Keywords: validation, constitutive relation error, stochastic model, polynomial chaos.

The importance of numerical simulations in industrial applications is constantly growing, one of challenges is model validation based on experimental data. Model updating methods are used in order to minimize the difference between tests and calculations by acting on the numerical model. A state-of-the-art review of these methods can be found in [1]. One of indirect or parametric methods is based on the "mechanics concept" of the constitutive relation error estimator introduced for quantifying the quality of finite element analyses. Equations and quantities of the problem are divided into a reliable group and another which is less reliable. A model error is constructed assuming that constitutive relations from the numerical model can be inaccurate and it can be calculated locally allowing badly modeled regions to be detected. Measured data (displacement, force, etc) are considered to be the less reliable quantities which lead to the construction of an error on the measurements added to model error. Its capability for model updating has been studied in [2] for the case where a deterministic reference is used (i.e. one set of measurements). Further developments are given in [3].

An extension of this approach to the stochastic case has been given and particularly applied for solving simple structural dynamics problems [4]. The case of uncertain measurements obtained from a family of quasi-identical structures needs to model parameters randomly. The main difficulty lies in the validation of the stochastic model since the problem consists in deriving an error measure which is zero if the model is "exact", i.e. if the results obtained from the model match the experimental data. Since the statistical description of the experimental data is often very limited, we propose to reconstruct these experimental data using the statistical mean value and the model. This reconstruction is based on a qualitative analysis of the error in the response in terms of two sources of errors which are the fluctuations of the parameters and the possible variations due to the approximate nature of the stochastic model itself.

In this paper, we go further considering the case of complex engineering structures for which there are many of stochastic variables. Polynomial chaos expansion [5] and reduced basis are used here to solve the stochastic problems involved in the error computation. An efficient reduced basis consists in a truncated modal basis [6] enriched with static responses associated to forces located at sensors, and to variable parameters (stiffness, mass, damping). The capabilities of this approach is illustrated through an industrial example.

A first part of this paper is dedicated to the theoretical aspects of model validation based on the extension of the error (model error and error measure) to the stochastic case. The following part deals with the discretization of the stochastic error, the use of a deterministic reduced basis adapted to such a model updating method and aimed to compute industrial problems. Moreover, the minimization of the stochastic error is described and proceeds by using the chaos polynomial expansion. Finally, we present the stochastic model updating method for the case of an industrial structure.

1

J. Mottershead, M. Friswell, "Model updating in structural dynamics: a survey", J. Sound Vib., 167 (2), 347-375, 1993. doi:10.1006/jsvi.1993.1340

2

P. Ladevèze, D. Nedjar, M. Reynier, "Updating of finite element models using vibrations tests", AIAA Journal, 32 (7), 1485-1491, 1994. doi:10.2514/3.12219

3

P. Ladevèze, A. Chouaki, "Application of a posteriori error estimation for structural model updating", Inverse Prob, 15, 49-58, 1999. doi:10.1088/0266-5611/15/1/009

4

P. Ladevèze, G. Puel, A. Deraemaeker, T. Romeuf, "Validation of structural dynamics models containing uncertainties", Comput. Methods Appl. Mech. Engrg, 195, 373-393, 2006. doi:10.1016/j.cma.2004.10.011

5

R. Ghanem, P. Spanos, "Stochastic Finite Elements: A Spectral Approach", Springer, Berlin, 1991.

6

A. Deraemaeker, P. Ladevèze, Ph. Leconte, "Reduced based for model updating in structural dynamics based on constitutive relation error", Comput. Methods Appl. Mech. Engrg, 191, 2427-2444, 2002. doi:10.1016/S0045-7825(01)00421-2

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