Calculation of design sensitivity for large-size transient dynamic problems using Krylov subspace-based model order reduction

Nowadays, transient dynamic responses of a large-size finite element (FE) model can be solved within a reasonable computation time owing to rapid improvement in both numerical schemes and computing resources. However, increasing demands for accurate simulation and complicated modeling have led to larger and more complex finite element models, which consequently result in considerably high computational cost. In addition, when structural optimizations include transient responses such as displacement, velocity, and acceleration, the optimizations often do not end within a reasonable process time because the large-size simulation must be repeated many times. In order to reduce the computational cost in this respect, model order reduction (MOR) for the original full-order model (FOM) can be used for the transient response simulation. In this paper, a transient dynamic response analysis using Krylov subspace-based MOR and its design sensitivity analysis with respect to sizing design variables is suggested as an approach to the handling of large-size finite element models. Large-size finite element models can incur the problem of a long computation time in gradient-based optimization iterations because of the need for repeated simulation of transient responses. In the suggested method, the reduced order models (ROMs) generated from the original FOMs using implicit moment-matching via the Arnoldi process are used to calculate the transient response and its design sensitivity. As a result, the speed of numerical computation for the transient response and its design sensitivity is maximized. Newmark's time integration method is employed to calculate transient responses and their design sensitivities. In the case of the transient sensitivity analysis, we apply a temporal discretization scheme to the design sensitivity equation derived by directly differentiating the governing equation with respect to design variables. This methodology has been programmed on the MATLAB with the FE information extracted from the FE package ANSYS. Two application examples are provided to demonstrate the numerical accuracy and efficiency of the suggested approach. The relative errors of transient response and design sensitivity between the FOMs and ROMs are also compared according to the orders of the reduced model. Calculation of transient dynamic responses and their sensitivities using Krylov subspace-based MOR shows a sizeable reduction in computation time and a good agreement with those provided by the FOM.