Keywords: model reduction, proper generalized decomposition, structural optimization.

In classical structural optimization methods, the primary problem, consists in the evaluation and minimization of a cost function. The structural problem is only secondary: it is considered only for the purpose of evaluating the cost function or as additional constraints. The objective of this work is to change this paradigm by bringing the structural problem out of the optimization loop. The structural problem has to be considered as the primary problem while the minimization of the cost function is the secondary one. In this way, the design parameters are no longer considered as parameters but as additional variables of an augmented structural problem; the resolution scheme consists in: (i) solving the augmented structural problem for every combination of design parameters, (ii) defining a cost function, (iii) minimizing it.

In this context, we investigate the potential efficiency of the so-called proper generalized decomposition (PGD) method [1] in the definition and resolution of structural optimization problems. In particular we use the ability of this method to solve high dimensionality problems to define and solve a thermo-mechanical problem where the geometrical optimization parameters are no longer treated as parameters of the problem but rather as additional variables of a generalized problem.

The particular problem considered here consists in the optimization of the layer thicknesses in a multilayer body with respect to arbitrary cost functions. For the sake of simplicity we consider a one dimensional steady-state heat transfer problem subjected to given temperatures on the boundaries. To apply the PGD method, layers thicknesses are transformed into variables of the problem. Then, using a specific change of variable, the separated form of the problem is established in terms of a geometrical scale parameter and of the layers thicknesses. The discretization of the problem is made via linear finite elements for all variables, and the solution for any combination of parameters is easily obtained (step (i)). The definition (step (ii)) and the minimization (step (iii)) of the cost function can be investigated a posterior.

Numerical results demonstrate the efficiency of the approach: only 937 differential problems (as compared to 15625 differential problems with classical methods) need to be solved to obtain a sufficient representation of the structural solution with respect to design variables. Furthermore, different cost functions as well as advanced optimization algorithms can be considered since the cost function and its gradient can be evaluated at a very low computational cost.

1

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

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