This version is available at https://strathprints.strath.ac.uk/44827/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. Abstract 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

It is now well established that discrete energy conservation/dissipation plays a key-role for the unconditional stability of time integration schemes in nonlinear elastodynamics. In this paper, from a rigorous conservation analysis of the Hilber–Hughes–Taylor time integration scheme [H. Hilber, T. Hughes, R. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engrg. Struct. Dynam. 5 (1977) 283–292], we propose an original way of introducing a controllable energy dissipation while conserving momenta in conservative strategies like [J. Simo, N. Tarnow, The discrete energy–momentum method: conserving algorithms for nonlinear elastodynamics, Z. Angew. Math. Phys. 43 (1992) 757–792]. Moreover, we extend the technique proposed in [O. Gonzalez, Exact energy and momentum conserving algorithms for general models in nonlinear elasticity, Comput. Methods Appl. Mech. Engrg. 190 (13–14) (2000) 1763–1783] to provide energy-controlling time integration schemes for frictionless contact problems enforcing the standard Kuhn–Tucker conditions at time discretization points. We also extend this technique to viscoelastic models. Numerical tests involving the impact of incompressible elastic or viscoelastic bodies in large deformation are proposed to confirm the theoretical analysis

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems. 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 which isolate the terms responsible for slow convergence. We give a general theoretical result and then some numerical examples on a heterogeneous elasticity problem.To cite this article: , C. R. Acad. Sci. Paris, Ser. I +++++ (+++++). RésuméUn moyen efficace pour obtenir des méthodes de décomposition de domaine extensibles (≪ scalable ≫ en anglais) est l'utilisation d'une grille grossière. Cependant, lorsque les coefficients deséquations présentent de grandes hétérogénéités, les méthodes usuelles tombent en défaut, surtout dans le cas des systèmes. Nous introduisons ici, au niveau variationnel, une grille grossière robuste même en présence de telles discontinuités. Pour cela, nous résolvons des problèmes aux valeurs propres généralisés locaux qui isolent les composantes de la solution nuisant a la convergence. Nous présentons un résultat théorique général puis quelques résultats numériques pour un problème d'élasticitéà coefficients discontinus. Pour citer cet article : , C. R. Acad. Sci. Paris, Ser. I +++++ (+++++).Preprint submitted to the Académie des sciences 6 juillet 2011Version française abrégée Ce travail s'intéresseà la résolution d'un système linéaire (3) issu de la discrétisation paréléments finis (2) d'un problème aux limites elliptique donné sous forme variationnelle (1) où les coefficients peuvent etre discontinus. Afin d'obtenir des méthodes de décomposition de domaine extensibles (robustes visà vis du nombre de sous-domaines), nous considérons des méthodesà deux niveaux [2]. Ces méthodes sont etroitement liées aux méthodes multigrilles et de déflation. Elles sont définies par deux ingrédients : une grille grossière V H composée de m vecteurs avec m petit devant la taille du problème initial et une formulation algébrique de la correction qui consistera ici en la méthode de Schwarzà deux niveaux (4) que l'on utilise comme préconditionneur pour un solveur de type gradient conjugué. Ce choix nous permet d'appliquer des résultats connus qui ramènent l'étude de la convergence de l'algorithmeà celle du conditionnement de l'opérateur préconditionné. La contribution clé de ce travail consiste en une définition systématique de la grille grossière fondée sur les plus basses fréquences de problèmes spectraux généralisés locaux (Définitions 2.3 et 2.4), voir aussi [7] et les références citées. Par rapportà [7], notre grille grossière présente l'avantage d'être construiteà partir de la matrice avant assemblage sans calculs supplémentaires de contributionsélémentaires. De plus, l'estimation ne dépend pas d'une hypothèse de stabilité d'u...

SUMMARYWe present a finite element discretization scheme for the compressible and incompressible elasticity problems that possess the following properties: (i) the discretization scheme is defined on a triangulation of the domain; (ii) the discretization scheme is defined-and is identical-in all spatial dimensions; (iii) the displacement field converges optimally with mesh refinement; and (iv) the inf-sup condition is automatically satisfied. The discretization scheme is motivated both by considerations of topology and analysis, and it consists of the combination of a certain mesh pattern and a choice of interpolation that guarantees optimal convergence of displacements and pressures. Rigorous proofs of the satisfaction of the inf-sup condition are presented for the problem of linearized incompressible elasticity. We additionally show that the discretization schemes can be given a compelling interpretation in terms of discrete differential operators. In particular, we develop a discrete analogue of the classical tensor differential complex in terms of which the discrete and continuous boundary-value problems are formally identical. We also present numerical tests that demonstrate the dimension-independent scope of the scheme and its good performance in problems of finite elasticity.

First, the present paper is concerned with the extension to linearized elastodynamics of the optimal results known in statics for the mortar method. It also analyzes and tests a new couple of displacements/Lagrange multipliers for the method, as proposed independently by F. Ben Belgacem [6] and the authors [21]. Finally, questions of practical implementation in the presence of curved interfaces are addressed and validated from the numerical point of view.