Publications

2015

• Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: the 1d case
  
  • Bécache Eliane
  • Bourgeois Laurent
  • Franceschini Lucas
  • Dardé Jérémi
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2015. In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical La-grange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations. 1. Introduction. The method of quasi-reversibility has now a quite long history since the pioneering book of Latt es and Lions in 1967 [1]. The original idea of these authors was, starting from an ill-posed problem which satisfies the uniqueness property, to introduce a perturbation of such problem involving a small positive parameter ε. This perturbation has essentially two effects. Firstly the perturbation transforms the initial ill-posed problem into a well-posed one for any ε, secondly the solution to such problem converges to the solution (if it exists) to the initial ill-posed problem when ε tends to 0. Generally, the ill-posedness in the initial problem is due to unsuitable boundary conditions. As typical examples of linear ill-posed problems one may think of the backward heat equation, that is the initial condition is replaced by a final condition, or the heat or wave equations with lateral Cauchy data, that is the usual Dirichlet or Neumann boundary condition on the boundary of the domain is replaced by a pair of Dirichlet and Neumann boundary conditions on the same subpart of the boundary, no data being prescribed on the complementary part of the boundary. (10.3934/ipi.2015.9.971)
  DOI : 10.3934/ipi.2015.9.971
• Generation of acoustic solitary waves in a lattice of Helmholtz resonators
  
  • Richoux Olivier
  • Lombard Bruno
  • Mercier Jean-François
  Wave Motion, Elsevier, 2015, 56, pp.85-99. This paper addresses the propagation of high amplitude acoustic pulses through a 1D lattice of Helmholtz resonators connected to a waveguide. Based on the model proposed by Sugimoto (J. Fluid. Mech., 244 (1992), 55-78), a new numerical method is developed to take into account both the nonlinear wave propagation and the different mechanisms of dissipation: the volume attenuation, the linear visco-thermic losses at the walls, and the nonlinear absorption due to the acoustic jet formation in the resonator necks. Good agreement between numerical and experimental results is obtained, highlighting the crucial role of the nonlinear losses. Different kinds of solitary waves are observed experimentally with characteristics depending on the dispersion properties of the lattice. (10.1016/j.wavemoti.2015.02.005)
  DOI : 10.1016/j.wavemoti.2015.02.005
• The topological derivative of stress-based cost functionals in anisotropic elasticity
  
  • Delgado Gabriel
  • Bonnet Marc
  Computers & Mathematics with Applications, Elsevier, 2015, 69, pp.1144-1166. The topological derivative of cost functionals J that depend on the stress (through the displacement gradient, assuming a linearly elastic material behavior) is considered in a quite general 3D setting where both the background and the inhomogeneity may have arbitrary anisotropic elastic properties. The topological derivative dJ(z) of J quantifies the asymptotic behavior of J under the nucleation in the background elastic medium of a small anisotropic inhomogeneity of characteristic radius a at a specified location z. The fact that the strain perturbation inside an elastic inhomogeneity remains finite for arbitrarily small a makes the small-inhomogeneity asymptotics of stress-based cost functionals quite different than that of the more usual displacement-based functionals. The asymptotic perturbation of J is shown to be of order O(a^3) for a wide class of stress-based cost functionals having smooth densities. The topological derivative of J, i.e. the coefficient of the O(a^3) perturbation, is established, and computational procedures then discussed. The resulting small-inhomogeneity expansion of J is mathematically justified (i.e. its remainder is proved to be of order o(a^3)). Several 2D and 3D numerical examples are presented, in particular demonstrating the proposed formulation of \dJ on cases involving anisotropic elasticity and non-quadratic cost functionals. (10.1016/j.camwa.2015.03.010)
  DOI : 10.1016/j.camwa.2015.03.010
• Solving multizone and multicrack elastostatic problems: a fast multipole symmetric Galerkin boundary element method approach
  
  • Trinh Quoc Tuan
  • Mouhoubi Saida
  • Chazallon Cyrille
  • Bonnet Marc
  Engineering Analysis with Boundary Elements, Elsevier, 2015, 50, pp.486-495. Symmetric Galerkin boundary element methods (SGBEMs) for three-dimensional elastostatic problems give rise to fully-populated (albeit symmetric) matrix equations, entailing high solution times for large models. This article is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for multi-zone elasticity problems with cracks. The subdomain coupling approach is based on a minimal set of interfacial unknowns (i.e. one displacement and one traction vector at any interfacial point) that are defined globally for the complete multizone configuration. Then, unknowns for each subdomain are defined in terms of the global unknowns, with appropriate sign conventions for tractions induced by subdo-main numbering. This formulation (i) automatically enforces the perfect-bonding transmission conditions between subdomains, and (ii) is globally symmetric. The subsequent FM-SGBEM basically proceeds by as-sembling contributions from each subregion, which can be computed by means of an existing single-domain FM-SGBEM implementation such as that previously presented by the authors (EABE, 36:1838-1847, 2012). Along the way, the computational performance of the FM-SGBEM is enhanced through (a) suitable storage of the near-field contribution to the SGBEM matrix equation and (b) preconditioning by means of nested GMRES. The formulation is validated on numerical experiments for 3D configurations involving many cracks and inclusions, and of sizes up to N ≈ 10 6 . (10.1016/j.enganabound.2014.10.004)
  DOI : 10.1016/j.enganabound.2014.10.004
• Qualitative modeling of the dynamics of detonations with losses
  
  • Faria Luiz
  • Kasimov Aslan
  Proceedings of the Combustion Institute, Elsevier, 2015, 35 (2), pp.2015-2023. (10.1016/j.proci.2014.07.006)
  DOI : 10.1016/j.proci.2014.07.006
• Stochastic Multi-Stage Optimization
  
  • Carpentier Pierre
  • Cohen Guy
  • Chancelier Jean-Philippe
  • de Lara Michel
  , 2015, 75. (10.1007/978-3-319-18138-7)
  DOI : 10.1007/978-3-319-18138-7
• Gaussian and non-Gaussian processes of zero power variation
  
  • Russo Francesco
  • Viens Frederi
  ESAIM: Probability and Statistics, EDP Sciences, 2015, 19 (9), pp.414-439. This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let $m$ be an odd integer. Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $\delta^{2}(r) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta(r) = o(r^{1/(2m)}) $. In the case of a Gaussian process with homogeneous increments, $\delta$ is $X$'s canonical metric and the condition on $\delta$ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. In the non-homogeneous Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô's formula is proved to hold for all functions of class $C^{6}$. (10.1051/ps/2014031)
  DOI : 10.1051/ps/2014031
• Numerical investigation of acoustic solitons
  
  • Lombard Bruno
  • Mercier Jean-François
  • Richoux Olivier
  Proceedings of the Estonian Academy of Sciences, Estonian Academy Publishers, 2015, 64 (3), pp.304-310. Acoustic solitons can be obtained by considering the propagation of large amplitude sound waves across a set of Helmholtz resonators. The model proposed by Sugimoto and his coauthors has been validated experimentally in previous works. Here we examine some of its theoretical properties: low-frequency regime, balance of energy, stability. We propose also numerical experiments illustrating typical features of solitary waves.
• The stochastic porous media equation in $\R^d$
  
  • Barbu Viorel
  • Röckner Michael
  • Russo Francesco
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2015, 103 (4), pp.1024-1052. Existence and uniqueness of solutions to the stochastic porous media equation $dX-\D\psi(X) dt=XdW$ in $\rr^d$ are studied. Here, $W$ is a Wiener process, $\psi$ is a maximal monotone graph in $\rr\times\rr$ such that $\psi(r)\le C|r|^m$, $\ff r\in\rr$, $W$ is a coloured Wiener process. In this general case the dimension is restricted to $d\ge 3$, the main reason being the absence of a convenient multiplier result in the space $\calh=\{\varphi\in\mathcal{S}'(\rr^d);\ |\xi|(\calf\varphi)(\xi)\in L^2(\rr^d)\}$, for $d\le2$. When $\psi$ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space $H^{-1}(\rr^d)$. If $\psi(r)r\ge\rho|r|^{m+1}$ and $m=\frac{d-2}{d+2}$, we prove the finite time extinction with strictly positive probability. (10.1016/j.matpur.2014.10.004)
  DOI : 10.1016/j.matpur.2014.10.004
• Stochastic analysis : A series of lectures
  
  • Flandoli Franco
  • Russo Francesco
  • Dalang Robert C.
  • Dozzi Marco
  , 2015, 68, pp.xiii + 393.
• Uniqueness results for 2D inverse Robin problems with bounded coefficient
  
  • Baratchart Laurent
  • Bourgeois Laurent
  • Leblond Juliette
  , 2015. We address in this work the uniqueness issue in the classical Robin inverse problem with the Laplace equation on a Dini-smooth planar domain, with uniformly bounded Robin coefficient and L2 Neumann data. We prove uniqueness of the Robin coefficient on a subpart of the boundary, given Cauchy data on the complementary part.
• Three-dimensional transient elastodynamic inversion using an error in constitutive relation functional
  
  • Bonnet Marc
  • Aquino Wilkins
  Inverse Problems, IOP Publishing, 2015, 31, pp.035010. This work is concerned with large-scale three-dimensional inversion under transient elastodynamic conditions by means of the modified error in constitutive relation (MECR), an energy-based, cost functional. In contrast to quasi-static or frequency-domain contexts, time-domain formulations have so far seen very limited investigation. A peculiarity of time-domain MECR formulations is that each evaluation involves the solution of two elastodynamic problems (one forward, one backward), which moreover are coupled (unlike the case of $L^2$ misfit functionals, where the forward state does not depend on the adjoint state). This coupling creates a major computational bottleneck, making MECR-based inversion difficult for spatially 2D or 3D configurations. To overcome this obstacle, we propose an approach whose main ingredients are (a) setting the entire computational procedure in a consistent time-discrete framework that incorporates the chosen time-stepping algorithm, and (b) using an iterative SOR-like method for the resulting stationarity equations. The resulting MECR-based inversion algorithm is formulated under quite general conditions, allowing for three-dimensional transient elastodynamics, straightforward use of available parallel solvers, a wide array of time-stepping algorithms commonly used for transient structural dynamics, and flexible boundary condition and measurement settings. The proposed MECR algorithm is then demonstrated on computational experiments involving 2D and 3D transient elastodynamics and up to over 500,000 unknown elastic moduli. (10.1088/0266-5611/31/3/035010)
  DOI : 10.1088/0266-5611/31/3/035010
• A method to build non-scattering perturbations of two-dimensional acoustic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Lunéville Éric
  • Mbeutcha Yves
  • Nazarov Sergei
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2015. We are interested in finding deformations of the rigid wall of a two-dimensional acoustic waveguide, which are not detectable in the far field, as they produce neither reflection nor conversion of propagative modes. A proof of existence of such invisible deformations has been presented in a previous paper. It combines elements of the asymptotic analysis for small deformations and a fixed-point argument. In the present paper, we give a systematic presentation of the method, and we prove that it works for all frequencies except a discrete set. A particular attention is devoted to the practical implementation of the method. The main difficulty concerns the building of a dual family to given oscillating functions. Advantages and limits of the method are illustrated by several numerical results. Copyright © 2015 John Wiley & Sons, Ltd. (10.1002/mma.3447)
  DOI : 10.1002/mma.3447
• Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perflectly matched layer methods
  
  • Nguyen Khac-Long
  • Treyssede Fabien
  • Hazard Christophe
  Journal of Sound and Vibration, Elsevier, 2015, 344, pp.pp.158-178. Among the numerous techniques of non destructive evaluation, elastic guided waves are of particular interest to evaluate defects inside industrial and civil elongated structures owing to their ability to propagate over long distances. However for guiding structures buried in large solid media, waves can be strongly attenuated along the guide axis due to the energy radiation into the surrounding medium, usually considered as unbounded. Hence, searching the less attenuated modes become necessary in order to maximize the inspection distance. In the numerical modeling of embedded waveguides, the main difficulty is to account for the unbounded section. This paper presents a numerical approach combining a semi-analytical finite element method and a perfectly matched layer (PML) technique to compute the so-called trapped and leaky modes in three-dimensional embedded elastic waveguides of arbitrary cross-section. Two kinds of PML, namely the Cartesian PML and the radial PML, are considered. In order to understand the various spectral objects obtained by the method, the PML parameters effects upon the eigenvalue spectrum are highlighted through analytical studies and numerical experiments. Then, dispersion curves are computed for test cases taken from the literature in order to validate the approach. (10.1016/j.jsv.2014.12.032)
  DOI : 10.1016/j.jsv.2014.12.032
• Intrinsic Finite Element Methods for the Computation of Fluxes for Poisson's Equation
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  • Sauter Stefan
  • Simian C
  Numerische Mathematik, Springer Verlag, 2015, pp.30. In this paper we consider an intrinsic approach for the direct computation of the fluxes for problems in potential theory. We develop a general method for the derivation of intrinsic conforming and non-conforming finite element spaces and appropriate lifting operators for the evaluation of the right-hand side from abstract theoretical principles related to the second Strang Lemma. This intrinsic finite element method is analyzed and convergence with optimal order is proved. (10.1007/s00211-015-0730-9)
  DOI : 10.1007/s00211-015-0730-9
• Modal method for 2D wave propagation in heterogeneous anisotropic media
  
  • Maurel Agnes
  • Mercier Jean-François
  • Félix Simon
  Journal of the Optical Society of America, Optical Society of America, 2015, 32 (5), pp.11. A multimodal method based on a generalization of the admittance matrix is used to analyze wave propagation in heterogeneous two-dimensional anisotropic media. The heterogeneity of the medium can be due to the presence of anisotropic inclusions with arbitrary shapes, to a succession of anisotropic media with complex interfaces between them, or both. Using a modal expansion of the wave field, the problem is reduced to a system of two sets of first-order differential equations for the modal components of the field, similar to the system obtained in the rigorous coupled wave analysis. The system is solved numerically, using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed. The convergence of the method is discussed, considering arrays of anisotropic inclusions with complex shapes, which tend to show that Li’s rules are not concerned within our approach. The method is validated by comparison with a subwavelength layered structure presenting an effective anisotropy at the wave scale. (10.1364/JOSAA.32.000979)
  DOI : 10.1364/JOSAA.32.000979
• Improved multimodal method for the acoustic propagation in waveguides with finite wall impedance
  
  • Félix Simon
  • Maurel Agnes
  • Mercier Jean-François
  Wave Motion, Elsevier, 2015, 54, pp.10. We address the problem of acoustic propagation in waveguides with wall impedance, or Robin, boundary condition. Two improved multimodal methods are developed to remedy the problem of the low convergence of the series in the standard modal approach. In the first improved method, the series is enriched with an additional mode, which is thought to be able to restore the right boundary condition. The second improved method consists in a reformulation of the expansions able to restore the right boundary conditions for any truncation, similar to polynomial subtraction technique. Surprisingly, the first improved method is found to be the most efficient. Notably, the convergence of the scattering properties is increased from N−1 in the standard modal method to N−3 in the reformulation and N−5 in the formulation with a supplementary mode. The improved methods are shown to be of particular interest when surface waves are generated near the impedance wall. (10.1016/j.wavemoti.2014.11.007)
  DOI : 10.1016/j.wavemoti.2014.11.007
• Wave propagation in a waveguide containing restrictions with circular arc shape
  
  • Félix Simon
  • Maurel Agnes
  • Mercier Jean-François
  Journal of the Acoustical Society of America, Acoustical Society of America, 2015, 137 (3), pp.7. A multimodal method is used to analyze the wave propagation in waveguides containing restrictions (or corrugations) with circular arc shapes. This is done using a geometrical transformation which transforms the waveguide with complex geometry in the real space to a straight waveguide in the transformed space, or virtual space. In this virtual space, the Helmholtz equation has a modified structure which encapsulates the complexity of the geometry. It is solved using an improved modal method, which was proposed in a paper by A. Maurel, J.-F. Mercier, and S. Félix [Proc. R. Soc. A 470, 20130743 (2014)], that increases the accuracy and convergence of usual multimodal formulations. Results show the possibility to solve the wave propagation in a waveguide with a high density of circular arc shaped scatterers. (10.1121/1.4913506)
  DOI : 10.1121/1.4913506
• Zubov's equation for state-constrained perturbed nonlinear systems
  
  • Grüne Lars
  • Zidani Hasnaa
  Mathematical Control and Related Fields, AIMS, 2015, 5 (1), pp.55-71. The paper gives a characterization of the uniform robust domain of attraction for a nite non-linear controlled system subject to perturbations and state constraints. We extend the Zubov approach to characterize this domain by means of the value function of a suitable in nite horizon state-constrained control problem which at the same time is a Lyapunov function for the system. We provide associated Hamilton-Jacobi-Bellman equations and prove existence and uniqueness of the solutions of these generalized Zubov equations. (10.3934/mcrf.2015.5.55)
  DOI : 10.3934/mcrf.2015.5.55
• Monotone numerical schemes and feedback construction for hybrid control systems
  
  • Ferretti Roberto
  • Zidani Hasnaa
  Journal of Optimization Theory and Applications, Springer Verlag, 2015, 165 (2), pp.507-531. Hybrid systems are a general framework which can model a large class of control systems arising whenever a collection of continuous and discrete dynamics are put together in a single model. In this paper, we study the convergence of monotone numerical approximations of value functions associated to control problems governed by hybrid systems. We discuss also the feedback reconstruction and derive a convergence result for the approximate feedback control law. Some numerical examples are given to show the robustness of the monotone approximation schemes. (10.1007/s10957-014-0637-0)
  DOI : 10.1007/s10957-014-0637-0
• Value iteration convergence of "-monotone schemes for stationary Hamilton-Jacobi equations
  
  • Bokanowski Olivier
  • Falcone Maurizio
  • Ferretti Roberto
  • Grüne Lars
  • Kalise Dante
  • Zidani Hasnaa
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (9), pp.4041 - 4070. We present an abstract convergence result for the xed point approximation of stationary Hamilton{Jacobi equations. The basic assumptions on the discrete operator are invariance with respect to the addition of constants, "-monotonicity and consistency. The result can be applied to various high-order approximation schemes which are illustrated in the paper. Several applications to Hamilton{Jacobi equations and numerical tests are presented. (10.3934/dcds.2015.35.4041)
  DOI : 10.3934/dcds.2015.35.4041
• La mécanique des Sphères de Lie: un futur pour la CAO?
  
  • Arber Christian
  • Jean Frédéric
  , 2015, 28. La plupart des pièces ou ensembles de pièces de la vie courante sont conçus maintenant en CAO (Conception Assistée par Ordinateur): voiture, avion bien sûr, mais aussi savon, emballage de savon, machine à emballer les savons,... On peut se douter que la modélisation géométrique y joue un grand rôle. Mais quelle géométrie? Et finalement quels ressorts mathématiques sont employés dans ce qui est utilisé tous les jours par des centaines de milliers de concepteurs à travers le monde (et des millions d'utilisateurs de jeux vidéo!)? A leur insu et pour leur bonheur. Géométrie cartésienne bien sûr, mais aussi de belles vieilles idées de la grande école de géométrie franco-allemande de la fin du XXIème siècle revivent actuellement. Rangées dans les réserves du musée de l'algèbre triomphante, elles sont exhumées régulièrement par quelques archéologues des mathématiques (les géomètres!). Maintenant combinées avec les progrès fantastiques portés par la géométrie Riemannienne sur les groupes de Lie, elles permettent des avancées importantes dans le codage d'un logiciel de CAO. En seront-elles le futur? On illustrera cela avec l'espace des Sphères de Lie, pour traiter le problème du solveur d'esquisses (construction naturelle de géométries sous contraintes), un des problèmes centraux de la modélisation.
• A modified error in constitutive equation approach for frequency-domain viscoelasticity imaging using interior data
  
  • Diaz Manuel I.
  • Aquino Wilkins
  • Bonnet Marc
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2015, 296, pp.129-149. This paper presents a methodology for the inverse identification of linearly viscoelastic material parameters in the context of steady-state dynamics using interior data. The inverse problem of viscoelasticity imaging is solved by minimizing a modified error in constitutive equation (MECE) functional, subject to the conservation of linear momentum. The treatment is applicable to configurations where boundary conditions may be partially or completely underspecified. The MECE functional measures the discrepancy in the consti-tutive equations that connect kinematically admissible strains and dynamically admissible stresses, and also incorporates the measurement data in a quadratic penalty term. Regularization of the problem is achieved through a penalty parameter in combination with the discrepancy principle due to Morozov. Numerical results demonstrate the robust performance of the method in situations where the available measurement data is incomplete and corrupted by noise of varying levels. (10.1016/j.cma.2015.07.025)
  DOI : 10.1016/j.cma.2015.07.025