Keywords: multiparametric strategy, proper generalized decomposition, model reduction.

Optimization studies consist of seeking the set of parameters which minimizes a cost function over a design space. The algorithms which are used to seek this minimum lead to multiple resolutions of the same problem with different sets of parameters. The objective of this work is to minimize the computational effort associated with the resolution of these similar problems. One idea consists of using model reduction techniques, also known as reduced-order modeling (ROM).

A first option consists in using what is called an a posteriori reduction method, often based on the proper orthogonal decomposition (POD) of an initial solution obtained through a first calculation (associated with the first set of parameters).

Then, the basis obtained from that decomposition is reused in order to project the equations of the problem for the subsequent sets of parameters. Another option, which is the subject of our study, consists of using an a priori model reduction technique. One uses proper generalized decomposition (PGD), which leads to very significant gains in terms of computation cost [2]. PGD, associated with the LATIN method, leads to an approximation of the solution with separated variables [3]. The LATIN method is an iterative resolution technique which consists of generating at each iteration a solution defined over the whole space and time domains, to which corrections are made at each iteration. These corrections are represented by a PGD approximation.

The multiparametric strategy proposed herein consists in reusing the initial basis issued from the first calculation (associated with the first set of parameters) and in enriching it, if necessary, by using the LATIN method to solve the problems associated with the other sets of parameters. Thus, one can expect to reduce computation times by using a "closer" solution obtained in a previous calculation. The efficiency of this technique has already been shown [1].

This paper highlights that in practical situations is required to enrich the basis constructed in a first calculation in order to better represent the solution of the problem associated with another parameter set.

Using the strategy presented, we compared the relevance of the POD and PGD bases as initial bases. Even though the coupling of POD and this strategy leads to a certain gain, the results observed seem to favour the choice of the PGD.

1

P-A. Boucard, L. Champaney, "A suitable computational strategy for the parametric analysis of problems with multiple contact", International Journal for Numerical Methods in Engineering, 57, 1259-1282, 2003. doi:10.1002/nme.724

2

F. Chinesta, A. Ammar, E. Cueto, "Recent advances and new challenges in the use of the Proper Generalized Decomposition for solving multidimensional models", Archive of Computational Methods in Engineering, available online, 2010. doi:10.1007/s11831-010-9049-y

3

P. Ladevèze, "Nonlinear Computational Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation", Springer Verlag, 1999.

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