Parallel-Vector Equation Solvers for Finite Element Engineering Applications

Medium: Buch

ISBN: 978-1-4613-5504-5

Verlag: Springer US

Erscheinungstermin: 25.09.2012

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1. Introduction.- 1.1 Parallel Computers.- 1.2 Measurements for Algorithms’ Performance.- 1.3 Vector Computers.- 1.4 Summary.- 1.5 Exercises.- 1.6 References.- 2. Storage Schemes for the Coefficient Stiffness Matrix.- 2.1 Introduction.- 2.2 Full Matrix.- 2.3 Symmetrical Matrix.- 2.4 Banded Matrix.- 2.5 Variable Banded Matrix.- 2.6 Skyline Matrix.- 2.7 Sparse Matrix.- 2.8 Detailed Procedures For Determining The Mapping Between 2-D Array and 1-D Array in Skyline Storage Scheme.- 2.9 Determination of the Column Height (ICOLH) of a Finite Element Model.- 2.10 Computer Implementation For Determining Column Heights.- 2.11 Summary.- 2.12 Exercises.- 2.13 References.- 3. Parallel Algorithms for Generation and Assembly of Finite Element Matrices.- 3.1 Introduction.- 3.2 Conventional Algorithm to Generate and Assemble Element Matrices.- 3.3 Node-by-Node Parallel Generation and Assembly Algorithms.- 3.4 Additional Comments on Baddourah-Nguyen’s (Node-by-Node) Parallel Generation and Assembly (G&A) Algorithm.- 3.5 Application of Baddourah-Nguyen’s Parallel G&A Algorithm.- 3.6 Qin-Nguyen’s G&A Algorithm.- 3.7 Applications of Qin-Nguyen’s Parallel G&A Algorithm.- 3.8 Summary.- 3.9 Exercises.- 3.10 References.- 4. Parallel-Vector Skyline Equation Solver on Shared Memory Computers.- 4.1 Introduction.- 4.2 Choleski-based Solution Strategies.- 4.3 Factorization.- 4.4 Solution of Triangular Systems.- 4.5 Force: A Portable, Parallel FORTRAN Language.- 4.6 Evaluation of Methods on Example Problems.- 4.7 Skyline Equation Solver Computer Program.- 4.8 Summary.- 4.9 Exercises.- 4.10 References.- 5. Parallel-Vector Variable Bandwidth Equation Solver on Shared Memory Computers.- 5.1 Introduction.- 5.2 Data Storage Schemes.- 5.3 Basic Sequential Variable Bandwidth Choleski Method.- 5.4Vectorized Choleski Code with Loop Unrolling.- 5.5 More on Force: A Portable, Parallel FORTRAN Language.- 5.6 Parallel-Vector Choleski Factorization.- 5.7 Solution of Triangular Systems.- 5.8 Relations Amongst the Choleski, Gauss and LDLT Factorizations.- 5.9 Factorization Based Upon “Look Backward” Versus “Look Forward” Strategies.- 5.10 Evaluation of Methods For Structural Analyses.- 5.11 Descriptions of Parallel-Vector Subroutine PVS.- 5.12 Parallel-Vector Equation Solver Subroutine PVS.- 5.13 Summary.- 5.14 Exercises.- 5.15 References.- 6. Parallel-Vector Variable Bandwidth Out-of-Core Equation Solver.- 6.1 Introduction.- 6.2 Out-of-Core Parallel/Vector Equation Solver (version 1).- 6.3 Out-of-Core Vector Equation Solver (version 2).- 6.4 Out-of-Core Vector Equation Solver (version 3).- 6.5 Application.- 6.6 Summary.- 6.7 Exercises.- 6.8 References.- 7. Parallel-Vector Skyline Equation Solver for Distributed Memory Computers.- 7.1 Introduction.- 7.2 Parallel-Vector Symmetrical Equation Solver.- 7.3 Numerical Results and Discussions.- 7.4 FORTRAN Call Statement to Subroutine Node.- 7.5 Summary.- 7.6 Exercises.- 7.7 References.- 8. Parallel-Vector Unsymmetrical Equation Solver.- 8.1 Introduction.- 8.2 Parallel-Vector Unsymmetrical Equation Solution Algorithms.- 8.3 Numerical Evaluations.- 8.4 A Few Remarks On Pivoting Strategies.- 8.5 A FORTRAN Call Statement to Subroutine UNSOLVER.- 8.6 Summary.- 8.7 Exercises.- 8.8 References.- 9. A Tridiagonal Solver for Massively Parallel Computers.- 9.1 Introduction.- 9.2 Basic Sequential Solution Procedures for Tridiagonal Equations.- 9.3 Cyclic Reduction Algorithm.- 9.4 Parallel Tridiagonal Solver by Using Divided and Conquered Strategies.- 9.5 Parallel Factorization Algorithm for Tridiagonal System of Equations UsingSeparators.- 9.6 Forward and Backward Solution Phases.- 9.7 Comparisons between Different Algorithms.- 9.8 Numerical Results.- 9.9 A FORTRAN Call Statement To Subroutine Tridiag.- 9.10 Summary.- 9.11 Exercises.- 9.12 References.- 10. Sparse Equation Solver with Unrolling Strategies.- 10.1 Introduction.- 10.2 Basic Equation Solution Algorithms.- 10.3 Storage Schemes for the Coefficient Stiffness Matrix.- 10.4 Reordering Algorithms.- 10.5 Sparse Symbolic Factorization.- 10.6 Sparse Numerical Factorization.- 10.7 Forward and Backward Solutions.- 10.8 Sparse Solver with Improved Strategies.- 11. Algorithms for Sparse-Symmetrical-Indefinite and Sparse-Unsymmetrical System of Equations.- 11.1 Introduction.- 11.2 Basic Formulation for Indefinite System of Linear Equations.- 11.3 Rotation Matrix [R] Strategies.- 11.4 Natural 2x2 Pivoting.- 11.5 Switching Row(s) and Column(s) During Factorization.- 11.6 Simultaneously Performing Symbolic and Numerical Factorization.- 11.7 Restart Memory Managements.- 11.8 Major Step-by-Step Procedures for Mixed Look Forward/ Backward, Sparse LDLT Factorization, Forward and Backward Solution With 2x2 Pivoting Strategies.- 11.9 Numerical Evaluations.- 11.10 Some Remarks on Unsymmetrical-Sparse System of Linear Equations.- 11.11 Summary.- 11.12 Exercises.- 11.13 References.