Appendix A: Singular Value Decomposition (SVD)

Singular Value Decomposition (SVD) has many important, practical applications: search engines 12', computational information retrieval 13', least square problems ['I, image compressing l4], etc. A general (square, or rectangular) matrix A can be decomposed as:

A = U C V ~ (A. 1 ) where

[h] = diagonal matrix (does have to be a square matrix)

uH = uT (for real matrix) [u] and [v] = unitary matrices = u-' I

The SVD procedures to obtain matrices [U], [XI, and [V] from a given matrix [A] can be illustrated by the following examples:

Example 1: Given [A] = [: i] T [I 2][1 31 [ 5 111

Step I: Compute A AH = A A = 3 4 2 4 = 1125

1de;tity Matrix

111 B. Noble, et a]., (2"d Edition), Amlied Linear Algebra, Prentice Hall. [21 M. W. Berry and M. Bworne, "Understanding Search Engines," SUM. 131 M. W. Berry, "Computational Information Retrieval," SIAM. [4] J. W. Demmel, "Applied Numerical Linear Algebra," SIAM.

516 Finite Element Methods: Parallel-Sparse Statics and Eigen-Solutions

Step 2: Compute the standard eigen-solution of A A T

... [ A A ~ - h ~ l i i = O L

Hence

or

Thus

Now

Hence

For

or

Let

So

h=15- 4221, then Eq(A.6) becomes:

Eq(A.16) can be normalized, so that u(') = 1, to obtain: H II

Similarly, for h=l5 -&,and let up) = I, then

Hence

(A.6)

04.7)

('4.8)

(A.9)

(A. 10)

(A. l l )

(A. 12)

(A.13)

(A. 14)

(A. 1 5)

(A. 16)

(A. 17)

(A. 1 8)

(A. 19)

Duc T. Nguyen 517

Thus

H T Step 3: Compute A A = A A = [I 2 4 31 [l 3 4 21 = I 4 2 0 [lo 141

The two eigen-values associated with ( A ~ A) can be computed as:

(A.22)

Hence

o = & = d(0.1339 29.87) = (0.3660 5.465) (A.23)

For h=15-andle let v:' = 1, then

For h=15+m,and le t vf' = 1, then

Thus

Therefore, the SVD of [A] can be obtained from Eq.(A.l) as:

Given [A] = 2 2 1: :1 2 4 4

Compute A * A ~ = 4 8 8 3x2 bl [4 8 X]

The corresponding eigen-values and eigen-vectors of Eq.(A.28) can be given as:

(h, h2 h 3 ) = ( 1 8 0 0 ) 3 0 = , / ~ = ( 3 & ? 0 0 ) (A.29)

Finite Element Methods: Parallel-Sparse Statics and Eigen-Solutions

(A.30)

Compute AH* A =[; ;] (A.31) 2x3 3x2

The corresponding eigen-values and eigen-vectors of Eq.(A.31) can be given as:

(4 h 2 ) = (I8 O) (A.32)

O=O, = & = & = 3 & (A.33)

Hence A = U z, V 3x2 3x3 3x2 2x2

Example 3: Given [A] = [i : i ] (A.37)

10 11 12 The SVD of A is:

A = u m H = umT =

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

Given SVD of A = 2 IL

Y Y X ][:-oo]-~ O 0 - 2"q5-y5 9 0 1

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 (1 991)

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)

0. 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 (1 993)

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. (1 984)

J. A. Adam, "The effect of surface curvature on wound healing in bone: 11. The critical size defect." Mathematical and Computer Modeling, 35 (2002), p 1085 - 1094.

W. Gropp, "Tutorial on MPI: The Message Passing Interface," Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439

522 Finite Element Methods: Parallel-Sparse Statics and Eigen-Solutions

SGI sparse solver library sub-routine, Scientific Computing Software Library (SCSL) User's Guide, document number 007-4325-001, published Dec. 30,2003

I. S. Duff and J. K. Reid, "MA47, a FORTRAN Code for Direct Solution of Indefinite Sparse Symmetric Linear Systems," RAL (Report) #95-001, Rutherford Appleton Laboratory, Oxon, OX1 1 OQX (Jan 1995)

G. Karypis and V. Kumar, "ParMETiS: Parallel Graph Partitioning and Sparse Matrix Ordering Library," University of Minnesota, CS Dept., Version 2.0 (1998)

J. W. H. Liu, "Reordering Sparse Matrices For Parallel Elimination," Technical Report #87-01, Computer Science, York University, North York, Ontario, Canada (1987)

D. T. Nguyen, G. Hou, B. Han, and H. Runesha, "Alternative Approach for Solving Indefinite Symmetrical System of Equation," Advances in Engineering Software, Vol. 31 (2000), pp. 581 - 584, Elsevier Science Ltd.

I. S. Duff, and G. W. Stewart (editors), Sparse Matrix Proceedings 1979, SIAM (1979)

I. S. Duff, R. G. Grimes, and J. G. Lewis, "Sparse Matrix Test Problems," ACM Trans. Math Software, 15, pp. 1 - 14 (1989)

G. H. Golub and C. F. VanLoan, "Matrix Computations," Johns Hopkins University Press, Baltimore, MD, 2nd edition (1989)

A. George and J. W. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall (1981)

E. Ng and B. W. Peyton, ''Block Sparse Choleski Algorithm on Advanced Uniprocessor Computer," SIAM J. of Sci. Comput., volume 14, pp. 1034 - 1056 (1993).

H. B. Runesha and D. T. Nguyen, "Vectorized Sparse Unsymmetrical Equation Solver for Computational Mechanics," Advances in Engr. Software, volume 31, nos. 8 - 9, pp. 563 - 570 (Aug. - Sept. 2000), Elsevier

J. A. George, "Nested Disection of a Regular Finite Element Mesh," SIAM J. Numer. Anal., volume 15, pp. 1053 - 1069 (1978)

I. S. Duff and J. K. Reid, 'The Design of MA48: A Code for the Direct Solution of Sparse Unsymmetric Linear Systems of Equations," ACM Trans. Math. Software., 22 (2): 187 - 226 (June 1996)

Duc T. Nguyen 523

I. S. Duff and J. Reid, "MA27: A Set of FORTRAN Subroutines for Solving Sparse Symmetric Sets of Linear Equations," AERE Technical Report, R-10533, Harwell, England (1982)

Nguyen, D. T., Bunting, C., Moeller, K. J., Runesha H. B., and Qin, J., "Subspace and Lanczos Sparse Eigen-Solvers for Finite Element Structural and Electromagnetic Applications," Advances in Engineering Sofhvare, volume 31, nos. 8 - 9, pages 599 - 606 (August - Sept. 2000)

Nguyen, D. T. and Arora, J. S., "An Algorithm for Solution of Large Eigenvalue Problems," Computers & Structures, vol. 24, no. 4, pp. 645 - 650, August 1986.

Arora, J. S. and Nguyen, D. T., "Eigen-solution for Large Structural Systems with Substructures," International Journal for Numerical Methods in Engineering, vol. 15, 1980, pp. 333 - 341.

Qin, J. and Nguyen, D. T., "A Vector Out-of-Core Lanczons Eigen- solver for Structural Vibration Problems," presented at the 35th Structures, Structural Dynamics, and Material Conference, Hilton Head, SC, (April 18 - 20, 1994).

K. J. Bathe, Finite Element Procedures, Prentice Hall (1996)

G. Golub, R. Underwood, and J. H. Wilkinson, 'The Lanczos Algorithm for Symmetric Ax=Lamda*Bx Problem," Tech. Rep. STAN-CS-72-720, Computer Science Dept., Stanford University (1972)

B. Nour-Omid, B. N. Parlett, and R. L. Taylor, "Lanczos versus Subspace Iteration for Solution of Eigenvalue Problems," IJNM in Engr., volume 19, pp. 859 - 871 (1983)

B. N. Parlett and D. Scott, "The Lanczos Algorithm with Selective Orthogonalization," Mathematics of Computation, volume 33, no. 145, pp. 217 - 238 (1979)

H.D. Simon, "The Lanczos Algorithm with Partial 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. 1 180 - 1 189 (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. 1 - 124 (1 994)

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 2 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 J. W. Dernmel, "SuperLU-DIST: A Scalable Distributed- Memory Sparse Direct Solver for Unsymmetric Linear Systems," ACM Trans. Mathematical Sofnyare, volume 29, no. 2, pp. 110 - 140 (June 2003)

A. D. Belegundu and T. R. Chandrupatla, Optimization Concepts and Applications in Engineering, Prentice-Hall (1999)

Duc T. Nguyen 525

[6.10] D. T. Nguyen and P. Chen, "Automated Procedures For Obtaining Generalized Inverse for FETI Formulation," Structures Technical Note ' 03-22-2004, Civil & Env. Engr. Dept. ODU, Norfolk, VA 23529

[6.11] M. Papadrakakis, S. Bitzarakis, and A. Kotsopulos, "Parallel Solution Techniques in Computational Structural Mechanics," B. H. V. Topping (Editor), Parallel and Distributed Processing for Computational Mechanics: Systems and Tools, pp. 180 - 206,

Saxe-Coburg Publication, Edinburgh, Scotland ( I 999)

L. Komzsik, P. Poschmann, and I. Sharapov, "A Preconditioning Technique for Indefinite Linear Systems," Finite Element in Analysis and Design, volume 26, pp. 253-258 (1997)

P. Chen, H. Runesha, D. T. Nguyen, P. Tong, and T. Y. P. Chang, "Sparse Algorithms for Indefinite System of Linear Equations," pp. 712 - 717, Advances in Computational Engineering Science, edited (1997) by S. N. Atluri and G. Yagawa, Tech. Science Press, Forsyth, Georgia

D. T. Nguyen, G. Hou, H. Runesha, and B. Han, "Alternative Approach for Solving Sparse Indefinite Symmetrical System of Equations," Advances in Engineering Software, volume 31 (8 - 9), pp. 581 - 584 (2000)

J. Qin, D. T. Nguyen, T. Y. P. Chang, and P. Tong, "Efficient Sparse Equation Solver With Unrolling Strategies for Computational Mechanics", pp. 676 - 681, Advances in Computational Engineering Science, edited (1997) by S. N. Atluri and G. Yagawa, Tech. Science Press, Forsyth, Georgia

A. George and J. W. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall (1981)

C. Farhat, M. Lesoinne, and K. Pierson, "A Scalable Dual-Primal Domain Decomposition 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] 1. 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. 5 - 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, 1 15 Algebraic equations, 8 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, 2 Belegundu, A.D., 521 Bending moment, 6 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, 5 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, 1 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), 41 7 Conjugate vectors, 496 Connectivity information, 186 Continuity condition, 8 Convergence, 6 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-sectional area, 17 Curvature, 5 Curve boundary, 44

Damping matrix, 356 Decompose (a matrix), 77 Deflection, 6 Degree-of-freedom, 10 Demmel, J.W., 515 Dependent variable field, 37 Dependent variable, 8 Derivatives, 6 Determinant, 270 Determinant, 50 Diagonal matrix, 16 Diagonal terms, 162 Differential equation, 4 DIPSS (MPI software), 401 Direct sparse equation solvers,

105 Dirichlet boundary conditions,

187,229 Discretized locations, 6 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, 41 7 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, 7-

9, 17 Euler-Bernoulli beam, 30 Extended GCR algorithm, 509 External virtual work, 10

Factorization, 1 10 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, 9 Finite element connectivity, 1 15 Finite element model (symmetrical

matrices), 183

Finite element model (unsymmetrical matrices), 21 9

Finite element stiffness equations, 12

Finite elements, 9 First sub-diagonal, 340 Fischer, P.F., 526 Floating sub-domains, 41 1, 456 Floating substructure, 427 Forcing function, 1 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, 2 Geometric stiffness matrix, 294 George, A., 525 Gippspool Stock Meiger, 1 15 Global coordinate reference, 12 Global coordinate references, 387 Global dof, 186 Gropp, W., 521

Heat, 8 Heath, M.T., 524 Hessenberg (form) matrix, 340 Hessenberg reduction, 377 Hinton, E., 521 Homogeneous equation, 501 Homogeneous form, 7

Duc T. Nguyen

Hooke's law, 10 Householder transformation, 341, 342,344 Huebner, K.H., 521

Identity matrix, 16 Ill-posed (matrix), 41 1 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, 7 Integrating by parts, 25 Integration by parts, 36 Integration, 4 Interface constraints, 467 Interior displacement, 380 Interior dof, 488 Interior load vectors, 386 Internal nodal load vector, 27 Internal virtual work, 10 Interpolant, 18 Interpolation function, 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), 41 1 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, 1 10,132 LDU, 110,114,168,172 Least square problems, 51 5 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, 1 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, 1 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, 2 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

Natural boundary conditions, 6-9, 17

Natural coordinate system, 44,48 Natural frequency, 14 Necessary condition, 491 Nested Dissection (ND), 1 15 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

Off-diagonal terms, 163 Omid, B.M., 523 Optimization problems, 490 Ordinary differential equations, 1 Orthogonal condition, 310 Orthonormality conditions, 272 Othogonalize (Lanczos vector),

315 Out-of-core memory, 160 Outward normal, 8 Overall boundary node numbering

system, 383 Overhead computational costs,

1 60 Owen, D.R., 521

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, 1 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, 5 Positive definite matrix, 13 Positive definite, 1 10 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, 41 1 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), 41 1 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, 31 6 Reid, J., 489 Relative error norm, 163 Remainder displacement, 487 Remainder dof, 465, 467,470 Re-ordering algorithms, 110, 11 7,

224 Re-orthogonalize, 361 Residual, 3 Reversed Cuthill-Mckee, 11 5 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, 8 SGl (parallel) sparse solver, 401 SGlIOrigin 2000, 91 SGl's unsymmetrical sparse

matrix storage scheme, 258 Shape functions, 9-1 0 Shear force, 6 Shifted eigen-problem, 304 Shifted eigen-problems, 274 Simon, H.D., 523 Simply supported beam, 1 Simply supports, 6 Simpson's integration rule, 53 Singular matrix, 13 Singular value decomposition

(SVD), 515 Skyline column storage, 78 Slope, 5, 6 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, 9 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, 1 0 Stride, 81

532 Finite Element Methods: Parallel-Sparse Statics and Eigen-Solutions

Strip mining, 81 Structural banded finite element

model, 402 Structural Engineering, 1, 6 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-1 0000 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, 1 18 Symbolic sparse assembly, 189 Symmetrical equation solver, 77 Symmetrical positive definite, 369 Symmetrical sparse assembly,

200 Symmetrical, 8 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, 31 5, 365

Truss 2-D by FETl formulation, 433

Truss finite element, 27, 184 Tungkahotara, S., 526 Twice differiable, 9

Unconstrained finite element model, 21 9

Uncoupling, 14 Uniform load, 1 Unitary m'atrices, 51 5 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, 1 14

Variational, 1 Vectorlcache computer, 360 Velocity, 13 Virtual displacement, 2 Virtual nodal displacement, 10 Virtual strain, 10 Virtual work equation, 11 Virtual work, 2

Watson, W.R., 524 Weak form, 8 Weak formulations, 6,7,32 Weighted integral statement, 8 Weighted residual, 32, 35 Weighting function, 3, 7 Weighting function, 9 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., Northrop- Grumman 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.