On the treatment of high-frequency issues in numerical simulation for dynamic systems by model order reduction via the proper orthogonal decomposition

• Published in: Computer methods in applied mechanics and engineering Vol. 325; pp. 139 - 154
• Main Authors: Deokar, R., Shimada, M., Lin, C., Tamma, K.K.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.10.2017; Elsevier BV
• Subjects: Computational dynamics; Computer simulation; Conservation laws; Dissipation; Dynamical systems; Finite element analysis; Finite element method; Integrators; Mathematical analysis; Mathematical models; Model reduction; Numerical analysis; Proper Orthogonal Decomposition; Robustness (mathematics); Studies; Time integration; Time integration; Computational dynamics; Proper orthogonal decomposition
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2017.07.003

A new numerical strategy to remedy high-frequency issues caused by finite element discretization in structural dynamic problems has been proposed. A noteworthy characteristic of this advocated approach is that it is based upon the use of the proper orthogonal decomposition (POD) incorporated in conjunction with implicit or explicit numerically non-dissipative time integration schemes to substantially improve or eliminate undesirable effects due to high-frequency instabilities. Original systems with high-frequency issues are reduced via POD based on an adequate choice of a numerically dissipative scheme so that the resulting reduced systems contain no high-frequency participation. This approach confers the inherent advantages that numerically non-dissipative mechanical integrators, e.g., energy–momentum conserving or variational integrators, can be used to solve the reduced systems, fulfilling the requisite conservation laws in the projected basis and therefore provides a robust simulation. Linear and nonlinear numerical applications are shown to demonstrate the benefits and feasibility of this technique. •A method to resolve numerical high frequency issues through model reduction is proposed.•The first step involves using a numerically dissipative scheme to obtain a reduced basis.•The method projects the dynamic system onto a low frequency dominated POD basis.•Physics of the resulting system is preserved using instead energy-momentum conserving algorithms.•Numerical examples successfully demonstrate the effectiveness of the proposed method.