Accelerated Subspace Iteration Method for ComputingNatural Frequencies and Mode Shapes of Structures

Accelerated Subspace Iteration Method for ComputingNatural Frequencies and Mode Shapes of Structures

한국전산구조공학회 2003 년도 가을 학술발표회 2003 년 10 월 11 일2003 년 10 월 11 일

Byoung-Wan Kim1), Chun-Ho Kim2), and In-Won Lee3)

1) Postdoc. Res., Dept. of Civil and Environmental Eng., KAIST2) Professor, Dept. of Civil Eng., Joongbu Univ.3) Professor, Dept. of Civil and Environmental Eng., KAIST

Byoung-Wan Kim1), Chun-Ho Kim2), and In-Won Lee3)

1) Postdoc. Res., Dept. of Civil and Environmental Eng., KAIST2) Professor, Dept. of Civil Eng., Joongbu Univ.3) Professor, Dept. of Civil and Environmental Eng., KAIST

2 2

Introduction Proposed method Numerical examples Conclusions

Contents

3 3

Introduction

Background

• Dynamic analysis or seismic design of structures Natural frequencies and mode shapes Eigenvalue analysis

• Dynamic analysis or seismic design of structures Natural frequencies and mode shapes Eigenvalue analysis

• Eigensolution methods- Subspace iteration method (Bathe & Wilson, 1972)- Lanczos method (Lanczos, 1950)

• Eigensolution methods- Subspace iteration method (Bathe & Wilson, 1972)- Lanczos method (Lanczos, 1950)

• Subspace iteration method- Widely used in structural problems- Various improved versions are developed.- Subspace iteration method with Lanczos starting subspace (Bathe & Ramaswamy, 1980)

• Subspace iteration method- Widely used in structural problems- Various improved versions are developed.- Subspace iteration method with Lanczos starting subspace (Bathe & Ramaswamy, 1980)

4 4

• In quantum problems, Grosso et al. (1993) proposedaccelerated Lanczos recursion with squared operator.

• In quantum problems, Grosso et al. (1993) proposedaccelerated Lanczos recursion with squared operator.

12

11

111

)(

nnnnntnn

nnnnnnn

bab

bab

fffEHf

ffHff

a, b = coefficientsf = basis functions for quantum systemsH = operatorEt = trial energy

a, b = coefficientsf = basis functions for quantum systemsH = operatorEt = trial energy

5 5

Objective

• To improve the subspace iteration method withLanczos starting subspace by applying the square technique togeneration of Lanczos vectors used as starting vectors

• To improve the subspace iteration method withLanczos starting subspace by applying the square technique togeneration of Lanczos vectors used as starting vectors

6 6

Proposed method Subspace iteration method with conventional

Lanczos starting subspace

ΛMΦKΦ

• Eigenproblem of structures• Eigenproblem of structures

mass matrix (n n) mass matrix (n n) stiffness matrix (n n) stiffness matrix (n n)

MK

diagonal matrix with eigenvalues (q q) diagonal matrix with eigenvalues (q q) eigenvectors set (n q) eigenvectors set (n q) Φ

Λ

n = system orderq = 2pp = no. of desired eigenpairs

n = system orderq = 2pp = no. of desired eigenpairs

7 7

• Generation of starting vectors by Lanczos algorithm• Generation of starting vectors by Lanczos algorithm

11~

iiiiii xxxx

ii MxKx 1

iTii Mxx

2/1)~~( iTii xMx

iii /~1 xx

][ 211 qxxxΦ

8 8

• Simultaneous inverse itertion and system reduction• Simultaneous inverse itertion and system reduction

kk MΦKΦ 11

111 kTkk ΦKΦK

111 kTkk ΦMΦM

• Eigensolution for reduced system and eigenvector calculation• Eigensolution for reduced system and eigenvector calculation

11111 kkkkk ΛQMQK

111 kkk QΦΦ

9 9

Subspace iteration method with proposedLanczos starting subspace

• Generation of starting vectors by proposed Lanczos algorithm• Generation of starting vectors by proposed Lanczos algorithm

11~

iiiiii yyyy

ii yMKy 21 )(

iTii yMy

2/1)~~( iTii yMy

iii /~1 yy

][ 211 qyyyΦ

10 10

• Simultaneous inverse itertion, system reduction,eigensolution for reduced system and eigenvector calculationare the same as the conventional method.

• Simultaneous inverse itertion, system reduction,eigensolution for reduced system and eigenvector calculationare the same as the conventional method.

• Squared dynamic matrix can separate Riz values more rapidly. Starting vectors are closer to exact eigenvector space. The number of iterations is reduced.

• Squared dynamic matrix can separate Riz values more rapidly. Starting vectors are closer to exact eigenvector space. The number of iterations is reduced.

11 11

Numerical examples

Structures

6

2

2 10||||

||||

i

iiii

K

MK

• Building structure with 1008 DOFs• Building structure with 5040 DOFs• Building structure with 1008 DOFs• Building structure with 5040 DOFs

Error measure for checking convergence

12 12

Building structure with 1008 DOFs

36 m

2 1 m

9 m6 m

• Geometry and material properties• Geometry and material properties

A = 0.01 m2

I = 8.310-6 m4

E = 2.11011 Pa = 7850 kg/m3

A = 0.01 m2

I = 8.310-6 m4

E = 2.11011 Pa = 7850 kg/m3

13 13

• Number of iterations• Number of iterations

ProposedConventional No. of desired eigenpairs

13 8 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

131210 5 914 7 32114161510 4 1 1 1 1 1 1

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100

14 14

• Computing time• Computing time

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0N o . o f d es i red eig en p airs

1

1 0

1 0 0

Com

puti

ng t

ime

(sec

) H = 9 9 .9 9 1 %

C o n v en tio n a lP ro p o sed

ss

ΜsssT

jTT

jT

j

p

jj

h

hH

)})({(

1

hj = modal contribution factorss = spatial load distribution vector

hj = modal contribution factorss = spatial load distribution vector

15 15

Building structure with 5040 DOFs

• Geometry and material properties• Geometry and material properties

A = 0.01 m2

I = 8.310-6 m4

E = 2.11011 Pa = 7850 kg/m3

A = 0.01 m2

I = 8.310-6 m4

E = 2.11011 Pa = 7850 kg/m3

5 0 m

3 0 m

2 1 0 m

16 16

• Number of iterations• Number of iterations

ProposedConventional No. of desired eigenpairs

45111111111111111111

81612 52313 7121313 715 3 1 1 1 1 1 1 1

10 20 30 40 50 60 70 80 90100110120130140150160170180190200

17 17

• Computing time• Computing time

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0N o . o f d es i red eig en p ai rs

1 0

1 0 0

1 0 0 0

1 0 0 0 0

Com

puti

ng t

ime

(sec

)

H = 9 9 .5 6 1 %

C o n v en tio n a lP ro p o sed

18 18

Conclusions

• Subspace iteration method with proposed Lanczos starting subspace has smaller number of iterations than the subspace iteration method with conventional Lanczos starting subspace because squared dynamic matrix in proposed algorithm can accelerate convergence.

• Since proposed method has less computing time than the conventional method when the number of desired eigenpairs is small, proposed method is practically efficient.

• Subspace iteration method with proposed Lanczos starting subspace has smaller number of iterations than the subspace iteration method with conventional Lanczos starting subspace because squared dynamic matrix in proposed algorithm can accelerate convergence.

• Since proposed method has less computing time than the conventional method when the number of desired eigenpairs is small, proposed method is practically efficient.