Keywords: Bayesian designs, inverse problem, Monte Carlo, metamodel.

The general framework of this work is to develop decision making tools for the robust design of mechanical structures in an uncertain environment. The behavior of real mechanical systems can be investigated in detail via model-based numerical simulations. However, the credibility of these results must first be established and eventually improved using experimental measurements from a real system in order to validate the model for a specific application. In structural dynamics, we are interested in predicting the dynamical properties of the structure where the measured data are quantities such as modal data or FRFs. If the model predictions are significantly different from those observed on the real structure, we must try to understand what physics in the model are poorly represented and then try to correct them. So, we are faced here with an inverse problem where, given observed data, we search for a better estimation of model parameter values. Since we need experimental data and given the cost of experimentation, it is important to evaluate which tests are the most useful, that is to say, the most informative. Moreover, data obtained after measurement contains uncertainties due to, for example, experimental conditions, the precision of the measuring instruments or the experimenter himself. Therefore, we must deal with two difficulties, namely, design informative measurements and take into account the experimental uncertainties.

Bayesian experimental design (BED), originally proposed by Lindley [1], offers a framework to solve our problem, since the purpose of BED is to maximize the information gain of an experiment where the outcomes are described by a probability density function (PDF). A utility function describing the goal of the experiment, in fact information gain, is selected first, and we look for the experiment maximizing this utility function. A prior knowledge on the parameters is combined with the PDF of the output of the test via the Bayes theorem to obtain a a posteriori PDF of the parameters. Classically speaking, the utility function uses an information measure such the Kullback-Leibler distance.

Hence, a probabilistic formulation on the data and model parameters is needed. In this paper, we use the general theory of inverse problems developed by Tarantola [2]. In this formulation, inverse theory is not strictly based on the Bayes theorem but on the notion of conjunction of states of information. A priori information concerning the model parameters, experimental information but also theoretical information are combined to produce a posteriori PDF on the data and model parameters. A Markov Chain Monte Carlo method, a Metropolis-Hastings algorithm [3], is used to sample an a posteriori distribution. Convergence issues are aborded with various convergence monitoring tools.

The drawback with Monte Carlo methods is the number of computations. To obtain convergence, many iterations are required with one or more modal analysis of the finite element model (FEM) for each iteration. When the FEM is time intensive, the proposed methodology is simply intractable. It is thus worthwhile to replace the exact FEM analysis by a metamodel approach in order to reduce computing times. In classical response surface methodologies, polynomial function are use to approximate responses of a model. Here we prefer to use neural networks which give very precise approximations.

We propose in this paper to use a reference FEM based on the designer's current knowledge to simulate data, then noise is added on these data, a posteriori PDFs associated with the parameters of interest are estimated using inverse theory, and finally the utility functions are computed. The process is repeated for various configurations or boundary conditions of the FEM to find experiments maximizing the information gain.

The proposed methodology is illustrated using an academic example.

1

D. V. Lindley. On a measure of information provided by an experiment. Annales of Mathematical Statistic, 27:986-1005, 1956. doi:10.1214/aoms/1177728069

2

A. Tarantola. Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia, 2005.

3

K. Mosegaard and A. Tarantola. Monte carlo sampling of solutions to inverse problems. Journal of Geophysical Research, 100(B7):12431-12447, 1995. doi:10.1029/94JB03097

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