2002 2002 4 13 Improved Lanczos Method for

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2002한국전산구조공학회 봄학술발표회 2002년 4월 13일 Improved Lanczos Method for the Eigenvalue Analysis of Structures

2002한국전산구조공학회 봄학술발표회 2002년 4월 13일 Improved Lanczos Method for the Eigenvalue Analysis of Structures Byoung-Wan Kim 1), Woon-Hak Kim 2) and In-Won Lee 3) 1) Graduate Student, Dept. of Civil and Environmental Eng. , KAIST 2) Professor, Dept. of Civil Engineering, Hankyong National Univ. 3) Professor, Dept. of Civil and Environmental Eng. , KAIST

Contents l Introduction l Matrix-powered Lanczos method l Numerical examples l Conclusions 2

Contents l Introduction l Matrix-powered Lanczos method l Numerical examples l Conclusions 2

l Introduction l Background • Dynamic analysis of structures - Direct integration method -

l Introduction l Background • Dynamic analysis of structures - Direct integration method - Mode superposition method Eigenvalue analysis • Eigenvalue analysis - Subspace iteration method - Determinant search method - Lanczos method • The Lanczos method is very efficient. 3

l Literature review • The Lanczos method was first proposed in 1950. • Erricson

l Literature review • The Lanczos method was first proposed in 1950. • Erricson and Ruhe (1980): Lanczos algorithm with shifting • Smith et al. (1993): Implicitly restarted Lanczos algorithm • Gambolati and Putti (1994): Conjugate gradient scheme in Lanczos method 4

 • In the fields of quantum physics, Grosso et al. (1993) modified Lanczos

• In the fields of quantum physics, Grosso et al. (1993) modified Lanczos recursion to improve convergence. 5

l Objective • Application of Lanczos method using the power technique to the eigenproblem

l Objective • Application of Lanczos method using the power technique to the eigenproblem of structures in structural dynamics Matrix-powered Lanczos method 6

l Matrix-powered Lanczos method l Eigenproblem of structure 7

l Matrix-powered Lanczos method l Eigenproblem of structure 7

l Modified Gram-Schmidt process of Krylov sequence 8

l Modified Gram-Schmidt process of Krylov sequence 8

l Modified Lanczos recursion 9

l Modified Lanczos recursion 9

l Reduced tridiagonal standard eigenproblem 10

l Reduced tridiagonal standard eigenproblem 10

l Summary of algorithm and operation count Operation Calculation Number of operations Factorization Iteration

l Summary of algorithm and operation count Operation Calculation Number of operations Factorization Iteration i = 1 ··· q Substitution Multiplication Reorthogonalization Multiplication Division Repeat Reduced eigensolution n = order of M and K, m = half-bandwidth of M and K q = the number of calculated Lanczos vectors or order of T sj = the number of iterations of jth step in QR iteration 11

l Numerical examples l Structures • Simple spring-mass system (Chen 1993) • Plan framed

l Numerical examples l Structures • Simple spring-mass system (Chen 1993) • Plan framed structure (Bathe and Wilson 1972) • Three-dimensional frame structure (Bathe and Wilson 1972) • Three-dimensional building frame (Kim and Lee 1999) l Physical error norm (Bathe 1996) 12

l Simple spring-mass system (DOFs: 100) • System matrices 13

l Simple spring-mass system (DOFs: 100) • System matrices 13

 • Number of operations No. of eigenpairs =1 =2 =3 =4 2 4

• Number of operations No. of eigenpairs =1 =2 =3 =4 2 4 6 8 10 38663 78922 120458 157649 214729 29823 58529 85712 117587 154418 26954 47567 73040 103055 138122 23653 44122 69391 99550 Failure in convergence due to numerical instability of high matrix power 14

l Plane framed structure (DOFs: 330) • Geometry and properties A = 0. 2787

l Plane framed structure (DOFs: 330) • Geometry and properties A = 0. 2787 m 2 I = 8. 631 10 -3 m 4 E = 2. 068 107 Pa = 5. 154 102 kg/m 3 15

 • Number of operations No. of eigenpairs =1 =2 =3 =4 6 12

• Number of operations No. of eigenpairs =1 =2 =3 =4 6 12 18 24 30 10908273 20855865 27029145 31581179 102944376 7429050 13578945 18676209 22516533 65994807 7072452 11688377 16508507 20164797 54112986 6633536 11237625 16047093 Failure in convergence due to numerical instability of high matrix power 16

l Three-dimensional frame structure (DOFs: 468) • Geometry and properties E = 2. 068

l Three-dimensional frame structure (DOFs: 468) • Geometry and properties E = 2. 068 107 Pa = 5. 154 102 kg/m 3 Column in front building Column in rear building All beams into x-direction All beams into y-direction : A = 0. 2787 m 2, I = 8. 631 10 -3 m 4 : A = 0. 3716 m 2, I = 10. 789 10 -3 m 4 : A = 0. 1858 m 2, I = 6. 473 10 -3 m 4 : A = 0. 2787 m 2, I = 8. 631 10 -3 m 4 17

 • Number of operations No. of eigenpairs =1 =2 =3 =4 10 20

• Number of operations No. of eigenpairs =1 =2 =3 =4 10 20 30 40 50 71602154 181780512 307269560 684162222 1024104917 50687925 124269611 215884077 453454527 656188310 48705515 116680070 192064376 378770940 553972908 46214349 108715163 182518601 356596304 504420108 18

l Three-dimensional building frame (DOFs: 1008) • Geometry and properties A = 0. 01

l Three-dimensional building frame (DOFs: 1008) • Geometry and properties A = 0. 01 m 2 I = 8. 3 10 -6 m 4 E = 2. 1 1011 Pa = 7850 kg/m 3 19

 • Number of operations No. of eigenpairs =1 =2 =3 =4 20 40

• Number of operations No. of eigenpairs =1 =2 =3 =4 20 40 60 80 100 395079020 1196316954 3045578295 3398746793 3536190824 278717178 801878160 1993108128 2509125474 3625240574 Failure in convergence due to numerical instability of high matrix power 20

l Conclusions • Matrix-powered Lanczos method has not only the better convergence but also

l Conclusions • Matrix-powered Lanczos method has not only the better convergence but also the less operation count than the conventional Lanczos method. • The suitable power of the dynamic matrix that gives numerically stable solution in the matrix-powered Lanczos method is the second power. 21