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 MotivationsMotivations IntroductionIntroduction –SKyline Matrix (SKM) Approach –General Sparse Matrix (GSM) Approach Different Procedures between SKM and GSM ApproachesDifferent Procedures between SKM and GSM Approaches Numerical Comparisons on Structural AnalysesNumerical Comparisons on Structural Analyses ConclusionsConclusions

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

Introduction to SKM and GSM Approaches (I) SKM ApproachSKM Approach –stores and computes items within skyline (still storing a number of zero items) GSM ApproachGSM Approach –only stores items that are required during matrix factorization

SKM ApproachSKM Approach –Simpler data structures –Usually costs more time and storage GSM ApproachGSM Approach –More complicated data structures –Usually costs less time and storage Introduction to SKM and GSM Approaches (II)

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

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

Numerical Comparisons on Structural Analyses Testing –Solving the equilibrium equations using direct method (LDL T 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

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

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

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

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

Conclusions General Sparse Matrix Approach reduces time and storage requirements in solving equilibrium equations using direct methods, especially when the finite element model is :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

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

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

Some Popular Packages SPARSPAKSPARSPAK –Book: George, A. and Liu, J. W. H., Computer Solution of Large Sparse Positive Definite Systems, Prentice- Hall, USA, 1981.George, A. and Liu, J. W. H., Computer Solution of Large Sparse Positive Definite Systems, Prentice- Hall, USA, – Harwell Subroutine LibraryHarwell Subroutine Library –Web site: –