large-scale structural analysis using general sparse matrix technique yuan-sen yang, shang-hsien...
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Large-scale Structural Analysis Using Large-scale Structural Analysis Using General Sparse Matrix TechniqueGeneral Sparse Matrix Technique
Yuan-Sen Yang, Shang-Hsien Hsieh, Yuan-Sen Yang, Shang-Hsien Hsieh,
Kuang-Wu Chou, Kuang-Wu Chou, andand I-Chau Tsai I-Chau Tsai
Department of Civil EngineeringDepartment of Civil Engineering
National Taiwan UniversityNational Taiwan University
Taiwan, R.O.C.Taiwan, R.O.C.
ContentsContents
• MotivationsMotivations• IntroductionIntroduction
– SKyline Matrix (SKM) ApproachSKyline Matrix (SKM) Approach
– General Sparse Matrix (GSM) ApproachGeneral Sparse Matrix (GSM) Approach
• Different Procedures between SKM and GDifferent Procedures between SKM and GSM ApproachesSM Approaches
• Numerical Comparisons on Structural AnNumerical Comparisons on Structural Analysesalyses
• ConclusionsConclusions
MotivationsMotivations
• Large-scale Structural AnalysesLarge-scale Structural Analyses– Cost lots of timeCost lots of time
– Require lots of memory storageRequire lots of memory storage
• SKM ApproachSKM Approach– Generally employed by many finite element Generally employed by many finite element
packagespackages
• GSM ApproachGSM Approach– Has been proposed for about 20 yearsHas been proposed for about 20 years
– Requires less time and storageRequires less time and storage
– Seldom employed by structural analysis packagesSeldom employed by structural analysis packages
IntroductionIntroduction to SKM and GSM to SKM and GSM Approaches Approaches (I)(I)
• SKM ApproachSKM Approach– stores and computes items withistores and computes items withi
n skyline (still storing a number n skyline (still storing a number of zero items)of zero items)
• GSM ApproachGSM Approach– only stores items that are requironly stores items that are requir
ed during matrix factorizationed during matrix factorization
S y m .
S y m .
• SKM ApproachSKM Approach– Simpler data structuresSimpler data structures
– Usually costs more time and storageUsually costs more time and storage
• GSM ApproachGSM Approach– More complicated data structuresMore complicated data structures
– Usually costs less time and storageUsually costs less time and storage
Introduction to SKM and GSM Introduction to SKM and GSM Approaches Approaches (II)(II)
Different Procedures between Different Procedures between SKM and GSM Approaches SKM and GSM Approaches (I)(I)
• Renumbering AlgorithmsRenumbering Algorithms– SKM: gather nonzero items closer to diagonalSKM: gather nonzero items closer to diagonal
– GSM: scatter nonzero items over the matrixGSM: scatter nonzero items over the matrix
• Symbolic FactorizationSymbolic Factorization– SKM: Not neededSKM: Not needed
– GSM: Needed (to predict the nonzero pattern of the GSM: Needed (to predict the nonzero pattern of the factorized matrix)factorized matrix)
Different Procedures between Different Procedures between SKM and GSM Approaches SKM and GSM Approaches (II)(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
Store as
Numerical Comparisons on Numerical Comparisons on Structural AnalysesStructural 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
Elapsed time(sec)
1.29
1.02
0.00
0.50
1.00
1.50
SKM GSM
Results of Numerical Comparisons Results of Numerical Comparisons (I)(I)
• Different Mesh SizesDifferent Mesh Sizes( R= GSM / SKM * 100%)( R= GSM / SKM * 100%)
Storage size(MBytes)
1.84
2.75
0.00
1.00
2.00
3.00
SKM GSM
Elapsed time(sec)
184.41
108.89
0.00
50.00
100.00
150.00
200.00
SKM GSM
Storage size(MBytes)
29.79
61.48
0.00
20.00
40.00
60.00
80.00
SKM GSM
R= 79%R= 79%
R= 59%R= 59%
R= 67%R= 67%
R= 48%R= 48%
960 BC elements960 BC elements2,160 D.O.F.‘s2,160 D.O.F.‘s
6,820 BC elements6,820 BC elements14,520 D.O.F.‘s14,520 D.O.F.‘s
Results of Numerical Comparisons Results of Numerical Comparisons (II)(II)
Elapsed time(sec)
112.58
61.79
0.00
50.00
100.00
SKM GSM
R= 55%R= 55%
Storage size(MBytes)
26.79
52.35
0.00
20.00
40.00
60.00
SKM GSM
Elapsed time(sec)
200.80
79.47
0.00
50.00
100.00
150.00
200.00
SKM GSM
Storage size(MBytes)
31.61
72.23
0.00
20.00
40.00
60.00
80.00
SKM GSM
R= 40%R= 40% R= 43%R= 43%
R= 51%R= 51%
• Branched structuresBranched structures
3,480 BC elements 3,480 BC elements 7,680 D.O.F.‘s7,680 D.O.F.‘s
5,100 BC elements5,100 BC elements11,232 D.O.F.‘s11,232 D.O.F.‘s
( R= GSM / SKM * 100%)( R= GSM / SKM * 100%)
Results of Numerical Comparisons Results of Numerical Comparisons (III)(III)
Elapsed time(sec)
14.5012.18
0.00
5.00
10.00
15.00
SKM GSM
Storage size(MBytes)
31.98
46.21
0.00
10.00
20.00
30.00
40.00
50.00
SKM GSM
R= 84% R= 69%
Elapsed time(sec)
73.18
46.44
0.00
20.00
40.00
60.00
80.00
SKM GSM
Storage size(MBytes)
31.75
62.89
0.00
20.00
40.00
60.00
80.00
SKM GSM
R= 63%R= 50%
• Different Aspect RatioDifferent Aspect Ratio
64,000 Truss elements46,743 D.O.F.‘s
39,200 Truss elements28,983 D.O.F.‘s
( R= GSM / SKM * 100%)
Results of Numerical Comparisons Results of Numerical Comparisons (IV)(IV)
(Ref: Hsieh and Abel ,1995)
(Ref: Wawrzynek ,1995)
Elapsed time(sec)
250.80
161.80
0.00
50.00
100.00
150.00
200.00
250.00
300.00
SKM GSM
Storage size(MBytes)
47.95
78.98
0.00
20.00
40.00
60.00
80.00
100.00
SKM GSM
Elapsed time(sec)
12.15
3.840.00
5.00
10.00
15.00
SKM GSM
Storage size(MBytes)
8.87
17.98
0.00
5.00
10.00
15.00
20.00
SKM GSM
R= 65% R= 61%
R= 32%R= 49%
• Meshes with High-order ElementsMeshes with High-order Elements
944 20-node solid elements18180 D.O.F. ‘s
504 20-node solid elements10044 D.O.F.‘s
( R= GSM / SKM * 100%)
ConclusionsConclusions
• General Sparse Matrix Approach reduces time General Sparse Matrix Approach reduces time and storage requirements in solving equilibrium and storage requirements in solving equilibrium equations using direct methods, especially when equations using direct methods, especially when the finite element model is :the finite element model is :– Large-scaleLarge-scale– With irregular shapes (e.g., w/ branches)With irregular shapes (e.g., w/ branches)– Not very slenderNot very slender
Future WorkFuture Work
• Applying General Sparse Matrix Technique on Applying General Sparse Matrix Technique on
– Parallel Finite Element AnalysisParallel Finite Element Analysis
• Matrix Static Condensation of SubstructuresMatrix Static Condensation of Substructures
– Numerical Structural DynamicsNumerical Structural Dynamics
• Mode Superposition Method (Eigen-solution Mode Superposition Method (Eigen-solution
Analysis)Analysis)
SuggestionsSuggestions
• Use one of the popular public general sparse Use one of the popular public general sparse matrix packagesmatrix packages
– For saving time on tedious codingFor saving time on tedious coding
– The results are usually more reliableThe results are usually more reliable
• Some popular packagesSome popular packages
– SPARSPAK (George and Liu,1981) SPARSPAK (George and Liu,1981)
– Harwell Subroutine Library (Duff,1996)Harwell Subroutine Library (Duff,1996)
Some Popular PackagesSome Popular Packages• SPARSPAKSPARSPAK
– Book:Book:• George, A. and Liu, J. W. H., George, A. and Liu, J. W. H., Computer Solution Computer Solution
of Large Sparse Positive Definite Systemsof Large Sparse Positive Definite Systems, , Prentice- Hall, USA, 1981.Prentice- Hall, USA, 1981.
– E-mail: E-mail: [email protected]@sparse1.uwaterloo.ca [email protected]@cs.yorku.ca
• Harwell Subroutine LibraryHarwell Subroutine Library– Web site: Web site: http://www.dci.clrc.ac.uk/Activity/HSLhttp://www.dci.clrc.ac.uk/Activity/HSL– E-mail: E-mail: [email protected]@rl.ac.uk