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Large-scale Structural Analysis Using General Sparse Matrix Technique

Large-scale Structural Analysis Using General Sparse Matrix Technique. Yuan-Sen Yang, Shang-Hsien Hsieh, Kuang-Wu Chou, and I-Chau Tsai Department of Civil Engineering National Taiwan University Taiwan, R.O.C. Contents. Motivations Introduction SKyline Matrix (SKM) Approach

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Large-scale Structural Analysis Using General Sparse Matrix Technique

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  1. Large-scale Structural Analysis Using General Sparse Matrix Technique Yuan-Sen Yang, Shang-Hsien Hsieh, Kuang-Wu Chou, and I-Chau Tsai Department of Civil Engineering National Taiwan University Taiwan, R.O.C.

  2. Contents • Motivations • Introduction • SKyline Matrix (SKM) Approach • General Sparse Matrix (GSM) Approach • Different Procedures between SKM and GSM Approaches • Numerical Comparisons on Structural Analyses • Conclusions

  3. Motivations • Large-scale Structural Analyses • Cost lots of time • Require lots of memory storage • SKM Approach • Generally employed by many finite element packages • GSM Approach • Has been proposed for about 20 years • Requires less time and storage • Seldom employed by structural analysis packages

  4. Introductionto SKM and GSM Approaches (I) • SKM Approach • stores and computes items within skyline (still storing a number of zero items) • GSM Approach • only stores items that are required during matrix factorization

  5. Introduction to SKM and GSM Approaches (II) • SKM Approach • Simpler data structures • Usually costs more time and storage • GSM Approach • More complicated data structures • Usually costs less time and storage

  6. Different Procedures between SKM and GSM Approaches (I) • Renumbering Algorithms • SKM: gather nonzero items closer to diagonal • GSM: scatter nonzero items over the matrix • Symbolic Factorization • SKM: Not needed • GSM: Needed (to predict the nonzero pattern of the factorized matrix)

  7. Different Procedures between SKM and GSM Approaches (II) • SKMApproach 24,840 nonzero items 81,504 nonzero items Store as 12-Story Building 612 BC Elements 182 Nodes 1,008 D.O.F’s (Ref: Hsieh,1995) • GSM Approach 66,204 nonzero items 24,840 nonzero items Store as

  8. Numerical Comparisons on Structural Analyses • Testing • Solving the equilibrium equations using direct method (LDLT factorization) • Measurements • Time requirement • Storage requirement • Computing Environment • Software: Windows NT; MS Visual C++ SPARSPAK Library (George and Liu, 1981) • Hardware: Pentium II-233 PC with 128 MB SDRAM

  9. Results of Numerical Comparisons (I) ( R= GSM / SKM * 100%) • Different Mesh Sizes R= 79% R= 67% 960 BC elements 2,160 D.O.F.‘s R= 59% R= 48% 6,820 BC elements 14,520 D.O.F.‘s

  10. Results of Numerical Comparisons (II) • Branched structures ( R= GSM / SKM * 100%) R= 55% R= 51% 3,480 BC elements 7,680 D.O.F.‘s R= 40% R= 43% 5,100 BC elements 11,232 D.O.F.‘s

  11. Results of Numerical Comparisons (III) • Different Aspect Ratio ( R= GSM / SKM * 100%) R= 84% R= 69% 64,000 Truss elements 46,743 D.O.F.‘s R= 63% R= 50% 39,200 Truss elements 28,983 D.O.F.‘s

  12. Results of Numerical Comparisons (IV) • Meshes with High-order Elements ( R= GSM / SKM * 100%) R= 65% R= 61% (Ref: Wawrzynek ,1995) 944 20-node solid elements 18180 D.O.F. ‘s R= 49% R= 32% (Ref: Hsieh and Abel ,1995) 504 20-node solid elements 10044 D.O.F.‘s

  13. Conclusions • General Sparse Matrix Approach reduces time and storage requirements in solving equilibrium equations using direct methods, especially when the finite element model is : • Large-scale • With irregular shapes (e.g., w/ branches) • Not very slender

  14. Future Work • Applying General Sparse Matrix Technique on • Parallel Finite Element Analysis • Matrix Static Condensation of Substructures • Numerical Structural Dynamics • Mode Superposition Method (Eigen-solution Analysis)

  15. Suggestions • Use one of the popular public general sparse matrix packages • For saving time on tedious coding • The results are usually more reliable • Some popular packages • SPARSPAK (George and Liu,1981) • Harwell Subroutine Library (Duff,1996)

  16. Some Popular Packages • SPARSPAK • Book: • George, A. and Liu, J. W. H., Computer Solution of Large Sparse Positive Definite Systems, Prentice- Hall, USA, 1981. • E-mail: jageorge@sparse1.uwaterloo.ca joseph@cs.yorku.ca • Harwell Subroutine Library • Web site: http://www.dci.clrc.ac.uk/Activity/HSL • E-mail: I.S.Duff@rl.ac.uk

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