Keywords: multilevel optimization, metamodel, cokriging, radial basis functions, multiparametric strategy, LATIN method, assemblies.

Optimization processes for assembly design are often relatively time consuming. In order to locate the global optimum of the objective function, the use of dedicated optimisers is inevitable but requires a large number of calculations. As a result of the nonlinearities related to friction or contact phenomena each evaluation is very time consuming. In this context the main purpose of this paper is to reduce the computation time. That is the reason why a two-level optimization process [1] is proposed. It is based on two tools: the first is a gradient-based metamodel and the second is a dedicated strategy to solve nonlinear problems. In this paper, the proposed study focuses on the computation cost to build a metamodel coupled with the mechanical solver. So as to reduce the computational cost for solving assembly problems, a multiparametric strategy [2] is presented. It relies on a feature of the LATIN method developed by Ladevèze [3] that when it is used with a reinitialisation process, allows the computation time to be significantly reduced. The performance of this method enables one to evaluate the gradients of the objective function very inexpensively. It is proposed to integrate gradients to build richer approximation of the objective function. Thus gradient-based metamodels are introduced and compared with classical non-gradient-based approximations. The formulations of gradient- [4] and non-gradient-based [5,6] kriging and radial basis functions are briefly presented and their qualities will be study on one- and two-dimensional analytical examples. The use of gradient information enables one to obtain a more accurate approximation for a same number of sample points. Finally metamodels are coupled with the multiparametric strategy in three- and four-dimensional examples. The strategy leads to a significant reduction of the computational time for building a gradient-based approximation with a similar quality than the non-gradient-based metamodel. In the context of the optimisation process these results will enable one to obtain an accurate optimum while reducing the computational time for the whole optimisation process.

1

G.M. Robinson, A.J. Keane, "A case for multi-level optimisation in aeronautical design", Aeronautical Journal, 103(1028), 481-485, 1999.

2

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(9), 1259-1281, 2003. doi:10.1002/nme.724

3

P. Ladevèze, "Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation", Springer Verlag, 1999.

4

S. Leary, A. Bhaskar, A. Keane, "Global approximation and optimization using adjoint computational fluid dynamics codes", AIAA Journal, 42(3), 631-641, 2004. doi:10.2514/1.9114

5

N.A.C. Cressie, "Statistics for spatial data", John Wiley & Sons, New York, 1993.

6

R. Hardy, "Multiquadric Equations of Topography and Other Irregular Surfaces", J. Geophys. Res., 76, 1905-1915, 1971. doi:10.1029/JB076i008p01905

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