Numerical analysis of two frictionless elastic-piezoelectric contact problems

In this work, we consider two frictionless contact problems between an elastic-piezoelectric body and an obstacle. The linear elastic-piezoelectric constitutive law is employed to model the piezoelectric material and either the Signorini condition (if the obstacle is rigid) or the normal compliance condition (if the obstacle is deformable) are used to model the contact. The variational formulations are derived in a form of a coupled system for the displacement and electric potential fields. An existence and uniqueness result is recalled. Then, a discrete scheme is introduced based on the finite element method to approximate the spatial variable. Error estimates are derived on the approximate solutions and, as a consequence, the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the performance of the algorithm.     

Formulation and analysis of two energy-consistent methods for nonlinear elastodynamic frictional contact problems

Energy-conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability. Furthermore, some physical phenomena (such as friction) can generate dissipation; then in this work, we present and analyse two energy-consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon. The first approach permits one to enforce, respectively, the Kuhn–Tucker and persistency conditions during each time step by combining an adapted continuation of the Newton method and a Lagrangean formulation. In addition the second method which is based on the work in [P. Hauret, P. Le Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact, Comput. Methods Appl. Mech. Eng. 195 (2006) 4890–4916] represents a specific penalisation of the unilateral contact conditions. Some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods.     

Numerical analysis of a bilateral frictional contact problem for linearly elastic materials

We consider a mathematical model which describes the contactbetween a linearly elastic body and an obstacle, the so-calledfoundation. The process is quasistatic and the contact is bilateral,i.e. there is no loss of contact during the process. The frictionis modelled with Tresca's law. The variational formulation ofthe problem is a nonlinear evolutionary inequality for the displacementfield which has a unique solution under certain assumptionson the given data. We study spatially semi-discrete and fullydiscrete schemes for the problem with finite-difference discretizationin time and finite-element discretization in space. The numericalschemes have unique solutions. We show the convergence of thescheme under the basic solution regularity. Under appropriateregularity assumptions on the solution, we derive optimal ordererror estimates. Finally, we present numerical results in thestudy of two-dimensional test problems.     

Analysis and numerical solution of a piezoelectric frictional contact problem

We consider a mathematical model which describes the frictional contact between an electro-elastic–visco-plastic body and a conductive foundation. The contact is modelled with normal compliance and a version of Coulomb’s law of dry friction, in which the stiffness and the friction coefficients depend on the electric potential. We derive a variational formulation of the problem and we prove an existence and uniqueness result. The proof is based on a recent existence and uniqueness result on history-dependent quasivariational inequalities obtained in [15]. Then we introduce a fully discrete scheme for solving the problem and, under certain solution regularity assumptions, we derive an optimal order error estimate. Finally, we present some numerical results in the study of a two-dimensional test problem which describes the process of contact in a microelectromechanical switch.     

Construction du préconditionneur Neumann–Neumann de décomposition de domaine de niveau 2 pour des problèmes élastodynamiques en grandes déformations

In this Note, we present the construction of a Neumann–Neumann domain decomposition preconditioner with scalability properties to solve nonlinear elastostatic and elastodynamic problems. By using the results given in Alart et al. [C. R. Acad. Sci. Paris, Ser. I 331 (2000) 399–404] and according to the work realized in Farhat et al. [Int. J. Numer. Methods Engrg. 38 (1995) 3831–3853] and Fragakis and Papadrakakis [Comput. Methods Appl. Mech. Engrg. 193 (2004) 4611–4662], we give the construction steps of the two level preconditioner which are based on the introduction of a coarse space especially adapted to finite deformations problems with static and dynamic process. To cite this article: M. Barboteu, C. R. Acad. Sci. Paris, Ser. I 340 (2005).