Publications

2021

• Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients
  
  • Bunoiu Renata
  • Chesnel Lucas
  • Ramdani Karim
  • Rihani Mahran
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2021, 30 (5), pp.1075-1119. In this work, we are interested in the homogenization of time-harmonic Maxwell's equations in a composite medium with periodically distributed small inclusions of a negative material. Here a negative material is a material modelled by negative permittivity and permeability. Due to the sign-changing coefficients in the equations, it is not straightforward to obtain uniform energy estimates to apply the usual homogenization techniques. The goal of this article is to explain how to proceed in this context. The analysis of Maxwell's equations is based on a precise study of two associated scalar problems: one involving the sign-changing permittivity with Dirichlet boundary conditions, another involving the sign-changing permeability with Neumann boundary conditions. For both problems, we obtain a criterion on the physical parameters ensuring uniform invertibility of the corresponding operators as the size of the inclusions tends to zero. In the process, we explain the link existing with the so-called Neumann-Poincaré operator, complementing the existing literature on this topic. Then we use the results obtained for the scalar problems to derive uniform energy estimates for Maxwell's system. At this stage, an additional difficulty comes from the fact that Maxwell's equations are also sign-indefinite due to the term involving the frequency. To cope with it, we establish some sort of uniform compactness result. (10.5802/afst.1694)
  DOI : 10.5802/afst.1694
• Optimal slip velocities of micro-swimmers with arbitrary axisymmetric shapes
  
  • Guo Hanliang
  • Zhu Hai
  • Liu Ruowen
  • Bonnet Marc
  • Veerapaneni Shravan
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2021, 910, pp.A26. This article presents a computational approach for determining the optimal slip velocities on any given shape of an axisymmetric micro-swimmer suspended in a viscous fluid. The objective is to minimize the power loss to maintain a target swimming speed, or equivalently to maximize the efficiency of the micro-swimmer. Owing to the linearity of the Stokes equations governing the fluid motion, we show that this PDE-constrained optimization problem reduces to a simpler quadratic optimization problem, whose solution is found using a high-order accurate boundary integral method. We consider various families of shapes parameterized by the reduced volume and compute their swimming efficiency. {Among those, prolate spheroids were found to be the most efficient micro-swimmer shapes for a given reduced volume. We propose a simple shape-based scalar metric that can determine whether the optimal slip on a given shape makes it a pusher, a puller, or a neutral swimmer.} (10.1017/jfm.2020.969)
  DOI : 10.1017/jfm.2020.969
• Optimal Ciliary Locomotion of Axisymmetric Microswimmers
  
  • Guo Hanliang
  • Zhu Hai
  • Liu Ruowen
  • Bonnet Marc
  • Veerapaneni Shravan
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2021, 927, pp.A22. Many biological microswimmers locomote by periodically beating the densely-packed cilia on their cell surface in a wave-like fashion. While the swimming mechanisms of ciliated microswimmers have been extensively studied both from the analytical and the numerical point of view, the optimization of the ciliary motion of microswimmers has received limited attention, especially for non-spherical shapes. In this paper, using an envelope model for the microswimmer, we numerically optimize the ciliary motion of a ciliate with an arbitrary axisymmetric shape. The forward solutions are found using a fast boundary integral method, and the efficiency sensitivities are derived using an adjoint-based method. Our results show that a prolate microswimmer with a 2:1 aspect ratio shares similar optimal ciliary motion as the spherical microswimmer, yet the swimming efficiency can increase two-fold. More interestingly, the optimal ciliary motion of a concave microswimmer can be qualitatively different from that of the spherical microswimmer, and adding a constraint to the ciliary length is found to improve, on average, the efficiency for such swimmers. (10.1017/jfm.2021.744)
  DOI : 10.1017/jfm.2021.744
• Variational Methods for Acoustic Radiation in a Duct with a Shear Flow and an Absorbing Boundary
  
  • Mercier Jean-François
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2021, 81 (6), pp.2658-2683. The well-posedness of the acoustic radiation in a 2D duct in presence of both a shear flow and an absorbing wall described by the Myers boundary condition is studied thanks to variational methods. Without flow the problem is found well-posed for any impedance value. The presence of a flow complicates the results. With a uniform flow the problem is proven to be always of the Fredholm type but is found well-posed only when considering a dissipative radiation problem. With a general shear flow, the Fredholm property is recovered for a weak enough shear and the dissipative radiation problem requires to introduce extra conditions to be well-posed: enough dissipation, a large enough frequency and non-intuitive conditions on the impedance value. (10.1137/20M1384026)
  DOI : 10.1137/20M1384026
• An Adaptive Eigenfunction Basis Strategy to Reduce Design Dimension in Topology Optimization
  
  • Sanders Clay
  • Bonnet Marc
  • Aquino Wilkins
  International Journal for Numerical Methods in Engineering, Wiley, 2021, 122, pp.7452-7481. The concept of adaptive eigenspace basis (AEB) has recently proved effective for solving medium imaging problems. In this work, we present an AEB strategy for design parameterization in topology optimization (TO) problems. We seek the density design field as a linear combination of eigenfunctions, computed for an elliptic operator defined over the structural domain, and solve for the associated eigenfunction coefficients. Restriction to this truncated eigenspace drastically reduces the design dimension and imposes implicit regularization upon the solution, removing the need for auxiliary filtering operations and design-variable bound constraints. We furthermore develop the basis adaptation scheme inherent in the AEB, which iteratively recomputes the eigenfunction basis to conform to the evolving density field, enabling further dimension reduction and acceleration of the optimization process. The known aptitude of the adapted eigenfunctions to approximate piecewise constant fields is especially useful for TO as relevant design subspaces can be given low-dimensional representations. We propose criteria for the selection of the basis dimension and demonstrate the use of basis function selection as means for length scale control. We compare performance of the AEB against conventional TO implementations in problems for static linear-elasticity, showing comparable structural solutions, computational cost benefits, and consistent design dimension reduction. K E Y W O R D S Adaptive eigenspace basis, Topology design, Dimensionality reduction Abbreviations: AEB, adaptive eigenspace basis; TO, topology optimization; EB, eigenspace basis. * Equally contributing authors. (10.1002/nme.6837)
  DOI : 10.1002/nme.6837
• A stable, unified model for resonant Faraday cages
  
  • Delourme Bérangère
  • Lunéville Éric
  • Marigo Jean-Jacques
  • Maurel Agnès
  • Mercier Jean-François
  • Pham Kim
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2021, 477 (2245), pp.20200668. We study some effective transmission conditions able to reproduce the effect of a periodic array of Dirichlet wires on wave propagation, in particular when the array delimits an acoustic Faraday cage able to resonate. In the study of Hewett & Hewitt (2016 Proc. R. Soc. A 472 , 20160062 ( doi:10.1098/rspa.2016.0062 )) different transmission conditions emerge from the asymptotic analysis whose validity depends on the frequency, specifically the distance to a resonance frequency of the cage. In practice, dealing with such conditions is difficult, especially if the problem is set in the time domain. In the present study, we demonstrate the validity of a simpler unified model derived in Marigo & Maurel (2016 Proc. R. Soc. A 472 , 20160068 ( doi:10.1098/rspa.2016.0068 )), where unified means valid whatever the distance to the resonance frequencies. The effectiveness of the model is discussed in the harmonic regime owing to explicit solutions. It is also exemplified in the time domain, where a formulation guaranteeing the stability of the numerical scheme has been implemented. (10.1098/rspa.2020.0668)
  DOI : 10.1098/rspa.2020.0668
• Numerical Analysis of a Method for Solving 2D Linear Isotropic Elastodynamics with Free Boundary Condition using Potentials and Finite Elements
  
  • Albella Martínez Jorge
  • Imperiale Sébastien
  • Joly Patrick
  • Rodríguez Jerónimo
  Mathematics of Computation, American Mathematical Society, 2021. When solving 2D linear elastodynamic equations in a homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition was treated In [3, 2]. However in [4] the case of free surface boundary condition was proven to be unstable if a straight- forward approach is used. Then an adequate functional framework as well as a time domain mixed formulation to circumvent these issues was presented. In this work we first review the formulation presented in [4] and propose a subsequent discretised formulation. We provide the complete stability analysis of the corresponding numerical scheme. Numerical results that illustrate the theory are also shown. (10.1007/s10915-018-0768-9)
  DOI : 10.1007/s10915-018-0768-9