Parallel-Vector Equation Solvers for Finite Element Engineering Applications

By: Duc T. Nguyen, Thai Nguyen Duc Thai Nguyen, Duc Thai Nguyen

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Hardcover | 30 November 2001

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372 Pages

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Despite the ample number of articles on parallel-vector computational algorithms published over the last 20 years, there is a lack of texts in the field customized for senior undergraduate and graduate engineering research. Parallel-Vector Equation Solvers for Finite Element Engineering Applications aims to fill this gap, detailing both the theoretical development and important implementations of equation-solution algorithms. The mathematical background necessary to understand their inception balances well with descriptions of their practical uses. Illustrated with a number of state-of-the-art FORTRAN codes developed as examples for the book, Dr. Nguyen's text is a perfect choice for instructors and researchers alike.

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You Can Find This Book In

Non-Fiction Computing & I.T.Computer Science Computer Architecture & Logic Design Parallel Processing Mathematics Calculus & Mathematical Analysis Numerical Analysis Differential Calculus & Equations Mathematical Theory of Computation Maths for Computer Scientists Engineering & Technology

Civil Engineering Applied Mathematics Computer Programming & Software Development Algorithms & Data Structures Technology in General Engineering in General

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Paperback

Published: 25th September 2012

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Introduction	p. 1
Parallel Computers	p. 1
Measurements for Algorithms' Performance	p. 2
Vector Computers	p. 3
Summary	p. 10
Exercises	p. 11
References	p. 11
Storage Schemes for the Coefficient Stiffness Matrix	p. 13
Introduction	p. 13
Full Matrix	p. 14
Symmetrical Matrix	p. 14
Banded Matrix	p. 14
Variable Banded Matrix	p. 14
Skyline Matrix	p. 15
Sparse Matrix	p. 16
Detailed Procedures For Determining The Mapping Between 2-D Array and 1-D Array in Skyline Storage Scheme	p. 17
Determination of the Column Height (ICOLH) of a Finite Element Model	p. 19
Computer Implementation For Determining Column Heights	p. 23
Summary	p. 25
Exercises	p. 26
References	p. 26
Parallel Algorithms for Generation and Assembly of Finite Element Matrices	p. 27
Introduction	p. 27
Conventional Algorithm to Generate and Assemble Element Matrices	p. 27
Node-by-Node Parallel Generation and Assembly Algorithms	p. 29
Additional Comments on Baddourah-Nguyen's (Node-by-Node) Parallel Generation and Assembly (GandA) Algorithm	p. 37
Application of Baddourah-Nguyen's Parallel GandA Algorithm	p. 38
Qin-Nguyen's GandA Algorithm	p. 41
Applications of Qin-Nguyen's Parallel GandA Algorithm	p. 46
Summary	p. 48
Exercises	p. 49
References	p. 50
Parallel-Vector Skyline Equation Solver on Shared Memory Computers	p. 51
Introduction	p. 51
Choleski-based Solution Strategies	p. 51
Factorization	p. 52
Basic sequential skyline Choleski factorization: computer code (version 1)	p. 55
Improved basic sequential skyline Choleski factorization: computer code (version 2)	p. 59
Parallel-vector Choleski factorization (version 3)	p. 60
Parallel-vector (with "few" synchronization checks) Choleski factorization (version 4)	p. 64
Parallel-vector enhancement (vector unrolling) Choleski factorization (version 5)	p. 66
Parallel-vector (unrolling) skyline Choleski factorization (version 6)	p. 69
Solution of Triangular Systems	p. 72
Forward solution	p. 72
Backward solution	p. 78
Force: A Portable, Parallel FORTRAN Language	p. 81
Evaluation of Methods on Example Problems	p. 82
Skyline Equation Solver Computer Program	p. 86
Summary	p. 86
Exercises	p. 87
References	p. 88
Parallel-Vector Variable Bandwidth Equation Solver on Shared Memory Computers	p. 91
Introduction	p. 91
Data Storage Schemes	p. 91
Basic Sequential Variable Bandwidth Choleski Method	p. 96
Vectorized Choleski Code with Loop Unrolling	p. 101
More on Force: A Portable, Parallel FORTRAN Language	p. 103
Parallel-Vector Choleski Factorization	p. 103
Solution of Triangular Systems	p. 108
Forward solution	p. 109
Backward solution	p. 112
Relations Amongst the Choleski, Gauss and LDL[superscript T] Factorizations	p. 115
Choleski (U[superscript T]U) factorization	p. 115
Gauss (with diagonal terms L[subscript ii]=1) LU factorization	p. 117
Gauss (LU) factorization with diagonal terms U[subscript ii]=1	p. 118
LDL[superscript T] factorization with diagonal term L[subscript ii]=1	p. 120
Similarities of Choleski and Gauss methods	p. 122
Factorization Based Upon "Look Backward" Versus "Look Forward" Strategies	p. 123
Evaluation of Methods For Structural Analyses	p. 129
High speed research aircraft	p. 130
Space shuttle solid rocket booster (SRB)	p. 131
Descriptions of Parallel-Vector Subroutine PVS	p. 134
Parallel-Vector Equation Solver Subroutine PVS	p. 136
Summary	p. 137
Exercises	p. 138
References	p. 139
Parallel-Vector Variable Bandwidth Out-of-Core Equation Solver	p. 141
Introduction	p. 141
Out-of-Core Parallel/Vector Equation Solver (version 1)	p. 141
Memory usage and record length	p. 142
A synchronous input/output on Cray computers	p. 144
Brief summary for parallel-vector incore equation solver on the Cray Y-MP	p. 145
Parallel-vector out-of-core equation solver on the Cray Y-MP	p. 146
Out-of-Core Vector Equation Solver (version 2)	p. 149
Memory usage	p. 149
Vector out-of-core equation solver on the Cray Y-MP	p. 149
Out-of-Core Vector Equation Solver (version 3)	p. 155
Application	p. 157
Version 1 performance	p. 157
Version 2 performance	p. 159
Version 3 performance	p. 160
Summary	p. 162
Exercises	p. 163
References	p. 163
Parallel-Vector Skyline Equation Solver for Distributed Memory Computers	p. 165
Introduction	p. 165
Parallel-Vector Symmetrical Equation Solver	p. 165
Basic symmetrical equation solver	p. 165
Parallel-vector performance improvement in decomposition	p. 166
Communication performance improvement in factorization	p. 176
Forward/backward elimination	p. 177
Numerical Results and Discussions	p. 181
FORTRAN Call Statement to Subroutine Node	p. 185
Summary	p. 187
Exercises	p. 188
References	p. 188
Parallel-Vector Unsymmetrical Equation Solver	p. 191
Introduction	p. 191
Parallel-Vector Unsymmetrical Equation Solution Algorithms	p. 191
Basic unsymmetric equation solver	p. 191
Detailed derivation for the [L] and [U] matrices	p. 193
Basic algorithm for decomposition of "full" bandwidth/column heights unsymmetrical matrix	p. 194
Basic algorithm for decomposition of "variable" bandwidths/column heights unsymmetrical matrix	p. 198
Algorithms for decomposition of "variable" bandwidths/column heights unsymmetrical matrix with unrolling strategies	p. 199
Parallel-vector algorithm for factorization	p. 200
Forward solution phase [L]{y}={b}	p. 202
Backward solution phase [U] {x{ = {y{	p. 204
Numerical Evaluations	p. 206
A Few Remarks On Pivoting Strategies	p. 211
A FORTRAN Call Statement to Subroutine UNSOLVER	p. 212
Summary	p. 214
Exercises	p. 214
References	p. 216
A Tridiagonal Solver for Massively Parallel Computers	p. 217
Introduction	p. 217
Basic Sequential Solution Procedures for Tridiagonal Equations	p. 217
Cyclic Reduction Algorithm	p. 221
Parallel Tridiagonal Solver by Using Divided and Conquered Strategies	p. 226
Parallel Factorization Algorithm for Tridiagonal System of Equations Using Separators	p. 229
Forward and Backward Solution Phases	p. 236
Forward solution phase: [L] {z} = {y}	p. 236
Backward solution phase: [U] {x} = {z}	p. 238
Comparisons between Different Algorithms	p. 239
Numerical Results	p. 240
A FORTRAN Call Statement To Subroutine Tridiag	p. 241
Summary	p. 244
Exercises	p. 244
References	p. 245
Sparse Equation Solver with Unrolling Strategies	p. 247
Introduction	p. 247
Basic Equation Solution Algorithms	p. 248
Choleski algorithm	p. 248
LDL[superscript T] algorithm	p. 249
Storage Schemes for the Coefficient Stiffness Matrix	p. 252
Reordering Algorithms	p. 254
Sparse Symbolic Factorization	p. 255
Sparse Numerical Factorization	p. 271
Forward and Backward Solutions	p. 278
Forward substitution phase	p. 279
Backward substitution phase	p. 279
Sparse Solver with Improved Strategies	p. 280
Finding master (or super) degree-of-freedom (dof)	p. 280
Sparse matrix (with unrolling strategies) times vector	p. 281
Modifications for the chained list array ICHAINL (-)	p. 288
Sparse numerical factorization with unrolling strategies	p. 289
Out-of-core sparse equation solver with unrolling strategies	p. 299
Numerical Performance of the Developed Sparse Equation Solver	p. 301
FORTRAN Call Statement to SPARSE Equation Solver	p. 306
Summary	p. 308
Exercises	p. 308
References	p. 309
Algorithms for Sparse-Symmetrical-Indefinite and Sparse-Unsymmetrical System of Equations	p. 311
Introduction	p. 311
Basic Formulation for Indefinite System of Linear Equations	p. 311
Rotation Matrix [R] Strategies	p. 318
Natural 2 x 2 Pivoting	p. 323
Switching Row(s) and Column(s) During Factorization	p. 325
Simultaneously Performing Symbolic and Numerical Factorization	p. 329
Restart Memory Managements	p. 329
Major Step-by-Step Procedures for Mixed Look Forward/Backward, Sparse LDL[superscript T] Factorization, Forward and Backward Solution With 2x2 Pivoting Strategies	p. 331
Numerical Evaluations	p. 332
Some Remarks on Unsymmetrical-Sparse System of Linear Equations	p. 334
Summary	p. 338
Exercises	p. 338
References	p. 338
Index	p. 341
Table of Contents provided by Syndetics. All Rights Reserved.

Parallel-Vector Equation Solvers for Finite Element Engineering Applications

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Hardcover

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