ADJOINT SENSITIVITY METHODS FOR TRANSIENT RESPONSES OF VISCOUSLY DAMPED SYSTEMS AND THEIR CONSISTENCY ISSUES

Design sensitivity analysis (DSA) of transient response is indispensable in a time domain gradient-based optimization algorithm. DSA usually only requires the differentiation with respect to certain design variables. But for the problem of transient response sensitivities, it also contains the process of time discretization. Therefore, the order of discretization and differentiation may also affect the results of the DSA. In this paper, two new DSA methods, namely the differentiate-then-discretize adjoint variable method (AVM) and the discretize-then-differentiate AVM method, are derived based on a modified precise integration method (MPIM) to compute the transient response sensitivities for viscously damped systems. The damping force of the viscously damped systems is assumed to be proportional to the instantaneous velocity. The equations of motion of the viscously damped systems are transformed into a state-space formulation and the transient responses are calculated by the MPIM. The differentiate-then-discretize AVM method firstly differentiates the augmented function constructed by the adjoint vectors and then discretizes the function at each time point based on the MPIM. On the contrary, the discretize-then-differentiate AVM method discretizes the augmented function built by the residual equation at each separated time point first and then differentiates the discrete augmented function to obtain the transient response sensitivities. Two numerical methods are presented to show the correctness and effectiveness of the proposed method. The performances of the proposed methods are also compared them with the conventional Newmark-based method. The results show that, when calculating the sensitivities of the transient responses for viscously damped systems, the time integration method, the time step size and the order of discretization and differentiation all have influences on the consistency error. By considering the accuracy, efficiency and consistency issue, the proposed MPIM-based differentiate-then-discretize AVM is more suitable than other compared methods for applying in gradient-based time domain optimizations for viscously damped systems.