Ideal minimal residual-based proper generalized decomposition for non-symmetric multi-field models – Application to transient elastodynamics in space-time domain

• Published in: Computer methods in applied mechanics and engineering Vol. 273; pp. 56 - 76
• Main Authors: Boucinha, L., Ammar, A., Gravouil, A., Nouy, A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.05.2014; Elsevier
• Subjects: Algorithms; Approximation; Decomposition; Elastodynamics; Engineering Sciences; Galerkin methods; Mathematical models; Mechanics; Minimal residual; Multi-field; Norms; Physics; Proper generalized decomposition (PGD); Proper orthogonal decomposition (POD); Representations; Separation of variables; Structural mechanics; Minimal residual; Separation of variables; Proper generalized decomposition (PGD); Elastodynamics; Proper orthogonal decomposition (POD); Multi-field; multi-field; minimal residual; separation of variables; elastodynamics; Proper Orthogonal Decomposition (POD); Proper Generalized Decomposition (PGD)
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2014.01.019

It is now well established that separated representations built with the help of proper generalized decomposition (PGD) can drastically reduce computational costs associated with solution of a wide variety of problems. However, it is still an open question to know if separated representations can be efficiently used to approximate solutions of hyperbolic evolution problems in space-time domain. In this paper, we numerically address this issue and concentrate on transient elastodynamic models. For such models, the operator associated with the space-time problem is non-symmetric and low-rank approximations are classically computed by minimizing the space-time residual in a natural L2 sense, yet leading to non optimal approximations in usual solution norms. Therefore, a new algorithm has been recently introduced by one of the authors and allows to find a quasi-optimal low-rank approximation a priori with respect to a target norm. We presently extend this new algorithm to multi-field models. The proposed algorithm is applied to elastodynamics formulated over space-time domain with the Time Discontinuous Galerkin method in displacement and velocity. Numerical examples demonstrate convergence of the proposed algorithm and comparisons are made with classical a posteriori and a priori approaches.