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Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate ability of proper generalized decomposition (PGD) to solve transient elastodynamic models in space–time domain. Classical methods use time integration schemes and an incremental resolution process. We propose here to use standard time integration methods in a non-incremental strategy. As a result, PGD gives a separated representation of the space–time solution as a sum of tensorial products of space and time vectors, that we interpret as space–time modes. Recent time integration schemes are based on multi-field formulations. In this case, separated representation can be constructed using state vectors in space and same vectors in time. However, we have experienced bad convergence order using this decomposition. Furthermore, temporal approximation must be the same for all fields. Thus, we propose an extension of classical separated representation for multi-field problems. This multi-field PGD (MF-PGD) uses space and time vectors that are different for each field. Calculation of decomposition is done using a monolithic approach in space and time, potentially allowing the use of different approximations in space and time. Finally, several simulations are performed with the transient elastodynamic problem with one dimension in space. Different approximations in time are investigated: Newmark scheme, single field time discontinuous Galerkin method and two fields time continuous and discontinuous Galerkin methods. [Copyright &y& Elsevier]