Improved Lanczos Method for the Eigenvalue Analysis of Structures

Improved Lanczos Method forthe Eigenvalue Analysis of Structures

Improved Lanczos Method forthe Eigenvalue Analysis of Structures

2002 한국전산구조공학회 봄학술발표회 2002 년 4 월 13 일2002 년 4 월 13 일

Byoung-Wan Kim1), Woon-Hak Kim2) and In-Won Lee3)

1) Graduate Student, Dept. of Civil and Environmental Eng., KAIST2) Professor, Dept. of Civil Engineering, Hankyong National Univ.3) Professor, Dept. of Civil and Environmental Eng., KAIST

Byoung-Wan Kim1), Woon-Hak Kim2) and In-Won Lee3)

1) Graduate Student, Dept. of Civil and Environmental Eng., KAIST2) Professor, Dept. of Civil Engineering, Hankyong National Univ.3) Professor, Dept. of Civil and Environmental Eng., KAIST

2 2

Introduction Matrix-powered Lanczos method Numerical examples Conclusions

Contents

3 3

Introduction

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.

• 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.

4 4

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

• 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

5 5

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

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

12

11

111

)(

nnnnntnn

nnnnnnn

fbfafEHfb

fbfaHffb

(shift)energy trial

operatorgiven

systems quantumfor functions basis

tscoefficien,

tE

H

f

ba

6 6

Objective

• Application of Lanczos method using the power techniqueto the eigenproblem of structures in structural dynamics

Matrix-powered Lanczos method

• Application of Lanczos method using the power techniqueto the eigenproblem of structures in structural dynamics


7 7

Eigenproblem of structure

niλ iii ,,1 MK

KM

K

M

and oforder

eigenpairth ),(

matrix stiffness symmetric

matrix mass symmetric

n

iλ ii


8 8

Modified Gram-Schmidt process of Krylov sequence

i

jjj

iδi

i

jjj

ii

υ

υ

10

11

10

11

))((

)(

xxMKx

xxMKx

shift,matrix, dynamic

vectorLanczos vector,trial

tcoefficieninteger, positive

1

0

MKKMK

xx

υ

9 9

11~

iiiiii βα xxxx

Modified Lanczos recursion

2/1

1

1

)~~(

~

)(

iTii

i

ii

iTii

iδ

i

β

β

α

xMx

xx

xMx

xMKx

10 10

Reduced tridiagonal standard eigenproblem

iδi

i λ~

)(

1~

T

qq

qq

δT

αβ

βα

βαβ

βα

1

11

221

11

1 )( XMKMXT

nqq ,][ 21 xxxX

11 11

Summary of algorithm and operation countOperation Calculation Number of operations

Factorization

Iteration i = 1 ··· q

Substitution

Multiplication

Multiplication

Reorthogonalization

Multiplication

Division

Repeat

Reduced eigensolution

TLDLK

ii xMKx )( 1

iTii xMx

11~

iiiiii xxxx

})12({ nnmni

nmnm )2/3()2/1( 2

}2)12({ nmmn nmn )12(

1)12( nmn

q

jjtotal jsqqnqqqnmqqqnmN

2

2222 610})2/17()2/3{()2/354()2/1(

n2

i

kkk

Tiii

1

)~(~~ xMxxxx

2/1)~~( iTii xMx

iii /~1 xx

iii ~

))/(1(~ T

n

q

jj qjs

2

2106

n = order of M and K, m = half-bandwidth of M and Kq = the number of calculated Lanczos vectors or order of Tsj = the number of iterations of jth step in QR iteration

n = order of M and K, m = half-bandwidth of M and Kq = the number of calculated Lanczos vectors or order of Tsj = the number of iterations of jth step in QR iteration

12 12

Numerical examples

6

2

2 10||||

||||

i

iiiiε

K

MK

• 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)

• 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)

Structures

Physical error norm (Bathe 1996)

13 13

Simple spring-mass system (DOFs: 100)

11

12

1

121

12

K

1

1

1

1

M

• System matrices• System matrices

14 14

Failure in convergence due tonumerical instability of high matrix power


• Number of operations• Number of operations

No. of eigenpairs = 1 = 2 = 3 = 4

2 4 6 810

38663 78922120458157649214729

29823 58529 85712117587154418

26954 47567 73040103055138122

23653441226939199550

0 2 4 6 8 1 0N o . o f eig en p airs

1 E 4

1 E 5

1 E 6

1 E 7

No.

of

oper

atio

ns

= 1 = 2 = 3 = 4

15 15

Plane framed structure (DOFs: 330)

• Geometry and properties• Geometry and properties

A = 0.2787 m2

I = 8.63110-3 m4

E = 2.068107 Pa = 5.154102 kg/m3

A = 0.2787 m2

I = 8.63110-3 m4

E = 2.068107 Pa = 5.154102 kg/m3

6 1 .0 m

30.5

m

16 16



612182430

10908273 20855865 27029145 31581179102944376

742905013578945186762092251653365994807

707245211688377165085072016479754112986

66335361123762516047093

0 6 1 2 1 8 2 4 3 0N o . o f eig en p airs

1 E 6

1 E 7

1 E 8

1 E 9

No.

of

oper

atio

ns

= 1 = 2 = 3 = 4



17 17

Three-dimensional frame structure (DOFs: 468)


E = 2.068107 Pa = 5.154102 kg/m3

E = 2.068107 Pa = 5.154102 kg/m3

: A = 0.2787 m2, I = 8.63110-3 m4: A = 0.2787 m2, I = 8.63110-3 m4

: A = 0.3716 m2, I = 10.78910-3 m4: A = 0.3716 m2, I = 10.78910-3 m4

: A = 0.1858 m2, I = 6.47310-3 m4: A = 0.1858 m2, I = 6.47310-3 m4

: A = 0.2787 m2, I = 8.63110-3 m4: A = 0.2787 m2, I = 8.63110-3 m4

Column in front buildingColumn in front building

Column in rear buildingColumn in rear building

All beams into x-directionAll beams into x-direction

All beams into y-directionAll beams into y-direction

4 5 .7 5 m

22.8

75 m

24.4

m

x y

zy

z

x R e a r

F ro n t

E le v a tio n P la n

18 18



1020304050

71602154 181780512 307269560 6841622221024104917

50687925124269611215884077453454527656188310

48705515116680070192064376378770940553972908

46214349108715163182518601356596304504420108

0 10 20 30 40 50N o . o f eig en p airs

1E 7

1E 8

1E 9

1E 1 0

No.

of

oper

atio

ns

= 1 = 2 = 3 = 4

19 19


Three-dimensional building frame (DOFs: 1008)

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

36 m

2 1 m

9 m6 m

20 20



20 40 60 80100

3950790201196316954304557829533987467933536190824

278717178 801878160199310812825091254743625240574

0 2 0 4 0 6 0 8 0 1 0 0N o . o f eig en p airs

1 E 8

1 E 9

1 E 1 0

1 E 1 1

No.

of

oper

atio

ns

= 1 = 2



21 21

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.

• 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.

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Improved Lanczos Method for the Eigenvalue Analysis of Structures