Publications

2019

• Wave propagation in periodic media : mathematical analysis and numerical simulation
  
  • Fliss Sonia
  , 2019.
• Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D
  
  • Pérez-Arancibia Carlos
  • Faria Luiz
  • Turc Catalin
  Journal of Computational Physics, Elsevier, 2019, 376, pp.411-434. We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Green’s third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary integral scan then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions. (10.1016/j.jcp.2018.10.002)
  DOI : 10.1016/j.jcp.2018.10.002
• Analysis of topological derivative as a tool for qualitative identification
  
  • Bonnet Marc
  • Cakoni Fioralba
  Inverse Problems, IOP Publishing, 2019, 35 (104007). The concept of topological derivative has proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. Although for the most part, this approach remains based on a heuristic interpretation of the topological derivative, a first attempt toward its mathematical justification was done in Bellis et al. (Inverse Problems 29:075012, 2013) for the case of isotropic media with far field data and inhomogeneous refraction index. Our paper extends the analysis there to the case of anisotropic scatterers and background with near field data. Topological derivative-based imaging functional is analyzed using a suitable factorization of the near fields, which became achievable thanks to a new volume integral formulation recently obtained in Bonnet (J. Integral Equ. Appl. 29:271-295, 2017). Our results include justification of sign heuristics for the topological derivative in the isotropic case with jump in the main operator and for some cases of anisotropic media, as well as verifying its decaying property in the isotropic case with near field spherical measurements configuration situated far enough from the probing region. (10.1088/1361-6420/ab0b67)
  DOI : 10.1088/1361-6420/ab0b67
• A Fourier-accelerated volume integral method for elastoplastic contact
  
  • Frérot Lucas
  • Bonnet Marc
  • Molinari Jean-François
  • Anciaux Guillaume
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2019, 351, pp.951-976. The contact of solids with rough surfaces plays a fundamental role in physical phenomena such as friction, wear, sealing, and thermal transfer. However, its simulation is a challenging problem due to surface asperities covering a wide range of length-scales. In addition, non-linear local processes, such as plasticity, are expected to occur even at the lightest loads. In this context, robust and efficient computational approaches are required. We therefore present a novel numerical method, based on integral equations, capable of handling the large discretization requirements of real rough surfaces as well as the non-linear plastic flow occurring below and at the contacting asperities. This method is based on a new derivation of the Mindlin fundamental solution in Fourier space, which leverages the computational efficiency of the fast Fourier transform. The use of this Mindlin solution allows a dramatic reduction of the memory in-print (as the Fourier coefficients are computed on-the-fly), a reduction of the discretization error, and the exploitation of the structure of the functions to speed up computation of the integral operators. We validate our method against an elastic-plastic FEM Hertz normal contact simulation and showcase its ability to simulate contact of rough surfaces with plastic flow. (10.1016/j.cma.2019.04.006)
  DOI : 10.1016/j.cma.2019.04.006
• An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation
  
  • Amlani Faisal
  • Chaillat Stéphanie
  • Loseille Adrien
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2019, 352 (1), pp.189-210. This paper presents an efficient algebraic preconditioner to speed up the convergence of Fast Multipole accelerated Boundary Element Methods (FM-BEMs) in the context of time-harmonic 3D wave propagation problems and in particular the case of highly non-uniform discretizations. Such configurations are produced by a recently-developed anisotropic mesh adaptation procedure that is independent of partial differential equation and integral equation. The new preconditioning methodology exploits a complement between fast BEMs by using two nested GMRES algorithms and rapid matrix-vector calculations. The fast inner iterations are evaluated by a coarse hierarchical matrix (H-matrix) representation of the BEM system. These inner iterations produce a preconditioner for FM-BEM solvers. It drastically reduces the number of outer GMRES iterations. Numerical experiments demonstrate significant speedups over non-preconditioned solvers for complex geometries and meshes specifically adapted to capture anisotropic features of a solution, including discontinuities arising from corners and edges. (10.1016/j.cma.2019.04.026)
  DOI : 10.1016/j.cma.2019.04.026
• Trapped modes in thin and infinite ladder like domains. Part 2 : asymptotic analysis and numerical application
  
  • Delourme Bérangère
  • Fliss Sonia
  • Joly Patrick
  • Vasilevskaya Elizaveta
  Asymptotic Analysis, IOS Press, 2019. We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a " thick graph " , namely a thin structure (the thinness being characterized by a small parameter ε > 0) whose limit (when ε tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter ε) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.
• On the convergence in $H^1$-norm for the fractional Laplacian
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2019, 57, pp.1723-1743. We consider the numerical solution of the fractional Laplacian of index $s \in (1/2, 1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space $\widetilde{H}^s(\Omega)$. For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in $H^1(\Omega)$. In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to $H^1(\Omega)$. A natural question is then whether one can obtain error estimates in $H^1(\Omega)$-norm, in addition to the classical ones that can be derived in the $\widetilde{H}^s(\Omega)$ energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes. (10.1137/18M1221436)
  DOI : 10.1137/18M1221436
• The eddy current model as a low-frequency, high-conductivity asymptotic form of the Maxwell transmission problem
  
  • Bonnet Marc
  • Demaldent Edouard
  Computers & Mathematics with Applications, Elsevier, 2019, 77 (8), pp.2145-2161. We study the relationship between the Maxwell and eddy current (EC) models for three-dimensional configurations involving bounded regions with high conductivity $\sigma$ in air and with sources placed remotely from the conducting objects, which typically occur in the numerical simulation of eddy current nondestructive testing (ECT) experiments. The underlying Maxwell transmission problem is formulated using boundary integral formulations of PMCHWT type. In this context, we derive and rigorously justify an asymptotic expansion of the Maxwell integral problem with respect to the non-dimensional parameter $\gamma:=\sqrt{\omega\varepsilon_{0}/\sigma}$. The EC integral problem is shown to constitute the limiting form of the Maxwell integral problem as $\gamma\to0$, i.e. as its low-frequency and high-conductivity limit. Estimates in $\gamma$ are obtained for the solution remainders (in terms of the surface currents, which are the primary unknowns of the PMCHWT problem, and the electromagnetic fields) and the impedance variation measured at the extremities of the excitating coil. In particular, the leading and remainder orders in $\gamma$ of the surface currents are found to depend on the current component (electric or magnetic, charge-free or not). These theoretical results are demonstrated on three-dimensional illustrative numerical examples, where the mathematically established estimates in $\gamma$ are reproduced by the numerical results. (10.1016/j.camwa.2018.12.006)
  DOI : 10.1016/j.camwa.2018.12.006
• Analysis of the error in constitutive equation approach for time-harmonic elasticity imaging
  
  • Aquino Wilkins
  • Bonnet Marc
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2019, 79, pp.822-849. We consider the identification of heterogeneous linear elastic moduli in the context of time-harmonic elastodynamics. This inverse problem is formulated as the minimization of the modified error in constitutive equation (MECE), an energy-based cost functional defined as an weighted additive combination $\mathcal{E}+\kappa\mathcal{D}$ of the error in constitutive equation (ECE) $\mathcal{E}$, expressed using an energy seminorm, and a quadratic error term $\mathcal{D}$ incorporating the kinematical measurements. MECE-based identification are known from existing computational evidence to enjoy attractive properties such as improved convexity, robustness to resonant frequencies, and tolerance to incompletely specified boundary conditions (BCs). The main goal of this work is to develop theoretical foundations, in a continuous setting, allowing to explain and justify some of the aforementioned beneficial properties, in particular addressing the general case where BCs may be underspecified. A specific feature of MECE formulations is that forward and adjoint solutions are governed by a fully coupled system, whose mathematical properties play a fundamental role in the qualitative and computational aspects of MECE minimization. We prove that this system has a unique and stable solution at any frequency, provided data is abundant enough (in a sense made precise therein) to at least compensate for any missing information on BCs. As a result, our formulation leads in such situations to a well-defined solution even though the relevant forward problem is not \emph{a priori} clearly defined. This result has practical implications such as applicability of MECE to partial interior data (with important practical applications including ultrasound elastography), convergence of finite element discretizations and differentiability of the reduced MECE functional. In addition, we establish that usual least squares and pure ECE formulations are limiting cases of MECE formulations for small and large values of $\kappa$, respectively. For the latter case, which corresponds to exact enforcement of kinematic data, we furthermore show that the reduced MECE Hessian is asymptotically positive for any parameter perturbation supported on the measurement region, thereby corroborating existing computational evidence on convexity improvement brought by MECE functionals. Finally, numerical studies that support and illustrate our theoretical findings, including a parameter reconstruction example using interior data, are presented. (10.1137/18M1231237)
  DOI : 10.1137/18M1231237