Multi-Objective Optimization Algorithms for Finite Element Model Updating. Ntotsios and...

Multi-Objective Optimization Algorithms for Finite Element Model Updating

University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory

E. Ntotsios, C. Papadimitriou University of Thessaly

Greece


Outline

Ø  STRUCTURAL IDENTIFICATION USING MEASURED MODAL DATA

§  Weighted Modal Residuals Framework

§  Multi-Objective Framework

§  Optimally Weighted Modal Residuals Method

Ø  COMPUTATIONAL ISSUES

§  Single-Objective Optimization

§  Multi-Objective Optimization

§  Gradient and Hessian of Objectives

Ø  ILUSTATIVE EXAMPLE

•  Structural Identification of a Full Scale Bridge Using Ambient

Vibration Measurements

Ø  CONCLUSIONS


Model Updating Issues

Ø  MODELLING ERROR

§  Assumptions used to describe a physical system by a model

§  Numerical errors (e.g. discretization of partial differential equations

of motion)

Ø  MEASUREMENT AND PROCESSING ERROR

•  Measurement of response time histories

•  Modal estimation from response time histories


Problem: Find values so that model predicted modal data are close to the measured modal data

Structural Identification - Formulation

2( ) ( )r rK Mω⎡ ⎤− =⎣ ⎦f 0θ θ

= Available Measured Modal Data ˆˆ{ , , 1, , ; 1, , }fr r DD r m k Nω= = =L L

= Modal data predicted by Model, solving the Eigenvalue Problem ( ) , ( ) , 1, ,r r r mω = Lfθ θ

= Class of Linear Models, Μ q = structural parameter set to be identified

Measure of fit (Modal Residuals):

Modal Frequencies ( ) 2

( ) 21

ˆ[ ( ) ]1( )ˆ[ ]

D

r

N kr r

kkD r

JNω

ω ωω=

−= ∑

θθ

Modeshapes

2( ) ( )

2( )1

ˆ( )1( )ˆ

D

r

k kNr r r

kkD r

JN

β

=

−= ∑φ

φ θ φθ

φ

= Number of available

Data sets DN

1, ,r m= K

For m modesà

maximum 2m objectives

q


Grouping of Modal Properties – General Case

1 2 3 1 2 3ˆ ˆ ˆ ˆˆ ˆ ˆ ˆm mω ω ω ωL Lf f f f

Kg g g1 2 n

( ) ( ) ( )Kθ θ θJ J J1 2 n

Modal Properties

Modal Groups

Modal Residuals


Grouping of Modal Properties –Special Case

1 2 3 1 2 3ˆ ˆ ˆ ˆˆ ˆ ˆ ˆm mω ω ω ωL Lf f f f

g g1 2

( ) ( )θ θJ J1 2

Modal Properties

Modal Groups

Modal Residuals

Modal Frequencies

11

( ) ( )r

m

rJ Jω

=

=∑θ θ

Modeshapes

21

( ) ( )r

m

rJ J

=

=∑ φθ θ


Weighted Modal Residuals Framework

Find that minimizes the weighted modal residuals:

1( ; ) ( )

n

i iiw J

== ∑J wθ θ n = number of modal groups

Optimal Solution depends on the value of the weights ˆ( )wθ

Values of weight factors affect optimal which in turn affects model predictions w

Problem: Find the most probable (optimal) weight values , based on the measured data

and the norms used to measure the fit between measured and model predicted modal

properties

w

θ

ˆ( )wθ

iw

11

n

iiw

=

=∑


Ø  Pareto optimal solutions (Set of alternative solutions).

Ø  All Pareto solutions are acceptable: The characteristics of the Pareto solutions are that the modal residuals cannot

be improved in any modal property without deteriorating the modal residuals in at least one other modal property.

Relation to “Weight Modal Residuals Framework” : Varying the values of the weights

from 0 to 1, Pareto optimal solutions are alternatively obtained.

Multi-Objective Framework

( ) ( ) ( )( )( ) , , ,KJ J J1 2 nJ =θ θ θ θ

Find that simultaneously minimizes the objectives

Objective Space Parameter Space

J1

J 2

x1

x 2

Pareto front

Pareto solutions

Equivalent Problem: Find the most probable Pareto point and optimal solution to be used for

model predictions, based on the measured data

θ

iw


Optimally Weighted Modal Residuals Method

Find that minimizes the weighted modal residuals:

1ˆ ˆ( ; ) ( )

n

i iiw J

== ∑θ θJ w

selecting the most preferred values of weights to be inversely proportional to the optimal

values of the modal residuals

ˆ ˆ ˆ( )prθ = θ w

ˆ , 1, ,ˆ ˆ( ( ))i

ii

w i nJα

= = Kθ w

The most preferred model for the most preferred weights are obtained by simultaneously

solving the above set of equality equations and the optimization problem

1( ) ln ( )

n

i ii

I Jα=

= ∑θ θ

Efficient Solution Strategy: Most preferred model minimizes the sum of the logarithms of

the residuals (Christodoulou and Papadimitriou 2007)

ˆ arg ( )pr I=θ

θ θmin

Most preferred model


" Gradient Based Methods §  Local methods - Cannot guarantee the estimation of global optimum §  Require user-defined initial estimates §  Fast convergence – exploit gradient information

" Evolution Strategies (ES) §  Global Methods §  Do not require user-defined initial estimates §  Very slow convergence in the neighborhood of the global optimum

" Hybrid Algorithms (Combine ES and Gradient methods) §  Exploit the advantages of ES and Gradient methods §  ES explore the parameter space and detect the neighborhood of the global optimum §  Gradient methods start from the best estimate of ES and use gradient information to

accelerate convergence to the global optimum

Computational Issues – Single Objective Optimization


" Strength Pareto Evolutionary Algorithms (SPEA) – [Zitzler and Thiele 1999] §  Random initialized population of search points in the parameter space which by

means of selection, mutation and recombination evolves towards better and better regions in the search space

§  Clustering techniques are used to uniformly distribute points along the Pareto front, provided that the values of objectives are of the same order of magnitude along the Pareto front

§  Require user-defined initial estimates §  Slow convergence in the neighborhood of the Pareto front

" Normal Boundary Intersection Method (NBI) – [Das and Dennis 1998] §  Deterministic algorithms based on gradient methods §  Produces an evenly spread of points along the Pareto front §  Fast convergence §  Computationally expensive for more than 3 objectives

Computational Issues – Multi Objective Optimization


Gradient of eigenvalue and eigenvector of a mode is computed using information from the

eigenvalue and eigenvector of the same mode

Nelson’s Method

Hessian of objectives

Computational Issues – Gradients of Objectives In order to guarantee the convergence of the gradient-based optimization methods, the

gradients of the objective functions with respect to the parameter set θ has to be estimated

accurately

Adjoint Formulation

The computational cost is independent of number of parameters.

For each mode a solution of a linear system of algebraic equations is required


35m 30m

U3RT 100

U3LL100

U3LV 100

B2RT 40

B2LV 40

B2RV 40 A2RV40

A2LV 40

A2RT 40

SRV 100

SLV 100

SRT 100

T1RT 100

U1RT 100

U1LV 100 U1LL 100

M2RV 10

M2 LL 10

M2RT 10

M2LV 10

U2LT 40

U2LL 40

U2LV 40

T3RT 100

35m

ΠΟΛΥΜΥΛΟΣ

Aκρόβαθρο T2 Aκρόβαθρο T1

Βάση Πυλώνα M2

30m -50

0

50

-760-755-750

-30-20-100

1

2

13 14

15 3

16

8

4 17

6

5

18

7

x-axis

19 9

20 21

22 11

10

12

23 24

y-axis

z-axis

Polymylos Bridge - Instrumentation

Instrumented with an array of 24 accelerometers optimally placed on the deck and the base of the

columns and bearings


No Identified Modes Hz Damping Ratios (%)

1 1st longitudinal 1.19 5.56 2 2nd transverse 1.12 1.97 3 1st bending (deck) 2.13 0.60 4 2nd bending (deck) 3.07 0.43 5 4th transverse 4.07 0.76 6 3rd bending (deck) 6.65 0.45

Polymylos Bridge – Operational Modal Analysis

Modal Identification Software

-50

0

50

-760-755-750

-30-20-10

0

6

8

7 1

9 10

Mode 1: 2.131 Hz, zeta=1.260%

3 2

x-axis

12

11 4

14

13 5

15

y-axisz-

axis

-50

0

50

-760-755-750

-30-20-10

0

6

8

7

1

10 9

Mode 2: 2.225 Hz, zeta=3.737%

3 2

x-axis

11

12 4

13 14 5

15

y-axis

z-ax

is

-50

0

50

-760-755-750

-30-20-10

0

6 7

8

1

10 9

Mode 3: 3.074 Hz, zeta=1.035%

2 3

x-axis

11

12

4

13

14

5

15

y-axis

z-ax

is

-50

0

50

-760-755-750

-30-20-10

0

8

6

7

1

9 10

2

Mode 4: 4.095 Hz, zeta=1.435%

3

x-axis

11 4 12

13 14 5

15

y-axis

z-ax

is


Polymylos Bridge – Finite Element Model

Finite Element Model Updating using Multi-Objective Identification

z x

y

Finite element model of 228 beam elements (1038 DOF)

1θ2θ2θ1J 2J

Model Updating Software

3 parameter FE model

θ1 θ1 θ2

θ3


Polymylos Bridge - Model Updating Results

3 parameters Multi-Objective model updating using 3 modes

5 10 15 20

0.20.40.60.8

11.21.41.61.8

22.22.42.62.8

33.23.43.63.8

# of solutions

θ va

lue

θ1θ2

θ3θw =1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140.062

0.064

0.066

0.068

0.07

0.072

J1

J 2

1 2 3 4 5 6 7

8 9

1011 12 13 14 15 16 17 18 19 20

Pareto Solutionsw=1

3.3 3.4 3.5 3.6 3.7 3.81

1.2

1.4

1.6

1.8

θ1

θ 2

1 2 3 4 5 6 7 8 9 10 1112

1314

15

16

17

181920

Pareto Solutionsw=1

3.3 3.4 3.5 3.6 3.7 3.80

0.2

0.4

0.6

0.8

1

θ1

θ 3

1 2 3 4 5 6 7 8 9 10 1112

1314

1516

1718

1920

Pareto Solutionsw=1

1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

θ2

θ 3

1 2 3 4 5 6 7 8 9101112

1314

1516

1718

1920

Pareto Solutionsw=1

θ1 (E bearings) 3.6872 θ2 (E deck) 1.1293 θ3 (E pier) 0.6425

J1(mode frequencies) 0.0347 J2 (mode shapes) 0.0635

Objective and parameters space

Equally weighted method parameter values

( ) 2

1 ( ) 21

ˆ[ ( ) ]1( )ˆ[ ]

DN kr r

kkD r

JN

ω ωω=

−= ∑

qq

2( ) ( )

2 2( )1

ˆ( )1( )ˆ

Dk kNr r r

kkD r

JN

β

=

−= ∑

qq

φ φ

φ


Conclusions Ø  Model updating algorithms were proposed to compute all Pareto optimal models

consistent with measured data and the norms used to measure the fit between the

measured and model predicted modal properties.

Ø  The equivalence between the multi-objective identification and the weighted modal

residuals method was established.

Ø  Hybrid algorithms based on evolution strategies and gradient methods are well-suited

optimization tools for solving the resulting optimization problem and identifying the global

optimum from multiple local ones.

Ø  NBI algorithms are well-suited multi-objective optimization tools for solving the multi-

objective identification problem. NBI effectively computes the useful identifiable part of

the Pareto front.

Ø  The computational cost for estimating analytically the gradients of the objectives is shown

to be independent of the number of structural model parameters.

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