Topology and shape optimization for elastoplastic structural response

Abstract

It is a common practice to base both, material topology optimization as well as a subsequent shape optimization on linear elastic response. However, in order to obtain a realistic design, it might be essential to base the optimization on a more realistic physical behavior, i.e. to consider geometrically or/and materially nonlinear effects. In the present paper, an elastoplastic von Mises material model with linear isotropic hardening/softening for small strains is used. The objective of the design problem is to maximize the structural ductility in the elastoplastic range while the mass in the design space is prescribed. With respect to the specific features of either topology or shape optimization, for example the number of optimization variables or their local–global influence on the structural response, different methods are applied. For topology optimization problems, the gradient of the ductility is determined by the variational adjoint approach. In shape optimization, the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach. The topology optimization problem is solved by an optimality criteria (OC) method and the shape optimization problem by a mathematical programming (MP) method. In topology optimization, a geometrically adaptive procedure is additionally applied in order to increase the efficiency and to avoid artificial stress singularities. The procedures are verified by 2D design problems under plane stress conditions.