A space mapping approach for the p-Laplace equation

Abstract

A main aspect in the design of passenger cars with respect to pedestrian safety is the energy absorption capability of the engine hood. Besides that, the hood has to fulfill several other requirements. The associated optimal problem is hard to solve numerically. That makes it necessary to develop easy and fast to solve prediction models with little loss in accuracy for optimization purpose. Current simulation tools combined with standard optimization software are not well suited to deal with the above mentioned needs. We will show the application of mathematical methods on a simplified model to reduce the optimization effort. Our goal is to minimize the mass of the plate. Due to the nonlinear structure of the elliptic partial differential equation, the numerical solution of the optimization problem governed by the PDE constraint is expensive, we consider a linear elliptic PDE as an alternative constraint. The space mapping technique is utilized to link the linear and nonlinear equations and drives the optimization iteration of the time intensive nonlinear equation using the fast linear equation. For this reason an efficient realization of the space mapping is utilized. Numerical examples are presented to illustrate the advantage of the proposed approach.