Solving dynamic contact problems with local refinement in space and time

Abstract

International audienceFrictional dynamic contact problems with complex geometries are a challenging task from the compu tational as well as from the analytical point of view since they generally involve space and time multi scale aspects. To be able to reduce the complexity of this kind of contact problem, we employ a non conforming domain decomposition method in space, consisting of a coarse global mesh not resolving the local struc ture and an overlapping fine patch for the contact computation. This leads to several benefits: First, we resolve the details of the surface only where it is needed, i.e., in the vicinity of the actual contact zone. Second, the subproblems can be discretized independently of each other which enables us to choose a much finer time scale on the contact zone than on the coarse domain. Here, we propose a set of interface conditions that yield optimal a priori error estimates on the fine meshed subdomain without any artificial dissipation. Further, we develop an efficient iterative solution scheme for the coupled problem that is robust with respect to jumps in the material parameters. Several complex numerical examples illustrate the performance of the new scheme

Boundary elements with mesh refinements for the wave equation

“…See also [5,27]. See [21,22] for the complementary problem of the tire dynamics in contact with the road.…”

A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties

“…Note that U and the iterates U n belong to the subspace U Ξ * ⊂ U Ξ , so that only the restriction to U Ξ * of operator A defined in (18) has to be analyzed.…”

“…where λ min (A) and λ max (A) are the lowest and largest eigenvalues of operator (18). In practice, λ max (A) is estimated using some power iterations (typically 2 or 3), each power iteration requiring the solution of one local problem (for computing Ψ(V )) and one global problem (for computing Φ(Ψ(V ), V )).…”

“…At the deterministic level, dedicated methods have met the demand of coupling numerical models at different scales and some have been extended to stochastic models. Among these deterministic methods, one can distinguish the mono-model methods based on adaptive mesh or enrichment techniques [10,11,12,13] from the multi-model methods based on patches as the global-local iterative methods proposed in [14,15,16,17,18] or the bridging methods proposed in [19,20,21] or in [22] with the Arlequin method. The latter has been exploited in the stochastic framework for deterministic-stochastic coupling in [23,24] for a homogenization purpose.…”

A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities

“…First, superposition methods, such as the method of finite element patches [34,53] and the method of harmonic patches [38], consist in adding a fine local correction to a coarse global solution. Second, surface coupling methods include the Chimera-Schwarz method [11,72], the Semi-Schwarz method [66], the Semi-Schwarz-Lagrange method [29,30,36,55] and the local multigrid method [35,64,65,67]. Both multiscale superposition and surface coupling methods are based on global-local iterative algorithms originally developed for domain decomposition methods or multigrid methods.…”