Design sensitivity analysis for rate-independent elastoplasticity

Abstract

A new incremental, direct differentiation method for design sensitivity analysis of structures with rate-independent elastoplastic behavior is presented. We formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions. The sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation. Combined with the inherent advantages of implicit integration strategies, this represents a significant improvement over previous sensitivity formulations for history-dependent materials. First-order sensitivity expressions involving the complete set of design variables, including shape design variables, are derived for a generic response functional. The reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step. No iterations are needed in the sensitivity computations. A numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem. Explicit sensitivities from the new method are confirmed by finite difference estimates.