Grégory Legrain

Low-Rank Approximate Inverse for Preconditioning Tensor-Structured Linear Systems

In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator. It provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators. These subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank-one corrections. In order to handle high-order tensors, approximate projections are computed in low-rank hierarchical Tucker subsets of the successive subspaces of operators. Some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step, where an optimal rank-one correction is searched as the tensor product of operators with the desired properties. Numerical examples illustrate the ability of this a...

Treatment of nearly-singular problems with the X-FEM

BackgroundIn recent years, lot of research have been conducted on fictitious domain approaches in order to simplify the meshing process for computed aided analysis. The behaviour of such non-conforming methods is studied in the case of the approximation of nearly singular solutions. Such solutions appear when problems involve singularities whose center are located outside (but close) of the domain of interest. These solutions are common in industrial structures that usually involve rounded re-entrant corners.MethodsThe performance of the finite element method is evaluated in this context by means of a simple unidimensional example. Both numerical and theoretical estimates are considered in order to assess the behaviour of the numerical approximation. It is demonstrated that despite being regular, the convergence of the approximation can be bounded to an algebraic rate that depends on the solution. Reasons for such behaviour are presented, and two complementary strategies are proposed in order to recover optimal convergence rates. The first strategy is based on a proper enrichment of the approximation thanks to the X-FEM, while the second is based on a proper mesh design that follows a geometric progression. Finally, the proposed strategies are extended and validated in 2D.ResultsThe performance of the two strategies is highlighted for both 1D and 2D examples. Both methods allow to recover proper convergence rates (optimal algebraic rate for h-convergence, exponential for p-convergence) in 1D and 2D.ConclusionsThe proposed strategies allow for a very accurate solution for such solutions. The enrichment strategy is valid for both h and p refinement, whereas the mesh-design strategy is only usable for p refinement. However, such enrichment functions can be tedious to derive.

Thermal properties of composite materials : effective conductivity tensor and edge effects

The homogenization theory is a powerful approach to determine the effective thermal conductivity tensor of heterogeneous materials such as composites, including thermoset matrix and fibres. Once the effective properties are calculated, they can be used to solve a heat conduction problem on the composite structure at the macroscopic scale. This approach leads to good approximations of both the heat flux and temperature in the interior zone of the structure, however edge effects occur in the vicinity of the domain boundaries. In this paper, following the approach proposed in [10] for elasticity, it is shown how these edge effects can be corrected. Thus an additional asymptotic expansion is introduced, which plays the role of a edge effect term. This expansion tends to zero far from the boundary, and is assumed to decrease exponentially. Moreover, the length of the edge effect region can be determined from the solution of an eigenvalue problem. Numerical examples are considered for a standard multilayered material. The homogenized solutions computed with a finite element software, and corrected with the edge effect terms, are compared to a heterogeneous finite element solution at the microscopic scale. The influences of the thermal contrast and scale factor are illustrated for different kind of boundary conditions.

Etude de la stabilité d’une formulation incompressible traitée par X-FEM

Le traitement de l’incompressibilité est un point-clé pour le dimensionnement des composants élastomères ou la simulation du processus de formage. L’utilisation de formulations mixtes permet d’éviter le phénomène de verrouillage (locking) de l’approximation éléments finis. Cependant, la stabilité de ces méthodes est conditionnée par la vérification de la condition inf-sup. Récemment, les approximations E.F. ont évolué avec l’introduction de la partition de l’unité. La méthode X-FEM (eXtended Finite Element Method) utilise ce concept pour éviter le maillage (et remaillage) des surfaces physiques du problème. Dans cet article, une stratégie est proposée pour la gestion des trous avec la méthode X-FEM dans le cas incompressible. Les applications numériques montrent que la convergence théorique des éléments finis est préservée, et que la condition inf-sup est vérifiée.