Block preconditioning for fault/fracture mechanics saddle-point problems

Penalty and regularization method for variational‐hemivariational inequalities with application to frictional contact

A scaled boundary finite element method for modelling crack face contact problems

There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost-effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well-posedness of such models. There is a need to extend the use of such sensors into high temperature environments and so this paper shows, for a broad class of models, the well-posedness of the magneto-electro-thermo-elastic problem. Due to its widespread use in the literature, we also show the well-posedness of the quasi-electrostatic case.

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On well-posedness for some thermo-piezoelectric coupling models

Abstract We present the theory of a new fractional Sobolev space in complete manifolds with variable exponent. As a result, we investigate some of our new space’s qualitative properties, such as completeness, reflexivity, separability, and density. We also show that continuous and compact embedding results are valid. We apply the conclusions of this study to the variational analysis of a class of fractional $p(z, \cdot )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo>)</mml:mo></mml:math> -Laplacian problems involving potentials with vanishing behavior at infinity as an application. 

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On a new fractional Sobolev space with variable exponent on complete manifolds

This work studies a mathematical model describing the static process of contact between a piezoelectric body and a thermally-electrically conductive foundation. The behavior of the material is modeled with a thermo-electro-elastic constitutive law. The contact is described by Signorini’s conditions and Tresca’s friction law including the electrical and thermal conductivity conditions. A variational formulation of the model in the form of a coupled system for displacements, electric potential, and temperature is derived. Existence and uniqueness of the solution are proved using the results of variational inequalities and a fixed point theorem.

Existence results for unilateral contact problem with friction of thermo-electro-elasticity

In this paper, we consider the penalty method to solve the unilateral contact with friction between an electro-elastic body and a conductive foundation. Mathematical properties, such as the existence of a solution to the penalty problem and its convergence to the solution of the original problem, are reported. Then, we present a finite elements approximation for the penalised problem and prove its convergence. Finally, we propose an iterative method to solve the resulting finite element system and establish its convergence.

On convergence of the penalty method for a static unilateral contact problem with nonlocal friction in electro-elasticity

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz's representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.

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A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

We consider a mathematical model which describes the frictional unilateral contact between a thermo-piezoelectric body and a rigid electrically conductive foundation. The thermo-piezoelectric constitutive law is assumed to be nonlinear and the contact is modeled with the Signorini condition, the nonlocal Coulomb friction law with slip dependent friction coefficient and the regularized electrical and thermal conductivity conditions. The variational form of this problem is a coupled system which consists of a nonlinear variational inequality for the displacement field and two nonlinear variational equations for the electric potential and the temperature. The existence of a unique weak solution to the problem is proved by using abstract results for elliptic variational inequalities and fixed point arguments.

/pdf/analysis-of-a-signorini-problem-with-nonlocal-friction-in-jggh0galy9.pdf

El-H. Essoufi

Papers

Existence results for unilateral contact problem with friction of thermo-electro-elasticity

On convergence of the penalty method for a static unilateral contact problem with nonlocal friction in electro-elasticity

A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

Analysis of a Signorini problem with nonlocal friction in thermo-piezoelectricity

Regular solution for the compressible Landau–Lifshitz–Bloch equation in a bounded domain of $$\mathbb {R}^{3}$$