김정호
Jeong-Ho Kim2017.11.23
동적해석을위한자유단경계기반의부구조를갖는모델축소기법
On the model reduction methods with free-interface substructuring for dynamic analysis
Ph.D. degree dissertation presentation
On the model reduction methods with free-interface substructuring for dynamic analysis
Contents
• Background
• History
• Contributions
• Future Works
2
On the model reduction methods with free-interface substructuring for dynamic analysis
Background
3
On the model reduction methods with free-interface substructuring for dynamic analysis
Development of design & FE analysesSimplification Diversity & Complexity
4
On the model reduction methods with free-interface substructuring for dynamic analysis
Various FE analyses in engineering fields
Analysis of complex & large structures Model updating & Design optimization
Experimental‐Numerical correlationStructural health monitoring
5
On the model reduction methods with free-interface substructuring for dynamic analysis
Wind turbine tower Vehicle manufacturing
Aircraft
Civil construction
Substructuring & Assembling
6
Global structure obtained through assembly of substructures
On the model reduction methods with free-interface substructuring for dynamic analysis
Needs for model reduction methods• Complex structures consisting of diverse parts
• Satisfying the requirements in local & global configuration
• Easy assembly & disassembly of the reduced models
• Frequent design modifications & re-analysis
• Different time consumption for each substructure
7
Model reduction method with solution reliability&
Effective substructuring techniques
On the model reduction methods with free-interface substructuring for dynamic analysis
History
8
On the model reduction methods with free-interface substructuring for dynamic analysis
Mode based reduction (Component mode synthesis)
DOF based reductionMaster node
)1(dominantΦ
)2(dominantΦ
)3(dominantΦ
Model reduction method
Retained substructual modes+
Interface handling (Fixed / Free)
Reduced transformation matrix
=
TMTΜ gT TKTK g
T,
Select master DOFs↓
Condense slave DOFs into Master DOFs
Reduced transformation matrix
=
TMTΜ gT TKTK g
T,
9
On the model reduction methods with free-interface substructuring for dynamic analysis 10
Mode based reduction (Component mode synthesis)
History
Craig & Bampton (CB), 1968
Rixen, 2004 (DCB method)Park, 2004 (F-CMS method)
Hurty, 1965
Fixed-interface CMS (Pioneer)
Goldman, 1969MacNeal, 1971 Craig & Chang, 1976
Rubin, 1975
Free-interface & Hybrid CMS
1960 1980 20001970
“Simplicity & robustness”
Free-interface CMS using 1st residual flexibility
“dual counterpart of CB method”
Bennighof, 2004Automated Multi-Level Substructuring
On the model reduction methods with free-interface substructuring for dynamic analysis 11
Mode based reduction (Component mode synthesis)
History
Enhanced CB / AMLS, HCB method
• CB / AMLS transformation + residual flexibility
• Considerably improved accuracy, inevitably increased computational costs
Algebraic dynamic condensation method
• Employing algebraic substructuring algorithm (Using METIS adopted in AMLS method)
• Static condensation → mode based interface reduction → dynamic condensation
• Submatrix operations, improving accuracy & computational efficiency of the IRS method
(2015 ~ 2017)
Boo et al., (2017)
Improvement on the fixed-interface based CMS methods
On the model reduction methods with free-interface substructuring for dynamic analysis 12
DOF based reduction
Guyan, 1965Iron, 1965
O’Callahan, 1989(IRS / SEREP)
Static condensation (Pioneer)
Firswell, 1995(Iterative IRS)
1960 1980 1990 20001970
Cho, 2007Bouhaddi & Fillod, 1996
Kaufman & Hall, 1968Ramsden & Stoker, 1969
Sowers, 1978
Dynamic condensation
Static condensation using flexibility matrix
DOF based reduction using substructuring
Hsiung & El-Sayed, 1990Static condensation with parallel processing
“Fixed-interface based substructuring”
History
On the model reduction methods with free-interface substructuring for dynamic analysis 13
History
Sub-domain reduction method in non-matched interface problemsCho MH et al., (2008)
• Two-level condensation scheme + penalty frame method for non-matched subdomains
• Construct interfacial stiffness matrix for coupling the substructures with non-matching mesh
Identification of structural systems using an iterative, improved method for system reduction
DOF based reduction
Cho MH et al., (2009)
• Two-level condensation scheme + fixed interface substructuring
• Improving computational efficiency of the Iterative IRS (IIRS) method
Applying fixed-interface substructuring to the DOF based reduction
Optimization of substructure, dynamic re-analysis, parametric reduced modeling..2010s ~
On the model reduction methods with free-interface substructuring for dynamic analysis
Contributions
14
On the model reduction methods with free-interface substructuring for dynamic analysis
Global equilibrium equation Substructuring (matrix permutation)
Fixed-interface
Substructure 1
Substructure 2
Fixed-interface based substructuring (CB, AMLS, …)
ggggg fuKuM
b
s
b
s
bTc
cs
b
s
bTc
cs
ff
uu
KKKK
uu
MMMM
15
)1(su
bu
)2(su
Substructuring techniques
“Primal assembly” of substructures
Unique set of interface DOFS
On the model reduction methods with free-interface substructuring for dynamic analysis
Substructuring techniques
16
Enhanced CB / AMLS method, (2015).
Fixed-interface based substructuring
Algebraic dynamic condensation method, (2017).
On the model reduction methods with free-interface substructuring for dynamic analysis
: interconnecting force
Free-interface
Substructure 1
Substructure 2
Free-interface based substructuring (DCB, F-CMS)
Local equilibrium equation “Dual assembly” of substructures
μ
Displacement compatibility
μB )(k
0f
μu
0BBK
μu
000M
T
[
)1(u
)2(u
17
: Lagrange multipliersμ
Force equilibrium
0ub
s TN
k
kb
k
1
)()(
)()()()()()( kkkkkk fμBuKuM
Substructuring techniques
On the model reduction methods with free-interface substructuring for dynamic analysis
Substructuring techniques
18
Brecher et al., (2016).
Various physical constraintsModal test of assembly parts
Blachowski et al. (2016).
Damage detection for bolted connection
Van der Valk, (2010).
Free-interface based substructuring
Dynamic behavior of substructures & Assembly of reduced model
New model reduction methods with free-interface substructuring are required to solve various structural problems !
On the model reduction methods with free-interface substructuring for dynamic analysis
Research Issues Solution reliability of the reduced model by the free-CMS method
• Topic 1. Error estimation method for the DCB method Relative eigenvalue error estimation Correction of the approximated eigenvalues Investigation on the performance of DCB method
Solution accuracy of the reduced model by the free-CMS method
• Topic 2. Improving the accuracy of the DCB method Considering the 2nd order residual flexibility + Interface reduction Discussion on the solution accuracy and efficiency of the present method
New DOF based reduction method with solution efficiency
• Topic 3. A dynamic condensation method with free-interface based substructuring Free-interface based substructuring Reduced models in assembly process
19
On the model reduction methods with free-interface substructuring for dynamic analysis
An error estimation method for dual Craig-Bampton method
20
Research topic I
On the model reduction methods with free-interface substructuring for dynamic analysis
Introduction
21
The verification of the solution reliabilityRelative eigenvalue errors
1
i
i
i
iii
: Exact eigenvalue (unknown)i
i : Approximated eigenvalue (Reduced method)
iggiigg )()( φMφK Global eigenvalue problem
Difficult to construct the entire finite element model
Large computational cost
Q. How to get the exact eigenvalues ?
On the model reduction methods with free-interface substructuring for dynamic analysis
Introduction
22
Error estimation methodsPriori bound for automated multi-level substructuring (AMLS)
ir
ii
Elssel & Voss, (2006)
r, : The smallest substructural eigenvalue in the residual parts
For the CB method,
Posteriori accurate error estimator for CB & F-CMS methodsKim et al., (2014, 2015)
Simplified error estimator for AMLS & CB method Boo et al., (2015, 2016)
Calculate the enhanced transformation matrix
iprgi
gT
rTipiprg
ig
TTipi )(1)()(1)(2 0 φTKMTφφTKMTφ
rT
Submatrix level computation & error control by the substructural contribution
n
kib
kc
krs
Tkc
Tibii
1
)()()( )(ˆˆ)( φMFMφFor the CB method,
On the model reduction methods with free-interface substructuring for dynamic analysis
Dual Craig-Bampton method
Substructural dynamic behavior
Static solutions (interface, flexibility) + Dynamic solutions (substructure)
,
Construct transformation matrix
jkkk
jjkk )()( )()()()()( φMφK ][ )()()( kkk ΘRΦ ][ )()()( k
rk
dk ΘΘΘ ,
)()()()()()()( kkkkkkk qΘαRμBKu
)()()()()()(1
)( kd
kd
kkkkk qΘαRμBFu
μqα
Tμu )(
)(
)(1
)(k
d
k
kk
I00BFΘR
T)()(
1)()(
)(1
kkkd
kk
,
Tkd
kd
kd
kk )()()()()(1
1
ΘΛΘKF
23
Flexibility mode (unit force for 1 interface DOF)
On the model reduction methods with free-interface substructuring for dynamic analysis
1. Second order dynamic residual flexibility
24
Employing Neumann series expansion
)(1)(2
)(1
)(1)()()( )( kj
kkkkr
kr
kr
kr
T
FFFΘIΛΘ
Tj kr
kr
kr
kj
)()()()( ΘΛΘF
The j-th order residual flexibility matrix
)()()()()()(2
)()()(1
)( kd
kd
kkkkkkkk qΘαRμBFμBFu )()()()()()(
1)( k
dk
dkkkkk qΘαRμBFu
Original DCB methodApproximated substructural displacement
The 2nd order residual flexibility matrix can be easily calculated by reusing )(1
kF)(
1)()(
1)(
2kkkk FMFF
Tkr
kr
kr
k )()()()(2
2
ΘΛΘF
Error estimator for DCB method
Tid
id
id
ii )()()()()(1
1
ΘΛΘKF
On the model reduction methods with free-interface substructuring for dynamic analysis
2. Improved transformation matrix
25
Error estimator for DCB method
μqα
Tμu )(
)(
)()( ~ k
d
k
kk
)()()(1
)(~ ka
kkk TTT
000BF00
T)()(
2)(kk
ka
I00BFΘR
T)()(
1)()(
)(1
kkkd
kk
,
Substructural transformation matrices
aTTT 1~
Global transformation matrices
,
,
I00ΨΘR
T 11
d
000Ψ00
T 2a
)()(1
)1()1(1
1ss NN BF
BFΨ
)()(2
)1()1(2
2ss NN BF
BFΨ , , ,
Additional transformation matrix
On the model reduction methods with free-interface substructuring for dynamic analysis
3. Formulation of error estimator
26
Relative eigenvalue error
Error estimator for DCB method
iggTigigg
Tig
i
)()()()(1 φMφφKφ iaiiig φTTφTφ )(~)( 1
iiTi φKφ 1i
Ti φMφ
ii
i
1 iagTaiag
Taiag
Tag
Ti
Tii φTKTTMTTKTTMTφ ]22[ 2
11
1
i
i
i
iii
: Exact eigenvalue (unknown)i
i : Approximated eigenvalue (DCB method)
Global eigenvalue problem
,,11 T000M
TM
T
11 T0BBK
TK
T
T,
Error estimator for the ith approximated eigenvalue with global matrix computation
On the model reduction methods with free-interface substructuring for dynamic analysis 27
Error estimator for DCB method
iagTaiag
Ti
Tii φTMTTMTφ ][ 2
1 i
si
φφ
φ
sN
k
kii
1
)( ikkTkT
iiikkTkT
iik
i )()()()( )()(4
)(2)()(3
)()( φBFBφφBFBφ
3. Formulation of error estimatorDecomposition of approximated eigenvector
Error estimator for the ith approximated eigenvalue with substructural independence
,Substructural part
Lagrange multiplier part
ss N
k
kkkkTkN
k
kkTk
1
)()(2
)()(1
)(
1
)()(3
)( BFMFBBFB
ss N
k
kkkkTkN
k
kkTk
1
)()(2
)()(2
)(
1
)()(4
)( BFMFBBFB
Tj kr
kr
kr
kj
)()()()( ΘΛΘF
Higher order residual flexibility matrices
Simply calculated by reusing & )(2
kF)(1
kF
On the model reduction methods with free-interface substructuring for dynamic analysis
Hyperboloid shell problem
Total DOFs = 4830 Retain modes = 54 (26+14+14)
28
Numerical examples
Reduced system size= 387 (8 %)
H
2�
3�
2/H
1�
0 5 10 15 20 25 30Mode number
10-8
10-6
10-4
10-2
100
102
ExactEstimated(Elssel & Voss)Estimated(Present)
On the model reduction methods with free-interface substructuring for dynamic analysis
Hyperboloid shell problem
29
Numerical examples
1
i
ii
Eigenvalue correction
sN
k
kii
1
)(
Modal assurance criterion (MAC)
On the model reduction methods with free-interface substructuring for dynamic analysis 30
Closure
1. An error estimation method is proposed for the DCB method.
2. Simplified formulation is well defined by using the substructural matrix computations
3. The relative eigenvalue error is accurately estimated by the present method.
4. The estimated error can be used to correct the approximated eigenvalues.
5. The error estimation of the approximated eigenvector also need to be investigated.
6. The approximated eigenpair (eigenvalue & vector) by the DCB method need to be
improved.
On the model reduction methods with free-interface substructuring for dynamic analysis
Improving the accuracy of the dual Craig-Bampton method
31
Research topic II
Kim JH, Kim J, Lee PS., Computers & Structures, 191, 22-32, Oct 2017.
On the model reduction methods with free-interface substructuring for dynamic analysis
Introduction
32
The dual Craig-Bampton (DCB) methodWeak compatibilities between neighboring substructures
Negative eigenvalues in low frequency range, “Spurious Mode”
)()()()()()(1
)( kd
kd
kkkkk qΘαRμBFu
Need to improve the accuracy of the reduced model (eigenvalues & eigenvectors)
In the insufficient reduction basis condition,
Minimize the additional computation costs
Dual Craig-Bampton with enrichment to avoid spurious modes Rixen, (2009)
Maintain the substructural independence of the original DCB method
Avoid the increment of reduction size
On the model reduction methods with free-interface substructuring for dynamic analysis
Improved DCB method1. Second order dynamic residual flexibility
33
Approximated substructural displacement)()()()()()(
2)()(
1)( k
dk
dkkkkkkk qΘαRμBFμBFu
)()()()()()(1
)( kd
kd
kkkkk qΘαRμBFu
Original DCB method
The 2nd order residual flexibility matrix can be easily calculated by reusing )(1
kF)(
1)()(
1)(
2kkkk FMFF
Tkr
kr
kr
k )()()()(2
2
ΘΛΘF
2. Construct transformation matrix
0I00ΘBFΘR
T)(
2)()(
1)()(
)(2
kkkkd
kk
,
ψμ
qα
u)(
)(
)(2
kd
k
k, ,
)()(2
)(2
kkk BFΘ , μψ
Badly scaled matrix
)(2
)(2
)(kk
k
uTμ
u
Define the additional generalized coordinates
Computational cost efficiency
Tid
id
id
ii )()()()()(1
1
ΘΛΘKF
On the model reduction methods with free-interface substructuring for dynamic analysis
2. Construct transformation matrix
34
1)()(2
)(2
kkk GΘΘ
2
)(2
22)(
2
21)(
2
)(
}{
}{
}{
bNk
k
k
k
θ0
θ
0θ
G
,
0I00ΘBFΘR
T)(
2)()(
1)()(
)(2
kkkkd
kk
Substructural transformation matrix
Improved DCB method
Normalization of 2nd order residual flexibility matrix term (L2-norm)
On the model reduction methods with free-interface substructuring for dynamic analysis
Improved DCB method3. Rayleigh-Ritz procedure
35
Substructural reduced system matrices
)(2
)()(
2)(
2k
kTkk T
000M
TM
)(
2)(
)()()(
2)(
2k
Tk
kkTkk T
0BBK
TK
,
ψμ
qα
u)(
)(
)(2
kd
k
k,
4. Improved reduced system matrices
MMMMM
MM
M
)()1(
)()(
)1()1(
2
s
ss
Nss
Ns
Ns
ss
KKKKK
KK
K
)()1(
)()(
)1()1(
2
s
ss
Nss
Ns
Ns
ss
,
ψμ
qα
qα
u)(
)(
)1(
)1(
2
s
s
Nd
N
d
,
Simple assemblage of substructural reduced matrices
1N2N
N
)()1(
1
)( ss
NN
k
k MMMM
,)()1(
1
)( ss
NN
k
k KKKK
On the model reduction methods with free-interface substructuring for dynamic analysis
Improved DCB method5. Interface reduction
36
Reduced eigenvalue problem
iii )()( 22 φMφK
1
)()()( 21 NφφφΦ
2,,1 Ni ,
ΦMΦM 22ˆ T ΦKΦK 22
ˆ T
6. Resultant reduced system matrices
,
Calculate the eigenvectors up to the N1-th mode
Same size with the original DCB method
On the model reduction methods with free-interface substructuring for dynamic analysis
• Without global matrix operations
• Substructural independence
• Efficient substructural operations
• Improved accuracy
• Non-matching mesh condition
Reduction procedure
37
Start
Substructure 1 ꞏꞏꞏꞏꞏSubstructure 2 Substructure aa sN
Construct)1()1()1( ,, BKM
Construct Construct)2()2()2( ,, BKM )()()( ,, sss NNN BKM
Calculate the substructural eigenvalue problem
Calculate the substructural eigenvalue problem
Calculate the substructural eigenvalue problem
Sort Calculate Calculate
Calculate Calculate Calculate
Calculate substructural reduced system matrices
Calculate substructural reduced system matrices
Calculate substructural reduced system matrices
Conduct interface reduction by using
)1(2
)1(2 , KM )2(
2)2(
2 , KM )(2
)(2 , ss NN KM
Solve the newly constructed reduced model
End
)1()1( , dΘR )2()2( , dΘR )()( , ss Nd
N ΘR
)1(2
)1(1 , FF )2(
2)2(
1 , FF )(2
)(1 , ss NN FF
Generate transformation matrix aaaaa
Generate transformation matrix aaaaa
Generate transformation matrix aaaaa)1(
2T )2(2T )(
2sNT
Assemble the reduced system matrices
22 , KM
2T̂
22ˆ,ˆ KM
On the model reduction methods with free-interface substructuring for dynamic analysis
NACA 2415 wing with ailerons problem
Total DOFs = 21098 Retain modes = 33 (15+9+9)
38
Numerical examples
Reduced system size= 243 (1.2 %)
X Y
Z
Y
Z
X
Y
H
W
L
Clamped
1st mode (Substructure 3)
2nd mode (Substructure 2)
Revolute joint
1
23
On the model reduction methods with free-interface substructuring for dynamic analysis
NACA 2415 wing with ailerons problem
Total DOFs = 21098 Retain modes = 33 (15+9+9)
39
Numerical examples
Reduced system size= 243 (1.2 %)
Modal assurance criterion (MAC)
On the model reduction methods with free-interface substructuring for dynamic analysis
NACA 2415 wing with ailerons problem
Total DOFs = 21098 Retain modes = 33 (15+9+9)
40
Numerical examples
Reduced system size= 243 (1.2 %)
Methods ItemsComputation times
[sec] Ratio[%]
DCB Substructural eigenvalue problem 9.44 11.97
1st order residual flexibility matrices 66.80 84.75
Reduced system matrices 2.59 3.28
Total 78.83 100.00
Improved
DCB
Substructural eigenvalue problem 9.44 11.97
1st order residual flexibility matrices 66.80 84.75
2nd order residual flexibility matrices 108.03 137.04
Reduced system matrices 5.18 6.57
Subtotal 189.45 240.33
Interface reduction 0.99 1.25
Total 190.44 241.58
Methods ItemsComputation times
[sec] Ratio[%]
DCB Substructural eigenvalue problem 9.12 0.34
1st order residual flexibility matrices 2698.08 99.58
Reduced system matrices 2.27 0.08
Total 2709.47 100.00
Improved
DCB
Substructural eigenvalue problem 9.12 0.34
1st order residual flexibility matrices 2698.08 99.58
2nd order residual flexibility matrices 104.42 3.85
Reduced system matrices 4.51 0.17
Subtotal 2816.13 103.94
Interface reduction 0.70 0.03
Total 2816.83 103.97
Inverse calculation (physically fixed substructure 1) Pseudo-inverse calculation (free boundary)
On the model reduction methods with free-interface substructuring for dynamic analysis 41
Closure
1. A new CMS method is proposed by improving the DCB method.
2. The accuracy of reduced models is remarkably improved without increasing reduced
matrix size.
3. Negative eigenvalues are avoided in lower modes.
4. An attractive solution for the applications of the DCB method.
5. Pseudo-inverse calculation has very large computational costs (inv. VS. p-inv.).
6. Parallel computation algorithm for the present method will be valuable.
On the model reduction methods with free-interface substructuring for dynamic analysis
A dynamic condensation method with free-interface based substructuring
42
Research topic III
On the model reduction methods with free-interface substructuring for dynamic analysis
Introduction
43
Dynamic condensation method + Free-interface substructuring
Dynamic condensation methodNo need to solve the substructural eigenvalue problems
Good accuracy
Free-interface substructuringNo need to construct assembled FE model
Independent FE model of substructure
Fully-decoupled substructural model reduction
Preserve the important physical information related to the selected DOFs
Expansion of measured DOFs for the FE model validation, health monitoring, etc.
On the model reduction methods with free-interface substructuring for dynamic analysis
Free-interface based Dynamic Condensation method
Eigenvalue problem for each substructure
1. Construct transformation matrix
44
Substructural Slave displacement vector with invoking harmonic response
Substructural master & slave DOFs (matrix permutation)
μu
000M
μu
0BBK )()(
)()(
)(
)()( kkk
k
Tk
kk
μuu
0000MM0MM
μuu
0BBBKKBKK
)(
)(
)()(
)()(
)()(
)(
)()(
)()()(
)()()(
km
ks
kmm
kms
ksm
kss
kkm
ks
Tkm
Tks
km
kmm
kms
ks
ksm
kss
])[()( )()()()()(1)()()()( μBuMKMKu ks
km
ksm
kksm
kss
kkss
ks
On the model reduction methods with free-interface substructuring for dynamic analysis
Approximated slave DOFs
1. Construct transformation matrix
45
Substructural Guyan & IRS transformation
Approximated substructural displacement vector
,
,
Unknown substructural eigenvalue
Free-interface based Dynamic Condensation method
μΘtuΘtuu )(][ )()()()()()()()()( kkkkm
ks
kks
ks
ks
)(1)()( )( ksm
kss
ks KKt
)(1)()( )( ks
kss
k BKt
)()( )()()(1)()( ks
kss
ksm
kss
ks tMMKΘ
)()(1)()( )( kkss
kss
k tMKΘ
μu
Tμ
uT
μuu
μuu
)()()(
)()(
0)(
)(
)(
)(k
mka
kk
mkkm
ks
km
ks
I00I
ttT )(
)()(
)(0
km
kks
k
0000
ΘΘT
)()(
)(
kks
ka
On the model reduction methods with free-interface substructuring for dynamic analysis
Guyan reduced substructural matrices (static condensation)
1. Construct transformation matrix
46
Unknown substructural eigenvalue
,with
,
Substructural transformation matrix
,
Free-interface based Dynamic Condensation method
μu
Mμ
uK
)()(
0)(
)()(
0
kmkk
kmk )(
0
)()(
0)(
0k
kTkk T
000M
TM
)(
0)(
)()()(
0)(
0k
Tk
kkTkk T
0BBK
TK
μu
Hμ
u )()(
)()(
kmk
kmk )(
01)(
0)( )( kkk KMH
μu
Tμ
uu )(
)(1
)(
)(k
mkkm
ks
)()(
)()()(
kkm
km
kmmk
HHHH
H,
I0ΨA
T)()(
)(1
kkk
)(
)()(
ˆk
m
ksk
It
A
0t
Ψ)(
)(ˆ k
k , ,
)()()()()()(ˆ km
kkmm
ks
ks
ks HΘHΘtt , )()()()()()(ˆ kkk
mk
skk
HΘHΘtt
On the model reduction methods with free-interface substructuring for dynamic analysis
2. Reduced substructural matrices
47
IRS reduced substructural matrices (dynamic condensation)
,
3. Resultant reduced system matrices
μu
u
u )(
)1(
1 sNm
m
Simple assemblage of substructural reduced matrices
,
111 fuKuM
)()1(
1
)( ss
NN
k
k MMMM
,)()1(
1
)( ss
NN
k
k KKKK
Free-interface based Dynamic Condensation method
)(1
)()(
1)(
1k
kTkk T
000M
TM
)(
1)(
)()()(
1)(
1k
Tk
kkTkk T
0BBK
TK
MMMMM0
M0M
M
)()1(
)()(
)1()1(
s
ss
Nmm
Nm
Nmm
mmm
KKKKK0
K0K
K
)()1(
)()(
)1()1(
s
ss
Nmm
Nm
Nmm
mmm
,
On the model reduction methods with free-interface substructuring for dynamic analysis
• Without global matrix operations
• Without substructural eigenvalue prob.
• Substructural independence
• Efficient substructural operations
• Non-matching mesh condition
• Master DOFs distribution
48
Start
Substructure 1 Substructure 2 Substructure aa
Construct Construct Construct
Select master and slave DOFs
Select master and slave DOFs
Select master and slave DOFs
Calculate Calculate Calculate
Calculate Calculate Calculate
Calculate substructural reduced system matrices
Calculate substructural reduced system matrices
Calculate substructural reduced system matrices
Assemble the reduced system matrices
Solve the newly constructed reduced model
End
Reduction procedure
On the model reduction methods with free-interface substructuring for dynamic analysis
Bended pipe problem
Non-matching mesh Case : Total DOFs = 14300 Retain modes = 1215 (8.4 %)
49
Numerical examples
1
3
2
0uu )2(1
)1(1
0uu )2(2
)1(2
0uuu )2(2
)2(1
)1(3 5.05.0
T
005.05.000100001
)2(B
T
000000100000000010000000001
)1(B
Displacement compatibility
0f
μu
0BBK
μu
000M
T
Force equilibrium
)2(
)1(
BB
B
On the model reduction methods with free-interface substructuring for dynamic analysis
Bended pipe problem
Non-matching mesh Case : Total DOFs = 14300 Retain modes = 1215 (8.4 %)
50
Numerical examples
0 2 4 6 8 10 12 14 16 18Mode number
10-10
10-8
10-6
10-4
10-2
100
Rel
ativ
e ei
genv
alue
erro
r
Present method
Modal assurance criterion (MAC)
On the model reduction methods with free-interface substructuring for dynamic analysis
Wind turbine problem (600kW wind turbine, rotor diameter = 39.76m)
Total DOFs = 94906 Reduced DOFs = 2287 (2.4 %)
51
Hub( 1 )
Blade-1( 2 )
Blade-2( 3 )
Blade-3( 4 )
(a)(b)
(c) Rotor hub (5DOF)
Turbine blade (6DOF)
Numerical examples
On the model reduction methods with free-interface substructuring for dynamic analysis
Wind turbine problem
Total DOFs = 94906 Reduced DOFs = 2287 (2.4 %)
52
Hub( 1 )
Blade-1( 2 )
Blade-2( 3 )
Blade-3( 4 )
(a)
Numerical examples
0 5 10 15 20 25Mode number
10-10
10-8
10-6
10-4
10-2
100
Rel
ativ
e ei
genv
alue
erro
r
Present method
On the model reduction methods with free-interface substructuring for dynamic analysis
Wind turbine problem
Total DOFs = 94906 Reduced DOFs = 2287 (2.4 %)
53
Method ItemsComputation times
[sec] Ratio [%]
Present
Substructural reduction (Guyan) 283.62 33.66
Substructural reduction (IRS) 558.85 66.33
Assemble reduced matrices 0.10 0.01
Total 842.57 100.00
Numerical examples
Modal assurance criterion (MAC)
On the model reduction methods with free-interface substructuring for dynamic analysis
Cable-stayed bridge problem
54
Numerical examples
X Y
Z
Girder : 504 8-node shell elementsTower : 50 3D 27-node solid elementsCable : 4 3D 2-node truss elementsTotal DOFs = 6878
Substructure
6 substructures assembly modelTotal DOFs = 42293
Global structure
On the model reduction methods with free-interface substructuring for dynamic analysis
Cable-stayed bridge problem
Total DOFs = 42293 Reduced DOFs = 3555 (8.4 %)
55
Numerical examples
Modal assurance criterion (MAC)
On the model reduction methods with free-interface substructuring for dynamic analysis
Cable-stayed bridge problem
Total DOFs = 42293 Reduced DOFs = 3555 (8.4 %)
56
Numerical examples
Method ItemsComputation times
[sec] Ratio [%]
IRS( ) Reduction procedure 826.76 100.00
Present( )
Matrix permutation 2.00 0.24
Substructural reduction (Guyan) 118.12 14.29
Substructural reduction (IRS) 161.04 19.48
Assemble reduced matrices 0.20 0.02
Total 281.37 34.03
14201 N
35551 N
Accuracy of eigensolutions in IRS method
On the model reduction methods with free-interface substructuring for dynamic analysis
Cable-stayed bridge problem
Total DOFs = 42293 Reduced DOFs = 3555 (8.4 %)
57
Numerical examples
1
6
• Master DOFs + Interface connectivity
• Quasi block-diagonal matrices
• Reduced model in each assembly stage
Sparsity pattern of reduced matrix
Reduced matrix by the IRS method
On the model reduction methods with free-interface substructuring for dynamic analysis 58
Closure
1. A new dynamic condensation method is proposed by using free-interface substructuring
2. The reduction procedure can be performed independently for each substructure
3. Obtain the reduced model for each assembly stage without re-analysis
4. Compared to the CMS method, the size of the reduced model is large.
5. Need to develop the interface reduction techniques for the present method
6. Parallel computation algorithm for the present method will be valuable
On the model reduction methods with free-interface substructuring for dynamic analysis
Future Works
59
On the model reduction methods with free-interface substructuring for dynamic analysis
Solution reliability of the model reduction methods
• Topic 1. Error estimation method for the DCB method Error estimation for eigenpair (eigenvalue & eigenvector) More efficient computation for higher-order residual flexibilities
Model reduction methods with free-interface substructuring
• Topic 2. Improving the accuracy of the DCB method
• Topic 3. A dynamic condensation method with free-interface based substructuring Optimized parallel computation algorithm for the present methods Application studies for various dynamic analyses
(Model updating, Multi-physics, etc.) Hybrid model reduction method (CMS + DOF reduction)
60
Future works
On the model reduction methods with free-interface substructuring for dynamic analysis
Improved DCB method
⁺ Relatively small size of reduced model (Rigid body modes + dominant modes)
⁺ Validation & correlation with the experimental modal analysis
⁺ Global damage detection
⁻ Generalized coordinates
⁻ Solution accuracy disturbance (pseudo-inverse computation), Computational costs
Model reduction methods with free-interface substructuring
⁺ Physical coordinates
⁺ Sensor positioning, measured DOFs
⁺ Local damage detection
⁻ Master DOFs selection (quantity and distribution)
⁻ Relatively large size of reduced model (selection of master DOFs)61
Pros and cons
On the model reduction methods with free-interface substructuring for dynamic analysis 62
Hybrid model reduction methodCable-stayed bridge problem
Total DOFs = 42293 Reduced DOFs = 2069 (4.9 %)
X Y
Z
X Y
Z
X Y
Z
X Y
Z
X Y
Z
X Y
Z
• DCB + Free-dynamic condensation• Global dynamic behavior (mode based)• Local master DOFs in large FE model
Sub #1, 2, 5, 6 : DCB (Nd = 16)Sub #3, 4 : Free-DC (Nm = 490)
감사합니다
On the model reduction methods with free-interface substructuring
for dynamic analysis
동적해석을위한자유단경계기반의부구조를갖는모델축소기법
Ph.D. degree dissertation presentation