The consistent tangent operator for design sensitivity analysis of history-dependent response

A new incremental direct differentiation method for design sensitivity analysis of structures with history-dependent material response is presented. The method uses the domain parameterization approach to represent design shape variations, and the consistent tangent operator to ensure that the sensitivity estimate is consistent with the underlying simulation model. The use of the consistent tangent operator represents an improvement, with respect to both accuracy and efficiency, over the semi-analytical methods and the previous incremental methods. First-order sensitivity expressions, involving the complete set of static design variables (including shape design variables), are derived for a generic response functional. Sensitivity computations are performed following each load step or time step in an incremental analysis procedure. These computations involve the assembly of a pseudo-load vector and an inexpensive solution operation using the reduced form of the consistent-tangent stiffness matrix generated in the analysis procedure. No iterations are needed in the sensitivity computations. Sensitivity expressions are presented for the case of Norton-Soderberg power-law creep materials, and a numerical example demonstrates the accuracy of the new sensitivity formulation. The sensitivities computed with the new method are verified by finite-difference sensitivity estimates that were obtained using careful convergence studies to determine suitable choices for the finite-difference increments.