Theoretical and Numerical Study of Nonsmooth Shape Optimization Applied to the Arch Problem

Abstract

Abstract The subject of this work is the nonsmooth optimization of the midsur-face of a particular thin shell, the arch. Within the framework of the continuous model, nondifferentiable cost is chosen as the maximum of the square norm of displacements. In the theoretical portion of this paper, the existence of subgradients for this cost is demonstrated. It is then shown how the subgradients can be computed using an adequate adjoint state method. In the numerical portion of the paper, the implementation steps of the subgradient subroutine are detailed. Finally, results of numerical optimization for a case of uniform load and a case of non-uniform load are presented. With the aim of obtaining the shape with the best mechanical behavior, a comparison is made between non-differentiable cost optima and optima obtained for differentiable costs (such as the L2-norm of displacements). When the load is uniform, these optima are almost the same, but when the load is non-uniform, the optimum for the non-differentiabl...