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7. Bernoulli Free Boundary Problems Under Uncertainty: The Convex Case

Author

Marc Dambrine, Helmut Harbrecht, and Benedicte Puig

Subjects
Computational Mathematics, Numerical Analysis, Applied Mathematics
Abstract

The present article is concerned with solving Bernoulli’s exterior free boundary problem in the case of an interior boundary that is random. We provide a new regularity result on the map that sends a parametrization of the inner boundary to a parametrization of the outer boundary. Moreover, assuming that the interior boundary is convex, also the exterior boundary is convex, which enables to identify the boundaries with support functions and to determine their expectations. We in particular construct a confidence region for the outer boundary based on Aumann’s expectation and provide a numerical method to compute it.

Published
2022
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12. A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations

Author

Marc Dambrine, Akhtar A. Khan, and Miguel Sama

Subjects
General Mathematics, General Engineering, General Physics and Astronomy
Abstract

Numerous applied models used in the study of optimal control problems, inverse problems, shape optimization, machine learning, fractional programming, neural networks, image registration and so on lead to stochastic optimization problems in Hilbert spaces. Under a suitable convexity assumption on the objective function, a necessary and sufficient optimality condition for stochastic optimization problems is a stochastic variational inequality. This article presents a new stochastic regularized second-order iterative scheme for solving a variational inequality in a stochastic environment where the primary operator is accessed by employing sampling techniques. The proposed iterative scheme, which fits within the general framework of the stochastic approximation approach, has its almost-sure convergence analysis given in a Hilbert space. We test the feasibility and the efficacy of the proposed stochastic approximation approach for a stochastic optimal control problem and a stochastic inverse problem, both associated with a second-order stochastic partial differential equation. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Published
2022

15. Shape sensitivity of eigenvalue functionals for scalar problems: computing the semi-derivative of a minimum

Author

Fabien Caubet, Marc Dambrine, Rajesh Mahadevan, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), and Universidad de Concepción - University of Concepcion [Chile]

Subjects
Control and Optimization, shape sensitivity analysis, Applied Mathematics, eigenvalues of elliptic operators, generalized boundary condition, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics::Spectral Theory, shape derivatives
Abstract

International audience; This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators involving a variety of boundary conditions. A uniform treatment of these problems is possible by considering them as a problem of calculating the semi-derivative of a minimum with respect to a parameter. The applicability of this approach, which can be traced back to the works of Danskin and Zolésio, to the treatment of eigenvalue problems (where the full shape derivative may not exist, due to multiplicity issues), has been illustrated by Zolésio. Despite this, some of the recent literature on the shape sensitivity of eigenvalue problems still continue to employ methods such as the material derivative method or Lagrangian methods which seem less adapted to this class of problems. The Delfour-Zolésio approach does not seem to be fully exploited in the existing literature: we aim to recall the importance and the simplicity of these ideas, by applying it to the analysis of the shape sensitivity for eigenvalue functionals for a class of elliptic operators in the scalar setting (Laplacian or diffusion in heterogeneous media), thus recovering known results in the case of Dirichlet or Neumann boundary conditions and obtaining new results in the case of Steklov or Wentzell boundary conditions.

Published
2022

16. Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

Author

Matteo Dalla Riva, Virginie Bonnaillie-Noël, Paolo Musolino, Marc Dambrine, Bonnaillie-Noel V., Dalla Riva M., Dambrine M., Musolino P., Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), Dipartimento di Matematica [Padova], Universita degli Studi di Padova, ANR-12-BS01-0021,ARAMIS,Analyse de méthodes asymptotiques robustes pour la simulation numérique en mécanique(2012), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Università degli Studi di Padova = University of Padua (Unipd)

Subjects
multiscale asymptotic expansion, multi-scale asymptotic expansion, Boundary (topology), 01 natural sciences, 35J25, 31B10, 45A05, 35B25, 35C20, Domain (mathematical analysis), Settore MAT/05 - Analisi Matematica, Situated, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Dirichlet problem, Laplace operator, real analytic continuation in Banach space, singularly perturbed perforated domain, Small hole, [MATH]Mathematics [math], 0101 mathematics, Representation (mathematics), Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, A domain, 010101 applied mathematics, Analysis
Abstract

For each pair (Formula presented.) of positive parameters, we define a perforated domain (Formula presented.) by making a small hole of size (Formula presented.) in an open regular subset (Formula presented.) of (Formula presented.) ((Formula presented.)). The hole is situated at distance (Formula presented.) from the outer boundary (Formula presented.) of the domain. Thus, when (Formula presented.) both the size of the hole and its distance from (Formula presented.) tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain (Formula presented.) and we denote its solution by (Formula presented.) Our aim is to represent the map that takes (Formula presented.) to (Formula presented.) in terms of real analytic functions of (Formula presented.) defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of (Formula presented.) to suitable subsets of (Formula presented.) we prove a global representation formula that holds on the whole of (Formula presented.) Such a formula allows us to rigorously justify multiscale expansions, which we subsequently construct.

Published
2021

17. A mathematical model for marine dinoflagellates blooms

Author

G. Vallet, Marc Dambrine, Bénédicte Puig, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
education.field_of_study, Extinction, biology, Applied Mathematics, 010102 general mathematics, Population, Dinoflagellate, biology.organism_classification, 01 natural sciences, 010101 applied mathematics, Oceanography, Discrete Mathematics and Combinatorics, Environmental science, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Finite time, education, Analysis, ComputingMilieux_MISCELLANEOUS
Abstract

We present a model for the life cycle of a dinoflagellate in order to describe blooms. We prove the mathematical well-posedness of the model and the possibility of extinction in finite time of the alga form meaning that the full population is under the cysts from.

Published
2021
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18. Shape Optimization for Composite Materials and Scaffold Structures

Author

Helmut Harbrecht, Marc Dambrine, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Optimal design, Scaffold, Materials science, Ecological Modeling, ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, General Physics and Astronomy, 010103 numerical & computational mathematics, General Chemistry, Microstructure, 01 natural sciences, Homogenization (chemistry), Computer Science Applications, 010101 applied mathematics, Modeling and Simulation, Shape optimization, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Composite material, ComputingMilieux_MISCELLANEOUS
Abstract

This article combines shape optimization and homogenization techniques by looking for the optimal design of the microstructure in composite materials and of scaffolds. The development of materials ...

Published
2020
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19. Oriented distance point of view on random sets

Author

Marc Dambrine, Bénédicte Puig, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Sequence, Control and Optimization, 010102 general mathematics, Open set, Hausdorff space, Boundary (topology), Signed distance function, 16. Peace & justice, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Control and Systems Engineering, Law of large numbers, Free boundary problem, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Central limit theorem, Mathematics
Abstract

Motivated by free boundary problems under uncertainties, we consider the oriented distance function as a way to define the expectation for a random compact or open set. In order to provide a law of large numbers and a central limit theorem for this notion of expectation, we also address the question of the convergence of the level sets of fn to the level sets of f when (fn) is a sequence of functions uniformly converging to f. We provide error estimates in term of Hausdorff convergence. We illustrate our results on a free boundary problem.

Published
2020
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20. Numerical solution of the homogeneous Neumann boundary value problem on domains with a thin layer of random thickness

Author

Bénédicte Puig, Isabelle Greff, Helmut Harbrecht, Marc Dambrine, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics and Astronomy (miscellaneous), [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 010103 numerical & computational mathematics, Boundary layer thickness, 01 natural sciences, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Neumann boundary condition, Free boundary problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, [MATH]Mathematics [math], 0101 mathematics, Mathematics, Numerical Analysis, Applied Mathematics, Mathematical analysis, Mixed boundary condition, Robin boundary condition, Computer Science Applications, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010101 applied mathematics, Thin layer equation, Computational Mathematics, Modeling and Simulation, Blasius boundary layer, Cauchy boundary condition, Random domain, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

ACL; International audience; The present article is dedicated to the numerical solution of homogeneous Neumann boundary value problems on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on the random domain can be transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Ventcell boundary condition and yields a second order accurate solution in the scale parameter epsilon of the layer's thickness. With the help of the Karhunen-Loeve expansion, we transform this random boundary value problem into a deterministic, parametric one with a possibly high-dimensional parameter y. Based on the decay of the random fluctuations of the layer's thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter y which are robust in the scale parameter epsilon. Numerical results validate our theoretical findings.

Published
2017
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21. A new method for the data completion problem and application to obstacle detection

Author

Marc Dambrine, Fabien Caubet, Helmut Harbrecht, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Basel], and University of Basel (Unibas)

Subjects
Laplace's equation, Cauchy problem, Computer science, Applied Mathematics, 010102 general mathematics, Inverse, Inverse problem, 01 natural sciences, 010101 applied mathematics, Obstacle, Obstacle problem, Applied mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, ComputingMilieux_MISCELLANEOUS
Abstract

The present article is devoted to the study of two well-known inverse problems, that is, the data completion problem and the inverse obstacle problem. The general idea is to reconstruct some boundary conditions and/or to identify an obstacle or void of different conductivity which is contained in a domain, from measurements of voltage and currents on the outer boundary of the domain. We focus here on Laplace's equation. First, we use a penalized Kohn-Vogelius functional in order to numerically solve the data completion problem, which consists in recovering some boundary conditions from partial Cauchy data. The functional to be minimized is quadratic, hence we compute its minimum by solving the linearized equation. Second, we propose to build an iterative method for the inverse obstacle problem based on the combination of the previously mentioned data completion subproblem and the so-called trial method. The underlying boundary value problems are efficiently computed by means of boundary integral equations and several numerical simulations show the applicability and feasibility of our new approach. For the numerical simulations, we focus on star-shaped domains in the two-dimensional case.

Published
2019
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22. An extremal eigenvalue problem for the Wentzell–Laplace operator

Author

Jimmy Lamboley, Djalil Kateb, and Marc Dambrine

Subjects
Pure mathematics, Conjecture, Applied Mathematics, 010102 general mathematics, Dimension (graph theory), Mathematics::Spectral Theory, 01 natural sciences, Upper and lower bounds, 010101 applied mathematics, Combinatorics, Operator (computer programming), Rayleigh–Faber–Krahn inequality, 0101 mathematics, Divide-and-conquer eigenvalue algorithm, Laplace operator, Mathematical Physics, Analysis, Eigenvalues and eigenvectors, Mathematics
Abstract

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Ventcel-Laplace operator of a domain $\Om$, involving only geometrical informations. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalue, and we conjecture a Faber-Krahn type inequality which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Ventcel eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.

Published
2016
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23. Minimization of the ground state of the mixture of two conducting materials in a small contrast regime

Author

Rajesh Mahadevan, Duver Quintero, Carlos Conca, and Marc Dambrine

Subjects
Dirichlet conditions, General Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, Geometry, 01 natural sciences, Homogenization (chemistry), 010101 applied mathematics, Elliptic operator, symbols.namesake, Surface-area-to-volume ratio, symbols, Shape optimization, 0101 mathematics, Ground state, Asymptotic expansion, Eigenvalues and eigenvectors, Mathematics
Abstract

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second-order approximation with respect to the mixture. We present numerical simulations in dimensions two and three to illustrate optimal distributions and the advantage of using a second-order method. Copyright © 2016 John Wiley & Sons, Ltd.

Published
2016
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24. A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

Author

Paolo Musolino, Virginie Bonnaillie-Noël, Matteo Dalla Riva, Marc Dambrine, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, The University of Tulsa, 800 South Tucker Drive, Tulsa, Oklahoma 74104, USA \& Department of Mathematics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, Aberystwyth University, `Progetto di Ateneo: Singular perturbation problems for differential operators -- CPDA120171/12' - University of Padova, HORIZON 2020 MSC EF project FAANon (grant agreement MSCA-IF-2014-EF - 654795) at the University of Aberystwyth, UK, `INdAM GNAMPA Project 2015 - Un approccio funzionale analitico per problemi di perturbazione singolare e di omogeneizzazione', ANR-12-BS01-0021,ARAMIS,Analyse de méthodes asymptotiques robustes pour la simulation numérique en mécanique(2012), European Project: 654795,H2020,H2020-MSCA-IF-2014,FAANon(2015), École normale supérieure - Paris (ENS-PSL), Bonnaillie-Noel V., Dalla Riva M., Dambrine M., Musolino P., Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Mathematics, The University of Tulsa, University of Tulsa, Dipartimento di Matematica [Padova], Universita degli Studi di Padova, and HORIZON 2020 MSC EF project FAANon (grant agreement MSCA-IF-2014-EF - 654795), `Progetto di Ateneo: Singular perturbation problems for differential operators -- CPDA120171/12' - University of Padova, `INdAM GNAMPA Project 2015 - Un approccio funzionale analitico per problemi di perturbazione singolare e di omogeneizzazione'

Subjects
Asymptotic analysis, General Mathematics, Boundary (topology), Asymptotic expansion, 01 natural sciences, 35J25, 31B10, 45A05, 35B25, 35C20, Mathematics - Analysis of PDEs, Settore MAT/05 - Analisi Matematica, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics (all), Mathematics - Numerical Analysis, 0101 mathematics, Mathematics, Dirichlet problem, Laplace's equation, Analytic continuation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, High Energy Physics::Phenomenology, Real analytic continuation in Banach space, Numerical Analysis (math.NA), Physics::Classical Physics, 010101 applied mathematics, asymptotic analysis, Laplace operator, Physics::Space Physics, Singularly perturbed perforated domain, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analytic function, Analysis of PDEs (math.AP)
Abstract

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain $\Omega_{\boldsymbol\varepsilon}$ obtained by making a small hole of size $\varepsilon_1 \varepsilon_2 $ in an open regular subset $\Omega$ of $\mathbb{R}^n$ at distance $\varepsilon_1$ from the boundary $\partial\Omega$. As $\varepsilon_1 \to 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $\boldsymbol\varepsilon \to (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $u_{\boldsymbol\varepsilon}$ the solution of a Dirichlet problem for the Laplace equation in $\Omega_{\boldsymbol\varepsilon}$. For a space dimension $n\geq 3$, we show that the function mapping $\boldsymbol\varepsilon$ to $u_{\boldsymbol\varepsilon}$ has a real analytic continuation in a neighborhood of $(0,0)$. By contrast, for $n=2$ we consider two different regimes: $\boldsymbol\varepsilon$ tends to $(0,0)$, and $\varepsilon_1$ tends to $0$ with $\varepsilon_2$ fixed. When $\boldsymbol\varepsilon\to(0,0)$, the solution $u_{\boldsymbol\varepsilon}$ has a logarithmic behavior; when only $\varepsilon_1\to0$ and $\varepsilon_2$ is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of $\varepsilon_1$. We also show that for $n=2$, the energy integral and the total flux on the exterior boundary have different limiting values in the two regimes. We prove these results by using functional analysis methods in conjunction with certain special layer potentials., Comment: combined with 1612.04637

Published
2018
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25. Incorporating knowledge on the measurement noise in electrical impedance tomography

Author

Bénédicte Puig, Helmut Harbrecht, Marc Dambrine, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Basel], and University of Basel (Unibas)

Subjects
Control and Optimization, Random field, Observational error, Computation, 010102 general mathematics, Inverse problem, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Control and Systems Engineering, Shape optimization, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Linear combination, Electrical impedance tomography, Algorithm, ComputingMilieux_MISCELLANEOUS, Mathematics, Voltage
Abstract

The present article is concerned with the identification of an obstacle or void of different conductivity which is included in a two-dimensional domain by measurements of voltage and currents at the boundary. In general, the voltage distribution is prescribed and hence deterministic. Whereas, the current distribution is measured and contains measurement errors. We assume that some information is given on these measurement errors which can be described by means of a random field. We exploit this extra knowledge by minimizing a linear combination of the expectation and the variance of the Kohn–Vogelius functional. It is shown how these ideas can be realized in numerical computations. By numerical results, the applicability and feasibility of our approach is demonstrated.

Published
2017
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26. A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid

Author

Djalil Kateb, Fabien Caubet, Marc Dambrine, Chahnaz Zakia Timimoun, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), Département de Mathématiques [Oran], and Université d'Oran 1 Ahmed Ben Bella [Oran]

Subjects
Hessian matrix, Control and Optimization, stationary Stokes problem, Boundary (topology), Type (model theory), 01 natural sciences, Regularization (mathematics), Domain (mathematical analysis), symbols.namesake, order two shape sensitivity, Discrete Mathematics and Combinatorics, Pharmacology (medical), Shape optimization, 0101 mathematics, Mathematics, 010102 general mathematics, Mathematical analysis, shape calculus, Inverse problem, 010101 applied mathematics, Modeling and Simulation, Bounded function, symbols, geometric inverse problem, Kohn-Vogelius functional, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 49Q10, 34A55, 49Q12, Analysis
Abstract

The aim of our work is to reconstruct an inclusion $\omega$ immersed in a fluid flowing in a larger bounded domain $\Omega$ via a boundary measurement on $\partial\Omega$. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem of reconstructing $\omega$ thanks to the tools of shape optimization by minimizing a Kohn-Vogelius type cost functional. We first characterize the gradient of this cost functional in order to make a numerical resolution. Then, in order to study the stability of this problem, we give the expression of the shape Hessian. We show the compactness of the Riesz operator corresponding to this shape Hessian at a critical point which explains why the inverse problem is ill-posed. Therefore we need some regularization methods to solve numerically this problem. We illustrate those general results by some explicit calculus of the shape Hessian in some particular geometries. In particular, we solve explicitly the Stokes equations in a concentric annulus. Finally, we present some numerical simulations using a parametric method.

Published
2013
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27. Interactions Between Moderately Close Inclusions for the Two-Dimensional Dirichlet–Laplacian

Author

Christophe Lacave, Virginie Bonnaillie-Noël, Marc Dambrine, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), ANR-13-BS01-0003,DYFICOLTI,DYnamique des Fluides, Couches Limites, Tourbillons et Interfaces(2013), ANR-12-BS01-0021,ARAMIS,Analyse de méthodes asymptotiques robustes pour la simulation numérique en mécanique(2012), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), École normale supérieure - Paris (ENS-PSL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Conformal map, 01 natural sciences, Dirichlet boundary conditions, Domain (mathematical analysis), symbols.namesake, 35J08, 35J05, Dimension (vector space), Dirichlet's principle, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, asymptotic expansion, conformal mapping, 35C20, Dirichlet conditions, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Spectral Theory, 35B25, 010101 applied mathematics, Computational Mathematics, Dirichlet laplacian, Dirichlet boundary condition, perforated domain, symbols, Asymptotic expansion, Analysis
Abstract

International audience; This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).

Published
2016
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28. Sard theorems for Lipschitz functions and applications in optimization

Author

Luc Barbet, Ludovic Rifford, Marc Dambrine, Aris Daniilidis, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Departament de Matemàtiques [Barcelona] (UAB), Universitat Autònoma de Barcelona (UAB), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Mathematics for Control, Transport and Applications (McTAO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)

Subjects
Discrete mathematics, Class (set theory), 021103 operations research, Clarke generalized Jacobian, General Mathematics, Lipschitz function, 010102 general mathematics, 0211 other engineering and technologies, Structure (category theory), Sard's theorem, 02 engineering and technology, AMS Subject Classification Primary 49J52 Secondary 58E05, 26B40, 16. Peace & justice, Lipschitz continuity, 01 natural sciences, Multi-objective optimization, semi-infinite optimization, Maxima and minima, Sard theorem, Countable set, Affine transformation, 0101 mathematics, [MATH]Mathematics [math], Mathematics
Abstract

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R d to R p that can be expressed as finite selections of C k functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet–Daniilidis–Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given.

Published
2016
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29. A first order approach for worst-case shape optimization of the compliance for a mixture in the low contrast regime

Author

Marc Dambrine, Antoine Laurain, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematical optimization, Asymptotic analysis, Control and Optimization, Level set method, Computation, education, 02 engineering and technology, 01 natural sciences, ComputingMethodologies_ARTIFICIALINTELLIGENCE, Displacement (vector), 0203 mechanical engineering, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Shape optimization, 0101 mathematics, [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS, health care economics and organizations, Mathematics, ANÁLISE ASSINTÓTICA, musculoskeletal system, Computer Graphics and Computer-Aided Design, Lamé parameters, Computer Science Applications, 010101 applied mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 020303 mechanical engineering & transports, surgical procedures, operative, Control and Systems Engineering, TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS, Displacement field, Asymptotic expansion, human activities, Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The purpose of this article is to propose a deterministic method for optimizing a structure considering its worst possible behaviour when a small uncertainty exists over its Lame parameters. The idea is to take advantage of the small parameter to derive an asymptotic expansion of the displacement and of the compliance with respect to the contrast in Lame coefficients. We are then able to compute the worst case design as post-treatment of the computation of the displacement field for the nominal parameters. The domain evolution resulting from the optimization is performed here using the level set method. The computational cost of our method remains of the same order as the cost of the optimization for a homogeneous material.

Published
2016
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30. On Bernoulli's free boundary problem with a random boundary

Author

Marc Dambrine, Michael H. Peters, Bénédicte Puig, Helmut Harbrecht, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, 0209 industrial biotechnology, Control and Optimization, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Boundary (topology), 02 engineering and technology, 01 natural sciences, 020901 industrial engineering & automation, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Free boundary problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, Boundary value problem, [MATH]Mathematics [math], 0101 mathematics, Mathematics, 010102 general mathematics, Mathematical analysis, Mixed boundary condition, Singular boundary method, Boundary knot method, Robin boundary condition, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Modeling and Simulation, Bernoulli's free boundary problem, random boundary, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Bernoulli process, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

ACL; International audience; This article is dedicated to the solution of Bernoulli's exterior free boundary problem in the situation of a random interior boundary. We provide the theoretical background that ensures the well-posedness of the problem under consideration and describe two different frameworks to define the expectation and the deviation of the resulting annular domain. The first approach is based on the Vorob'ev expectation, which can be defined for arbitrary sets. The second approach is based on the particular parametrization. In order to compare these approaches, we present analytical examples for the case of a circular interior boundary. Additionally, numerical experiments are performed for more general geometric configurations. For the numerical approximation of the expectation and the deviation, we propose a sampling method like the Monte Carlo or the quasi-Monte Carlo quadrature. Each particular realization of the free boundary is then computed by the trial method, which is a fixed-point-like iteration for the solution of Bernoulli's free boundary problem.

Published
2016
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31. Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations

Author

Djalil Kateb and Marc Dambrine

Subjects
Shape perturbation, Applied Mathematics, Mathematical analysis, Boundary (topology), Mixed boundary condition, Robin boundary condition, Boundary conditions in CFD, Free boundary problem, Neumann boundary condition, Cauchy boundary condition, Boundary value problem, Analysis, Ventcel’s boundary value problem, Mathematics
Abstract

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of the coefficient. This is achieved when physical situations are considered. Nevertheless, situations where this condition is violated appeared in several recent works where absorbing boundary conditions or equivalent boundary conditions on rough surfaces are sought for numerical purposes. The well-posedness of such problems was recently investigated: up to a countable set of parameters, existence and uniqueness of the solution for the Ventcel boundary value problem holds without the sign condition. However, the values to be avoided depend on the domain where the boundary value problem is set. In this work, we address the question of the persistency of the solvability of the boundary value problem under domain deformation.

Published
2012
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32. An Lp theory of linear elasticity in the half-space

Author

Chérif Amrouche, Marc Dambrine, and Yves Raudin

Subjects
Asymptotic analysis, Singular perturbation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Linear elasticity, Half-space, 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, General theory, Neumann boundary condition, Uniqueness, 0101 mathematics, Analysis, Mathematics
Abstract

Motived by the boundary values problem solved by correctors in the asymptotic analysis of singular perturbation of the domain, we consider Navier equations of linear elasticity in the half-space. We present a general theory of existence and uniqueness in the L p setting: we consider the weak solutions, the strong solutions and also very weak solutions.

Published
2012
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33. Mini-Workshop: Nonlinear Least Squares in Shape Identification Problems

Author

Frank Hettlich, Roland Potthast, and Marc Dambrine

Subjects
Levenberg–Marquardt algorithm, Iteratively reweighted least squares, Recursive least squares filter, Identification (information), Non-linear least squares, General Medicine, Total least squares, Non-linear iterative partial least squares, Algorithm, Least squares, Mathematics
Published
2011
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34. Effect of micro-defects on structure failure

Author

Virginie Bonnaillie-Noël, Grégory Vial, Marc Dambrine, Sébastien Tordeux, Delphine Brancherie, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mécanique et Technologie ( LMT ), École normale supérieure - Cachan ( ENS Cachan ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques et de leurs Applications [Pau] ( LMAP ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ), Advanced 3D Numerical Modeling in Geophysics ( Magique 3D ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire de Mécanique et Technologie (LMT), École normale supérieure - Cachan (ENS Cachan)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Advanced 3D Numerical Modeling in Geophysics (Magique 3D), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Asymptotic analysis, Singular perturbation, Work (thermodynamics), Structure (category theory), [ SPI.MECA.STRU ] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanics of the structures [physics.class-ph], 010103 numerical & computational mathematics, 01 natural sciences, Exact geometry, [ PHYS.MECA.STRU ] Physics [physics]/Mechanics [physics]/Mechanics of the structures [physics.class-ph], strong discontinuity, [PHYS.MECA.STRU]Physics [physics]/Mechanics [physics]/Structural mechanics [physics.class-ph], [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, singular perturbation, Mathematics, Coupling, Mechanical Engineering, Mathematical analysis, multi-scale asymptotic analysis, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], failure, 010101 applied mathematics, Discontinuity (linguistics), Mechanics of Materials, [SPI.MECA.STRU]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph], Modeling and Simulation, Micro defects, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; This work aims at taking into account the influence of geometrical defects on the behavior till complete failure of structures. This is achieved without any fine description of the exact geometry of the perturbations. The proposed strategy is based on two approaches : asymptotic analysis of Navier equations and strong discontinuity approach.; L'objectif de ce travail est de prendre en compte l'influence de la présence de défauts géométriques sur le comportement à rupture des structures et ce, sans description fine de la géométrie particulière des perturbations. L'approche proposée s'appuie sur deux outils : une analyse asymptotique des équations de Navier et l'utilisation de modèles à discontinuité forte.

Published
2010
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35. On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems

Author

Marc Dambrine and Djalil Kateb

Subjects
Dirichlet problem, Control and Optimization, Applied Mathematics, Mathematical analysis, symbols.namesake, Thermal conductivity, Dirichlet eigenvalue, Dirichlet boundary condition, symbols, Ball (mathematics), Differentiable function, Boundary value problem, Eigenvalues and eigenvectors, Mathematics
Abstract

We consider a two-phase problem in thermal conductivity: inclusions filled with a material of conductivity σ 1 are layered in a body of conductivity σ 2. We address the shape sensitivity of the first eigenvalue associated with Dirichlet boundary conditions when both the boundaries of the inclusions and the body can be modified. We prove a differentiability result and provide the expressions of the first and second order derivatives. We apply the results to the optimal design of an insulated body. We prove the stability of the optimal design thanks to a second order analysis. We also continue the study of an extremal eigenvalue problem for a two-phase conductor in a ball initiated by Conca et al. (Appl. Math. Optim. 60(2):173–184, 2009) and pursued in Conca et al. (CANUM 2008, ESAIM Proc., vol. 27, pp. 311–321, EDP Sci., Les Ulis, 2009).

Published
2010
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36. INTERACTIONS BETWEEN MODERATELY CLOSE INCLUSIONS FOR THE LAPLACE EQUATION

Author

Virginie Bonnaillie-Noël, Grégory Vial, Sébastien Tordeux, Marc Dambrine, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Mathématiques pour l'Industrie et la Physique (MIP), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques et de leurs Applications [Pau] ( LMAP ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ), Mathématiques pour l'Industrie et la Physique ( MIP ), Université Toulouse 1 Capitole ( UT1 ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), and Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Laplace's equation, Singular perturbation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Boundary (topology), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], First order, 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Modeling and Simulation, 0101 mathematics, Inclusion (mineral), Asymptotic expansion, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω0. This question has been studied extensively for a single inclusion or well-separated inclusions. In two-dimensional situations, we investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω0. We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.

Published
2009
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37. On the ersatz material approximation in level-set methods

Author

Djalil Kateb, Marc Dambrine, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), and Université de Technologie de Compiègne (UTC)

Subjects
Work (thermodynamics), Mathematical optimization, Control and Optimization, Level set method, education, Stability (learning theory), 010103 numerical & computational mathematics, Approx, ComputingMethodologies_ARTIFICIALINTELLIGENCE, 01 natural sciences, Second order analysis, Level set, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Shape optimization, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, health care economics and organizations, Mathematics, musculoskeletal system, 010101 applied mathematics, Computational Mathematics, surgical procedures, operative, Control and Systems Engineering, TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS, Method of steepest descent, human activities
Abstract

The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approx- imation (Allaire et al., J. Comput. Phys. 194 (2004) 363-393), a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limi- tations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error committed by using the ersatz material approximation and, on a model case, explain that they amplifies instabilities by a second order analysis of the objective function.

Published
2009
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38. A remark on precomposition on H1/2(S1) and ε-identifiability of disks in tomography

Author

Djalil Kateb and Marc Dambrine

Subjects
Pure mathematics, Applied Mathematics, Mathematical analysis, Inverse, Conformal map, Unit disk, Sobolev space, symbols.namesake, Fourier analysis, symbols, Uniqueness, Diffeomorphism, Fourier series, Analysis, Mathematics
Abstract

We consider the inverse conductivity problem with one measurement for the equation div ( ( σ 1 + ( σ 2 − σ 1 ) χ ω ) ∇ u ) = 0 determining the unknown inclusion ω included in Ω. We suppose that Ω is the unit disk of R 2 . With the tools of the conformal mappings, of elementary Fourier analysis and by studying how W 1 , ∞ ( S 1 , S 1 ) diffeomorphisms act by precomposition on the Sobolev space H 1 / 2 ( S 1 ) , we show how to approximate the Dirichlet-to-Neumann map when the original inclusion ω is a e-approximation of a disk. This enables us to give some uniqueness and stability results.

Published
2008
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39. Shape Methods for the Transmission Problem with a Single Measurement

Author

Lekbir Afraites, Marc Dambrine, and Djalil Kateb

Subjects
Control and Optimization, Current (mathematics), Mathematical analysis, Inverse, Conformal map, Inverse problem, Computer Science Applications, Signal Processing, Point (geometry), Shape optimization, Anomaly (physics), Boundary element method, Analysis, Mathematics
Abstract

In the current work, we consider the inverse conductivity problem of recovering inclusion with one measurement. First, we use conformal mapping techniques for determining the location of the anomaly and estimating its size. We then get a good initial guess for quasi-Newton type method. The inverse problem is treated from the shape optimization point of view. We give a rigorous proof for the existence of the derivative of the state function and of shape functionals. We consider both least squares fitting and Kohn and Vogelius functionals. For the numerical implementation, we use a parameterization of shapes coupled with a boundary element method. Several numerical examples indicate the superiority of the Kohn and Vogelius functional over least squares fitting.

Published
2007
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40. Conformal mapping and inverse conductivity problem with one measurement

Author

Djalil Kateb and Marc Dambrine

Subjects
Control and Optimization, Extremal length, Series (mathematics), Conformal field theory, Mathematical analysis, Boundary (topology), Conformal map, Function (mathematics), Inverse problem, Computational Mathematics, Riemann hypothesis, symbols.namesake, Control and Systems Engineering, symbols, Mathematics
Abstract

This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.

Published
2007
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41. A multiscale correction method for local singular perturbations of the boundary

Author

Grégory Vial, Marc Dambrine, Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire de Mathématiques Appliquées [Compiegne] ( LMAC ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Laplace's equation, Numerical Analysis, Singular perturbation, Fictitious domain method, Applied Mathematics, Numerical analysis, 010102 general mathematics, Mathematical analysis, Boundary (topology), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Computational Mathematics, Shape optimization, 35J25, Modeling and Simulation, Multiscale asymptotic analysis, Patch of elements, 0101 mathematics, Asymptotic expansion, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, Energy functional, Mathematics
Abstract

International audience; In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uε of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uε based on a multiscale superposition of the unperturbed solution u0 and a profile defined in a model domain. We conclude with numerical results.

Published
2007
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42. Artificial conditions for the linear elasticity equations

Author

Grégory Vial, Virginie Bonnaillie-Noël, Frédéric Hérau, Marc Dambrine, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon, Département de Mathématiques et Applications - ENS Paris ( DMA ), École normale supérieure - Paris ( ENS Paris ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques et de leurs Applications [Pau] ( LMAP ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques Jean Leray ( LMJL ), Université de Nantes ( UN ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Paris (ENS-PSL), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Singular perturbation, Spectral theory, Linear elasticity equations, artificial boundary conditions, Dirichlet-to-Neumann map, 35J47, 35J57, 35P10, 35S15, 47A10, 47G30, 65N20, Ventcel condition, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Neumann boundary condition, Countable set, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Fourier series, singular perturbation, ComputingMilieux_MISCELLANEOUS, Mathematics, Algebra and Number Theory, Applied Mathematics, Linear elasticity, Mathematical analysis, Mixed boundary condition, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], spectral theory, Computational Mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

34 pages; In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the well-posedness except for a countable set of parameters. A perturbation argument allows to consider near-circular domains. We complete the analysis by some numerical simulations.

Published
2015
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43. About stability of equilibrium shapes

Author

Michel Pierre and Marc Dambrine

Subjects
Numerical Analysis, Pure mathematics, Partial differential equation, Applied Mathematics, Order (ring theory), Stability (probability), Computational Mathematics, Modeling and Simulation, Calculus, A priori and a posteriori, Shape optimization, Differentiable function, Analysis, Energy functional, Second derivative, Mathematics
Abstract

We discuss the stability of "critical" or "equilibrium" shapes of a shape-dependent energy functional. We analyze a problem arising when looking at the positivity of the second derivative in order to prove that a critical shape is an optimal shape. Indeed, often when positivity -or coercivity- holds, it does for a weaker norm than the norm for which the functional is twice differentiable and local optimality cannot be a priori deduced. We solve this problem for a particular but significant example. We prove "weak-coercivity" of the second derivative uniformly in a "strong" neighborhood of the equilibrium shape.

Published
2000
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44. Shape optimization methods for the Inverse Obstacle Problem with generalized impedance boundary conditions

Author

Djalil Kateb, Fabien Caubet, Marc Dambrine, Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Hessian matrix, generalized impedance boundary conditions, Boundary (topology), identifiability result, 010103 numerical & computational mathematics, Ventcel boundary conditions, 01 natural sciences, Theoretical Computer Science, symbols.namesake, 49Q10, 49Q12, 34A55, 35Q93, Free boundary problem, Shape optimization, Uniqueness, Boundary value problem, 0101 mathematics, Mathematical Physics, Mathematics, Applied Mathematics, Mathematical analysis, Inverse problem, shape calculus, Computer Science Applications, 010101 applied mathematics, Signal Processing, Obstacle problem, symbols, geometric inverse problem, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

International audience; We aim to reconstruct an inclusion ω immersed in a perfect fluid flowing in a larger bounded domain Ω via boundary measurements on ∂Ω. The obstacle ω is assumed to have a thin layer and is then modeled using generalized boundary conditions (precisely Ventcel boundary conditions). We first obtain an identifiability result (i.e. the uniqueness of the solution of the inverse problem) for annular configurations through explicit computations. Then, this inverse problem of reconstructing ω is studied thanks to the tools of shape optimization by minimizing a least squares type cost functional. We prove the existence of the shape derivatives with respect to the domain ω and characterize the gradient of this cost functional in order to make a numerical resolution. We also characterize the shape Hessian and prove that this inverse obstacle problem is unstable in the following sense: the functional is degenerated for highly oscillating perturbations. Finally, we present some numerical simulations in order to confirm and extend our theoretical results.

Published
2013

45. The Morse-Sard theorem for Clarke critical values

Author

Marc Dambrine, Aris Daniilidis, Luc Barbet, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Institut pluridisciplinaire de recherche appliquée dans le domaine du génie pétrolier (IPRADDGP), Departament de Matemàtiques [Barcelona] (UAB), and Universitat Autònoma de Barcelona (UAB)

Subjects
Lebesgue measure, Euclidean space, General Mathematics, 010102 general mathematics, Null (mathematics), Mathematical analysis, Sard's theorem, Zero (complex analysis), Lipschitz continuity, 01 natural sciences, 010101 applied mathematics, Combinatorics, Set (abstract data type), Countable set, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics
Abstract

The Morse–Sard theorem states that the set of critical values of a C k smooth function defined on a Euclidean space R d has Lebesgue measure zero, provided k ≥ d . This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of C k functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null.

Published
2013
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46. On the necessity of Nitsche term. Part II: An alternative approach

Author

Pierre Villon, Marc Dambrine, G. Dupire, Jean-Paul Boufflet, Heuristique et Diagnostic des Systèmes Complexes [Compiègne] (Heudiasyc), Université de Technologie de Compiègne (UTC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), and Roberval (Roberval)

Subjects
Numerical Analysis, Non-matching grid, Discretization, Applied Mathematics, Mathematical analysis, Finite elements, 010103 numerical & computational mathematics, Bilinear form, 01 natural sciences, Finite element method, Domain (mathematical analysis), Mathematics::Numerical Analysis, 010101 applied mathematics, Computational Mathematics, symbols.namesake, Nitsche method, Minimum bounding box, Dirichlet boundary condition, symbols, Shape optimization, Boundary value problem, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; The aim of this article is to explore the possibility of using a family of fixed finite element shape functions that does not match the domain to solve a boundary value problem with Dirichlet boundary condition. The domain is embedded in a bounding box and the finite element approximation is associated to a regular structured mesh of the box. The shape of the domain is independent of the discretization mesh. In these conditions, a meshing tool is never required. This may be especially useful in the case of evolving domains, for example shape optimization or moving interfaces. Nitsche method has been intensively applied. However, Nitsche is weighted with the mesh size h and therefore is a purely discrete point of view with no interpretation in terms of a continuous variational approach associated with a boundary value problem. In this paper, we introduce an alternative to Nitsche method which is associated with a continuous bilinear form. This extension has strong restrictions: it needs more regularity on the data than the usual method. We prove the well-posedness of our formulation and error estimates. We provide numerical comparisons with Nitsche method.

Published
2012
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47. Artificial boundary conditions to compute correctors in linear elasticity

Author

Delphine Brancherie, Virginie Bonnaillie-Noël, Grégory Vial, Marc Dambrine, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Roberval (Roberval), Université de Technologie de Compiègne (UTC), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Roberval, Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques et de leurs Applications [Pau] ( LMAP ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), and Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Numerical Analysis, linear elasticity equations, Numerical analysis, Mathematical analysis, Linear elasticity, 010103 numerical & computational mathematics, Mixed boundary condition, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, Computer Science::Digital Libraries, Robin boundary condition, 010101 applied mathematics, Boundary conditions in CFD, Neumann boundary condition, Free boundary problem, Boundary value problem, 0101 mathematics, transparent boundary conditions, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

We present the derivation of a transparent boundary condition of order two to solve the equations of linear elasticity in a half plane. The resolution of the boundary value problem leads to a noncoercive variational formulation. We also present some numerical examples.

Published
2012
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48. Small Defects in Mechanics

Author

Virginie Bonnaillie-Noël, Marc Dambrine, Grégory Vial, Theodore E. Simos, George Psihoyios, Ch. Tsitouras, and Zacharias Anastassi

Subjects
Approximation theory, Feature (computer vision), Computer science, Computation, Applied mathematics, Boundary value problem, Navier–Stokes equations, Asymptotic expansion, Domain (mathematical analysis)
Abstract

In this paper, we present a method to compute rapidly the solution of the Navier equation in domains with small inclusions close to each other. The main feature of our method is the use of a coarse description of the geometry. The computation relies on asymptotic expansion and computation of profiles, which are solution of a problem posed in unbounded domain. We propose and compare several artificial boundary conditions to compute these profiles efficiently.

Published
2011
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49. Detecting an obstacle immersed in a fluid by shape optimization methods

Author

Marc Dambrine, Mehdi Badra, Fabien Caubet, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Hessian matrix, 01 natural sciences, Regularization (mathematics), Order two shape sensitivity, symbols.namesake, Operator (computer programming), Geometric inverse problem, Stationary Stokes problem, Neumann boundary condition, Shape optimization, 0101 mathematics, Mathematics, AMS : 35R30, 49Q10, 35Q30, (76D07), Applied Mathematics, 010102 general mathematics, Mathematical analysis, 16. Peace & justice, Sensitivity with respect to the domain, 010101 applied mathematics, Parameter identification problem, Nonlinear system, Shape calculus, Modeling and Simulation, symbols, Identifiability, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

International audience; The paper presents a theoretical study of an identification problem by shape optimization methods. The question is to detect an object immersed in a fluid. Here, the problem is modeled by the Stokes equations and treated as a nonlinear least-squares problem. We consider both the Dirichlet and Neumann boundary conditions. Firstly, we prove an identifiability result. Secondly, we prove the existence of the first order shape derivatives of the state, we characterize them and deduce the gradient of the least-squares functional. Moreover, we study the stability of this setting. We prove the existence of the second order shape derivatives and we give the expression of the shape Hessian. Finally, the compactness of the Riesz operator corresponding to this shape Hessian is shown and the ill-posedness of the identification problem follows. This explains the need of regularization to numerically solve this problem.

Published
2011

50. On the necessity of Nitsche term

Author

Marc Dambrine, Pierre Villon, G. Dupire, Jean-Paul Boufflet, Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), Heuristique et Diagnostic des Systèmes Complexes [Compiègne] (Heudiasyc), Université de Technologie de Compiègne (UTC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), and Roberval (Roberval)

Subjects
Well-posed problem, Dirichlet problem, Numerical Analysis, Numerical linear algebra, Non-matching grid, Discretization, Applied Mathematics, Finite elements, 010103 numerical & computational mathematics, 16. Peace & justice, computer.software_genre, 01 natural sciences, Finite element method, 010101 applied mathematics, Computational Mathematics, Minimum bounding box, Calculus, Applied mathematics, Shape optimization, Boundary value problem, 0101 mathematics, computer, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; The aim of this article is to explore the possibility of using a family of fixed finite elements shape functions to solve a Dirichlet boundary value problem with an alternative variational formulation. The domain is embedded in a bounding box and the finite element approximation is associated to a regular structured mesh of the box. The shape of the domain is independent of the discretization mesh. In these conditions, a meshing tool is never required. This may be especially useful in the case of evolving domains, for example shape optimization or moving interfaces. This is not a new idea, but we analyze here a special approach. The main difficulty of the approach is that the associated quadratic form is not coercive and an inf-sup condition has to be checked. In dimension one, we prove that this formulation is well posed and we provide error estimates. Nevertheless, our proof relying on explicit computations is limited to that case and we give numerical evidence in dimension two that the formulation does not provide a reliable method. We first add a regularization through a Nitscheterm and we observe that some instabilities still remain. We then introduce and justify a geometrical regularization. A reliable method is obtained using both regularizations.

Published
2010
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