Advanced Computational Methods in Science and Engineering (eBook)

Foreword 5
Preface by the editors 6
Contents 8
A Model-Order Reduction Approach toParametric Electromagnetic Inversion 10
1 Introduction 10
2 Integral representations and their discretized counterparts 12
3 Reduced-order models for the scattered field 15
4 The reduced-order model objective function 18
4.1 Multiple frequencies 20
5 Numerical experiments 21
References 28
Shifted-Laplacian Preconditioners for Heterogeneous Helmholtz Problems 29
1 Introduction 29
2 The Helmholtz equation and a seismic application 31
2.1 Mathematical problem definition 32
2.2 Discretization 34
2.2.1 Validation of the discretization 35
3 Iterative solution method 36
3.1 Multigrid for the Helmholtz equation 37
3.2 Shifted Laplacian preconditioned Krylov subspace method 38
3.3 Fourier analysis 39
3.4 Multigrid preconditioner 43
3.5 AMG type interpolation 44
4 Numerical experiments 46
4.1 Homogeneous problem 46
4.1.1 Second- and fourth-order discretizations 46
4.1.2 Fixed wavenumber, increasing mesh sizes 48
4.2 The wedge problem 49
4.3 The Sigsbee problem 49
5 Conclusion 51
Acknowledgment 51
References 52
On Numerical Issues in Time Accurate LaminarReacting Gas Flow Solvers 55
1 Introduction 56
2 Numerical modeling of chemical vapor deposition 57
2.1 Transport model for the gas species 59
2.2 Modeling of surface chemistry 60
3 Chemistry models 61
3.1 Chemistry model I: 7 species and 5 gas phase reactions 61
3.2 Chemistry model II: 17 species and 26 gas phase reactions 63
4 Numerical difficulties 65
4.1 Positivity conservation of mass fractions 65
4.1.1 Positive time integration 66
4.1.2 Positivity for Euler Forward (EF) and Euler Backward (EB) 67
4.1.3 Positive time integration continued: general remarks 68
4.2 Comparison of some stiff ODE methods 68
4.2.1 Numerical results 70
5 Design of the Euler Backward solver 73
5.1 Globalized Inexact Projected Newton methods 73
5.2 Preconditioned Krylov solver 74
6 Numerical results 76
6.1 Discussion on the integration statistics 77
6.2 Model validation and time accurate solutions 79
7 Conclusions 82
References 85
Parallel Scientific Computing on LooselyCoupled Networks of Computers 87
1 Introduction 88
2 The problem 89
3 The basics: iterative methods 91
3.1 Simple iterations 92
3.2 Impatient processors: asynchronism 93
4 Acceleration: subspace methods 95
4.1 Hybrid methods: best of both worlds 97
4.2 Some experimental results 99
5 Efficient numerical algorithms in Grid computing 100
5.1 Grid middleware 101
5.1.1 Brief description of GridSolve 101
5.1.2 Brief description of CRAC 103
5.2 Target hardware 104
5.3 Parallel iterative methods: building blocks revisited 105
5.3.1 Matrix–vectormultiplication 105
5.3.2 Vector operations 106
5.3.3 Preconditioning step 107
5.3.4 Convergence detection 107
5.4 Applications 108
5.5 Advanced techniques 108
6 Concluding remarks and further reading 109
References 111
Data Assimilation Algorithms for Numerical Models 115
1 Introduction 115
2 Variational data assimilation 117
2.1 Data assimilation formulated as a minimization problem 117
2.2 The adjoint model 118
2.3 Discussion 119
3 Kalman filtering 120
3.1 The linear Kalman filter 120
3.2 Kalman filtering for large-scale systems 122
3.2.1 Square root filtering 122
3.2.2 Classical Ensemble Kalman filter 123
3.2.3 Reduced Rank Square Root Kalman filter 124
3.2.4 Discussion 126
4 A software environment for data assimilation: COSTA 127
4.1 COSTA components 128
4.2 The COSTA software 129
5 Applications in coastal sea modeling 131
5.1 Storm surge prediction using Kalman filtering 131
5.2 Using an RRSQRT Kalman filter to assimilate HF radar datainto a 3D coastal ocean model 135
5.2.1 Introduction 135
5.2.2 The TRIWAQ model for the IJmond region 136
5.2.3 Kalman filtering for large 3D models 138
5.2.4 Results 140
6 Applications in atmospheric chemistry modeling 140
6.1 Eulerian chemistry transport model EUROS 141
6.2 State space representation 141
6.3 Ground based observations 142
6.4 Results 143
6.5 Performance of RRSQRT algorithm, ENKF algorithm andCOFFEE (Complementary Orthogonal subspace Filter ForEfficient Ensembles) algorithm 145
7 Conclusions 146
References 148
Radial Basis Functions for InterfaceInterpolation and Mesh Deformation 151
1 Introduction 151
2 Non-matching meshes 152
2.1 Conservative and consistent coupling approach 153
2.1.1 Conservative approach 154
2.1.2 Consistent approach 155
2.2 Radial basis function interpolation (RBFI) 156
2.2.1 Radial basis functions 157
2.2.2 Conservative approach 158
2.3 Analytical test problems 159
2.3.1 Transferring a smooth field 161
2.3.2 Transferring a non-smooth field 162
2.4 Quasi-1D FSI problem 164
2.4.1 Flow equations 165
2.4.2 Structure equations 165
2.4.3 Coupling procedure 166
2.4.4 Results 166
3 Mesh movement based on radial basis function interpolation 168
3.1 Radial basis function interpolation 169
3.2 Mesh quality metrics 172
3.3 2D mesh movement 173
3.3.1 Test case 1: Rotation and translation 173
3.3.2 Test case 2: Rigid body Rotation 176
3.3.3 Test case 3: Airfoil flap 177
3.3.4 Test case 4: Flow around airfoil 178
3.4 Importance of smooth mesh deformation for higher ordertime-integration 179
3.5 3D mesh deformation 181
3.6 Rotation and translation of 3D block 181
3.6.1 Flutter of the AGARD 445.6 wing 182
4 Conclusions 183
References 184
Least-Squares Spectral Element Methods inComputational Fluid Dynamics 187
1 Introduction 187
2 The least-squares formulation 188
2.1 Rayleigh-Ritz method 188
2.2 Galerkin formulations 191
2.3 Least-squares formulation 192
3 Spectral element methods 194
4 Convergence and a priori error estimates 195
5 Incompressible flows 198
5.1 Governing equations 198
5.2 The first order formulation of the Navier-Stokes equations 199
5.3 Linearization of the non-linear terms 199
5.4 Steady flow around a cylinder at low Reynolds 200
5.5 Comparison of numerical data with experimental data 201
5.5.1 S/d: LSQSEM vs. Experiment 201
5.6 Unsteady flow around a cylinder at low Reynolds 205
5.6.1 Outflow boundary condition 205
5.6.2 Base pressure coefficient 206
5.6.3 Drag and lift coefficients 207
6 Compressible flows 208
6.1 Compressible flow over a circular bump 209
6.1.1 General geometry and boundary conditions 209
6.1.2 Results for transonic flow 211
6.1.3 Results for supersonic flow 213
7 Miscellaneous topics 214
7.1 hp-adaptive LSQSEM 214
7.1.1 The mortar element method 214
7.1.2 The error estimator 215
7.1.3 Estimation of the Sobolev regularity 216
7.1.4 Application to the space-time linear advection equation 218
7.1.5 Illustration of an hp-adaptive strategy 219
7.2 Direct Minimization 220
7.2.1 Conventional least-squares finite element method 220
7.2.2 DirectMinimization - LSQSEM-DM 221
7.2.3 Global QR 223
7.2.4 The Poisson equation 224
7.3 Application of LSQSEM to viscoelastic fluids 227
8 Further reading 231
References 231
Finite-Volume Discretizations and ImmersedBoundaries 236
1 Introduction 236
2 Model equation and target problems 239
2.1 Standard finite-volume discretization 240
2.2 Initial and boundary conditions 240
2.3 Standard finite-volume schemes 241
3 Fluxes with embedded moving-boundary conditions 246
3.1 Higher-order accurate embedded-boundary fluxes 247
3.1.1 Cell-face states 247
3.1.2 Net cell fluxes 249
3.2 Spatial monotonicity domains and limiters 254
3.2.1 Spatial monotonicity domain and limiter for cell-face state 254
3.2.2 Limiter for cell-face state 256
3.2.3 Spatial monotonicity domain and limiter for cell-face state 256
4 Temporal discretization 258
4.1 TVD conditions and time step 259
4.1.1 TVD conditions for limiter function 261
4.1.2 TVD conditions for limiter function 263
4.2 Local adaptivity in time 267
5 Numerical examples 270
6 Extension to more general cases 271
6.1 Extension to higher dimensions 272
6.2 Higher-order accuracy in time 272
7 Conclusion 273
Appendix 273
References 274
Large Eddy Simulation of TurbulentNon-Premixed Jet Flames with a High OrderNumerical Method 276
1 Introduction 276
2 Governing equations 278
3 Chemistry model 280
4 Numerical method 281
4.1 The derivative 282
4.2 Time discretization 285
4.3 The discretization of the Navier-Stokes equations for theSANDIA flame D problem 285
5 Boundary conditions 287
5.1 Parallel implementation 288
6 Simulation details 289
7 Summary and conclusions 293
8 Acknowledgements 293
References 293
A Suite of Mathematical Models for BoneIngrowth, Bone Fracture Healing andIntra-Osseous Wound Healing 295
1 Introduction 296
2 The bone-ingrowth model 298
2.1 The mechanical model 299
2.1.1 The elasticity domain 301
2.1.2 The porous tantalum 302
2.2 The biological part 303
2.3 The numerical method for the ingrowth model 306
2.4 Numerical experiments on the ingrowth model 309
3 The fracture healing model due to Bailon-Plaza 311
4 The model due to Adam 314
4.1 The model equations 315
5 Conclusions 316
References 317
Numerical Modeling of the ElectromechanicalInteraction in MEMS 321
1 Introduction 321
1.1 Electromechanical coupling 322
1.2 A one-dimensional example 324
1.3 Numerical modeling 327
2 Solution techniques 329
3 Finding the pull-in curve 331
3.1 Voltage stepping 331
3.2 Path following methods 332
3.3 Displacement stepping 334
3.4 Charge stepping 335
3.5 General remarks 337
3.6 Example 338
4 Coupled eigenfrequencies 341
4.1 Staggered method 342
4.2 Monolithic method 343
4.3 Example 344
5 Summary and conclusions 346
References 346
Simulation of Progressive Failure in CompositeLaminates 349
1 Introduction 349
2 Continuum damage 352
2.1 Material degradation 353
2.2 Consistent tangent 355
2.3 Regularization 356
2.4 Notched plate 358
2.5 Off-axis tensile test 362
3 Phantom node method 365
3.1 Kinematical and equilibrium relations 365
3.2 Crack propagation 367
3.3 Cohesive law 368
3.4 Consistent tangent 372
3.5 Off-axis tensile test 372
4 Conclusions 374
Acknowledgments 375
References 375
Numerical Modeling of Wave Propagation,Breaking and Run-Up on a Beach 378
1 Introduction 378
2 Governing equations 382
3 Numerical framework 384
3.1 Space discretization 384
3.1.1 Grid schematization 384
3.1.2 Location of grid variables 385
3.1.3 Space discretization of global continuity equation 387
3.1.4 Space discretization of local continuity equation 387
3.1.5 Space discretization of horizontal momentum equation 388
3.1.6 Space discretization of vertical momentum equation 389
3.2 Time integration 390
3.3 Solution method 392
4 Numerical experiments 395
4.1 Regular wave breaking on a slope 396
4.2 Periodic wave run-up on a planar beach 397
4.3 Regular breaking waves over a submerged bar 398
4.4 Irregular wave breaking in a laboratory barred surf zone 399
4.5 Deformation of waves by an elliptic shoal on sloped bottom 401
5 Conclusions 404
References 405
Hybrid Navier-Stokes/DSMC Simulations of GasFlows with Rarefied-Continuum Transitions 407
1 Introduction 407
2 From Boltzmann to Navier-Stokes 410
3 The hybrid numerical method 413
3.1 Introduction 413
3.2 Finite volume scheme for compressible Navier-Stokesequations 413
3.2.1 Finite volume discretization 413
3.2.2 Time discretization 414
3.2.3 The MUSCL discretization scheme 415
3.2.4 Chapman-Enskog split fluxes 415
3.3 Direct Simulation Monte Carlo scheme for rarefied gas flowsimulations 417
3.3.1 Initialization 418
3.3.2 Particle movement 418
3.3.3 Particle collisions 419
3.3.4 Sampling 421
3.4 Dynamic coupling of Navier-Stokes and DSMC solvers 422
3.4.1 Breakdown parameter 422
3.4.2 Steady-state formulation 422
3.4.3 Unsteady formulation 424
4 Results and discussion 425
4.1 Unsteady shock-tube problem 425
4.1.1 Sensitivity to numerical parameters 428
4.2 Rarefied Poiseuille flow 430
4.3 Steady-state jet expanding in a low pressure chamber 433
5 Conclusion 437
References 438
Multi-Scale PDE-Based Design of HierarchicallyStructured Porous Catalysts 440
1 Introduction 441
2 Implementation framework 441
3 PDE-based design of hierarchically structured porous catalysts 444
4 Conclusions 453
References 454
From Molecular Dynamics and ParticleSimulations towards Constitutive Relations forContinuum Theory 456
1 Introduction 457
2 The soft particle molecular dynamics method 457
2.1 Discrete particle model 458
2.2 Equations of motion 458
2.3 Normal contact force laws 459
2.3.1 Linear normal contact model 459
2.3.2 Adhesive, elasto-plastic normal contact model 460
2.3.3 Long range normal forces 461
2.4 Tangential forces and torques in general 462
2.4.1 Sliding 462
2.4.2 Objectivity 462
2.4.3 Rolling 463
2.4.4 Torsion 464
2.4.5 Summary 464
2.5 The tangential force- and torque-models 464
2.5.1 Sliding/sticking friction model 464
2.5.2 Rolling resistance model 466
2.5.3 Torsion resistance model 466
2.6 Background friction 466
2.7 Example: tension test simulation results 467
2.7.1 Model parameters for tension 468
2.7.2 Compressive and tensile strength 470
3 Hard sphere molecular dynamics 471
3.1 Smooth hard sphere collision model 472
3.2 Event-driven algorithm 472
4 The link between ED and DEM via the TC model 474
5 The stress in particle simulations 476
5.1 Dynamic stress 476
5.2 Static stress from virtual displacements 476
5.3 Stress for soft and hard spheres 477
6 2D simulation results 477
6.1 The equation of state from ED 478
6.2 Quasi-static DEM simulations 479
6.2.1 Model parameters 480
6.2.2 Boundary conditions 480
6.2.3 Initial configuration and compression 481
6.2.4 Compression and dilation 481
6.2.5 Stress tensor 483
7 Larger computational examples 485
7.1 Free cooling and cluster growth (ED) 485
7.1.1 Initial configuration 485
7.1.2 System evolution 486
7.1.3 Discussion 487
7.2 3D (ring) shear cell simulation 487
7.2.1 Model system 487
7.2.2 Material and system parameters 488
7.2.3 Shear deformation results 488
7.2.4 Discussion 491
8 Conclusion 492
References 492
Editorial Policy 496
General Remarks 497
Lecture Notes in Computational Scienceand Engineering 498
Monographs in Computational Scienceand Engineering 500
Texts in Computational Scienceand Engineering 501