A robust and generalized effective DAE framework encompassing different methods, algorithms, and model order reduction for linear and nonlinear second order dynamical systems

Abstract

For linear and nonlinear dynamical problems, we propose the novel development and advancement of the well-known Generalized Single Step Single Solve (GS4) family of second order time-accurate algorithms encompassing the entire class of LMS methods developed over the past 50 years or so with/without controllable numerical dissipation in conjunction with the Differential Algebraic Equation (DAE) framework and Proper Orthogonal Decomposition (POD). Unlike traditional practices which have severe limitations in the ability to provide generality and flexibility and a wide variety of choices to the analyst without losing order of time accuracy, the proposed implementation uniquely not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains in a single body, but also additionally allows the selection of different space discretized methods such as the Finite Element Method (FEM) and particle methods, and other spatial methods as well in a single analysis. Moreover, the addition of the POD further provides reduction in computational times. Furthermore, unlike existing state of the art, the present framework readily permits a wide array of implicit–implicit, implicit–explicit, and explicit–explicit couplings and the integration of such a technology is of interest here. Consequently, the present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration schemes in different subdomains in a selective manner together with Reduced Order Modeling (ROM) to optimize capturing the local and global features of the representative physics. The ROM also employs an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to the prescribed accuracy requirements. Such a novelty, and computational features, are not possible with existing state of art and numerical illustrations validate the claim.