State-space based discretize-then-differentiate adjoint sensitivity method for transient responses of non-viscously damped systems

Abstract

In this paper, a new design sensitivity analysis (DSA) method for transient responses of non-viscously damped systems is developed. The damping force of the non-viscously damped systems depends on the past history of motion and is represented by convolution integrals over suitable kernel functions. The equations of motion are first transformed into a state-space formulation and the time-history analysis of the responses relies on a modified precise integration method (MPIM). The residual equation in each discrete time step is achieved by a recurrence formula of the MPIM. The sensitivities of the transient responses are calculated with an adjoint variable method (AVM). In particular, a discretize-then-differentiate approach is adopted for the DSA. The accuracy, consistency, implementation effort and computational complexity are discussed. Two numerical examples are illustrated to show the performances of the proposed method compared with other two methods. The results indicate that, when calculating the sensitivities of the transient responses for non-viscously damped systems based on the MPIM, the order of differentiation and discretization has no obvious effect on the consistency. The proposed state-space based discretize-then-differentiate AVM is more efficient than other two methods.