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

Lucas Boucinha (1, 2) , Amine Ammar (3) , Anthony Gravouil (1, 4, 2) , Anthony Nouy (5)

1 LaMCoS - Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne]
2 MIMESIS - Mécanique Multiéchelle pour les solides
3 LAMPA - Laboratoire Angevin de Mécanique, Procédés et InnovAtion
4 IUF - Institut universitaire de France
5 GeM - Institut de Recherche en Génie Civil et Mécanique

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.

Mots clés: multi-field; minimal residual; Proper Generalized Decomposition (PGD); Proper Orthogonal Decomposition (POD); separation of variables; elastodynamics

Domaines: Mécanique des structures [physics.class-ph]; Mécanique des structures [physics.class-ph]

Lucas Boucinha, Amine Ammar, Anthony Gravouil, Anthony Nouy. Ideal minimal residual-based proper generalized decomposition for non-symmetric multi-field models - Application to transient elastodynamics in space-time domain. Computer Methods in Applied Mechanics and Engineering, 2014, 273, pp.56-76. ⟨10.1016/j.cma.2014.01.019⟩. ⟨hal-00952496⟩