Abderrahmane Bendali

The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation

A model problem in the scattering of a time-harmonic wave by an obstacle coated with a thin penetrable shell is examined. In previous studies, the contrast coefficients of the thin shell are assumed to tend to infinity in order to compensate for the thickness considered. In this paper, these coefficients are assumed to remain finite. Such a treatment leads to a singular perturbation term that creates a typical difficulty for the asymptotic analysis of the problem with respect to the thickness of the coating. As a result, the asymptotic analysis is essentially based on a suitable handling of the stability of the solution relative to the thickness. As a consequence, it is shown how effective boundary conditions which can be substituted to the thin shell can then be obtained and analyzed in a simple way.

A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three dimensional domains

Abstract A few well-posed variational problems are constructed, whose solutions are vector potentials of the velocity field of the three-dimensional Stokes problem. A choice of the adequate boundary conditions to be imposed is performed in a systematic manner even in the case of not simply connected domains. The test functions do not have to be divergence free in the formulations given here. It is then seen that a similar approach is well suited to the treatment of the non-linear Navier-Stokes problem.

A boundary-element solution of the Leontovitch problem

A boundary-element method is introduced for solving electromagnetic scattering problems in the frequency domain relative to an impedance boundary condition (IBC) on an obstacle of arbitrary shape. The formulation is based on the field approach; namely, it is obtained by enforcing the total electromagnetic field, expressed by means of the incident field and the equivalent electric and magnetic currents and charges on the scatterer surface, to satisfy the boundary condition. As a result, this formulation is well-posed at any frequency for an absorbing scatterer. Both of the equivalent currents are discretized by a boundary-element method over a triangular mesh of the surface scatterer. The magnetic currents are then eliminated at the element level during the assembly process. The final linear system to be solved keeps all of the desirable properties provided by the application of this method to the usual perfectly conducting scatterer; that is, its unknowns are the fluxes of the electric currents across the edges of the mesh and its coefficient matrix is symmetric.

ANALYTIC PRECONDITIONERS FOR THE BOUNDARY INTEGRAL SOLUTION OF THE SCATTERING OF ACOUSTIC WAVES BY OPEN SURFACES

This study is devoted to some numerical issues in the boundary integral solution of the scattering of an acoustic wave by an open surface. More precisely, it deals with the construction of a cheap analytical preconditioner to enhance the iterative solving of this kind of equation. Detailed attention is paid to bring out the reasons that make this construction much more difficult than for closed surfaces. This preconditioner is carefully tested and compared to two more usual ones for two and three dimensional problems. It is shown that this preconditioner provides a cheap and efficient tool making reliable the iterative solving. The discussion also precisely brings out the issues where further studies are still needed to improve its efficiency.

Asymptotic analysis of the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal coated with a thin dielectric shell

Our purpose is to investigate the effect of a thin dielectric coating on the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal. An asymptotic expansion of the solution of a model problem is performed relatively to the thickness of the dielectric shell. A suitable scaling as well as an adapted writing of the the curl operator inside the thin shell make it possible to achieve an asymptotic expansion of the electromagnetic field at any order. Special stability estimates for Maxwell’s equations relatively to the thickness of the dielectric shell then yield optimal error estimates outside the obstacle.

Non-overlapping Domain Decomposition Method for a Nodal Finite Element Method

A new approach is proposed for constructing nonoverlapping domain decomposition procedures for solving a linear system related to a nodal finite element method. It applies to problems involving either positive semi-definite or complex indefinite local matrices. The main feature of the method is to preserve the continuity requirements on the unknowns and the finite element equations at the nodes shared by more than two subdomains and to suitably augment the local matrices. We prove that the corresponding algorithm can be seen as a converging iterative method for solving the finite element system and that it cannot break down. Each iteration is obtained by solving uncoupled local finite element systems posed in each subdomain and, in contrast to a strict domain decomposition method, is completed by solving a linear system whose unknowns are the degrees of freedom attached to the above special nodes.

Extension to Nonconforming Meshes of the Combined Current and Charge Integral Equation

We bring out some mathematical properties of the current and charge boundary integral equation when it is posed on a surface without geometrical singularities. This enables us to show that it is then possible to solve this equation by a boundary element method that requires no interelement continuity. In particular, this property allows the use of meshes on various parts of the surface obtained independently of each other. The extension to surfaces with geometrical singularities showed that acute dihedral angles can lead to inaccuracies in the results. We built a two-dimensional version of this equation which brought out that the wrong results are due to spurious oscillations concentrating around the singular points of the geometry. Noticing that the system linking the current and the charge is a saddle-point problem, we have tried augmenting the approximation of the charge to stabilize the numerical scheme. We show that this stabilization procedure, when coupled with a refinement of the mesh in the proximity of the geometrical singularities, obtained by a simple subdivision of the triangles, greatly reduces the effect of these instabilities.

Impedance boundary conditions for the scattering of time-harmonic waves by rapidly varying surfaces

A method to build impedance boundary conditions incorporating the effect of rapid variations of a perfectly conducting surface on the scattering of a scalar, E-polarized, time-harmonic electromagnetic wave is presented. The amplitude and the extent of the variations are assumed to be comparable to each other and small as compared to the wavelength. The derivation of the impedance boundary conditions is based on a decomposition of the field in two parts. The first part describes the overall behavior of the wave and the second one deals with its small scale variations. The effective boundary conditions are rigorously constructed for periodic surfaces presenting a large-scale global periodicity to suppress the boundary effects and a small local period to describe the rapid variations. Numerical examples prove that the method can even be heuristically extended to more general problems. In this respect, there are reported some results related to the numerical treatment of small details on a smooth surface and of rough surfaces without resorting to refined meshes

Non-reflecting boundary conditions for waveguides

New non-reflecting boundary conditions are introduced for the solution of the Helmholtz equation in a waveguide. These boundary conditions are perfectly transparent for all propagating modes. They do not require the determination of these propagating modes but only their propagation constants. A quasi-local form of these boundary conditions is well suited as terminating boundary condition beyond finite element meshes. Related convergence properties to the exact solution and optimal error estimates are established.

Mathematical Justification of the Rayleigh Conductivity Model for Perforated Plates in Acoustics

This paper is devoted to the mathematical justification of the usual models predicting the effective reflection and transmission of an acoustic wave by a low porosity multiperforated plate. Some previous intuitive approximations require that the wavelength be large compared with the spacing separating two neighboring apertures. In particular, we show that this basic assumption is not mandatory. Actually, it is enough to assume that this distance is less than a half-wavelength. The main tools used are the method of matched asymptotic expansions and lattice sums for the Helmholtz equations. Some numerical experiments illustrate the theoretical derivations.