Publications

2023

• Scattering in a partially open waveguide: the forward problem
  
  • Bourgeois Laurent
  • Fliss Sonia
  • Fritsch Jean-François
  • Hazard Christophe
  • Recoquillay Arnaud
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2023, 88, pp.102-151. This paper is dedicated to an acoustic scattering problem in a two-dimensional partially open waveguide, in the sense that the left part of the waveguide is closed, that is with a bounded cross-section, while the right part is bounded in the transverse direction by some Perfectly Matched Layers that mimic the situation of an open waveguide, that is with an unbounded cross-section. We prove well-posedness of such scattering problem and exhibit the asymptotic behaviour of the solution in the longitudinal direction with the help of the Kondratiev approach. Having in mind the numerical computation of the solution, we also propose some transparent boundary conditions in such longitudinal direction, based on Dirichlet-to-Neumann operators. After proving that such artificial conditions actually enable us to approximate the exact solution, some numerical experiments illustrate the quality of such approximation. (10.1093/imamat/hxad004)
  DOI : 10.1093/imamat/hxad004
• Analysis of time-harmonic Maxwell impedance problems in anisotropic media
  
  • Chicaud Damien
  • Ciarlet Patrick
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2023, 55 (3), pp.1969-2000. We consider the time-harmonic Maxwell's equations in anisotropic media. The problem to be solved is an approximation of the diffraction problem, or scattering from bounded objects, that is usually set in some exterior domain in $\mathbb{R}^3$. We consider perfectly conducting objects, so the equations are supplemented with a Dirichlet boundary condition on those objects, and we truncate the exterior domain by imposing an impedance condition on an artificial boundary, to model an approximate radiation condition. The resulting problem is then posed in a bounded domain, with Dirichlet and impedance boundary conditions. In this work, we focus on the mathematical meaning of the impedance condition, precisely in which function space it holds. This relies on a careful analysis of the regularity of the traces of electromagnetic fields, which can be derived thanks to the study of the regularity of the solution to second-order surface PDEs. Then, we prove well-posedness of the model, and we determine the a priori regularity of the fields in the domain and on the boundaries, depending on the geometry, the coefficients and the data. Finally, the discretization of the formulations is presented, with an approximation based on edge finite elements. Error estimates are derived, and a benchmark is provided to discuss those estimates. (10.1137/22M1485413)
  DOI : 10.1137/22M1485413
• Wave propagation in one-dimensional quasiperiodic media
  
  • Amenoagbadji Pierre
  • Fliss Sonia
  • Joly Patrick
  Communications in Optimization Theory, Mathematical Research Press, 2023. This work is devoted to the resolution of the Helmholtz equation −(µ u) − ρ ω 2 u = f in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed in [10, 19] for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation. (10.48550/arXiv.2301.01159)
  DOI : 10.48550/arXiv.2301.01159
• A complex-scaled boundary integral equation for time-harmonic water waves
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Faria Luiz
  • Pérez‐Arancibia Carlos
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2023, 84 (4), pp.1532-1556. This paper presents a novel boundary integral equation (BIE) formulation for the two-dimensional time-harmonic water-waves problem. It utilizes a complex-scaled Laplace's free-space Green's function, resulting in a BIE posed on the infinite boundaries of the domain. The perfectly matched layer (PML) coordinate stretching that is used to render propagating waves exponentially decaying, allows for the effective truncation and discretization of the BIE unbounded domain. We show through a variety of numerical examples that, despite the logarithmic growth of the complex-scaled Laplace's free-space Green's function, the truncation errors are exponentially small with respect to the truncation length. Our formulation uses only simple function evaluations (e.g. complex logarithms and square roots), hence avoiding the need to compute the involved water-wave Green's function. Finally, we show that the proposed approach can also be used to find complex resonances through a \emph{linear} eigenvalue problem since the Green's function is frequency-independent. (10.48550/arXiv.2310.04127)
  DOI : 10.48550/arXiv.2310.04127
• Variational methods for solving numerically magnetostatic systems
  
  • Ciarlet Patrick
  • Jamelot E.
  Advances in Computational Mathematics, Springer Verlag, 2023. In this paper, we study some techniques for solving numerically magnetostatic systems. We consider fairly general assumptions on the magnetic permeability tensor. It is elliptic, but can be nonhermitian. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements, or on continuous Lagrange finite elements. For the first type of discretization, we rely on the design of a new, mixed variational formulation that is obtained with the help of $T$-coercivity. The numerical method can be related to a perturbed approach for solving mixed problems in electromagnetism. For the second type of discretization, we rely on an augmented variational formulation obtained with the help of the Weighted Regularization Method.
• An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients
  
  • Ciarlet Patrick
  • Lassounon David
  • Rihani Mahran
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2023, 61 (3), pp.1316-1339. In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with for instance an electric permittivity that is strictly negative in the metal or metamaterial. The method is based on an optimal control reformulation of the problem. Contrary to other existing approaches, the convergence of this method is proved without any restrictive condition. In particular, no condition is imposed on the a priori regularity of the solution to the problem, and no condition is imposed on the meshes, other than that they fit with the interface between the two media. Our results are illustrated by some (2D) numerical experiments. (10.1137/22M1495998)
  DOI : 10.1137/22M1495998
• Quantifying mixing in arbitrary fluid domains: a Padé approximation approach
  
  • Anderson Thomas G
  • Bonnet Marc
  • Veerapaneni Shravan
  Numerical Algorithms, Springer Verlag, 2023, 93, pp.441-458. We consider the model problem of mixing of passive tracers by an incompressible viscous fluid. Addressing questions of optimal control in realistic geometric settings or alternatively the design of fluid-confining geometries that successfully effect mixing requires a meaningful norm in which to quantify mixing that is also suitable for easy and efficient computation (as is needed, e.g., for use in gradient-based optimization methods). We use the physically inspired reasonable surrogate of a negative index Sobolev norm over the complex fluid mixing domain Ω, a task which could be seen as computationally expensive since it requires the computation of an eigenbasis for L2(Ω) by definition. Instead, we compute a representant of the scalar concentration field in an appropriate Sobolev space in order to obtain an equivalent definition of the Sobolev surrogate norm. The representant, in turn, can be computed to high-order accuracy by a Padé approximation to certain fractional pseudo-differential operators, which naturally leads to a sequence of elliptic problems with an inhomogeneity related to snapshots of the time-varying concentration field. Fast and accurate potential theoretic methods are used to efficiently solve these problems, with rapid per-snapshot mix-norm computation made possible by recent advances in numerical methods for volume potentials. We couple the methodology to existing solvers for Stokes and advection equations to obtain a unified framework for simulating and quantifying mixing in arbitrary fluid domains. We provide numerical results demonstrating the convergence of the new approach as the approximation order is increased. (10.1007/s11075-022-01423-7)
  DOI : 10.1007/s11075-022-01423-7
• Scattering in a partially open waveguide: the inverse problem
  
  • Bourgeois Laurent
  • Fritsch Jean-François
  • Recoquillay Arnaud
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2023, 17 (2), pp.463-469. In this paper we consider an inverse scattering problem which consists in retrieving obstacles in a partially embedded waveguide in the acoustic case, the measurements being located on the accessible part of the structure. Such accessible part can be considered as a closed waveguide (with a finite cross section), while the embedded part can be considered as an open waveguide (with an infinite cross section). We propose an approximate model of the open waveguide by using Perfectly Matched Layers in order to simplify the resolution of the inverse problem, which is based on a modal formulation of the Linear Sampling Method. Some numerical results show the efficiency of our approach. This paper can be viewed as a continuation of the article [11], which was focused on the forward problem. (10.3934/ipi.2022052)
  DOI : 10.3934/ipi.2022052
• H-matrix accelerated FEM-BEM coupling for dynamic analysis of naval structures in pulsating potential fluids
  
  • Mavaleix-Marchessoux Damien
  • Bonnet Marc
  • Chaillat Stéphanie
  • Leblé Bruno
  , 2021. This article addresses one of the components of our ongoing work towards an efficient computational modeling methodology for evaluating all effects on a submerged structure of a remote underwater explosion. Following up on a previous study devoted to computing the transient acoustic fields induced by the shock wave initially sent by the blast on a rigid submarine, we focus here on the second stage of the underwater event, namely solving the transient fluid-structure interaction (FSI) between the structure and the incompressible potential flow induced by the delayed, and slower, oscillations of the gas bubble created by the remote blast. The boundary element method (BEM) is the best-suited approach for handling potential flow problems in large fluid domains (idealized as unbounded), whereas the finite element method (FEM) naturally applies to the transient structure analyses. To perform the FEM-BEM coupling we use a sub-cycling approach that alternates fluid and solid analyses with Neumann boundary conditions. The transient nature of the coupled analysis and the recourse to sub-cycling together make the overall procedure rely on a large number of BEM potential flow solutions, while the complexities of the wet surface and of the solid transient response imply a need for large BE models for the flow potential. This combination of reasons mandates accelerating the BE component. Accordingly, our main contribution is to study the feasibility and effectiveness of coupling the Hierarchical-matrix accelerated BEM (H-BEM) and the FEM for the FSI problems of interest. In particular, we show that the same integral operators can be used at all time instants in spite of the expected global motion of the submerged structure, a feature that the H-BEM can exploit to full advantage. The proposed original treatment is validated against analytical solutions for the case of a motionless or mobile rigid spherical immersed object, and then tested on a complex configuration representative of target applications.
• Shape optimization of peristaltic pumps transporting rigid particles in Stokes flow
  
  • Bonnet Marc
  • Liu Ruowen
  • Veerapaneni Shravan
  • Zhu Hai
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2023, 45 (1), pp.B78-B106. This paper presents a computational approach for finding the optimal shapes of peristaltic pumps transporting rigid particles in Stokes flow. In particular, we consider shapes that minimize the rate of energy dissipation while pumping a prescribed volume of fluid, number of particles and/or distance traversed by the particles over a set time period. Our approach relies on a recently developed fast and accurate boundary integral solver for simulating multiphase flows through periodic geometries of arbitrary shapes. In order to fully capitalize on the dimensionality reduction feature of the boundary integral methods, shape sensitivities must ideally involve evaluating the physical variables on the particle or pump boundaries only. We show that this can indeed be accomplished owing to the linearity of Stokes flow. The forward problem solves for the particle motion in a slip-driven pipe flow while the adjoint problems in our construction solve quasi-static Dirichlet boundary value problems backwards in time, retracing the particle evolution. The shape sensitivities simply depend on the solution of one forward and one adjoint (for each shape functional) problems. We validate these analytic shape derivative formulas by comparing against finite-difference based gradients and present several examples showcasing optimal pump shapes under various constraints. (10.1137/21M144863X)
  DOI : 10.1137/21M144863X
• A posteriori error estimates for mixed finite element discretizations of the Neutron Diffusion equations
  
  • Ciarlet Patrick
  • Do Minh Hieu
  • Madiot François
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2023, 57 (1), pp.1-27. We analyse a posteriori error estimates for the discretization of the neutron diffusion equations with mixed finite elements. We provide guaranteed and locally efficient estimators on a base block equation, the one-group neutron diffusion equation. We pay particular attention to AMR strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications. We exhibit a robust marker strategy for this specific constraint, the direction marker strategy. (10.1051/m2an/2022078)
  DOI : 10.1051/m2an/2022078
• Modelling of the fatigue cracking resistance of grid reinforced asphalt concrete by coupling fast BEM and FEM
  
  • Dansou Anicet
  • Mouhoubi Saida
  • Chazallon Cyrille
  • Bonnet Marc
  Road Materials and Pavement Design, Taylor & Francis, 2023, 24, pp.631-652. We present a computational modeling approach aimed at investigating the effect of fiber grid reinforcement on crack opening displacement and fatigue crack propagation. Grid reinforcements are modeled using elastic membrane finite elements, while the cracked concrete is treated using a symmetric boundary element method (BEM), which in particular allows easy geometrical modelling and meshing of cracks. The BEM is accelerated by the fast multipole method, allowing the handling of potentially large BEM models entailed by three-dimensional configurations hosting multiple cracks. Fatigue crack growth is modelled using the Paris law. The proposed computational approach is first verified on a reinforced cracked beam, and then applied to a three-dimensional configuration featuring a grid-reinforced asphalt pavement. (10.1080/14680629.2022.2029755)
  DOI : 10.1080/14680629.2022.2029755