Patrice Hauret

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.

Energy-Consistent CoRotational Schemes for Frictional Contact Problems

In this paper, we consider the unilateral frictional contact problem of a hyperelastic body in the case of large displacements and small strains. In order to retain the linear elasticity framework, we decompose the deformation into a large global rotation and a small elastic displacement. This corotational approach is combined with a primal-dual active set strategy to tackle the contact problem. The resulting algorithm preserves both energy and angular momentum.

Achieving Robustness Through Coarse Space Enrichment in the Two Level Schwarz Framework

As many DD methods the two level Additive Schwarz method may suffer from a lack of robustness with respect to coefficient variation. This is the case in particular if the partition into is not aligned with all jumps in the coefficients. The theoretical analysis traces this lack of robustness back to the so called stable splitting property. In this work we propose to solve a generalized eigenvalue problem in each subdomain which identifies which vectors are responsible for violating the stable splitting property. These vectors are used to span the coarse space and taken care of by a direct solve while all remaining components behave well. The result is a condition number estimate for the two level method which does not depend on the number of subdomains or any jumps in the coefficients.

Nonlinear schemes and multiscale preconditioners for time evolution problems in constrained structural dynamics

Publisher Summary Many modern structures, such as fiber-reinforced materials or granular structures, involve small scales substructures, whose evolution must be carefully controlled. A good prediction of their time response, of the local stress distribution, of local energy balance, and of the coupling among different substructures is important to assess the long-term stability and robustness of the global system. In particular, when designing tires, the detailed dynamic study of the surface structures is important for performance and durability assessment. The correct mechanical description of such systems is very demanding, both in space and in time resolution. A brute force description would require to compute the time evolution of several millions degrees of freedom over several thousand time steps. Moreover, because of large deformation, contact, or kinematic constraints—such as incompressibility— these systems have a highly nonlinear behavior, which affect the global conservation properties of most linear schemes.