Topological derivatives for shell structure optimization considering crash loadcases with material nonlinearities and large deformations

Abstract

AbstractThe topological derivatives for shell structures in dynamic loadcases, which are often used in lightweight designs, are developed. These sensitivities consider the highly nonlinear material behavior and large deformations of the structure. The aim of the investigation is the support of an efficient topology optimization for crashloaded structures. The considered functionals can in general be described by displacements, velocities and accelerations. In particular, the semi-analytical sensitivity calculations for the internal deformation energy and for single point displacements in arbitrary directions in the shell structure are presented. These functionals cover the most important functions for the development of crash structures. Using the material derivative in combination with the adjoint method, a terminal value problem has to be solved. Depending on whether first differentiation and then discretization in the time domain is performed or first discretization and then differentiation, a separate solution scheme for the adjoint is derived. Both schemes are presented in a comparable description. For the numerical evaluation of the topological derivatives, the basic equations for large deformed shell elements are included as well as a phenomenological material interpolation, which represents the tangential material behavior resulting from plastic strain and isotropic hardening. All assumptions made for the derivation are described in detail and their influences are discussed. The elaborated equations and the procedure for the calculation of the topological derivatives are applied, discussed and checked for plausibility on two clearly defined loading conditions for the internal deformation energy and single point displacements on different locations.