Low-cost formulation-free sensitivity analysis for path-dependent topology optimization using fixed-point automatic differentiation

Abstract A low-cost formulation-free sensitivity analysis method based on fixed-point automatic differentiation (AD) is presented for the topology optimization of multiphysics and path-dependent problems. With the conventional adjoint variable method, any change to the constitutive law, material interpolation, objective function, or coupling with another physical field requires the adjoint recursion and partial derivatives with respect to the design variables to be derived again by symbolic expansion, and the formulation and its verification must be redone. With the proposed method, the adjoint recursion and partial derivatives are obtained automatically so long as the forward analysis can be evaluated as the subject of AD. Thus, no formulation specific to a physical model is needed. To avoid the increasing memory consumption from AD recording every operation of the forward analysis, the proposed method fixes a consistent tangent at the converged point of each load step. It uses reverse-mode AD to record only once a map consisting of the evaluation of the residual, the update of the solution, and the update of the internal variables. Finally, it propagates the sensitivity of the internal variables through a backward recursion between the steps. Thus, the tape held at any one time is that of a single analysis step and is independent of the number of load steps. The proposed method was applied to the topology optimization of a thermo-elastoplastic multi-material problem based on a von Mises elastoplastic model coupled with steady-state heat conduction. In numerical examples, the sensitivities obtained by the proposed method agreed with those of the analytical adjoint variable method and reverse-mode AD recording every operation to within the roundoff error, and the computation time and peak memory of one gradient evaluation grew with the number of elements and number of load steps at the same rate as the analytical adjoint variable method while remaining lower than those of reverse-mode AD. Two-dimensional optimization yielded the same design regardless of the sensitivity analysis method, and three-dimensional optimization showed the practicality of the proposed method.