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Topological Sensitivity Analysis for a Nonlinear Case : the p-Poisson Problem

1. Abstract The topological derivative allow us to quantify the sensitivity of the problem when the domain under consideration is perturbed by the introduction of a hole. Extension of the topological derivative in order to include arbitrary shaped holes or inclusion and its applications to Laplace, Poisson, Helmoltz, Navier, Stokes and Navier-Stokes equations can be found in the literature. However, only linear problems or when the nonlinear term is a compact perturbation of the principal part of the operator have been considered. In the present paper, we propose a numerical procedure to estimate the topological derivative for a nonlinear case: the p-Poisson problem, considering the complementary dissipation energy as the cost function and the state equation in its weak form as the constraint. This problem is interesting because we are dealing with a case in which the nonlinear term involves the principal part of the operator. Finally, the obtained topological derivative is used to find the best place to insert holes in bars of different cross-sections submitted to torsion effects for several values of p. 2.