ung-Wan Kim, Hyung-Jo Jung and In-Won Lee uctural Dynamics & Vibration Control Lab. artment of Civil & Environmental Engineering, KAIST Matrix Power Lanczos Method and Its ication to the Eigensolution of Struct AIST-Kyoto Univ. Joint Seminar n Earthquake Engineering Feb. 25, 2002.
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Byoung-Wan Kim, Hyung-Jo Jung and In-Won Lee Structural Dynamics & Vibration Control Lab.
KAIST-Kyoto Univ. Joint Seminar on Earthquake Engineering. Feb. 25, 2002. Matrix Power Lanczos Method and Its Application to the Eigensolution of Structures. Byoung-Wan Kim, Hyung-Jo Jung and In-Won Lee Structural Dynamics & Vibration Control Lab. - PowerPoint PPT Presentation
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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
• 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
qq
qq
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.