Adjoint Method for Topological Derivatives for Optimization Tasks with Material and Geometrical Nonlinearities

In this paper the Topological Derivative for crash loaded structures is derived with the adjoint sensitivity analysis. The main idea of the adjoint sensitivity analysis is to circumvent the direct calculation of the sensitivity of the displacement field. Instead, the adjoint equilibrium equation has to be solved. In this approach, material derivation and partial integration in the time domain are applied to the Topological Derivative. This ensures, that the inertial effects are kept, as they are important for a reliable crash simulation. The result is a backward integration scheme for the adjoint state.