Parametric solutions involving geometry: A step towards efficient shape optimization

Optimization of manufacturing processes or structures involves the optimal choice of many parameters (process parameters, material parameters or geometrical parameters). Usual strategies proceed by defining a trial choice of those parameters and then solving the resulting model. Then, an appropriate cost function is evaluated and its optimality checked. While the optimum is not reached, the process parameters should be updated by using an appropriate optimization procedure, and then the model must be solved again for the updated process parameters. Thus, a direct numerical solution is needed for each choice of the process parameters, with the subsequent impact on the computing time. In this work we focus on shape optimization that involves the appropriate choice of some parameters defining the problem geometry. The main objective of this work is to describe an original approach for computing an off-line parametric solution. That is, a solution able to include information for different parameter values and also allowing to compute readily the sensitivities. The curse of dimensionality is circumvented by invoking the Proper Generalized Decomposition (PGD) introduced in former works, which is applied here to compute geometrically parametrized solutions