Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

© EDP Sciences, SMAI 2011This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in Rn (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := Rn\￣Ω. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincar´e-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart- Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory

Nonstandard Finite Element Methods

[no abstract available

Numerical Methods for Fluid-Structure Interaction - A Review

The interactions between incompressible fluid flows and immersed structures are nonlinear multi-physics phenomena that have applications to a wide range of scientific and engineering disciplines. In this article, we review representative numerical-methods based on conforming and non-conforming meshes that are currently available for computing fluid-structure interaction problems, with an emphasis on some of the recent developments in the field. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study in fluid-structure interaction

Robin Schwarz algorithm for the NICEM Method: the Pq finite element case

In Gander et al. [2004] we proposed a new non-conforming domain decomposition paradigm, the New Interface Cement Equilibrated Mortar (NICEM) method, based on Schwarz type methods that allows for the use of Robin interface conditions on non-conforming grids. The error analysis was done for P1 finite elements, in 2D and 3D. In this paper, we provide new numerical analysis results that allow to extend this error analysis in 2D for piecewise polynomials of higher order and also prove the convergence of the iterative algorithm in all these cases.Comment: arXiv admin note: substantial text overlap with arXiv:0705.028

hp-BEM for contact problems and extended Ms-FEM in linear elasticity

[no abstract

Dual-dual formulation for a contact problem with friction

A variational inequality formulation is derived for some frictional contact problems from linear elasticity. The formulation exhibits a two-fold saddle point structure and is of dual-dual type, involving the stress tensor as primary unknown as well as the friction force on the contact surface by means of a Lagrange multiplier. The approach starts with the minimization of the conjugate elastic potential. Applying Fenchel's duality theory to this dual minimization problem, the connection to the primal minimization problem and a dual saddle point problem is achieved. The saddle point problem possesses the displacement field and the rotation tensor as further unknowns. Introducing the friction force yields the dual-dual saddle point problem. The equivalence and unique solvability of both problems is shown with the help of the variational inequality formulations corresponding to the saddle point formulations, respectively.This work is supported by the German Research Foundation within the priority program 1180 Prediction and Manipulation of Interactions between Structure and Process

Efficient finite element methods for solving high-frequency time-harmonic acoustic wave problems in heterogeneous media

This thesis focuses on the efficient numerical solution of frequency-domain wave propagation problems using finite element methods. In the first part of the manuscript, the development of domain decomposition methods is addressed, with the aim of overcoming the limitations of state-of-the art direct and iterative solvers. To this end, a non-overlapping substructured domain decomposition method with high-order absorbing conditions used as transmission conditions (HABC DDM) is first extended to deal with cross-points, where more than two subdomains meet. The handling of cross-points is a well-known issue for non-overlapping HABC DDMs. Our methodology proposes an efficient solution for lattice-type domain partitions, where the domains meet at right angles. The method is based on the introduction of suitable relations and additional transmission variables at the cross-points, and its effectiveness is demonstrated on several test cases. A similar non-overlapping substructured DDM is then proposed with Perfectly Matched Layers instead of HABCs used as transmission conditions (PML DDM). The proposed approach naturally considers cross-points for two-dimensional checkerboard domain partitions through Lagrange multipliers used for the weak coupling between subproblems defined on rectangular subdomains and the surrounding PMLs. Two discretizations for the Lagrange multipliers and several stabilization strategies are proposed and compared. The performance of the HABC and PML DDM is then compared on test cases of increasing complexity, from two-dimensional wave scattering in homogeneous media to three-dimensional wave propagation in highly heterogeneous media. While the theoretical developments are carried out for the scalar Helmholtz equation for acoustic wave propagation, the extension to elastic wave problems is also considered, highlighting the potential for further generalizations to other physical contexts. The second part of the manuscript is devoted to the presentation of the computational tools developed during the thesis and which were used to produce all the numerical results: GmshFEM, a new C++ finite element library based on the application programming interface of the open-source finite element mesh generator Gmsh; and GmshDDM, a distributed domain decomposition library based on GmshFEM.Cette thèse porte sur la résolution numérique efficace de problèmes de propagation d'ondes dans le domaine fréquentiel avec la méthode des éléments finis. Dans la première partie du manuscrit, le développement de méthodes de décomposition de domaine est abordé, dans le but de surmonter les limitations des solveurs directs et itératifs de l'état de l'art. À cette fin, une méthode de décomposition de domaine sous-structurée sans recouvrement avec des conditions absorbante d'ordre élevé utilisées comme conditions de transmission (HABC DDM) est d'abord étendue pour traiter les points de jonction, où plus de deux sous-domaines se rencontrent. Le traitement des points de jonction est un problème bien connu pour les HABC DDM sans recouvrement. La méthodologie proposée mène à une solution efficace pour les partitions en damier, où les domaines se rencontrent à angle droit. La méthode est basée sur l'introduction de variables de transmission supplémentaires aux points de jonction, et son efficacité est démontrée sur plusieurs cas-tests. Une DDM sans recouvrement similaire est ensuite proposée avec des couches parfaitement adaptées au lieu des HABC (DDM PML). L'approche proposée prend naturellement en compte les points de jonction des partitions de domaine en damier par le biais de multiplicateurs de Lagrange couplant les sous-domaines et les couches PML adjacentes. Deux discrétisations pour les multiplicateurs de Lagrange et plusieurs stratégies de stabilisation sont proposées et comparées. Les performances des DDM HABC et PML sont ensuite comparées sur des cas-tests de complexité croissante, allant de la diffraction d'ondes dans des milieux homogènes bidimensionnelles à la propagation d'ondes tridimensionnelles dans des milieux hautement hétérogènes. Alors que les développements théoriques sont effectués pour l'équation scalaire de Helmholtz pour la simulation d'ondes acoustiques, l'extension aux problèmes d'ondes élastiques est également considérée, mettant en évidence le potentiel de généralisation des méthodes développées à d'autres contextes physiques. La deuxième partie du manuscrit est consacrée à la présentation des outils de calcul développés au cours de la thèse et qui ont été utilisés pour produire tous les résultats numériques : GmshFEM, une nouvelle bibliothèque d'éléments finis C++ basée sur le générateur de maillage open-source Gmsh ; et GmshDDM, une bibliothèque de décomposition de domaine distribuée basée sur GmshFEM