Résumés des Rapports de Recherche parus en 2002 ENIT

Résumés des Rapports de Recherche parus en 2002
ENIT-LAMSIN
RR 02-01
A Recovery of cracks from incomplete boundary data
Alain Cimetière(1) , Franck Delvare(2) , Mohamed Jaoua(3) , Moez Kallel(3) , F. Pons(1)
(1) Université de Poitiers & ENSMA, Laboratoire de Modélisation Mécanique et de Mathématiques Appliquées, Boulevard Pierre
et Marie Curie, BP 30179, 86962 Chasseneuil Futuroscope Cedex, France.
e-mail : [email protected]
(2) ENSI de Bourges, Laboratoire Energétique Explosions Structures, 10 Boulevard Lahitolle, 18020 Bourges Cedex, France.
e-mail : [email protected], [email protected]
(3) LAMSIN, Ecole Nationale d’Ingénieurs de Tunis, B.P. 37, 1002 Tunis, Tunisie.
e-mail : [email protected], [email protected]
Abstract : We are interested in this paper in recovering line segment (or planar) cracks from incomplete
boundary measurements. Most recovery algorithms, and especially fast ones based on the reciprocity gap,
however need a full set of boundary data, which compels us to extending the available data prior to achieving
recovery. Both the extension and recovery problems being severely ill posed, their stacking might drive the
whole process to a blowing up, unless the instabilities are properly handled. Numerical results actually prove
the so-built composite algorithm to hold good accuracy and robustness features.
Keywords : Geometrical inverse problems, 2D Laplace operator, cracks detection and recovery, data completion,
Cauchy problem, regularization
AMS subject classification : 65N21, 41A29, 68W25
RR 02-02
Stabilization and control for the subcritical semilinear wave equation
B. Dehman(1) , G. Lebeau(2) , E. Zuazua(3)
(1)FST, 1060 Tunis, Tunisie
e-mail : [email protected]
(2) Centre de Mathématique, Ecole polytechnique, Palaiseaau, France.
e-mail : [email protected]
(3)Departamento de Matematicas, Facultad de Ciencias, Madrid, Spain
e-mail : [email protected]
Abstract : In this paper, we analyse the exponential decay property of solutions of the semilinear wave equation
in R3 with a damping term which is effective on the exterior of a ball. Under suitable and natural assumptions
on the nonlinearity we prove that nonlinearity is subcritical at infinity. Subcritically means, roughly speaking,
that the nonlinearity grows at infinity at most as a power p < 5. The method of proof combines classical energy
estimates for the energy localized in the exterior of a ball, Strichart’s estimates and results bu P. Gérard on
microlocal defect measures and linearizable sequences. We also give an application to the controllability of the
semilinear wave equation under the same growth condition on the nonlinearity but provided the nonlinearity
has been cut-off away from the boundary.
Keywords :semilinear waves, stabilization, microlocal defect measures, exact controllability AMS subject
classification : 35B35, 35L05
RR 02-03
Regularization of parameter estimation by adaptive discretization using refinement and coarsening indicators
H. Benameur(1) , B. Kaltenbacher(2)
(1)FSB et Enit-Lamsin, Tunis, Tunisie
e-mail : [email protected]
(2) SFB 013, Johannes Kepler University of Linz
e-mail : [email protected]
Abstract : This paper is concerned with the ill-posed problem of identifying a hydraulic transmissivity in an
isotropic and confined aquifer in two space dimensions. To define a regularization by adaptive discretization of
the parameter, we use refinement and coarsening or removing degrees of freedom to a current set of parameters.
The direct problem is discretized by a mixed hybrid finite element method. A combination of the direct problem
discretization with adaptive discretization for the transmissivity allows us to prove convergence of the algorithm
as the mesh size goes to zero, stability of the finite-dimensional approximation, and convergence for noisy data
with an appropriate stopping rule
Keywords :Parameter estimation, adaptive discretization, refinement indicators, regularisation, hydraulic
transmissivity AMS subject classification : 65N21, 65N12, 35R25
RR 02-04
Singular Perturbation for Dirichlet Boundary Control of Elliptic Problems
Faker Ben Belgacem(1) , Henda El Fekih(2) , Hajer Metoui(2)
(1) MIP (UMR CNRS 5640). Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France.
e-mail : [email protected]
(2) LAMSIN, Ecole Nationale d’Ingénieurs de Tunis, B.P. 37, 1002 Tunis, Tunisie.
e-mail : [email protected], [email protected]
Abstract : A current procedure that takes into account the Dirichlet boundary condition with non-smooth data
is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure
is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem
where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap
between the penalized and the non-penalized boundary controls for the small penalization parameter. Some
numerical results are reported on to highlight the reliability of such an approach.
Keywords : Boundary control problems, non-smooth Dirichlet condition, Robin penalization, singularly
perturbed problem.
AMS subject classification : 49N05, 49N10, 34D15
RR 02-05
On handling the boundary conditions at infinity for some exterior problems by the alternating Schwarz method?
Faker Ben Belgacem(1) , Michel Fournié(1) , Nabil Gmati(2) , Faten Jelassi(2)
(1) UMR CNRS 5640, Laboratoire de Mathématiques pour l’Industrie et la Physique,
118 Route de Narbonne, 31062 Toulouse Cedex, France.
e-mail : [email protected], [email protected]
(2) LAMSIN, Ecole Nationale d’Ingénieurs de Tunis, B.P. 37, 1002 Tunis, Tunisie.
e-mail : [email protected], [email protected]
Abstract : We propose an iterative algorithm to solve the exterior Helmholtz and Poisson problems. It can
be written as an alternating Schwarz method which allows us to state a geometrical convergence result in the
elliptic case.
Keywords : Schwarz method, integral formulae, coupling FEM/BEM method
AMS subject classification : 65N30, 65N38, 65N55
RR 02-06
The topological asymptotic expansion for the Quasi-Stokes problem
Maatoug Hassine(1) , Mohamed Masmoudi(2)
(1) LAMSIN, Ecole Nationale d’Ingénieurs de Tunis, B.P. 37, 1002 Tunis, Tunisie.
e-mail : [email protected]
(2) MIP, UMR 5640, Université de Paul Sabatier, 118 route de Narbonne, 31032 Toulouse cedex4, France.
e-mail : [email protected]
Abstract : In this paper, we propose a topological sensitivity analysis for the Quasi-Stokes equations. It consists
in an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The
leading term of this expansion is related to the principal part of the operator. The theoretical part of this work
is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize
the locations of a fixed number of air injectors in an eutrophized lake.
Keywords : topological optimization, topological sensitivity, Quasi-Stokes equations, topological gradient,
shape optimization.
AMS subject classification : 49Q10, 49Q12, 74P05, 74P10, 74P15
RR 02-07
Simulation of muffler’s by a homogenized finite element method
A.S Bonnet-Ben Dhia(1) , Dorra Drissi(2) , Nabil Gmati(2)
(1) ENSTA-SMP, UMR 2706 du CNRS, 32 Boulevard Victor 75739 Paris, Cedex 15.
e-mail : [email protected]
(2) LAMSIN, Ecole Nationale d’Ingénieurs de Tunis, B.P. 37, 1002 Tunis, Tunisie.
e-mail : [email protected], [email protected]
Abstract : In this work, we are interested in the modelling of the acoustic attenuation of exhaust mufflers
including perforated ducts, and its numerical computation. The study is worked out in harmonic time regime,
for the two-dimensional case. The hole diameter and the center-to-center distance between consecutive holes
are supposed of same order, and small compared to the size of the muffler. The formulation is derived by using
multiscale techniques and matching the asymptotic expansions. The numerical method couples finite elements
in the muffler with modal decomposition in the inlet and the outlet of the duct.
Keywords : Muffler, perforated duct, Helmholtz equation, homogenization, finite elements.
AMS subject classification :
RR 02-08
Scattering and exponential decay of the local energy for the solutions of semilinear and subcritical wave equation
outside convex obstacle
Ahmed Bchatnia(1,2) , Moez Daoulatli(1)
(1) LAMSIN, Ecole Nationale d’Ingénieurs de Tunis, B.P. 37, 1002 Tunis, Tunisie.
e-mail : [email protected]
(2) Faculté des Sciences de Tunis, Département de Mathématiques, 1060 Tunis, Tunisie.
e-mail : [email protected]
Abstract : In this paper, we prove a Scattering theorem for the wave equation with localized subcritical
semilinearity outside convex obstacle; then we deduce the exponential decay of local energy. The proof relies
on generalized Strichartz estimates, and microlocal defect measures.
Keywords : emilinear waves, scattering, microlocal defect measures, exponential decay.
AMS subject classification : 35B35, 35L05