in-won lee, professor, pe in-won lee, professor, pe structural dynamics & vibration control lab....

In-Won Lee, Professor, PEIn-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab.Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology

International Conference on Numerical Methods & Computational MechanicsInternational Conference on Numerical Methods & Computational MechanicsThe University of Miskolc, HungaryThe University of Miskolc, HungaryAugust 26, 1998August 26, 1998

Efficient Free Vibration Analysis of Large Structures with Proportional and Non-Proportional Dampers

Structural Dynamics & Vibration Control Lab., KAIST, Korea

2

Problem Definition

Proposed Method

Numerical Examples

Conclusions

OUTLINE


3

PROBLEM DEFINITION

Dynamic Equation of Motion

where : Mass matrix, Positive definite

: Damping matrix

: Stiffness matrix, Positive semi-definite

: Displacement vector

: Load vector

: Order of K, C and M ( = 1,000 ~ 100,000)

)()()()( tftuKtuCtuM

M

C

K)(tu

)(tfn

(1)


4

Methods of Dynamic Analysis Step by step integration method Mode superposition method

Mode Superposition Method Free vibration analysis must be first performed. Most of computation time is required for free

vibration analysis.

An efficient solution technique is required !!!An efficient solution technique is required !!!


5

Condition of Proportional Damping

Ex. : Rayleigh Damping

CMKKMC 11

KMC


6

),...,2,1( npiMK iii

i

Eigenvalue Problem ( Proportionally Damped Case )

(3): Orthogonality of eigenvector

: ith eigenvalue(real)

: ith eigenvector(real)

: Number of eigenpairs to be sought

i

),...,2,1,( pjiM ijjTi

p

where

(2)


7

Current Methods for Proportionally Damped Case Subspace iteration method Determinant search method Householder-QR-inverse iteration method

Techniques Used by Commercial Programs ABAQUS - Subspace iteration method ADINA - Subspace iteration method

- Determinant search method ANSYS - Subspace iteration method - Householder-QR method NASTRAN - Givens method - Inverse power method SAP Series - Subspace iteration method


8

npiBA iii 2,,2,1 (4)

(5)

: Orthogonality of eigenvector

pjiB ijjTi ,,2,1,

: ith eigenvalue(complex conjugate)

: ith eigenvector(complex conjugate)

: Number of eigenpairs to be sought

i

ii

ii

where

M

KA

0

0

0M

MCB

Eigenvalue Problem ( Non-Proportionally Damped Case )

p


9

Current Methods for Non-Proportionally Damped Case

Transformation method: Kaufman (1974)

Perturbation method: Meirovitch et al (1979)

Vector iteration method: Gupta (1974; 1981)

Subspace iteration method: Leung (1995)

Lanczos method: Chen (1993)

Efficient Methods


10

PROPOSED METHOD Find p Smallest Eigenpairs

p 21

iii BA Solve

ijjTi B Subject to

For i iand pi ,,2,1

: close or multiple roots

BA

pT IB

where

p ,,, 21

pdiag ,,, 21

If p=1, then distinct root


11

For Proportionally Damped Case

(real)

For Non-Proportionally Damped Case

(complex conjugate)

KA

0M

MCB

ii

ii

M

KA

0

0

MB

ii


12

Relations between and Vectors in the Subspace of

BA

p ,,, 21

pdiag ,,, 21 where

X

(6)

(7)

(8)

XZ

pT IBXX

Let be the vectors in the subspace of and be orthonormal with respect to , then

pxxxX ,,, 21

(9)

(10)

B

X


13

ZDZ

where AXXdddD Tp ,,, 21 : Symmetric

Let (12)

Introducing Eq.(9) into Eq.(6)

BXZAXZ (11)

BXDZAXZ

BXDAX

or piBXdAx ii ,,2,1

Then

or

(13)

(14)

(15)


14

Multiple or Close Eigenvalues

Multiple eigenvalues case : is a diagonal matrix. Eigenvalues :

Eigenvectors :

Close eigenvalues case : is not a diagonal matrix. Solve the small standard eigenvalue problem.

Get the following eigenpairs.

Eigenvalues :

Eigenvectors :

ZDZ

D

D

XZ

DX

(12)

(9)


15

Find the Vectors in the Subspace of the Eigenvectors .

Rotate the Vectors in the Subspace to Find the Eigenvectors.

Strategy

),...,2,1( pixi ),...,2,1( pii


16

pidXBxA ki

kki ,,2,10)1()1()1(

pkTk IXMX )1()1( )(

(16)

(17)

)()()1( ki

ki

ki ddd

)()()1( ki

ki

ki xxx

)1()1(2

)1(1

)1( ,...,, kp

kkk xxxX

,)(kid )(k

ix

where

: unknown incremental values

(18)

(19)

(20)

Newton-Raphson Technique


17

)()()()( ki

kki

ki dXBAxr where : residual vector

)()()()()()( ki

ki

kki

kii

ki rdBXxBdxA

0)( )()( ki

Tk xBX

(21)

(22)

Introducing Eqs.(18) and (19) into Eqs.(16) and (17) and neglecting nonlinear terms

Matrix form of Eqs.(21) and (22)

pi

r

d

x

X

BXBdA ki

ki

ki

Tk

kkii

,,2,1

00B)(

)(

)(

)(

)(

)()(

(23)

Coefficient matrix : • Symmetric• Nonsingular


18

Modified Newton-Raphson Technique

Coefficient matrix : • Symmetric• Nonsingular

pi

rdx

XBXBdA k

ik

i

ki

Tk

kii

,...,,

B)(

)(

)(

)(

)(

)()(

21

00

0

)()()1( ki

ki

ki ddd

)()()1( ki

ki

ki xxx

(24)

(19)

(18)


19

Intermediate results by Subspace iteration method : Proportionally damped case Determinant search method

Results by Approximate Solution Methods such as Static or dynamic condensation method Lanczos method : Non-Proportionally damped case

Starting Eigenpairs


20

Step

Step 2: Solve for and )(kix )(k

id

00B)(

)(

)(

)(

)(

)()0( ki

ki

ki

Tk

kii r

d

x

X

BXBdA

Step 3: Compute)()()1( k

ik

ik

i ddd

)()()1( ki

ki

ki xxx

Step 1: Start with approximate eigenpairs ,)0(X )0(D

,)()0( kiix pid k

iii ,,2,1)()0(


21

Step 4: Check the error norm.

Error norm =

If the error norm is more than the tolerance, then go to Step 2 and if not, go to Step 5.

Step 5: Check if is a diagonal matrix, go to

Step 6, if not, go to Step 7.

2

)(2

)()()(

ki

ki

kki

xA

dXBxA

D


22

Step 7: Close case

Step 6: Multiple case

XD

Go to step 8.

Go to step 8.

ZDZ

XZ

Step 8: Check the error norm.

2

2

i

iii

A

BA

Error norm =

Stop !


23

NUMERICAL EXAMPLES: Proportionally Damped Case Structures

Three-dimensional framed structure(distinct) Simply-supported rectangular plate(multiple & close) Cooling tower(multiple)

Analysis Methods Proposed method Subspace iteration method Determinant search method

Comparisons CPU time Convergence

IRIS4D20-S17 with 10 MIPS, 0.9 MFLOPS


24

Three-Dimensional Framed Structure (Distinct Case)

Elevation Plan

Material Property Young’s modulus : 2.068E10 Pa

Mass density : 5.154E2 kg/m3

- Column in Front Building

I : 8.631E-3 m4 , A : 0.2787 m2

- Column in Rear Building

I : 10.787E-3 m4 , A : 0.3716 m2

- All Beams into x-Direction

I : 6.473E-3 m4 , A : 0.6906 m2

- All Beams into y-Direction

I : 8.631E-3 m4 , A : 0.2787 m2

System Data Number of equations : 468

Number of matrix elements : 42498

Maximum half-bandwidth : 138

Mean half-bandwidth : 91


25

Mode Number Eigenvalues

1 0.424

2 0.550

3 0.838

4 1.070

5 1.600

6 2.147

7 2.528

8 3.302

9 3.755

10 4.372

Eigenvalues (Distinct), 3-D. Frame


26

Analysis p = 10 p = 15

Method EN=10-6 EN=10-9 EN=10-6 EN=10-9

Proposed

Method

133.9

(1.0)

144.4

(1.0)

193.8

(1.0)

217.4

(1.0)

Subspace

Iteration

209.1

(1.6)

245.1

(1.7)

621.4

(3.2)

723.9

(3.3)

Determinant

Search

619.1

(4.6)

636.2

(4.4)

1,090.7

(5.6)

1,111.4

(5.1)

)1(

)()1(

k

i

ki

ki

Starting values : Subspace iteration method Relative error = 10-1

Relative error =

Error norm = 2

)(2

)()( )(k

i

ki

ki

K

MK

p = No. of eigenpairs

Solution Time (sec), 3-D. Frame


27

1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5Itera tio n N u m b er

1 .0 E -1 1

1 .0E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0E + 0E

rror

Nor

m

Convergence of the 12th eigenpair3-D. framed structure (distinct)

: Proposed Method : Subspace Iteration Method (q=2p) : Determinant Search Method

Error Limit


28

Simply-Supported Rectangular Plate

Material Properties

Young’s Modulus: 2.0E11 Pa

Mass Density: 7.850E3 kg/m3

Poisson Ratio: 0.3

Thickness: 0.01m

System Data

Number of Equations: 701

Number of Matrix Elements: 62,301

Maximum Half Bandwidths: 133

Mean Half Bandwidths: 89

(a) Multiple eigenvalues (b) Close eigenvalues


29


1 4.44

2 29.14

3 29.14

4 73.66

5 130.48

6 130.48

7 208.66

8 208.66

9 400.98

10 441.73

Eigenvalues (Multiple), Square Plate


30


Method EN=10-6 EN=10-9 EN=10-6 EN=10-9

Proposed

Method

138.1

(1.0)

172.4

(1.0)

213.5

(1.0)

245.9

(1.0)

Subspace

Iteration

221.0

(1.6)

274.6

(1.6)

487.4

(2.3)

607.2

(2.5)

Determinant

Search

1,072.4

(7.8)

1,118.6

(6.5)

1,570.2

(7.4)

1,618.8

(6.6)

)1(

)()1(

k

i

ki

ki


Relative error =

Error norm = 2

)(2

)()( )(k

i

ki

ki

K

MK


Solution Time (sec), Square Plate


31

1 2 3 4 5 6 7 8 9 1 0 1 1Ite ra tio n N u m b er

1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

Convergence of the 8th eigenpairSquare plate (multiple)


Error Limit


32


1 4.07

2 27.24

3 28.17

4 69.62

5 125.93

6 128.48

7 199.86

8 200.93

9 362.45

10 384.55

Eigenvalues (Close), Plate


33

)1(

)()1(

k

i

ki

ki


Relative error =

Error norm = 2

)(2

)()( )(k

i

ki

ki

K

MK


Solution Time (sec), Plate


Method EN=10-6 EN=10-9 EN=10-6 EN=10-9

Proposed

Method

156.5

(1.0)

177.4

(1.0)

237.9

(1.0)

290.8

(1.0)

Subspace

Iteration

169.7

(1.1)

291.9

(1.7)

350.9

(1.5)

742.6

(2.6)

Determinant

Search

813.2

(5.2)

832.1

(4.7)

1,146.7

(4.8)

1,194.1

(4.1)


34

1 3 5 7 9 1 1 1 3 1 5Itera tio n N u m b er

1 .0 E -1 1

1 .0E -1 0

1 .0E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

Convergence of the 8th eigenpairPlate (close)


Error Limit


35

Material Properties

Young’s Modulus: 4.32E8 lb/ft2

Mass Density: 4.66 slug/ft3

Poisson Ratio: 0.15

Shell Thickness: 0.583 ft

System Data

Number of Equations: 2,448

Number of Matrix Elements: 490,572

Maximum Half Bandwidths: 2,358

Mean Half Bandwidths: 201

Cooling Tower(Multiple Case)

Elevation Plan


36


1 2.323

2 2.323

3 2.428

4 2.428

5 2.571

6 2.571

7 3.302

8 3.302

9 4.015

10 4.015

Eigenvalues (Multiple), Cooling Tower


37

)1(

)()1(

k

i

ki

ki


Relative error =

Error norm = 2

)(2

)()( )(k

i

ki

ki

K

MK


Solution Time (sec), Cooling Tower


Method EN=10-6 EN=10-9 EN=10-6 EN=10-9

Proposed

Method

2,785.8

(1.0)

3,067.7

(1.0)

5,104.2

(1.0)

5,576.5

(1.0)

Subspace

Iteration

4,584.9

(1.6)

6,182.5

(2.0)

6,383.6

(1.3)

15,829.3

(2.8)

Determinant

Search

No Sol. No Sol. No Sol. No Sol.


38

1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 21 23 2 5 27 2 9Itera tio n N u m b er

1 .0 E -1 1

1 .0E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0E + 0E

rror

Nor

m

Convergence of the 10th eigenpairCooling tower (multiple)

: Proposed Method : Subspace Iteration Method (q=2p)

Error Limit


39

NUMERICAL EXAMPLES:Non-Proportionally Damped Case Structures

Cantilever beam(distinct) Grid structure(multiple) Three-dimensional framed structure(close)

Analysis Methods Proposed method Subspace iteration method (Leung 1988) Lanczos method (Chen 1993)

Comparisons Solution time(CPU) Convergence

Convex with 100 MIPS, 200 MFLOPS


40

Cantilever Beam with Lumped Dampers (Distinct Case)

1 2 3 4 99 100 101

C

5

Material PropertiesTangential Damper :c = 0.3

Rayleigh Damping : = = 0.001

Young’s Modulus :1000

Mass Density :1

Cross-section Inertia :1

Cross-section Area :1

System DataNumber of Equations :200

Number of Matrix Elements :696

Maximum Half Bandwidths :4

Mean Half Bandwidths :4


41

Proposed MethodMode

Number

Error Norm ofStarting Eigenpair(Lanczos method)

Number ofIterations

Eigenvalue Error Norm

12345678910

0.234069E-030.234069E-030.192793E-030.192793E-030.148072E-040.148072E-040.619329E-020.619329E-020.377004E+000.377004E+00

1111111111

-2.57457 + j 3.17201-2.57457 - j 3.17201-1.53800 + j 18.3566-1.53800 - j 18.3566-1.69581 + j 39.6477-1.69581 - j 39.6477-2.43492 + j 61.0104-2.43492 - j 61.0104-3.78360 + j 82.3222-3.78360 - j 82.3222

0.232258E-070.232258E-070.428428E-090.428428E-090.727353E-100.727353E-100.204824E-100.204824E-100.997372E-070.997372E-07

Results of Cantilever Beam Structure (Distinct)

Number of Lanczos vectors = 20


42

Method CPU time in seconds Ratio

Proposed method(Lanczos method + Iteration scheme)

Subspace Iteration Method

76.10(10.42 + 65.64)

100.94

1.00

1.33

CPU Time for 10 Lowest Eigenpairs, Cantilever Beam


43

2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 11 0

N u m b er o f G en era ted L a n czos V ecto rs

1 .0 E -8

1 .0E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0E + 0E

rror

Nor

m

E rror L im it

Convergence by Lanczos method(Chen 1993)Cantilever beam (distinct)

Starting values of proposed method

: 1st, 2nd eigenpairs : 3rd, 4th eigenpairs : 5th, 6th eigenpairs : 7th, 8th eigenpairs : 9th, 10th eigenpairs


44

1 2 3 4 5

Itera tio n N u m b er

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it

Convergence of the 1st eigenpairCantilever beam (distinct)



45

1 2 3 4 5 6 7 8 9

Ite ra tio n N u m b er

1 .0 E -11

1 .0E -1 0

1 .0E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rror L im it

Convergence of the 5th eigenpairCantilever beam (distinct)



46

Grid Structure with Lumped Dampers (Multiple Case)

Material Properties

Tangential Damper :c = 0.3

Rayleigh Damping : = = 0.001

Young’s Modulus :1,000

Mass Density :1

Cross-section Inertia :1

Cross-section Area :1

System Data

Number of Equations :590

Number of Matrix Elements :8,115


Mean Half Bandwidths :[email protected]=10

[email protected]=

10


47

Proposed MethodMode

Number


Number ofIterations


123456789101112

0.467621E-070.325701E-050.467621E-070.325701E-050.170965E-060.823644E-060.170965E-060.823644E-060.250670E-040.786392E-000.250670E-040.786392E-00

010100001111

-0.09590 + j 8.66792-0.09590 + j 8.66792-0.09590 - j 8.66792-0.09590 - j 8.66792-0.60556 + j 15.5371-0.60556 +j 15.5371-0.60556 - j 15.5371-0.60556 - j 15.5371-0.57725 + j 20.7299-0.57725 + j 20.7299-0.57725 - j 20.7299-0.57725 - j 20.7299

0.467621E-070.805278E-100.467621E-070.805278E-100.170965E-060.823644E-060.170965E-060.823644E-060.344167E-100.432693E-090.344167E-100.432693E-09

Results of Grid Structure (Multiple)



48




872.67(214.28 + 658.39)

3,096.62

1.00

3.55

CPU Time for 12 Lowest Eigenpairs, Grid Structure


49

24 3 6 4 8 6 0 7 2 84 96 10 8

N u m b er o f G en era ted L a n czos V ec to rs

1 .0 E -10

1 .0E -9

1 .0E -8

1 .0 E -7

1 .0E -6

1 .0 E -5

1 .0 E -4

1 .0E -3

1 .0 E -2

1 .0E -1

1 .0E + 0E

rror

Nor

m

E rro r L im it

Convergence by Lanczos method(Chen 1993)Grid structure (multiple)

: 1st, 3rd eigenpairs : 2nd, 4th eigenpairs : 5th, 7th eigenpairs : 6th, 8th eigenpairs : 9th, 11th eigenpairs : 10th, 12th eigenpairs



50

Convergence of the 2nd eigenpairGrid structure (multiple)


1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 13 1 4 1 5


1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it


51

Convergence of the 9th eigenpairGrid structure (multiple)


1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 3 3 3 5 3 7 3 9 4 1 4 3 4 5 4 7 4 9


1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it


52

Three-Dimensional Framed Structure with Lumped Dampers(Close Case)

[email protected]=6.02

6@3=

18

2@3=6

[email protected]=18.06

12@3=

36


53

Material Properties

Lumped Damper :c = 12,000.0

Rayleigh Damping : =-0.1755 = 0.02005

Young’s Modulus :2.1E+11

Mass Density :7,850

Cross-section Inertia :8.3E-06

Cross-section Area :0.01

System Data

Number of Equations :1,128

Number of Matrix Elements :135,276


Mean Half Bandwidths :120


54

Proposed MethodMode

Number


Number ofIterations


123456789101112

0.579542E-060.579542E-060.567549E-060.567549E-060.929150E-050.929150E-050.767207E-030.767207E-030.910611E-000.910611E-000.920451E-000.920451E-00

00001133

11111111

-0.13763 + j 3.08907-0.13763 - j 3.08907-0.13803 + j 3.09109-0.13803 - j 3.09109-3.52574 + j 2.20649-3.52574 - j 2.20649-0.24236 + j 4.16556-0.24236 – j 4.16556-1.64294 + j 7.02958-1.64294 - j 7.02958-1.65070 + j 7.03590-1.65070 - j 7.03590

0.579542E-060.579542E-060.567549E-060.567549E-060.421992E-090.421992E-090.614880E-060.614880E-060.624657E-060.624657E-060.660196E-060.660196E-06

Results of Three-Dimensional Framed Structure (Close)



55




8,335.20(918.15 + 7,417.05)

9,644.75

1.00

1.16

CPU Time for 12 Lowest Eigenpairs, 3-D. Framed Structure


56

Convergence by Lanczos method(Chen 1993)3-D. framed structure (close)

: 1st, 2nd eigenpairs : 3rd, 4th eigenpairs : 5th, 6th eigenpairs : 7th, 8th eigenpairs

: 9th, 10th eigenpairs : 11th, 12th eigenpairs


2 4 3 6 4 8 6 0 7 2 8 4 96 10 8

N u m b er o f G en era ted L a n czos V ecto rs

1 .0 E -8

1 .0E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0E + 0E

rror

Nor

m

E rror L im it


57

Convergence of the 9th eigenpair3-D. framed structure (close)


1 3 5 7 9 1 1 1 3 1 5 1 7 19 2 1 23 2 5 27 2 9


1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0E -2

1 .0E -1

1 .0 E + 0E

rror

Nor

m

E rror L im it


58

CONCLUSIONS

The proposed method is simple guarantees numerical stability converges fast.

An efficient solution technique !


59

Thank you for your attention.


60

0 2 4 6 8


1 .0 E -11

1 .0E -1 0

1 .0E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it

Convergence of the 3rd eigenpairCantilever beam (distinct)

: Proposed Method : Subspace Iteration Method


61

0 2 4 6 8 1 0 12 1 4


1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it




62

0 2 4 6 8 1 0 1 2 14 16 1 8


1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0E -6

1 .0E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rror L im it




63

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0


1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it

Convergence of the 10th eigenpairGrid structure (multiple)



64

0 2 4 6 8 1 0 12 1 4


1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it




65

0 2 4 6 8 10 12 1 4 1 6


1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0E -2

1 .0E -1

1 .0 E + 0E

rror

Nor

m

E rror L im it




66

0 2 4 6 8 1 0 12 1 4 1 6 1 8 20 2 2 24 2 6 2 8 3 0 3 2


1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0E -2

1 .0E -1

1 .0 E + 0E

rror

Nor

m

E rror L im it



in-won lee, professor, pe in-won lee, professor, pe structural dynamics & vibration control lab....

Documents