The document discusses multi-objective optimization algorithms for finite element model updating using measured modal data. It presents different frameworks for structural identification, including weighted modal residuals and multi-objective formulations. Computational issues related to single-objective and multi-objective optimization are discussed. An example application to identify the parameters of a full-scale bridge model using ambient vibration data is also outlined.

1. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Multi-Objective Optimization Algorithms for Finite Element
Model Updating
E. Ntotsios, C. Papadimitriou
University of Thessaly
Greece

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

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

4. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Structural Identification - Formulation
D = {ωr , f ˆr , r = 1,L , m; k = 1,L , N D }
ˆ = Available Measured Modal Data
Μ = Class of Linear Models, q = structural parameter set to be identified
ωr (θ ) , f r (θ ) , r = 1,L , m = Modal data predicted by Model, solving the Eigenvalue Problem
⎡ K (θ ) − ωr2 M (θ )⎤ f r = 0
⎣ ⎦
Problem: Find q values so that model predicted modal data are close to the measured modal data
Measure of fit (Modal Residuals):
ND
1 [ω r (θ ) − ω r( k ) ]2
ˆ
Modal Frequencies Jωr (θ ) = ∑ [ω (k ) ]2 r = 1, K , m
ND k =1 ˆr
2 N D = Number of available
ND
(k )
β φ (θ ) − φˆ (k )
1 r r r Data sets
Modeshapes Jφr (θ ) = ∑ 2
ND k =1 ˆ
φr( k ) For m modesà
maximum 2m objectives

5. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Grouping of Modal Properties – General Case
Modal Properties ˆ
ω1 ˆ
ω2 ˆ
ω3 L ˆ
ωm fˆ
1 f ˆ2 f ˆ3 L f ˆm
Modal Groups g1 g2 K gn
Modal Residuals J 1 (θ ) J 2 (θ ) K J n (θ )

6. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Grouping of Modal Properties –Special Case
Modal Properties ˆ
ω1 ˆ
ω2 ˆ
ω3 L ˆ
ωm fˆ
1 f ˆ2 f ˆ3 L f ˆm
Modal Groups g1 g2
Modal Residuals J 1 (θ ) J 2 (θ )
m
m
J1 (θ ) = ∑ J ωr (θ ) J 2 (θ ) = ∑ Jφr (θ )
r =1
r =1
Modal Frequencies Modeshapes

7. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Weighted Modal Residuals Framework
Find θ that minimizes the weighted modal residuals:
n
J (θ ; w ) = ∑ wi J i (θ ) n = number of modal groups
i =1
n
∑w =1
i =1
i
ˆ
Optimal Solution θ ( w ) depends on the value of the weights wi
Values of weight factors w affect optimal θˆ ( w ) which in turn affects model predictions
Problem: Find the most probable (optimal) weight values ˆ
w , based on the measured data
and the norms used to measure the fit between measured and model predicted modal
properties

8. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Multi-Objective Framework
Find θ that simultaneously minimizes the objectives
J (θ )= (J 1 (θ ),J 2 (θ ),K ,J n (θ ))
Ø Pareto optimal solutions (Set of alternative solutions).
Pareto solutions
2
2
J
x
Objective Space Pareto front Parameter Space
J x
1 1
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 wi
from 0 to 1, Pareto optimal solutions are alternatively obtained.
Equivalent Problem: Find the most probable Pareto point and optimal solution to be used for
model predictions, based on the measured data

9. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Optimally Weighted Modal Residuals Method
Most preferred model
Find θˆpr = θˆ( w ) that minimizes the weighted modal residuals:
ˆ
n
ˆ ˆ
J (θ ; w ) = ∑ wi J i (θ )
i =1
selecting the most preferred values of weights to be inversely proportional to the optimal
values of the modal residuals
αi
ˆ
wi = , i = 1,K , n
ˆ( w ))
J i (θ ˆ
The most preferred model for the most preferred weights are obtained by simultaneously
solving the above set of equality equations and the optimization problem
Efficient Solution Strategy: Most preferred model minimizes the sum of the logarithms of
the residuals (Christodoulou and Papadimitriou 2007)
n
θˆpr = arg min I (θ ) I (θ ) = ∑ αi ln J i (θ )
θ i =1

10. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Computational Issues – Single Objective Optimization
" 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

11. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Computational Issues – Multi 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

12. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
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
Nelson’s Method
Gradient of eigenvalue and eigenvector of a mode is computed using information from the
eigenvalue and eigenvector of the same mode
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
Hessian of objectives

13. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Polymylos Bridge - Instrumentation
Instrumented with an array of 24
accelerometers optimally placed
on the deck and the base of the
columns and bearings
2
14 1
13
3
15
B2RV 40 M2RV 10 A2RV40
5 16
SRV 100 17
T3RT 100 M2RT 10 T1RT 100 9
B2RT 40 A2RT 40 4
19 18
SRT 100 10
ΠΟΛΥΜΥΛΟΣ 21 20
M2 LL 10 12
U3LL100 22
U1LV 100 02423 6
M2LV 10 11
B2LV 40 A2LV 40 U1LL 100
SLV 100 -10 7 8 50
U3LV 100 Βάση U1RT 100 z-axis
U3RT 100 U2LV 40 -20
Πυλώνα M2
0
Aκρόβαθρο T2 U2LL 40
-30
35m 35m 30m Aκρόβαθρο T1
30m U2LT 40 -50
-750
-755 x-axis
-760
y-axis

14. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Polymylos Bridge – Operational Modal Analysis
Damping
No Identified Modes Hz
Ratios (%)
Modal Identification Software 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
1
Mode 1: 2.131 Hz, zeta=1.260% Mode 2: 2.225 Hz, zeta=3.737%
7
6 6
4
11
5 8 8
13
2
3 3
2
10
9 9
10
12 1 4 12
7
0 14
0 5 14 11
-10 15 50 -10 15
13 50
z-axis
z-axis
-20 -20
0 0
-30 -30
-50 -50
-750
-755 x-axis -750
-755 x-axis
-760 -760
y-axis y-axis
6
Mode 3: 3.074 Hz, zeta=1.035% Mode 4: 4.095 Hz, zeta=1.435%
1
7 6
1
4
5
13 11 8 7
3 2
2 3
9 12 8
10 4 10
9
11
12
5
0 0 13
14 14
-10 15 50 -10 15 50
z-axis
z-axis
-20 -20
0 0
-30 -30
-50 -50
-750
-755 x-axis -750
-755 x-axis
-760 -760
y-axis y-axis

15. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Polymylos Bridge – Finite Element Model
Finite Element Model Updating using Multi-Objective Identification
Finite element model of 228
beam elements (1038 DOF) 3 parameter FE model
J2
θ12
1
z θ1 θ2
x θ1
y θ3
Model Updating Software

16. University of Thessaly
Department of Mechanical and Industrial Engineering
System Dynamics Laboratory
Polymylos Bridge - Model Updating Results
3 parameters Multi-Objective model updating using 3 modes
ND
1 [ωr (q ) − ωr k ) ]2
ˆ(
J1 (q ) = ∑
ND [ωr k ) ]2
ˆ(
Objective and parameters space
k =1
2
ND ˆ
β r( k )φr (q ) − φr( k )
1
0.072 1.8 J 2 (q ) = ∑ 2
Pareto Solutions 20
19 ND k =1 ˆ
φr( k )
0.07 w=1 18
1 1.6
2 17
0.068 3 16 Equally weighted method parameter values
4 Pareto Solutions
1.4
θ1 (E bearings) 3.6872
2
5
2
15
θ
J
6 w=1
0.066
7
8 14 θ2 (E deck) 1.1293
9 1.2 13
0.064
10
11 12 12 θ3 (E pier) 0.6425
13 14 10 11
15 16 17 18 19 20 7 8
9 J1(mode frequencies) 0.0347
0.062 1 123 4 5 6
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 3.3 3.4 3.5 3.6 3.7 3.8 J2 (mode shapes) 0.0635
J θ
1 1 3.8
3.6
3.4
1 1
3.2
θ
3 1
Pareto Solutions
12 3 4 5 25
14
3 2.8 θ
2
0.8 6 7 0.8 67 w=1 2.6
8 9 89 θ
10 10 2.4 3
11 11 2.2 θw =1
12 12
θ value
0.6 13 0.6 13 2
Pareto Solutions 1.8
14 14
3
3
θ
θ
w=1 15 15 1.6
0.4 16 0.4 16 1.4
17 17 1.2
18 18 1
19 19
0.2 20 0.2 20 0.8
0.6
0.4
0 0 0.2
3.3 3.4 3.5 3.6 3.7 3.8 1 1.2 1.4 1.6 1.8
5 10 15 20
θ θ # of solutions
1 2

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