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Byoung-Wan Kim, Hyung-Jo Jung and In-Won LeeStructural Dynamics & Vibration Control Lab.Department of Civil & Environmental Engineering, KAIST
Byoung-Wan Kim, Hyung-Jo Jung and In-Won LeeStructural Dynamics & Vibration Control Lab.Department of Civil & Environmental Engineering, KAIST
Matrix Power Lanczos Method and ItsApplication to the Eigensolution of Structures
Matrix Power Lanczos Method and ItsApplication to the Eigensolution of Structures
KAIST-Kyoto Univ. Joint Seminaron Earthquake Engineering
Feb. 25, 2002.Feb. 25, 2002.
2 2
Introduction Matrix power 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
• Kim and Lee (1999):Lanczos-based algorithm for nonclassical damping system
• 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
• Kim and Lee (1999):Lanczos-based algorithm for nonclassical damping system
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
number step Lanczos
(shift)y trialenerg
operatorgiven
system quantumfor functions basis
tscoefficien,
n
E
H
sf
ba
t
6 6
Objective
• Application of Lanczos method using the power techniqueto the eigensolution in structural dynamics
Matrix power Lanczos method
• Application of Lanczos method using the power techniqueto the eigensolution in structural dynamics
Matrix power Lanczos method
7 7
Eigenproblem of structure
niλ iii ,,1 MK
KM
K
M
and oforder
eigenpairth ),(
matrix stiffness symmetric
matrix mass symmetric
n
iλ ii
Matrix power Lanczos method
8 8
Modified Gram-Schmidt orthogonalization
(i + 1) Lanczos vectors withi-iterated Krylov sequence
i
jjj
ii υ
10
11 )( xxMKx
i
jjj
iδi υ
10
11 ))(( xxMKx
(i + 1) Lanczos vectors with i-iterated Krylov sequence
• Conventional Gram-Schmidt process:• Conventional Gram-Schmidt process:
• Gram-Schmidt process with power technique:• Gram-Schmidt process with power technique:
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
δT
q
αβ
βα
βαβ
βα
nq
1
11
221
11
1
21
)(
,][
XMKMXT
xxxX
11 11
Summary of algorithm and operation count
Operation 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
12 12
n = order of M and Km = halfband-width 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 Km = halfband-width of M and Kq = the number of calculated Lanczos vectors or order of Tsj = the number of iterations of jth step in QR iteration
13 13
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)
14 14
Simple spring-mass system (DOFs: 100)
11
12
1
121
12
K
1
1
1
1
M
• System matrices• System matrices
15 15
Failure in convergence due tonumerical instability of high matrix power
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
16 16
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
17 17
• Number of operations• Number of operations
No. of eigenpairs = 1 = 2 = 3 = 4
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
Failure in convergence due tonumerical instability of high matrix power
Failure in convergence due tonumerical instability of high matrix power
18 18
Three-dimensional frame structure (DOFs: 468)
• Geometry and properties• Geometry and properties
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
19 19
• Number of operations• Number of operations
No. of eigenpairs = 1 = 2 = 3 = 4
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
20 20
• Geometry and properties• Geometry and properties
Three-dimensional building frame (DOFs: 1008)
36 m
2 1 m
9 m6 m
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
21 21
• Number of operations• Number of operations
No. of eigenpairs = 1 = 2 = 3 = 4
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
Failure in convergence due tonumerical instability of high matrix power
Failure in convergence due tonumerical instability of high matrix power
22 22
Conclusions
• The convergence of matrix power Lanczos method is better than that of the conventional Lanczos method.
• The optimal power of dynamic matrix that reduces the number of operations and gives numerically stable solution in matrix power Lanczos method is the second power.
• The convergence of matrix power Lanczos method is better than that of the conventional Lanczos method.
• The optimal power of dynamic matrix that reduces the number of operations and gives numerically stable solution in matrix power Lanczos method is the second power.