Comparison of Several Error Metrics for FE Model Updating

Comparison of Several Error Metrics for FE Model Updating

Anders T. Johansson a Thomas J.S. Abrahamsson a Fred van Keulen b a

b

Department of Applied Mechanics, Chalmers University of Technology, ¨ SE-412 96 Goteborg, Sweden

Structural Optimization and Computations Mechanics Group, Delft University of Technology, P.O. box, NL-2628 CD Delft, The Netherlands

ABSTRACT The choice of error metric is one of the most crucial choices of model updating. Many articles have been published advocating the use of certain such metrics, as have comparative studies of various methods based on different error metrics. This article describes the differences between a number of error metrics and their corresponding objective functions using many different evaluation methods, such as plots of objective functions, Cramer-Rao lower bounds and condition numbers of the Hessian. This approach has the advantage that it gives the user a way of assessing error metrics in terms of solvability of the optimization problem and identifiability of parameters of the model updating problem.

1

Introduction

Model updating has been subjected to much research in recent years. Consequently, quite a number of updating formulations and algorithms can be found in literature. For algorithms, the nominal iterative first-order Taylor approximation with subsequent least squares is often used, but a number of other solution methods have been proposed, such as Neural Networks (see [14] and [17]) and multi-objective optimization (see [11] and [8]). The most important difference in updating is, however, the difference in metric used in the formulation of the discrepancy between measured and analytical models - the error metric. For this reason, the error metric is often of prime concern in new model updating algorithms (see [1]), and the properties of the error metric in question are often elaborated upon. However, the evaluation of the error metric often tends to be biased towards the interest of the author in question. To the authors’ knowledge, there is a lack of consistent quantitative assessment of error metrics used within model updating. In such an assessment, a number of different error metrics were compared from the three basic viewpoints of Optimization, Identifiability and Statistics.

2 Studied model

The evaluation was carried out using an eight degree of freedom mass-spring system with fourteen springs known as the Kabe model, which was introduced by Kabe [10] and has been used as a demonstrative example of model updating by, for instance, Bolsman [2], Ebeling [4] and Johansson [9]. This system is inherently ill-conditioned in that the stiffness of the springs differ by as much as a factor of 104 . Also, the system has closely spaced eigenfrequencies, as can be seen in table 1. The model is parameterized so that the first 14 parameters correspond to stiffness parameters, θi,(k) = ki /ki∗ , and the following eight correspond to mass parameters, θi,(m) = mi /m∗i , in accordance with figure 1. Here, ki∗ and m∗i denotes the stiffness and mass parameter values for the nominal model.

4.88

5.05

5.06

5.15

5.44

5.66

6.17

6.67

Table 1 Eigenvalues of the Kabe-model in Hz.

k13 k1 m1

k3 m2

k5

k7

m3

k9

m5

m4

m6

m7

k11

k2

k4

k

k10

k6

8

m8

k14

k12

Fig. 1. The Kabe model eight degrees of freedom spring and mass system. k1∗ = 1.5, k2∗ = 1000, k3∗ = 10, k4∗ = 1000, ∗ ∗ ∗ ∗ ∗ k5∗ = 100, k6∗ = 900, k7∗ = 100, k8∗ = 900, k9∗ = 100, k10 = 1000, k11 = 10, k12 = 1000, k13 = 2, k14 = 1.5, m∗1 = 0.001, ∗ ∗ ∗ ∗ ∗ ∗ ∗ m2 = 1, m3 = 1, m4 = 1, m5 = 1, m6 = 1, m7 = 1, m1 = 0.002. 3

Error Metrics

Eight error metrics have been chosen for comparison. They are: 1) Eigenvalue Errors (i.e. [5]), calculated as the difference in eigenvalues for the measured and the analytical model, so that the vectorized error metric becomes:  ε=

