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**FINITE** **ELEMENT** METHODS: Parallel-Sparse **Statics** **and** Eigen-Solutions **FINITE** **ELEMENT** METHODS: Parallel-Sparse **Statics** **and** Eigen-Solutions Duc Thai Nguyen Old Dominion University Norfolk, Virginia - **Springer** Prof Duc Thai Nguyen 135 Kaufman Old Dominion University Department of Civil & Environmental Engineering Multidisc Parallel-Vector Comp Ctr Norfolk VA 23529 **Finite** **Element** Methods: Parallel-Sparse **Statics** **and** Eigen-Solutions Library of Congress Control Number: 2005937075 ISBN 0-387-29330-2 e-ISBN 0-387-30851-2 ISBN 978-0-387-29330-1 Printed on acid-free paper O **2006** **Springer** Science+Business Media, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage **and** retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks **and** similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed in the United States of America To Dac K Nguyen Thinh T Thai Hang N Nguyen Eric N D Nguyen **and** Don N Nguyen Contents A Review of Basic **Finite** **Element** Procedures Introduction Numerical Techniques for Solving Ordinary Differential Equations (ODE) 1.3 Identifying the "Geometric" versus "Natural" Boundary Conditions 6 1.4 The Weak Formulations 1.5 Flowcharts for **Statics** **Finite** **Element** Analysis 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-Dimensional Rod **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 1.1 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 59 1.15 Summary 1.16 Exercises 59 1.1 1.2 Simple MPI/FORTRAN Applications 2.1 2.2 2.3 2.4 2.5 2.6 -63 Introduction 63 Computing Value of "d'by Integration 63 Matrix-Matrix Multiplication 68 MPI **Parallel** 110 72 Unrolling Techniques 75 **Parallel** Dense Equation Solvers 77 2.6.1 Basic Symmetrical Equation Solver 77 78 2.6.2 **Parallel** Data Storage Scheme 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 Vlll 2.6.6 **Parallel** "Block" Factorization 83 85 2.6.7 "Block" Forward Elimination Subroutine 86 2.6.8 "Block" Backward Elimination Subroutine 2.6.9 "Block" Error Checking Subroutine 88 91 2.6.10 Numerical Evaluation 95 2.6.11 Conclusions 2.7 DevelopingDebugging **Parallel** MPI Application Code on Your Own 95 Laptop 2.8 Summary 103 2.9 Exercises 103 105 Direct **Sparse** Equation Solvers Introduction 105 105 **Sparse** Storage Schemes Three Basic Steps **and** Re-Ordering Algorithms .110 Symbolic Factorization with Re-Ordering Column Numbers 118 132 **Sparse** Numerical Factorization Super (Master) Nodes (Degrees-of-Freedom) 134 Numerical Factorization with Unrolling Strategies 137 137 ForwardBackward **Solutions** with Unrolling Strategies Alternative Approach for Handling an Indefinite Matrix 154 Unsymmetrical Matrix Equation Solver 165 180 Summary Exercises 181 **Sparse** Assembly Process 183 Introduction 183 183 A Simple **Finite** **Element** Model (Symmetrical Matrices) **Finite** **Element** **Sparse** Assembly Algorithms for 188 Symmetrical Matrices 189 Symbolic **Sparse** Assembly of Symmetrical Matrices 192 Numerical **Sparse** Assembly of Symmetrical Matrices 200 Step-by-step Algorithms for Symmetrical **Sparse** Assembly 219 A Simple **Finite** **Element** Model (Unsymmetrical Matrices) Re-Ordering Algorithms 224 229 Imposing Diricblet Boundary Conditions 254 Unsymmetrical **Sparse** Equations Data Formats .259 Symbolic **Sparse** Assembly of Unsymmetrical Matrices Numerical **Sparse** Assembly of Unsymmetrical Matrices 260 Step-by-step Algorithms for Unsymmetrical **Sparse** Assembly **and** 260 Unsymmetrical **Sparse** Equation Solver A Numerical Example 265 265 Summary Exercises 266 E ses i c r exE ses i c r exE ix Generalized Eigen-Solvers 269 Introduction 269 269 A Simple Generalized Eigen-Example 271 Inverse **and** Forward Iteration Procedures 274 Shifted Eigen-Problems 276 Transformation **Methods** 286 Sub-space Iteration Method 290 Lanczns Eigen-Solution Algorithms 290 5.7.1 Derivation of Lanczos Algorithms 295 5.7.2 Lanczos Eigen-Solution Error Analysis 302 5.7.3 Sturm Sequence Check .306 5.7.4 Proving the Lanczos Vectors Are M-Orthogonal 308 5.7.5 "Classical" Gram-Schmidt Re-Orthogonalization 314 5.7.6 Detailed Step-by-step Lanczos Algorithms 316 5.7.7 Educational Software for Lanczos Algorithms 5.7.8 Efficient Software for Lanczos Eigen-Solver 336 339 Unsymmetrical Eigen-Solvers 5.8 5.9 Balanced Matrix 339 5.10 Reduction to Hessenberg Form 340 341 5.1 QR Factoruat~on 341 5.12 Householder QR Transformation 348 5.13 "Modified" Gram-Schmidt Re-Orthogonalization 350 5.14 QR Iteration for Unsymmetrical Eigen-Solutions 5.15 QR Iteration with Shifts for Unsymmetrical Eigen-Solutions 353 5.16 Panel Flutter Analysis 355 5.17 Block Lanczos Algorithms 365 366 5.17.1 Details of "Block Lanczos" Algorithms 5.17.2 A Numerical Example for "Block Lanczos" Algorithms .371 377 5.18 Summary 5.19 Exercises 378 5.1 5.2 5.3 5.4 5.5 5.6 5.7 **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 F o r m n l a t i ~ n **'** ~ ~~~." 409 Preconditioned Conjugate Projected Gradient (PCPG) of the Dual Interface problem 16.41 417 Automated Procedures for Computing Generalized Inverse 422 **and** Rigid Body Motions 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 463 488 Multi-Level Sub-Domains **and** Multi-Frontal Solver [6.'3 Iterative Solution with Successive Right-Hand Sides [623m1 -490 Summary 510 Exercises 510 Appendix A Singular Value Decomposition (SVD) 515 521 References Index 527 **Finite** **element** **methods** (FEM) **and** associated computer software have been widely accepted as one of the most effective, general tools for solving large-scale, practical engineering **and** science applications It is no wonder there is a vast number of excellent textbooks in FEM (not including hundreds of journal articles related to FEM) written in the past decades! While existing FEM textbooks have thoroughly discussed different topics, such as linear/nonlinear, static/dynamic analysis, with varieties of 1-Dl2-Dl3-D **finite** **element** libraries, for thermal, electrical, contact, **and** electromagnetic applications, most (if not all) current FEM textbooks have mainly focused on the developments of "finite **element** libraries," how to incorporate boundary conditions, **and** some general discussions about the assembly process, solving systems of "banded" (or "skyline") linear equations For implicit **finite** **element** codes, it is a well-known fact that efficient equation **and** eigen-solvers play critical roles in solving large-scale, practical engineeringlscience problems **Sparse** matrix technologies have evolved **and** become mature enough that all popular, commercialized FEM codes have inserted **sparse** solvers into their software Furthermore, modern computer hardware usually has multiple processors; clusters of inexpensive personal computers (under WINDOWS, or LINUX environments) are available for **parallel** computing purposes to dramatically reduce the computational time required for solving large-scale problems Most (if not all) existing FEM textbooks discuss the assembly process **and** the equation solver based on the "variable banded" (or "skyline") strategies Furthermore, only limited numbers of FEM books have detailed discussions about Lanczos eigen-solvers or explanation about domain decomposition (DD) **finite** **element** formulation for **parallel** computing purposes This book has been written to address the concerns mentioned above **and** is intended to serve as a textbook for graduate engineering, computer science, **and** mathematics students A number of state-of-the-art FORTRAN software, however, have been developed **and** explained with great detail Special efforts have been made by the author to present the material in such a way to minimize the mathematical background requirements for typical graduate engineering students Thus, compromises between rigorous mathematics **and** simplicities are sometimes necessary The materials from this book have evolved over the past several years through the author's research work **and** graduate courses (CEE7151815 = **Finite** **Element** I, CEE695 = **Finite** **Element** **Parallel** Computing, CEE7 111811 = **Finite** **Element** 11) at Old Dominion University (ODU) In Chapter 1, a brief review of basic **finite** **element** xii procedures for Linear/Statics/Dynamics analysis is given One, two, **and** threedimensional **finite** **element** types are discussed The weak formulation is emphasized **Finite** **element** general field equations are derived, isoparametric formulation is explained, **and** Gauss Quadrature formulas for efficient integration are discussed In this chapter, only simple (non-efficient) **finite** **element** assembly procedures are explained Chapter illustrates some salient features offered by Message Passing Interface (MPI) FORTRAN environments Unrolling techniques, efficient usage of computer cache memory, **and** some basic MPYFORTRAN applications in matrix linear algebra operations are also discussed in this chapter Different versions of direct, "SPARSE" equation solvers' strategies are thoroughly discussed in Chapter The "truly sparse" **finite** **element** "assembly process" is explained in Chapter Different versions of the Lanczos algorithms for the solution of generalized eigenequations (in a **sparse** matrix environment) are derived in Chapter Finally, the overall **finite** **element** domain decomposition computer implementation, which can exploit "direct" **sparse** matrix equation, eigen-solvers, **sparse** assembly, "iterative" solvers (for both "symmetrical" **and** "unsymmetrical" systems of linear equations), **and** **parallel** processing computation, are thoroughly explained **and** demonstrated in Chapter Attempts have been made by the author to explain some difficult concepts/algorithms in simple language **and** through simple (hand-calculated) numerical examples Many FORTRAN codes (in the forms of main program, **and** sub-routines) are given in Chapters - Several large-scale, practical engineering problems involved with several hundred thousand to over million degree-offreedoms (don have been used to demonstrate the efficiency of the algorithms discussed in this textbook This textbook should be useful for graduate students, practicing engineers, **and** researchers who wish to thoroughly understand the detailed step-by-step algorithms used during the **finite** **element** (truely sparse) assembly, the "direct" **and** "iterative" **sparse** equation **and** eigen-solvers, **and** incorporating the DD formulation for efficient **parallel** computation The book can be used in any of the following "stand-alone" courses: (a) Chapter can be expanded (with more numerical examples) **and** portions of Chapter (only cover the **sparse** formats, **and** some "key components" of the **sparse** solver) can he used as a first (introductive type) course in **finite** **element** analysis at the senior undergraduate (or 1" year graduate) level (b) Chapters 1, 3.4, **and** can be used as a "stand-alone" graduate course such as "Special Topics in FEM:Sparse Linear **Statics** **and** Eigen-Solutions." (c) Chapters 1, 2, 3, 4, **and** can be used as a "stand-alone" graduate course, such as "Special Topics in FEM: **Parallel** **Sparse** Linear **Statics** Solutions." (d) Chapters 2, 3, **and** 5, **and** portions of Chapter 6, can be used as a "standalone" graduate course such as "High Performance **Parallel** Matrix Computation." Due T Nguyen Examule 4: (Relationship between SVD **and** generalized inverse) "Let the m x n matrix A of rank k have the SVD A = U Z V ~ w; i t h q to2 t- >o,)O Then the generalized inverse A+ of A is the nxm matrix A+ = VZ+ U* ;where Z+ **and** E is the kxk diagonal matrix, with IL Given SVD of A = Y Y X ][:-oo]-~O - 2"q5-y References J N Reddy, An lntroduction to the **Finite** **Element** Method, 2"d edition, McGraw-Hill(1993) K J Bathe, **Finite** **Element** Procedures, Prentice Hall (1996) K H Huebner, The **Finite** **Element** Methodfor Engineers, John Wiley & Sons (1975) T R Chandrupatla **and** A.D Belegundu, Introduction to **Finite** Elements in Engineering, Prentice-Hall (1991) D S Burnett, **Finite** **Element** Analysis: From Concepts to Applications, Addison-Wesley Publishing Company (1987) M A Crisfield, Nonlinear **Finite** **Element** Analysis of Solids **and** Structures, volume 2, John Wiley & Sons (2001) C Zienkiewicz, The **Finite** **Element** Method, 31d editiion, McGraw-Hill(1977) D R Owen **and** E Hinton, **Finite** Elements in Plasticity: Theory **and** Practice, Pineridge Press Limited, Swansea, UK (1980) D T Nguyen, Parallel-Vector Equation Solvers for **Finite** **Element** Engineering Applications, Kluwer/Plenum Publishers (2002) J Jin, The **Finite** **Element** Method in Electromagnetics, John Wiley & Sons (1993) P P Sivester **and** R L Ferrari, **Finite** Elements for Electrical Engineers, 3* edition, Cambridge University Press (1996) R D Cook, Concepts **and** Applications of **Finite** **Element** Analysis, 2" edition, John Wiley & Sons (1981) S Pissanetzky, **Sparse** Matrix Technology, Academic Press, Inc (1984) J A Adam, "The effect of surface curvature on wound healing in bone: 11 The critical size defect." 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Reorthogonalization", Mathematics of Computation, 42, no 165, pp 115-142 (1984) J J Dongarra, C B Moler, J R Bunch, **and** G W Stewart, LlNPACK Users' Guide, SIAM, Philadelphia (1979) S Rahmatalla **and** C C Swan, "Continuum Topology Optimization of Buckling-Sensitive Structures," AIAA Journal, volume 41, no 6, pp 1180 - 1189 (June 2003) 524 **Finite** **Element** Methods: Parallel-Sparse **Statics** **and** Eigen-Solutions W H Press, B P Flannery, S A Teukolsky, **and** W T Vetterling, Numerical Recipes (FORTRAN Version), Cambridge University Press (1989) M T Heath, Scientific Computing: An Introductory Survey, McGraw-Hill(1997) Tuna Baklan, "CEE7111811: Topics in **Finite** **Element** Analysis," Homework #5, Old Dominion University, Civil & Env Engr Dept., Norfolk, VA (private communication) W R Watson, "Three-Dimensional Rectangular Duct Code with Application to Impedance Eduction," AIAA Journal, 40, pp 217-226 (2002) D T Nguyen, S Tungkahotara, W R Watson, **and** S D Rajan "Parallel **Finite** **Element** Domain Decomposition for StructuraV Acoustic Analysis," Journal of Computational **and** Applied Mechanics, volume 4, no 2, pp 189 - 201 (2003) C Farhat **and** F X Roux, "Implicit **Parallel** Processing in Structural Mechanics," Computational Mechanics Advances, volume 2, pp - 124 (1994) D T Nguyen **and** P Chen, "Automated Procedures for Obtaining Generalized Inverse for FETI Formulations," Structures Research Technical Note No 03-22-2004, Civil & Env Engr Dept., Old Dominion University, Norfolk, VA 23529 (2004) C Farhat, M Lesoinne, P LeTallec, K Pierson, **and** D Rixen, "FETI-DP: A Dual-Primal Unified FETI Method- Part I: A Faster Alternative to the Level FETI Method," IJNME, volume 50, pp 1523 - 1544 (2001) R Kanapady **and** K K Tamrna, "A Scalability **and** Space/Time Domain Decomposition for Structural Dynamics - Part I: Theoretical Developments **and** **Parallel** Formulations," Research Report UMSI 20021 188 (November 2002) X S Li **and** 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Method," Numerical Linear Algebra with Applications, volume 7, pp 687 - 714 (2000) Nguyen, D T., "Multilevel Structural Sensitivity Analysis," Computers & Structures Journal, volume 25, no 2, pp 191 - 202, April 1987 **Finite** **Element** Methods: Parallel-Sparse **Statics** **and** Eigen-Solutions [6.19] S J Kim, C S Lee, J H Kim, M Joh, **and** S Lee, "ISAP: A High Performance **Parallel** **Finite** **Element** Code for Large-Scale Structural Analysis Based on Domain-wise Multifrontal Technique," proceedings of Super Computing, Phoenix, AZ (November 15 - 21,2003) [6.20] J H Kim, **and** S J Kim, "Multifrontal Solver Combined with Graph Patitioners," AIAA Journal, volume 37, no 8, pp 964 - 970 (Aug 1999) [6.21] Duff **and** J Reid, 'The Multifrontal Solution of Indefinite **Sparse** Symmetric Linear Systems," Association for Computing Machinery Transactions Mathematical Sofiware, volume 9, pp 302 - 325 (1983) I6.22) B M Iron, "A Frontal Solution Program for **Finite** **Element** Analysis," IJNME, volume 2, pp - 32 (1970) [6.23] F J Lingen, "A Generalized Conjugate Residual Method for the Solution of Non-Symmetric Systems of Equations with Multiple Right-Hand Sides," IJNM in Engr., volume 44, pp 641 - 656 (1999) [6.24] P F Fischer, "Projection Techniques for Iterative Solution of Ax = b with Successive Right-Hand Sides," ICASE Report # 93-90, NASA LaRC, Hampton, VA [6.25] S Tungkahotara, D T Nguyen, W R Watson, **and** H B Runesha, "Simple **and** Efficient **Parallel** Dense Equation Solvers," 9fi International Conference on Numerical **Methods** **and** Computational Mechanics, University of Miskolc, Miskolc, Hungary (July 15 - 19, 2002) Index ABAQUS, 105 Absolute displacement, 163 Acoustic **finite** **element** model, 400 Adam, J.A., 521 Aerodynamic equations, 356 Aerodynamic influence, 356 Adjacency array, 115 Algebraic equations, Arora, J.S., 523 Assembled, 12 Assembly procedures, 22 Axial displacement, 18 Axial reaction, 24 Axially distributed load, 17 Axially loaded rod, 17 Balanced matrix, 339 Banded **sparse** matrix, 401 Basic matrices, 40 Bathe, K.J., 521 Beam deflection, Belegundu, A.D., 521 Bending moment, Berry, M.W., 515 Block column storage, 79 Block forward elimination, 85 Block Lanczos algorithms, 365, 366,368,371 Body force, 35 Boeing's **sparse** indefinite equation solver, 163 Boolean transformation, 414, 466 Boundary conditions, 1, Boundary displacement, 380 Boundary dof, 488 Boundary force, 35 Brick element, 33 Buckling analysis, 294 Burnett, D.S., 521 Bworne, M., 515 Cache, 80 Chandrupatla, T.R., 521 Chen, P., 524,525 Choleski factorization, 295, 369 Classical Gram-Schmidt, 308 Colloquation, Compact column storage, 107 Compact row storage, 105 Compatibility requirements, 465 Compilation, 101 Complete polynomial, 18 Conjugate direction, 497 Conjugate gradient method, 394 Conjugate Projected Gradient (CPG), 417 Conjugate vectors, 496 Connectivity information, 186 Continuity condition, Convergence, Cook, R.D., 521 Coordinate transformation, 11 Corner dof, 466,471 Corner nodes, 465 Corner point, 465 Cray-C90, 77 Crisfield, M.A., 521 Critical oscillation, 355 Cross-sectionalarea, 17 Curvature, Curve boundary, 44 Damping matrix, 356 Decompose (a matrix), 77 Deflection, Degree-of-freedom, 10 Demmel, J.W., 515 Dependent variable field, 37 Dependent variable, Derivatives, Determinant, 270 Determinant, 50 Diagonal matrix, 16 Diagonal terms, 162 Differential equation, DIPSS (MPI software), 401 Direct **sparse** equation solvers, 105 Dirichlet boundary conditions, 187,229 Discretized locations, Displacement compatibility, 27 528 **Finite** **Element** Methods: ParalIe1-Sparse **Statics** **and** Eigen-Solutions Distributed loads, 21 Domain decomposition (DD) 379, 382 Dongara, J.J., 523 DOT product operations, 77 Dual DD formulation, 464 Dual interface problem, 417 Duff, I.,489 Dynamic pressure, 355,356 Dynamical equilibrium equations, 13,14 Dynamics, 13 Effective boundary stiffness (load), 380, 381 Eigen-matrix, 270 Eigen-solution error analysis, 295 Eigen-values matrix, 277 Eigen-values, 14, 15 Eigen-vectors, 14, 15 **Element** connectivity matrix, 187 **Element** local coordinate, 11 **Element** mass matrix, 13 **Element** shape function, 31,33 **Element** stiffness matrix, 21, 26 Energy approach, 20 Equivalent joint loads, 21,26, 31 Error norms computation, 89 Essential boundary conditions, 79, 17 Euler-Bernoullibeam, 30 Extended GCR algorithm, 509 External virtual work, 10 Factorization, 110 Farhat, C., 524,525 FETl domain decomposition (DD), 409 FETI-1 algorithms, 414 FETI-DP formulation, 463 FETI-DP step-by-step procedures, 472 Field equations, 35 Fill-in terms, 114 **Finite** **element** analysis, **Finite** **element** connectivity, 115 **Finite** **element** model (symmetrical matrices), 183 **Finite** **element** model (unsymmetrical matrices), 219 **Finite** **element** stiffness equations, 12 **Finite** elements, First sub-diagonal, 340 Fischer, P.F., 526 Floating sub-domains, 411, 456 Floating substructure, 427 Forcing function, Forth order differential equation, 30 FORTRAN-90,63 Forward substitution of blocks, 86 Forwardtbackward elimination, 78 Frame **finite** element, 29 Free vibration, 14 Galerkin, 1, 7,9 Gauss quadrature formulas, 56 Gauss quadrature, 51 Gaussian elimination, 340 Generalized Conjugate Residual (GCR) algorithms, 503 Generalized coordinates, 45 Generalized eigen-equation, 365 Generalized eigen-solvers, 269 Generalized eigen-value problem, 14 Generalized inverse, 404,427, 456 Generalized Jacobi method, 284 Geometric boundary conditions, Geometric stiffness matrix, 294 George, A., 525 Gippspool Stock Meiger, 115 Global coordinate reference, 12 Global coordinate references, 387 Global dof, 186 Gropp, W., 521 Heat, Heath, M.T., 524 Hessenberg (form) matrix, 340 Hessenberg reduction, 377 Hinton, E., 521 Homogeneous equation, 501 Homogeneous form, Duc T Nguyen Hooke's law, 10 Householder transformation, 341, 342,344 Huebner, K.H., 521 Identity matrix, 16 Ill-posed (matrix), 411 Incomplete factorized, 133 Incomplete Choleski factorization, 394 lncore memory requirements, 162 lndefinite (matrix), 410 lndefinite linear system, 456 lndefinite matrix, 154 lndefinite matrix, 294 lndefinite matrix, 456 Independent variables, 10 Initial conditions, 1, 14, 16 Integral form, Integrating by parts, 25 Integration by parts, 36 Integration, Interface constraints, 467 Interior displacement, 380 Interior dof, 488 Interior load vectors, 386 Internal nodal load vector, 27 Internal virtual work, 10 Interpolant, 18 Interpolationfunction, 19,26 Inverse (and forward) iteration procedures, 271 Irons, B.M., 489 lsoparametric bar element, 45 lsoparametric formulation, 44 Iterative solver, 416 Jacobi method, 277,305 Jacobian matrix, 44 Jacobian, 46 Kernel (of a matrix), 411 Kim, J.H., 526 Kinetic energy, 13 Komzsik, L., 525 Lagrange multiplier method, 460 Lagrange multipliers, 163 Lagrangian function, 410 Lanczos eigen-algorithms, 305 Lanczos eigen-solver, 336 Lanczos vectors, 290,294,296 Lanczos vectors, 306 Large amplitude vibration, 357 Lanczos eigen-solution, 290 LDL Transpose, 110,132 LDU, 110,114,168,172 Least square problems, 515 Li, X.S., 524 Linearly independent vectors, 502 Linearly independent, 15 Lingen, F.J., 526 Liu, W., 525 Lowest eigen-pairs, 359 Lumped mass matrix, 294 MA28 unsymmetrical **sparse** solver, 108 MA28,415 MA47,415 Mass matrices, 272 Mass, 13 Material matrix, 10, 20, 21 Mathematical operator, MATLAB (software), 4,425 Matrix notations, 4, 19 Matrix times matrix, 77 Matrix-matrix multiplication, 81 Message Passing Interface (MPI), 63 METiS, 115,224,305 Minimize residual, Mixed direct-iterative solvers, 393 Mixed **finite** **element** types, 207 ModifGCR algorithm, 509 Modified Gram-Schmidt, 348 Modified minimum degree (MMD), 163 Moment of inertia, Moment of inertia, 379 M-orthogonal, 306 M-orthonormality, 294 MPI-BCAST, 67 MPI-COMM-RANK, 67 MPI-COMM-SIZE, 65,67 MPI-DOUBLE-PRECISION, 66 530 **Finite** **Element** Methods: Parallel-Sparse **Statics** **and** Eigen-Solutions MPI-FILE-CLOSE, 73 MPI-FILE-OPEN, 73 MPI-FILE-READ, 73 MPI-FILE-SET-VIEW, 73 MPI-FILE-WRITE, 73 MPI-FINALIZE, 65, 67 MPI-INIT, 65,67 MPI-RECV, 71 MPI-REDUCE, 66 MPI-SSEND, 71 MPI-WTIME, 70 MSC-NASTRAN, 105 Multi-level substructures, 488 Multiple Minimum Degree (MMD), 115 Multipliers, 143 Off-diagonalterms, 163 Omid, B.M., 523 Optimization problems, 490 Ordinary differential equations, Orthogonal condition, 310 Orthonormality conditions, 272 Othogonalize (Lanczos vector), 315 Out-of-core memory, 160 Outward normal, Overall boundary node numbering system, 383 Overhead computational costs, 160 Owen, D.R., 521 Natural boundary conditions, 6-9, 17 Natural coordinate system, 44,48 Natural frequency, 14 Necessary condition, 491 Nested Dissection (ND), 115 Nested dissection (ND), 163 Nguyen, D.T., 339,521,523-526 Nguyen-Runesha's unsymmetrical **sparse** matrix storage scheme, 256 Noble, B., 515 Nodal displacement, 10 Nodal loads, 10 Non-homogeneous,7 Non-linear flutter analysis, 357 Nonlinear, 39 Non-singular, 13,24 Non-trivial solution, 14 Normalized, 15 Normalized eigen-matrix, 16 Normalized eigen-vector, 15,271, 299 Numerical integration, 44 Numerical recipe, 339 Numerical **sparse** assembly of unsymmetrical matrices, 260 Numerical **sparse** assembly, 192, 201 Panel flutter, 355 Papadrakakis, M., 525 **Parallel** (MPI) Gram-Schmidt QR, 361 **Parallel** block factorization, 83 **Parallel** Choleski factorization, 80 **Parallel** computer, 64 **Parallel** dense equation solvers, 77 **Parallel** 110, 72 Parlett, B.N., 523 Partial derivatives, 10 Partial differential equations, PCPG iterative solver, 457 Pissanetzsky, S., 521 Pivoting (2x2), 154 Plane cross-section, 30 Plane element, 47 Plane isoperimetric element, 47 Polak-Rebiere algorithm, 498 Polynomial function, 52 Polynomial, Positive definite matrix, 13 Positive definite, 110 Positive definite, 155 Potential energy, 469 Preconditioned conjugate gradient (D.D.), 396,397 Preconditioning matrix, 393 Prescribed boundary conditions, 386 Off-diagonal term, 107 Duc T Nguyen Press, Flannery, Teukolsky **and** Vetterling, 339 Primal DD formulation, 464 Primary dependent function, 32 Primary variable, 9, 18 Processor, 64 Projected residual, 421 Proportional damping matrix, 13 Pseudo force, 411 Pseudo rigid body motion, 412 Qin, J., 523 QR algorithm, 340 QR factorization, 341 QR iteration with shifts, 353 QR iteration, 350 QR, 361 Quadratic solid element, 44 Quadrilateral element, 47 Range (of a matrix), 411 Rank (of a matrix), 415 Rayleigh Ritz, 59 Rectangular element, 42 Reddy, J.N., 42-44,521 Reduced eigen-equation, 361 Reduced eigen-problem, 287 Reduced stiffness, mass matrices, 287 Reduced tri-diagonal system, 316 Reid, J., 489 Relative error norm, 163 Remainder displacement, 487 Remainder dof, 465, 467,470 Re-ordering algorithms, 110, 117, 224 Re-orthogonalize, 361 Residual, Reversed Cuthill-Mckee, 115 Right-hand-side columns, 160 Rigid body displacement, 405 Rod **finite** element, 17 Runesha, H.B., 523 Saxpy operations, 76 Saxpy unrolling strategies, 141 Scalar field problem, 47 Scalar product operations, 80 Schur complement 380,462 Scott, D., 523 Search direction, 398, 491, 494 Secondary variable, SGl (parallel) **sparse** solver, 401 SGlIOrigin 2000, 91 SGl's unsymmetrical **sparse** matrix storage scheme, 258 Shape functions, 9-10 Shear force, Shifted eigen-problem, 304 Shifted eigen-problems, 274 Simon, H.D., 523 Simply supported beam, Simply supports, Simpson's integration rule, 53 Singular matrix, 13 Singular value decomposition (SVD), 515 Skyline column storage, 78 Slope, 5, Solid elements, 33 **Sparse** assembly of rectangular matrix, 392 **Sparse** assembly of unsymmetrical matrices, 259 **Sparse** assembly, 183 **Sparse** eigen-solution, 17 **Sparse** matrix time vector, 398 **Sparse** storage scheme, 105 Sparse, 13 Standard eigen-problem, 296, 299,316 Standard eigen-value problems, 269 Static condensation, 295 Statics, Steepest descent direction, 494 Step-by-step optimization procedures, 491 Step-size, 491 Stiffness matrix, 11 Strain energy density, 20 Strain-displacement relationships, 10,20,46,50 Stress, 10 Stress-strain relationship, 10 Stride, 81 , 532 **Finite** **Element** Methods: Parallel-Sparse **Statics** **and** Eigen-Solutions Strip mining, 81 Structural banded **finite** **element** model, 402 Structural Engineering, 1, Structural problem, 10 Sturm sequence, 302,304 Sub-domains, 9,381 Subspace iteration method, 286 Substructures, 381 Sub-structuring numbering system, 383 Successive right-hand-sides,490 Sun-10000 processors, 401 Super (k-th) row, 143 Super (master) nodes, 134 Super linear speed-up, 401 Super row, 180 Support boundary condition, 386 Supported node, 24 Symbolic factorization, 118 Symbolic **sparse** assembly, 189 Symmetrical equation solver, 77 Symmetrical positive definite, 369 Symmetrical **sparse** assembly, 200 Symmetrical, System effective boundary load vector, 389 System global coordinate, 11 System mass matrix, 13 System stiffness equations, 12 Tamma, K.K., 524 Tangent stiffness matrix, 294 Taylor, R.L., 523 Tetrahedral, 31 Thickness of plate (or shell), 379 Three-node element, 45 Transformation methods, 276, 277 Transposing (a **sparse** matrix), 130 Transverse deflection, 30 Transverse distributed loads, 30 Trapezoid integration rule, 52 Triangular area, 39 Triangular element, 39,41, 205 Tri-diagonal matrix, 291, 315, 365 Truss 2-D by FETl formulation, 433 Truss **finite** element, 27, 184 Tungkahotara, S., 526 Twice differiable, Unconstrained **finite** **element** model, 219 Uncoupling, 14 Uniform load, Unitary m'atrices, 515 Unknown displacement vector, 12 Unrolling numerical factorization, 137 Unrolling of loops, 82 Unrolling techniques, 76 Unstable, 23 Unsymmetrical eigen-solver, 339, 354 Unsymmetrical equation solver, 168 Unsymmetrical matrix, 166, 167 Unsymmetrical **sparse** assembly, 230 Upper I-lessenberg matrix, 340, 359 Upper triangular matrix, 77 Upper triangular, 114 Variational, Vectorlcache computer, 360 Velocity, 13 Virtual displacement, Virtual nodal displacement, 10 Virtual strain, 10 Virtual work equation, 11 Virtual work, Watson, W.R., 524 Weak form, Weak formulations, 6,7,32 Weighted integral statement, Weighted residual, 32, 35 Weighting function, 3, Weighting function, Weighting residual, 3, 25 Young modulus, 2,10,117 Duc T Nguyen Zienkewicz, O.C., 521 ZPSLDLT (SGI subroutine), 401 **FINITE** **ELEMENT** METHODS: PARALLEL-SPARSE **STATICS** **AND** EIGEN-SOLUTIONS Duc T Nguyen Dr Duc T Nguyen is the founding Director of the Institute for Multidisciplinary Parallel-Vector Computation **and** Professor of Civil **and** Environmental Engineering at Old Dominion University His research work in **parallel** procedures for computational mechanics has been supported by NASA Centers, AFOSR, CIT, Virginia Power, NSF, Lawrence Livermore National Laboratory, Jonathan Corp., NorthropGrumman Corp., **and** Hong Kong University of Science **and** Technology He is the recipient of numerous awards, including the 1989 Gigaflop Award presented by Cray Research Incorporated, the 1993 Tech Brief Award presented by NASA Langley Research Center for his fast Parallel-Vector Equation Solvers, **and** Old Dominion University, 2001 A Rufus Tonelson distinguished faculty award Dr Nguyen has been listed among the Most Highly Cited Researchers in Engineering in the world ... element only 10 Finite Element Methods: Parallel- Sparse Statics and Eigen- Solutions The unknown primary function (say, deflection function) f(xi) at any location within a finite element can be... [ c l = a1[Kl + a W I where a, and a2are constant coefficients, and Eq.(1.90) can be generalized to: Finite Element Methods: Parallel- Sparse Statics and Eigen- Solutions with the following initial... Compute the 1'' normalized eigen- vector Similarly, one obtains: (1.1 11) 9:) as: 16 Finite Element Methods: Parallel- Sparse Statics and Eigen- Solutions Thus, the normalized eigen- matrix [

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