arXiv subjects

Eric Bonnetier

Publications and source records attributed to Eric Bonnetier.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic expansion of the voltage potential under small Robin perturbations of its boundary conditions

Inspired by questions related to inverse problems and shape optimization, we derive an asymptotic expansion of the voltage potential, solution to a model elliptic second-order partial differential equation, under small perturbations of its boundary conditions. More precisely, the homogeneous Dirichlet or homogeneous Neumann boundary condition in a fixed, reference configuration of the problem is replaced by a Robin boundary condition with admittance $k_\varepsilon > 0$ on a ``small'' subset $\omega_\varepsilon$ of the boundary of the ambient domain, vanishing at the limit $\varepsilon \to 0$. In each of these two situations, a general asymptotic formula is established for the voltage potential, which rests on minimal assumptions about the shape of the vanishing subset $\omega_\varepsilon$ and the parameter $k_\varepsilon$. The scalings of these expansions, capturing the intensity of the perturbation, are measured by new quantities called ``Robin-Dirichlet'' capacity or a ``Robin-Neumann'' capacity, which depend on the geometry of $\omega_\varepsilon$ and on the value of the parameter $k_\varepsilon$. We analyze how these quantities compare to more classical measures of the ``smallness'' of $\omega_\varepsilon$, such as the capacity or the Neumann capacity of $\omega_\varepsilon$, according to the behavior of $k_\varepsilon$ as $\varepsilon \to 0$.

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arXiv2025

Numerical shape and topology optimization of regions supporting the boundary conditions of a physical problem

This article deals with a particular class of shape and topology optimization problems: the optimized design is a region $G$ of the boundary $\partial \Omega$ of a given domain $\Omega$, which supports a particular type of boundary conditions in the considered physical problem. In our analyses, we develop adapted versions of the notions of shape and topological derivatives, which are classically tailored to functions of a ``bulk'' domain. This leads to two complementary notions of derivatives for a quantity of interest $J(G)$ depending on a region $G \subset \partial \Omega$: on the one hand, we elaborate on the boundary variation method of Hadamard for evaluating the sensitivity of $J(G)$ with respect to ``small'' perturbations of the boundary of $G$ within $\partial \Omega$. On the other hand, we use techniques from asymptotic analysis to appraise the sensitivity of $J(G)$ with respect to the addition of a new connected component to the region $G$, shaped as a ``small'' surface disk. The calculation of both types of derivatives raises original difficulties, which are carefully detailed in a simple mathematical setting based on the conductivity equation. We notably propose formal arguments to calculate our derivatives with a minimum amount of technicality, and show how they can be generalized to handle more intricate problems, arising for instance in the contexts of acoustics and structural mechanics, respectively governed by the Helmholtz and linear elasticity equations. In numerical applications, our derivatives are incorporated into a recent algorithmic framework for tracking arbitrarily dramatic motions of a region $G$ within a fixed ambient surface, which combines the level set method with remeshing techniques to offer a clear, body-fitted discretization of the evolving region. Finally, various 3d numerical examples are presented to illustrate the salient features of our analysis.

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arXiv2024

Networks of sign-changing metamaterials: existence and spectral properties

We study composite assemblages of dielectrics and metamaterials with respectively positive and negative material parameters. In the continuum case, for a scalar equation, such media may exhibit so-called plasmonic resonances for certain values of the (negative) conductivity in the metamaterial. This work investigates such resonances, and the associated eigenfunctions, in the case of composite conducting networks. Unlike the continuous media, we show a surprising specific dependence on the geometry of the network of the resonant values. We also study how the problem is affected by the choice of boundary conditions on the external nodes of the structure.

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arXiv2023

An Open Waveguide with a Thin High Contrast Core Layer: Asymptotic Analysis and Inverse Detection Problem

We investigate the Helmholtz equation in a two dimensional open waveguide with a thin and high contrast core layer. We develop an asymptotic analysis of the Green function of the problem, and through it we identify and characterize the appearance of resonant frequencies. For waves originating outside of the core, the waveguide response at these resonant frequencies is vastly different than the response at non-resonant frequencies. Using this phenomenon and multifrequency measurements containing the first resonance, we propose, theoretically analyze, and numerically validate a reconstruction algorithm to identify the location, thickness and index of refraction of the core layer.

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arXiv2022

Lamb modes and Born approximation for small shape defects inversion in elastic plates

The aim of this work is to present theoretical tools to study wave propagation in elastic waveguides and perform multi-frequency scattering inversion to reconstruct small shape defects in a 2D and 3D elastic plate. Given surface multi-frequency wavefield measurements, we use a Born approximation to reconstruct localized defect in the geometry of the plate. To justify this approximation, we introduce a rigorous framework to study the propagation of elastic wavefield generated by arbitrary sources. By studying the decreasing rate of the series of inhomogeneous Lamb mode, we prove the well-posedness of the PDE that model elastic wave propagation in 2D and 3D planar waveguides. We also characterize the critical frequencies for which the Lamb decomposition is not valid. Using these results, we generalize the shape reconstruction method already developed for acoustic waveguide to 2D elastic waveguides and provide a stable reconstruction method based on a mode-by-mode spacial Fourier inversion given by the scattered field.

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arXiv2022

The Helmholtz problem in slowly varying waveguides at locally resonant frequencies

This article aims to present a general study of the Helmholtz problem in slowly varying waveguides. This work is of particular interest at locally resonant frequencies, where a phenomenon close to the tunnel effect for Schr\"odinger equation in quantum mechanics can be observed. In this situation, locally resonant modes propagate in the waveguide under the form of Airy functions. Using previous mathematical results on the Schr\"odinger equation, we prove the existence of a unique solution to the Helmholtz source problem with outgoing conditions in such waveguides. We provide an explicit modal approximation of this solution, as well as a control of the approximation error in H1loc. The main theorem is proved in the case of a waveguide with a monotonously varying profile and then generalized using a matching strategy. We finally validate the modal approximation by comparing it to numerical solutions based on the finite element method.

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arXiv2022

Small defects reconstruction in waveguides from multifrequency one-side scattering data

Localization and reconstruction of small defects in acoustic or electromagnetic waveguides is of crucial interest in nondestructive evaluation of structures. The aim of this work is to present a new multi-frequency inversion method to reconstruct small defects in a 2D waveguide. Given one-side multi-frequency wave field measurements of propagating modes, we use a Born approximation to provide a L2-stable reconstruction of three types of defects: a local perturbation inside the waveguide, a bending of the waveguide, and a localized defect in the geometry of the waveguide. This method is based on a mode-by-mode spacial Fourier inversion from the available partial data in the Fourier domain. Indeed, in the available data, some high and low spatial frequency information on the defect are missing. We overcome this issue using both a compact support hypothesis and a minimal smoothness hypothesis on the defects. We also provide a suitable numerical method for efficient reconstruction of such defects and we discuss its applications and limits.

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arXiv2021

Approximation of multiphase mean curvature flows with arbitrary nonnegative mobilities

This paper is devoted to the robust approximation with a variational phase field approach of multiphase mean curvature flows with possibly highly contrasted mobilities. The case of harmonically additive mobilities has been addressed recently using a suitable metric to define the gradient flow of the phase field approximate energy. We generalize this approach to arbitrary nonnegative mobilities using a decomposition as sums of harmonically additive mobilities. We establish the consistency of the resulting method by analyzing the sharp interface limit of the flow: a formal expansion of the phase field shows that the method is of second order. We propose a simple numerical scheme to approximate the solutions to our new model. Finally, we present some numerical experiments in dimensions 2 and 3 that illustrate the interest and effectiveness of our approach, in particular for approximating flows in which the mobility of some phases is zero.

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arXiv2021

Small perturbations in the type of boundary conditions for an elliptic operator

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this ``background'' situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a ``small'' subset $\omega_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a ``small'' subset $\omega_\varepsilon $ of the Dirichlet boundary. The relevant quantity that measures the ``smallness'' of the subset $\omega_\varepsilon $ differs in the two cases: while it is the harmonic capacity of $\omega_\varepsilon $ in the former case, we introduce a notion of ``Neumann capacity'' to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general $\omega_\varepsilon $, under the sole assumption that it is ``small'' in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset $\omega_\varepsilon $ is a vanishing surfacic ball.

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arXiv2020

A shear-compression damage model for the simulation of underground mining by block caving

Block caving is an ore extraction technique used in the copper mines of Chile. It uses gravity to ease the breaking of rocks, and to facilitate the extraction from the mine of the resulting mixture of ore and waste. To simulate this extraction process numerically and better understand its impact on the mine environment, we study 3 variational models for damage, based on the gradient damage model of Pham and Marigo (2010). In these models, the damage criterion may exhibit an anisotropic dependence on the spherical and deviatoric parts of the stress tensor. We report simulations that satisfactorily represent the expected evolution of the stress field in a block caving operation.

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arXiv2019

Stability for quantitative photoacoustic tomography revisited

This paper is concerned with the stability issue in determining absorption and diffusion coefficients in quantitative photoacoustic imaging. Assuming that the optical wave is generated by point sources in a region where the optical coefficients are known, we derive pointwise H{\"o}lder stability estimate of the inversion. This result shows that the reconstruction of the optical coefficients is stable in the region close to the optical illumination sources and deteriorate exponentially far away. Our stability estimate is therefore in accordance with known experimental observations. Mathematics subject classification : 35R30.

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arXiv2018

An inverse problem for an electroseismic model describing the coupling phenomenon of electromagnetic and seismic waves

The electroseismic model describes the coupling phenomenon of the electromagnetic waves and seismic waves in fluid immersed porous rock. Electric parameters have better contrast than elastic parameters while seismic waves provide better resolution because of the short wavelength. The combination of theses two different waves is prominent in oil exploration. Under some assumptions on the physical parameters, we derived a H\"older stability estimate to the inverse problem of recovery of the electric parameters and the coupling coefficient from the knowledge of the fields in a small open domain near the boundary. The proof is based on a Carleman estimate of the electroseismic model.

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arXiv2018

The plasmonic resonances of a bowtie antenna

Metallic bowtie-shaped nanostructures are very interesting objects in optics, due to their capability of localizing and enhancing electromagnetic fields in the vicinity of their central neck. In this article, we investigate the electrostatic plasmonic resonances of two-dimensional bowtie-shaped domains by looking at the spectrum of their Poincar\'e variational operator. In particular, we show that the latter only consists of essential spectrum and fills the whole interval $[0,1]$. This behavior is very different from what occurs in the counterpart situation of a bowtie domain with only close-to-touching wings, a case where the essential spectrum of the Poincar\'e variational operator is reduced to an interval strictly contained in $[0,1]$. We provide an explanation for this difference by showing that the spectrum of the Poincar\'e variational operator of bowtie-shaped domains with close-to-touching wings has eigenvalues which densify and eventually fill the remaining intervals as the distance between the two wings tends to zero.

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arXiv2017

Characterization of the essential spectrum of the Neumann-Poincar\'e operator in 2D domains with corner via Weyl sequences

The Neumann-Poincar\'e (NP) operator naturally appears in the context of metamaterials as it may be used to represent the solutions of elliptic transmission problems via potentiel theory. In particular, its spectral properties are closely related to the well-posedness of these PDE's, in the typical case where one considers a bounded inclusion of homogeneous plasmonic metamaterial embedded in a homogeneous background dielectric medium. In a recent work, M. Perfekt and M. Putinar have shown that the NP operator of a 2D curvilinear polygon has an essential spectrum, which depends only on the angles of the corners. Their proof is based on quasi-conformal mappings and techniques from complex-analysis. In this work, we characterize the spectrum of the NP operator for a 2D domain with corners in terms of elliptic corner singularity functions, which gives insight on the behavior of generalized eigenmodes.

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arXiv2017

Homogenization of the eigenvalues of the Neumann-Poincar\'e operator

In this article, we investigate the spectrum of the Neumann-Poincar\'e operator associated to a periodic distribution of small inclusions with size $\varepsilon$, and its asymptotic behavior as the parameter $\varepsilon$ vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues $0$ and $1$, and of a subset which stays bounded away from $0$ and $1$ uniformly with respect to $\varepsilon$. This non trivial part is the reunion of the \textit{Bloch spectrum}, accounting for the collective resonances between collections of inclusions, and of the \textit{boundary layer spectrum}, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light about the homogenization of the voltage potential $u_\varepsilon$ caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possible negative) conductivity $a$, surrounded by a dielectric medium, with unit conductivity. In particular, we prove that the limit behavior of $u_\varepsilon$ is strongly related to the (possibly ill-defined) homogenized diffusion matrix predicted by the homogenization theory in the standard elliptic case. Additionally, we prove that the homogenization of $u_\varepsilon$ is always possible when $a$ is either positive, or negative with a `small' or `large' modulus.

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arXiv2016

Superlensing using hyperbolic metamaterials: the scalar case

This paper is devoted to superlensing using hyperbolic metamaterials: the possibility to image an arbitrary object using hyperbolic metamaterials without imposing any conditions on size of the object and the wave length. To this end, two types of schemes are suggested and their analysis are given. The superlensing devices proposed are independent of the object. It is worth noting that the study of hyperbolic metamaterials is challenging due to the change type of modelling equations, elliptic in some regions, hyperbolic in some others.

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arXiv2014

Numerical simulation of a class of models that combine several mechanisms of dissipation: fracture, plasticity, viscous dissipation

We study a class of time evolution models that contain dissipation mech- anisms exhibited by geophysical materials during deformation: plasticity, viscous dissipation and fracture. We formally prove that they satisfy a Clausius-Duhem type inequality. We describe a semi-discrete time evolu- tion associated with these models, and report numerical 1D and 2D traction experiments, that illustrate that several dissipation regimes can indeed take place during the deformation. Finally, we report 2D numerical simulation of an experiment by Peltzer and Tapponnier, who studied the indentation of a layer of plasticine as an analogue model for geological materials.

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arXiv2011

Inverse diffusion from knowledge of power densities

This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography. We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.

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