Time integration with proper generalized decomposition for efficient time response analysis in nonlinear dynamical systems

Abstract

This paper presents a new framework that aims to improve the efficiency of time response analysis for nonlinear dynamical systems by combining conventional time integration methods with proper generalized decomposition (PGD). The PGD approach utilizes low-dimensional subspaces of the time response to approximate the solution as a low-order separated representation of spatial and temporal components, with the Galerkin projection employed to formulate subproblems for each component. The subproblem for spatial basis is viewed as computing a reducedorder criterion, and the temporal problem projected to a subspace spanning this criterion uses time integration to obtain time coefficients. During the time integration, the spatial modes obtained from the calculation of the previous step are used as a reduced basis, and additional spatial modes are added until the residual of equations of motion satisfy the target tolerance. Numerical examples demonstrate that the proposed method allows significant computational savings compared to conventional time integration methods while accurately reflecting the nonlinear behavior.