Publications

2023

• Wave propagation in one-dimensional quasiperiodic media
  
  • Amenoagbadji Pierre
  • Fliss Sonia
  • Joly Patrick
  Communications in Optimization Theory, Mathematical Research Press, 2023, 17. 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. (10.23952/cot.2023.17)
  DOI : 10.23952/cot.2023.17
• 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
• 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