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