Loading...Finite Element Methods – Parallel Sparse Statics and Eigen Solutions
Duc Thai Nguyen
Old Dominion University
Norfolk, Virginia
Contents
1. A Review of Basic Finite Element Procedures 1
1.1 Introduction . 1
1.2 Numerical Techniques for Solving Ordinary Differential
Equations (ODE) 1
1.3 Identifying the “Geometric” versus “Natural” Boundary
Conditions 6
1.4 The Weak Formulations 6
1.5 Flowcharts for Statics Finite Element Analysis 9
1.6 Flowcharts for Dynamics Finite Element Analysis 13
1.7 Uncoupling the Dynamical Equilibrium Equations 14
1.8 One-Dimensional Rod Finite Element Procedures . 17
1.8.1 One-DimensionalRod Element Stiffness Matrix . 18
1.8.2 Distributed Loads and Equivalent Joint Loads ., 21
1.8.3 Finite Element Assembly Procedures .22
1.8.4 Imposing the Boundary Conditions . 24
1.8.5 Alternative Derivations of System of Equations
from Finite Element Equations . 25
1.9 Truss Finite Element Equations 27
1.10 Beam (or Frame) Finite Element Equations 29
1.11 Tetrahedral Finite Element Shape Functions 31
1.12 Finite Element Weak Formulations for General 2-D
Field Equations 35
1.13 The Isoparametric Formulation .44
1.14 Gauss Quadrature 5 1
1.15 Summary . 59
1.16 Exercises . 59
2. SimpleMPI/FORTRAN Applications . -63
2.1 Introduction . 63
2.2 Computing Value of “d’by Integration 63
2.3 Matrix-Matrix Multiplication 68
2.4 MPI Parallel 110 . 72
2.5 Unrolling Techniques 75
2.6 Parallel Dense Equation Solvers .77
2.6.1 Basic Symmetrical Equation Solver .77
2.6.2 Parallel Data Storage Scheme . 78
2.6.3 Data Generating Subroutine . 80
2.6.4 Parallel Choleski Factorization . 80
2.6.5 A Blocked and Cache-Based Optimized Matrix-Matrix Multiplication 81
2.6.5.1 Loop Indexes and Temporary Array Usage .81
2.6.5.2 Blocking and Strip Mining .82
2.6.5.3 Unrolling of Loops .82 .
V l l l
2.6.6 Parallel “Block” Factorization 83
2.6.7 “Block” Forward Elimination Subroutine 85
2.6.8 “Block” Backward Elimination Subroutine . . 86
2.6.9 “Block” Error Checking Subroutine 88
2.6.10 Numerical Evaluation . 91
2.6.11 Conclusions 95
2.7 DevelopingDebugging Parallel MPI Application Code on Your Own
Laptop 95
2.8 Summary 103
2.9 Exercises . 103
3. Direct Sparse Equation Solvers 105
Introduction 105
Sparse Storage Schemes 105
Three Basic Steps and Re-Ordering Algorithms .110
Symbolic Factorization with Re-Ordering Column Numbers 118
Sparse Numerical Factorization 132
Super (Master) Nodes (Degrees-of-Freedom) . 134
Numerical Factorization with Unrolling Strategies 137
ForwardBackward Solutions with Unrolling Strategies 137
Alternative Approach for Handling an Indefinite Matrix 154
Unsymmetrical Matrix Equation Solver 165
Summary . 180
Exercises 181
4. Sparse Assembly Process . 183
Introduction 183
A Simple Finite Element Model (Symmetrical Matrices) 183
Finite Element Sparse Assembly Algorithms for
Symmetrical Matrices . 188
Symbolic Sparse Assembly of Symmetrical Matrices 189
Numerical Sparse Assembly of Symmetrical Matrices . 192
Step-by-step Algorithms for Symmetrical Sparse Assembly . 200
A Simple Finite Element Model (Unsymmetrical Matrices) 219
Re-Ordering Algorithms 224
Imposing Diricblet Boundary Conditions . 229
Unsymmetrical Sparse Equations Data Formats 254
Symbolic Sparse Assembly of Unsymmetrical Matrices 259
Numerical Sparse Assembly of Unsymmetrical Matrices 260
Step-by-step Algorithms for Unsymmetrical Sparse Assembly and
Unsymmetrical Sparse Equation Solver 260
A Numerical Example 265
Summary 265
Exercises Exercises Exercises E 266ix
5. Generalized Eigen-Solvers 269
5.1 Introduction . 269
5.2 A Simple Generalized Eigen-Example . 269
5.3 Inverse and Forward Iteration Procedures 271
5.4 Shifted Eigen-Problems 274
5.5 Transformation Methods . 276
5.6 Sub-space Iteration Method 286
5.7 Lanczns Eigen-Solution Algorithms . 290
5.7.1 Derivation of Lanczos Algorithms 290
5.7.2 Lanczos Eigen-Solution Error Analysis 295
5.7.3 Sturm SequenceCheck 302
5.7.4 Proving the Lanczos Vectors Are M-Orthogonal .306
5.7.5 “Classical” Gram-Schmidt Re-Orthogonalization 308
5.7.6 Detailed Step-by-step Lanczos Algorithms 314
5.7.7 Educational Software for Lanczos Algorithms . . 316
5.7.8 Efficient Software for Lanczos Eigen-Solver 336
5.8 Unsymmetrical Eigen-Solvers . 339
5.9 Balanced Matrix . 339
5.10 Reduction to Hessenberg Form . 340
5.11 QR Factoruat~on . . 341
5.12 Householder QR Transformation 341
5.13 “Modified” Gram-Schmidt Re-Orthogonalization . 348
5.14 QR Iteration for Unsymmetrical Eigen-Solutions . 350
5.15 QR Iteration with Shifts for Unsymmetrical Eigen-Solutions . 353
5.16 Panel Flutter Analysis . 355
5.17 Block Lanczos Algorithms . 365
5.17.1 Details of “Block Lanczos” Algorithms . 366
5.17.2 A Numerical Example for “Block Lanczos” Algorithms 371
5.18 Summary . 377
5.19 Exercises . 378
6. Finite Element Domain Decomposition Procedures 379
Introduction 379
A Simple Numerical Example Using Domain Decomposition
(DD) Procedures . 382
Imposing Boundary Conditions on “Rectangular” Matrices K$! .390
How to Construct Sparse Assembly of “Rectangular” Matrix K$; . 392
Mixed Direct-Iterative Solvers for Domain Decomposition . 393
Preconditioned Matrix for PCG Algorithm with DD Formulation . 397
Generalized Inverse . . 404
FETI Domain Decomposition Formnlati~n’~.~~~.” . 409
Preconditioned Conjugate Projected Gradient (PCPG) ofthe Dual Interface problem 16.41 . 417
Automated Procedures for Computing Generalized Inverse
and Rigid Body Motions 422
Numerical Examples of a 2-D Truss by FETI Formulation .433
A Preconditioning Technique for Indefinite Linear S tern’^.”^ . 459
ETI-DP Domain Decomposition Formulation ,6.6,6.,Y 463
Multi-Level Sub-Domains and Multi-Frontal Solver [6.’3 488
Iterative Solution with SuccessiveRight-Hand Sides [623m1 . -490
Summary . 510
Exercises 510
Appendix A Singular Value Decomposition (SVD) . 515
References 521
Index . 527
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