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
Jan 20, 2015
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
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
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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
=
=∑
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
" 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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
" 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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
University of Thessaly Department of Mechanical and Industrial Engineering System Dynamics Laboratory
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
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
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
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
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=
−= ∑
2( ) ( )
2 2( )1
ˆ( )1( )ˆ
Dk kNr r r
kkD r
JN
β
=
−= ∑
φ φ
φ
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.