lift distribution in a steady (or so-called equilibrium) maneuver or in the special case of cruise. Thus, however accurate and sophisticated any aerodynamic calculations that are carried out, the final in-flight shape could be in error if the structure is modelled inaccurately; drag penalties could result and the aircraft range could reduce. Usually static aero-elastic effects can also lead to a reduction in the effectiveness of the control surfaces and eventually to the phenomenon of control reversal; here, for example, the aileron has the opposite effect to that intended because the rolling moment it generates is negated by the wing twist that accompanies the control rotation. There is also the potentially disastrous phenomenon of divergence to consider, where the wing twist can increase without limit when the aerodynamic pitching moment on the wing due to twist exceeds the structural restoring moment. It is important to recognize that the lift distribution and divergence are influenced by the trim of the aircraft, so strictly speaking the wing cannot be treated on its own. Dynamic Aero-elasticity Dynamic aero-elasticity is concerned with the oscillatory effects of the Aero-Elastic interactions, and the main area of interest is the potentially catastrophic phenomenon of flutter. This instability involves two or more modes of vibration and arises from the unfavorable coupling of aerodynamic, inertial and elastic forces. It means that the structure can effectively extract energy from the air stream. The most difficult issue when seeking to predict the flutter phenomenon is that of the unsteady nature of the aerodynamic forces and moments generated when the aircraft oscillates, and the effect the motion has on the resulting forces, particularly in the transonic regime. The presence of flexible modes influences the dynamic stability modes of the rigid aircraft and so affects the flight dynamics. Also of serious concern is the potential unfavorable interaction of the flight control system with the flexible aircraft, considered in the topic of aero-servo-elasticity (also known as structural coupling). Aero elastic considerations influence the aircraft design process in a number of ways. Within the design flight envelope, it must be ensured that flutter and divergence cannot occur and that the aircraft is sufficiently controllable. The in-flight wing shape influences drag and performance and so must be accurately determined requiring careful consideration of the jig shape used in manufacture. The aircraft handling is affected by the aero-elastic deformations, especially where the flexible modes are close in frequency to the rigid body modes. Single Discretization Methods These software packages mainly rely on the Finite Element Method or similar commonplace numerical methods for simulating coupled physics; thermal stress, electromechanical interaction, fluid structure interaction (FSI), fluid flow with heat transport and chemical reactions, electromagnetic fluids (magneto-hydrodynamics or plasma), electromagnetically induced heating. In many cases, to get accurate results, it is important to include mutual dependencies where the material properties significant for one field (such as the electric field) vary with the value of another field (such as temperature) and vice versa. Multiple Discretization Methods There are cases where each subset of partial differential equations has different mathematical behavior, for example when compressible fluid flow is coupled with structural analysis or heat transfer. To perform an optimal simulation in those cases, a different discretization procedure must be applied to each subset. For example, the compressible flow is discretized with a finite volume method and the conjugate heat transfer with a finite element analysis. Another example is the use of electromagnetic or electrostatic methods combined with Direct simulation Monte Carlo, where the particles may interact with an electromagnetic (EM) field or other fields, with each other, and with
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fluids evolved by finite volume or other methods. The particles interact with the EM fields through the charges and currents they create and by being accelerated by the EM field. Particles collide with each other, and they collide with fluids. Aero-Acoutics In developing numerical methods for sound generation and propagation problems, it is natural to try to adapt methods used generally in CFD. To reliably do so, however, we must first examine those characteristics of sound generation and propagation problems that are likely to pose a challenge to traditional methods. The generation of acoustic waves by fluid motion is, by its nature, an unsteady process; steady flows generate no sound. Turbulence modeling, leading to RANS, unsteady RANS, and LES, filters small spatial and high frequency fluctuations from the solution; the impact of such filtering on sound generation has not yet been characterized in any systematic way. Most computational results for sound generation, therefore, have used DNS, where all relevant scales of motion are resolved. Use of LES for aeroacoustics is under active development. Acoustic waves may propagate coherently, with very low attenuation due to viscous effects, over long distances in the flow. Artificial dissipation and dispersion at a level that may be tolerable for hydrodynamic fluctuations can lead to unacceptable attenuation of acoustic waves. As previously mentioned, even loud flows radiate a very small fraction of their total energy as sound. That is, acoustic (radiation) efficiency is invariably very low. At low Mach number, acoustic inefficiency can be viewed as the result of a delicate cancellation process of equal but nearly opposite sources (e.g. dipole, quadrupole, etc.). Numerical errors that may upset this delicate balance can therefore lead to serious overestimates of sound generation.. For example, truncation of a computational domain with artificial boundary conditions is a primary cause of such errors. As will be discussed in Section 4, one route to efficient and accurate solution to these difficulties is the use of high-order and optimized finite difference (FD) schemes. It should be kept in mind that even with high-order (or indeed spectral) methods, LES and DNS at low to moderate Reynolds numbers requires large numbers of computational nodes (106 and up). Furthermore, it is surprisingly difficult to maintain high accuracy and computational efficiency for flows in complex geometry (i.e. with unstructured or overlapping body-fitted coordinates). Thus complex geometry codes can only obtain good accuracy by increasing the resolution (hopefully adaptively). Complex geometry calculations are then typically limited to either linearized flow or unsteady RANS calculations, such that the computational resources required are not too large. These do not provide an unambiguous estimate of the sound sources or the radiated field, and engineering judgment must then dictate how useful data can be extracted for a given problem. In either case (DNS or modeled equations), one must proceed cautiously, always keeping in mind errors and uncertainties associated with the results.
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2 Multiphysics Flow 2.1 What Constitutes Multiphysics? Multiphysics system consists of more than one module governed by its own principles for evolution or equilibrium, typically conservation or constitutive laws. A major classification in such systems is whether the coupling occurs in the bulk (e.g., through source terms or constitutive relations that are active in the overlapping domains of the individual components) or whether it occurs over an idealized interface that is lower dimensional or a narrow buffer zone (e.g., through boundary conditions that transmit fluxes, pressures, or displacements). Typical examples of bulk-coupled multi-physics systems with their own extensively developed literature include radiation with hydrodynamics in astrophysics (radiation-hydrodynamics, or “rad-hydro”), electricity and magnetism with hydrodynamics in plasma physics (magneto hydrodynamics), and chemical reaction with transport in combustion or subsurface flows (reactive transport). Typical examples of interfacecoupled multi-physics systems are ocean-atmosphere dynamics in geophysics, fluid-structure dynamics in Aero-Elasticity. Beyond these classic multi-physics systems are many others that share important structural features. Success in simulating forward models leads to ambitions for inverse problems, sensitivity analysis, uncertainty quantification, model-constrained optimization, and reduced-order modeling, which tend to require many forward simulations. In these advances, the physical model is augmented by variables other than the primitive quantities in which the governing equations are defined. These variables may be probability density functions, sensitivity gradients, Lagrange multipliers, or coefficients of system-adaptive bases. Equations that govern the evolution of these auxiliarydependent variables are often derived and solved together with some of the physical variables. When the visualization is done in situ with the simulation, additional derived quantities may be carried along. Error estimation fields in adaptive meshing applications may constitute yet more. Though the auxiliary variables may not be “physical” in the standard sense, they give the overall simulation the structure of multi-physics. In another important class of systems that might fall under the title of multi-physics by virtue of being multiscale, the same component is described by more than one formulation, typically with a computationally defined boundary or transition zone between the domains of applicability. We refer, for example, to field-particle descriptions of N-body systems in celestial mechanics or molecular dynamics, in which the gravitational or electrical forces between particles that are not immediate neighbors are mediated by a field that arises from the particles themselves. Typically, each particle defines a partition of the domain into “near” and “far” for this purpose, and the decomposition is strict. Another example is provided by atomistic-continuum models of solids, such as are used in crack propagation. In this case, the atomistic and continuum models both hold in a zone of finite thickness. Recent schemes based on projective integration keep both fine and coarse models simultaneously in the picture and pass between the different “physics” for reasons of computational complexity. Specifically, they compute as much as possible with the coarse model, which may have constitutive terms that are difficult to derive from first principles but can be computed locally in time and space by taking suitable windowed averages or moments of the dependent variables of the fine model, which are closer to first principles. In these models, the coarse-to-fine transformation is called “lifting,” and the fine-to-coarse transformation is called “restriction.” Lifting may mean populating an ensemble of particles according to a distribution, and restriction may mean averaging. Still other systems may have a multi-physics character by virtue of being multi-rate or multiresolution. A chemical kinetics model may treat some components as being in equilibrium, idealizing a fast relaxation down to a constraint manifold on which other components vary more slowly. Some phenomena may be partitioned mathematically by projections onto wavelet bases of different
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frequency or wavenumber properties that are naturally treated differently. Stretching the semantics of multi-physics still further, we may distinguish only between different mathematical formulations or even just different discretization of what is essentially the same physical model. An example is grafting a continuum-based boundary-element model for the far field onto a finite-element model for the near field. Systems of partial differential equations (PDEs) of different types (e.g., ellipticparabolic, elliptic-hyperbolic, or parabolic-hyperbolic) for the same component may be thought of as multi-physics because each of the classical PDE archetypes represents a different physical phenomenon. Even a single equation with terms of different types represents a multi-physics model because each term must often be handled through separate discretization or solver methods.
2.2 Simulation Strategies for Multiphysics Problems There is a growing interest in analysis and efficient simulation of coupled multi-physics problems. This is enabled by continuing increases in computing power. As a result, both steady-state and transient calculations of coupled systems are increasingly sought. Multi-physics treats simulations that involve multiple physical models or multiple simultaneous physical phenomena. For example, combining chemical kinetics and fluid mechanics or combining finite elements with molecular dynamics. Multi-physics typically involve solving coupled systems of partial differential equations. Many physical simulations involve coupled systems, such as electric and magnetic fields for electromagnetism, pressure and velocity for sound, or the real and the imaginary part of the quantum mechanical wave function. Another case is the mean field approximation for the electronic structure of atoms, where the electric field and the electron wave functions are coupled algorithms may now exploit partitioning and iterative decoupling of the systems for various reasons. There are cases where each subset of partial differential equations has different mathematical behavior, for example when compressible fluid flow is coupled with structural analysis or heat transfer. To perform an optimal simulation in those cases, a different discretization procedure must be applied to each subset. For example, the compressible flow is discretized with a finite volume method and the conjugate heat transfer with a finite element analysis. Fluidyn-MP is one of the example of commercially available software package for simulating Multi-physics engineering problems using Multiple Discretization Methods2. Other notables included the MPCCI®. There are in general three methods of solution can be identified for Multiphysics problem.
2.4
Methodologies
Beside direct Monolithic schemes, there are multitudes of coupling schemes available through the literature and most fall into a weakly or strongly coupled partition schemes. Both refer to the case when the each domain is discretized separately and coupled using a synchronized procedure both in time and space. The weakly-coupled uses a single sub-iteration to advance both fluid and solid, where strong-coupling employs number of sub-iteration to enforce the conversation laws. Most use an under-relaxation factor to accelerate the convergence rate, and stability. The analogy could best demonstrated by a matrix notation to illustrate its modularity3. For a coupling which involves Fluid/Structure/Thermal/Electromagnetic interaction, the resulting matrix system displayed in Eq. 2.1 where the subscripts S, F, T, E stand for Structure/Fluid/Thermal/Electromagnetic fields and U are the vector of unknowns. The right hand side is the residuals resulting from fluxes. The diagonal sub-matrices are coefficients as if each sub-domain solved independently. The off-diagonal subwikipedia Lohner, Rainald, Cebral, Juan and Yang, Chi, “ Extending the Range and Applicability of the Loose Coupling Approach for FSI Simulation” 2 3
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matrices represent the interface interactions among different domains. Similarly, other modes such as Acoustics, Vibration or Controls could also be added.
K SS K FS K TS K ES
K SF K FF
K ST K FT
K TF K EF
K TT K ET
K SE ΔU S R S K FE ΔU F R F K TE ΔU T R T K EE ΔU E R E
Eq. 2.1
The analogy best described with the management of a Supervisor with different domain codes and depicted in Figure 2.1. Among other duties performed by Supervisor can be grouped as:
Global parameters Communicating with server and Domain Codes Coupling Schemes Convergence criteria Step size time management (Dual Times?) Maximum number of sub-iterations allowed Initial relaxation Termination flag Tag the interface region (Grid Motion, Deformation and Non-Matching interface) Quantities to be exchanged and when Feedback and monitor the result Minimize Uncertainties Job status and termination
Figure 2.1
Supervisor interaction with different domain codes
There are in general three approaches can be identified for Multiphysics problem. For example for fluid-solid interaction (FSI), these are Partitioned methods, Monolithic methods, and unified single
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method. While the Monolithic and Portioned used extensively, unified single, is based on continuum mechanics formulation for fluids and structures where both continua can be solved using the momentum and continuity equation. The difference between the two continua lies in the constitutive equations4.
2.3 PDE-Based Multiphysics Applications A broad range of multi-physics simulations are under way in the computational science community, as researchers increasingly confront questions about Figure 2.2 Classification of FSI Methods complex physical and engineered systems characterized by multiple, interacting physical processes that have traditionally been considered separately. Some large-scale PDE-based multi-physics applications are fluid-structure interaction (FSI), fission reactor fuel performance, reactor core modeling, crack propagation, DNA sequencing, fusion, subsurface science, hydrology, climate, radiation hydrodynamics, geodynamics, and accelerator design, to name the few. Coupling Strong vs. Weak Coupling of Physical Model We use strong (versus weak) coupling to refer to strong interactions between different physics models that are intrinsic between the different physics in a natural process. A strong coupling could be due to large overlap of geometric domains or due to a strong influence of one model on another. For example, in climate modeling, the coupling between the ocean and sea-ice models is strong, while the coupling between ocean and land models is weak. Mathematically, the off-diagonal block of the Jacobian matrix of a strongly coupled multi-physics model may be nearly full or may be sparse but contain relatively larger entries. In contrast, a weakly coupled multi-physics model may contain relatively few or relatively small off-diagonal entries. Tight vs. Loose Coupling of Numerical Models We use tight (versus loose) coupling to refer to the algorithmic aspect of multi-physics coupling schemes (or numerical models) in terms of whether the state variables across different models are well synchronized. A tightly coupled scheme (sometimes referred to as a strongly coupled scheme) would keep all the state variables as synchronized as possible across different models at all times, whereas a loosely coupled scheme may allow the state variables to be shifted by one time step or be staggered by a fraction of time steps. For example, a monolithic fluid-structure approach and a full iterative scheme are tightly coupled, whereas a sequential staggered scheme is loosely coupled. Note that there is not a direct one-to-one correspondence between strong (versus weak) physical coupling and tight (versus loose) numerical coupling, since a tight or a loose numerical coupling scheme may be used for strongly or weakly coupled physical models.
C. G. Giannopapa and G. Papadakis, “Indicative Results and Progress On The Development of The Unified Single Solution Method For Fluid-Structure Interaction Problems”, DOI: 10.1115/PVP2007-26420, January 2007. 4
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Error and Residual Error and residual are related concepts that are sometimes not sufficiently carefully distinguished. Operationally and mathematically their distinctions should be preserved. Given a putative solution, the error is a measure of the distance of that putative solution from the true solution. The residual is a measure of how well the putative solution solves the intended equation. The residual is always computable, whereas the error usually is not. In systems that possess multiple roots or are illconditioned, a small or even zero residual does not necessarily imply a small error. Most iterative methods for the solution of linear or nonlinear systems employ, in an inner loop, a map from a residual to an estimated error. Solving exactly for the error in this case is as difficult as the original problem. Solving approximately may be easy, in which case a convergent iteration is often possible. When the system is nonlinear, a Newton iteration accompanied by an approximate linear solution with the Jacobian matrix maps the nonlinear residual to an estimated error. Explicit Methods In an explicit method, future states are given by a formula involving known computed quantities that can be evaluated directly. The simplest example is forward Euler: yn+1 = yn + Δt F(yn). Explicit methods are inexpensive per iteration because they require a fixed number of function evaluations to complete the step; however, they typically have severe stability restrictions and, therefore, are suited only for non-stiff problems. Implicit Methods A fully implicit method advances the solution in time by using current information and an inversion process with no explicit steps. The simplest example is backward Euler: yn+1 = yn + Δt F(yn+1). Fully implicit methods are relatively expensive because of the need to solve (non)linear systems at each step; however, they have favorable stability properties. Multiscale Model A multiscale model of a physical system finds use when important features and processes occur at multiple and widely varying physical scales within a problem. (These scales are frequently expressed as spatial and/or temporal scales.) In multiscale models, a solution to the behavior of the system as a whole is aided by computing the solution to a series of sub problems within a hierarchy of scales. At each level in the hierarchy, a sub problem concerns itself with a range of the physical domain appropriate to the scale at which it operates. An example of a multiscale model is described where the development of cracks in solid materials is computed through a successive hierarchy of models: fast-acting, close-range atomic force calculations on the smallest microscopic scale through to macroscopic continuum models on the largest macroscopic scale. Macro-Scale The largest physical scale within a physical system or field of interest. Where the microscale and mesoscale frequently concern themselves with constituent quanta, such as individual atoms, molecules, or groups of such, the macroscale frequently deals with the continuum and bulk properties of a system. For example, in atmospheric science, the macroscale is at the scale of the largest features within weather or climate modeling, which are 1000 km; in engineering fields, the macroscale is often given by scale of the entire device of interest, which, depending on the device, can range from nanometers to kilometers.
Meso-Scale A physical scale intermediate to the largest and smallest scales existing within a physical system of interest. Like all physical systems with multiple scales, models at the mesoscale may represent one or more levels of a multiscale model or may represent the scale of a standalone model in its own right. The term mesoscale modeling first found use in meteorology for the modeling of severe storm
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systems on a local scale. Reliable storm forecasts cannot be achieved with standard continental-scale weather models, where grid sizes are typically of a scale larger than the size of the storm being modeled. Micro-Scale The smallest physical scale within a physical system or field of interest. Calculations at the microscale frequently treat fundamental system constituents and interactions. Models at the microscale may exist as standalone or as a level within a multiscale model. Examples of the microscale in chemistry and materials science usually refer to quantum-mechanical, atomic-scale interactions ( ͠ 10 nm). In meteorology, the microscale range is on the order of 1–105 cm and treats phenomena such as tornados, thunderstorms, and cloud-scale processes. Operator Splitting Multiphysics operator splitting is a loose coupling scheme that evolves one time step for each physics component in sequence. Schur Complement The Schur complement of a square diagonal sub-block of a square matrix is, itself, a square matrix that represents the original linear system after unknowns corresponding to the complementary nonsingular diagonal sub block have been eliminated. Static condensation of a finite-element stiffness matrix is an instance; here, typically, interior degrees of freedom of an element are eliminated. The concept generalizes algebraically. Semi-Implicit Methods Inclusion of both implicit and explicit elements gives rise to semi-implicit methods. Sub-Cycling Given two time-dependent variables w1 and w2, assume their time evolution is coupled between time levels tn-1 and tn. Let ui ≈ wi for i = 1, 2,,,, be decoupled approximations satisfying
u1 F1 (u1 , u 2 (t n 1 )) 0 t
,
u 2 F2 (u1 (t n 1 ),u 2 ) 0 t
Eq. 2.2
It is said that u2 is sub-cycled M times relative to u1 if u2(tn) is approximated by any time stepping method requiring M time steps to evolve from tn-1 to tn.
2.4 Attributes of Coupling Process as Devised by Supervisor Dual Time Step For a coupled FSI with sub-iterations, there are three distinctive time steps to consider. There are fluid and structure time steps (ΔtF, ΔtS), as well as, the coupling time step (ΔtC). The coupling time step controls the temporal or sub-iteration of the overall system and it could devised using an existing algorithms. It is recommended that its order of accuracy should be matched by order of accuracy of the coupling systems, or
O(tc ) Min (O(t F ) , O(t S ))
Eq. 2.3
Other considerations could be the relative choices of the step times, where for a transient simulation one solver could requires smaller time steps for greater accuracy. Forcing the same on the other solver might yield an inefficient process. Other considerations could be devised as choices are
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each time steps (constant, linear, adaptive), as well as, the number of time step iterations for each domain simulations within each sub-iteration. Other stringent time steps control algorithm may be introduced based on controlling accuracy as determined by truncation error estimates5 as
e 1 Δt c n en eu Max (e u , e T ) , e u tolu
KP
1 en u
, e
KI
KD
e n 12 2 Δt Prev e 2 n
U n U n 1
eT , eT tolT
Un
T
, e
T n T n 1 Tn
Eq. 2.4 And Δtc represents the new time-step size, ΔtPrev is the time-step size at the previous step, en is the measure of the change of the quantities of interest in time K P, K I, and K D are the parameters and tolu and tolT are user supplied tolerances corresponding to the normalized changes in velocities and temperature vectors, respectively. Iteration Number Each code computes its part of the problem with data exchanges at pre-defined intervals. For transient cases there could be several exchanges as prescribed by time differentiation. The data exchange could be managed on either before or after each iteration as devised previously. If convergence is not achieved and NMAX (maximum iteration) is not reached, a new time step would be devised using a second-order backward implicit – forward explicit scheme suggested by (Oden & Roache -1993) as appears as
f
n 1 i, j
f i,nj /Δt Ln 1
Ln 1 Ln
F
n 1 i, j
Fi,nj1 f i,nj
f i,nj /t OLD Ln
Δt f i,nj f i,nj1 Δt OLD
Eq. 2.5
With F signifying the projected value of quantity f on new time. The interface quantities updated for new iteration. Completion The computation is terminated by either the user-defined convergence criteria, or when NMAX is reached. Initialization The perspective domain codes initialize their data. These could include setting the initial conditions and timer or iterative flags on each domain codes. At this stage, a suitable coupling technique could be chosen with corresponding time step size. The interface region and the exchanges quantities should be specified and tagged as the send/receive on each code. For couplings using underrelaxation, a relaxation method (i.e., Aitken) could be chosen with a suitable initial value should be elected. The residual stooping criteria ε and maximum coupling numbers (NMAX) are also among values to be chosen at this stage.
Gustafsson K. , “Control theoretic techniques for step size selection in explicit Runge–Kutta methods”, ACM Transactions on Mathematical Software 1991; 17:533–554. 5
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Uncertainties One of duties of the supervisor would be to minimize the uncertainties due to coupling schemes. These could be uncertainties relevant to interpolation, spatial and temporal discretization, and simulation parameters. The overall uncertainty could be devised as
(U s ) 2 (U I ) 2 (U G ) 2 (U t ) 2 (U p ) 2
Eq. 2.6
Where I, G, t and P denote Interpolation, Grid, temporal and other parameters. The aim is to keep the values at %95 confidence level. A general approach would be to use the first-order sensitivity of the dependent parameters w.r.t. independent ones as:
P P(x)
2 P U p Ux x
1/2
Eq. 2.7
Where P is the dependent parameter and x are vectors of the independent ones. Ux are the uncertainties.
2.5 Interface Coupling Schemes
Fixed point This is a sub-set of Dirichlet-Neumann scheme when the structure response to fluid forces are significant (i.e., fluid is the dominant)6. It is the algorithm implemented for current investigation, complemented with a strongly-coupled sub-iteration. The interface Γ coupling operation could be organized iteratively as
X Γ,n 1 f (X Γ,n ) , n 1, 2, 3,....
Eq. 2.8
In the hope of converging x = f(x) on a continuous f. The interface function f could be denoted as the structure/fluid operator f = SΓ-1FΓ and consequently Eq. 2.8 could be readily written as
~ 1 X Γ,n 1 SΓ FΓ (X Γ,n ) With
x Γ, n+1
Eq. 2.9
being any projected dependent variable (i.e., structure displacement). The iterative
scheme could continue until a pre-defined convergence criteria is satisfied for residuals Cells
rΓ,n 1
~ X Γ,n 1 - X Γ,n ,
R Γ,n 1
r
Γ, n 1
1
L ref
Eq. 2.10
Subjected to RΓ, n+1
