Abdeljalil Nachaoui

An alternating method for an inverse Cauchy problem

In this paper, an iterative boundary element method based on our relaxed algorithm introduced in [8] is used to solve numerically a class of inverse boundary problems. A dynamical choice of the relaxation parameter is presented and a stopping criterion based on our theoretical results is used. The numerical results show that the algorithm produces a reasonably approximate solution and improves the rate of convergence of Kozlovs scheme [10].

An iterative approach to the solution of an inverse problem in linear elasticity

This paper presents an iterative alternating algorithm for solving an inverse problem in linear elasticity. A relaxation procedure is developed in order to increase the rate of convergence of the algorithm and two selection criteria for the variable relaxation factors are provided. The boundary element method is used in order to implement numerically the constructing algorithm. We discuss this implementation, mention the use of Krylov methods to solve the obtained linear algebraic systems of equations and investigate the convergence and the stability when the data is perturbed by noise.

An Improved Implementation of An Iterative Method in Boundary Identification Problems

In this paper, an inverse problem of determining geometric shape of a part of the boundary of a bounded domain is considered. Based on a conjugate gradient method, employing the adjoint equation to obtain the descent direction, an identification scheme is developed. The implementation of the method based on the boundary element method (BEM) is also discussed. This method solves the inverse boundary problem accurately without a priori information about the unknown shape to be estimated.

A numerical study of filtration problem in inhomogeneous dam with discontinuous permeability

In this paper, we deal with an optimal shape design approach for the numerical realization of the flow problem of incompressible liquid through an inhomogeneous porous medium (say dam), with a class of permeability coefficients. Our interest is in studying numerical approximation of the solution of the optimal shape design problem proposed in [C. R. Acad. Sci. Paris Ser. I Math. 329 (10) (1999) 933-938]. We consider discretization of this problem based on linear finite elements. We prove the existence of the solution of the discrete problem. We study the convergence of the discrete problem. We present some results on numerical experiments including comparisons to the results obtained by a variational approach.

A shape optimization formulation of weld pool determination

In this paper, we propose a shape optimization formulation for a problem modeling a process of welding. We show the existence of an optimal solution. The finite element method is used for the discretization of the problem. The

Shape optimization for a simulation of a semiconductor problem

This work gives a study of the Regier’s model by using the shape optimization techniques. Two formulations of this model are proposed with appropriate boundary conditions of the MESFET transistor. In each formulation the existence of a free boundary, separating the depletion and neutrality of charge regions, is proved with some hypothesis. The shape gradient is calculated for each formulation to approach the solution. To approximate the free boundary, two algorithms are presented. Numerical results obtained by implementing the last algorithm prove that this shape optimization techniques provide a reasonably smooth free boundary.

Iterative solution of the drift-diffusion equations

The drift-diffusion model can be described by a nonlinear Poisson equation for the electrostatic potential coupled with a system of convection-reaction-diffusion equations for the transport of charge. We use a Gummel-like process [10] to decouple this system. Each of the obtained equations is discretised with the finite element method. We use a local scaling method to avoid breakdown in the numerical algorithm introduced by the use of Slotboom variables. Solution of the discrete nonlinear Poisson equation is accomplished with quasi-Newton methods. The nonsymmetric discrete transport equations are solved using an incomplete LU factorization preconditioner in conjunction with some robust linear solvers, such as (CGS), (BI-CGSTAB) and (GMRES). We investigate the behaviour of these iterative methods to define the effective strategy for this class of problems.

Quasi‐variational inequality and shape optimization for solution of a free boundary problem

Electrical potentials in a junction field transistor can be calculated using a simplified model based on a complete depletion assumption. This gives rise to a free boundary problem. We show here how we can approximate this problem with a quasi‐variational inequality technique and the shape optimization method. A detailed analysis of these methods is presented. Using some numerical experiments we compare our results with the solution of the discrete drift‐diffusion system, accomplished with a Gummel‐like algorithm. The numerical results suggest that the methods proposed here work successfully and that the shape optimization technique provides a reasonably free boundary without excessive iterations.

Sufficient Conditions for Converging Drift-Diffusion Discrete Systems. Application to The Finite Element Method

In this paper we generalize the abstract results of Mock and Marcowich [13, 12] for convergence of discrete Van Roosbroeck systems [12, 13, 17], to the case when the solutions are typically in W1,4-e and not in H2. These conditions are verified on finite element discretizations. Error estimates are derived when the solution is unique. Due to the singularity at the flat angles, these estimates in the H1 norm are only O(h1/2). The techniques that are presented are broad and may be applied to other type of discretizations.

Non-linear eigenvalue problems for p -Laplacian with variable domain

In this work we consider some eigenvalue problems for p-Laplacian with variable domain. Eigenvalues of this operator are taken as a functional of the domain. We calculate the first variation of this functional, using the obtained formula investigate behavior of the eigenvalues when the domain varies. Then we consider one shape optimization problem for the first eigenvalue, prove the necessary condition of optimality relatively domain, offer an algorithm for the numerical solution of this problem.