Publications

2024

• The scattering phase: seen at last
  
  • Galkowski Jeffrey
  • Marchand Pierre
  • Wang Jian
  • Zworski Maciej
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2024, 84 (1), pp.246-261. The scattering phase, defined as $ \log \det S ( \lambda ) / 2\pi i $ where $ S ( \lambda ) $ is the (unitary) scattering matrix, is the analogue of the counting function for eigenvalues when dealing with exterior domains and is closely related to Krein's spectral shift function. We revisit classical results on asymptotics of the scattering phase and point out that it is never monotone in the case of strong trapping of waves. Perhaps more importantly, we provide the first numerical calculations of scattering phases for non-radial scatterers. They show that the asymptotic Weyl law is accurate even at low frequencies and reveal effects of trapping such as lack of monotonicity. This is achieved by using the recent high level multiphysics finite element software FreeFEM. (10.1137/23M1547147)
  DOI : 10.1137/23M1547147
• Radial perfectly matched layers and infinite elements for the anisotropic wave equation
  
  • Halla Martin
  • Kachanovska Maryna
  • Wess Markus
  , 2025, pp.3171-3216. We consider the scalar anisotropic wave equation. Recently a convergence analysis for radial perfectly matched layers (PML) in the frequency domain was reported and in the present article we continue this approach into the time domain. First we explain why there is a good hope that radial complex scalings can overcome the instabilities of PML methods caused by anisotropic materials. Next we discuss some sensitive details, which seem like a paradox at the first glance: if the absorbing layer and the inhomogeneities are sufficiently separated, then the solution is indeed stable. However, for more general data the problem becomes unstable. In numerical computations we observe instabilities regardless of the position of the inhomogeneities, although the instabilities arise only for fine enough discretizations. As a remedy we propose a complex frequency shifted scaling and discretizations by Hardy space infinite elements or truncation-free PMLs. We show numerical experiments which confirm the stability and convergence of these methods. (10.1137/24M1636551)
  DOI : 10.1137/24M1636551
• Towards high-performance linear potential flow BEM solver with low-rank compressions
  
  • Ancellin Matthieu
  • Marchand Pierre
  • Dias Frédéric
  Energies, MDPI, 2024, 17 (2), pp.372. The interaction of water waves with floating bodies can be modelled with linear potential flow theory, numerically solved with the Boundary Element Method (BEM). This method requires the construction of dense matrices and the resolution of the corresponding linear systems. The cost in time and memory of the method grows at least quadratically with the size of the mesh and the resolution of large problems (such as large farms of wave energy converters) can thus be very costly. Approximating some blocks of the matrix by data-sparse matrices can limit this cost. While matrix compression with low-rank blocks has become a standard tool in the larger BEM community, the present paper provides its first application (to our knowledge) to linear potential flows. In this paper, we assess that low-rank blocks can efficiently approximate interaction matrices between distant meshes when using the Green function of linear potential flow. Due to the complexity of this Green function, a theoretical study is difficult and numerical experiments are used to test the approximation method. Typical results on large arrays of floating bodies show that 99% of the accuracy can be reached with 10% of the coefficients of the matrix. (10.3390/en17020372)
  DOI : 10.3390/en17020372
• Modified error-in-constitutive-relation (MECR) framework for the characterization of linear viscoelastic solids
  
  • Bonnet Marc
  • Salasiya Prasanna
  • Guzina Bojan B.
  Journal of the Mechanics and Physics of Solids, Elsevier, 2024, 190, pp.105746. We develop an error-in-constitutive-relation (ECR) approach toward the full-field characterization of linear viscoelastic solids described within the framework of standard generalized materials. To this end, we formulate the viscoelastic behavior in terms of the (Helmholtz) free energy potential and a dissipation potential. Assuming the availability of full-field interior kinematic data, the constitutive mismatch between the kinematic quantities (strains and internal thermodynamic variables) and their ``stress'' counterparts (Cauchy stress tensor and that of thermodynamic tensions), commonly referred to as the ECR functional, is established with the aid of Legendre-Fenchel gap functionals linking the thermodynamic potentials to their energetic conjugates. We then proceed by introducing the modified ECR (MECR) functional as a linear combination between its ECR parent and the kinematic data misfit, computed for a trial set of constitutive parameters. The affiliated stationarity conditions then yield two coupled evolution problems, namely (i) the forward evolution problem for the (trial) displacement field driven by the constitutive mismatch, and (ii) the backward evolution problem for the adjoint field driven by the data mismatch. This allows us to establish compact expressions for the MECR functional and its gradient with respect to the viscoelastic constitutive parameters. For generality, the formulation is established assuming both time-domain (i.e. transient) and frequency-domain data. We illustrate the developments in a two-dimensional setting by pursuing the multi-frequency MECR reconstruction of (i) piecewise-homogeneous standard linear solid, and (b) smoothly-varying Jeffreys viscoelastic material. (10.1016/j.jmps.2024.105746)
  DOI : 10.1016/j.jmps.2024.105746
• Variational methods for solving numerically magnetostatic systems
  
  • Ciarlet Patrick
  • Jamelot Erell
  Advances in Computational Mathematics, Springer Verlag, 2024, 50 (1), pp.5. 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. (10.1007/s10444-023-10089-1)
  DOI : 10.1007/s10444-023-10089-1
• A new class of uniformly stable time-domain Foldy-Lax models for scattering by small particles. Acoustic sound-soft scattering by circles. Extended version
  
  • Kachanovska Maryna
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2024. In this work we study time-domain sound-soft scattering by small circles. Our goal is to derive an asymptotic model for this problem valid when the size of the particles tends to zero. We present a systematic approach to constructing such models, based on a well-chosen Galerkin discretization of a boundary integral equation. The convergence of the method is achieved by decreasing the asymptotic parameter rather than increasing the number of basis functions. For the case of circular obstacles, we prove the second-order convergence of the field error with respect to the particle size. Our findings are illustrated with numerical experiments. (10.1137/22M149551)
  DOI : 10.1137/22M149551
• Solvability results for the transient acoustic scattering by an elastic obstacle
  
  • Bonnet Marc
  • Chaillat Stéphanie
  • Nassor Alice
  Journal of Mathematical Analysis and Applications, Elsevier, 2024, 536 (128198). The well-posedness of the linear evolution problem governing the transient scattering of acoustic waves by an elastic obstacle is investigated. After using linear superposition in the acoustic domain, the analysis focuses on an equivalent causal transmission problem. The proposed analysis provides existence and uniqueness results, as well as continuous data-to-solution maps. Solvability results are established for three cases, which differ by the assumed regularity in space on the transmission data on the acoustic-elastic interface Γ. The first two results consider data with "standard" H −1/2 (Γ) and improved H 1/2 (Γ) regularity in space, respectively, and are established using the Hille-Yosida theorem and energy identities. The third result assumes data with L 2 (Γ) regularity in space and follows by Sobolev interpolation. Obtaining the latter result was motivated by the key role it plays (in a separate study) in the justification of an iterative numerical solution method based on domain decomposition. A numerical example is presented to emphasize the latter point. (10.1016/j.jmaa.2024.128198)
  DOI : 10.1016/j.jmaa.2024.128198
• The T-coercivity approach for mixed problems
  
  • Barré Mathieu
  • Ciarlet Patrick
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2024, 362, pp.1051-1088. Classically, the well-posedness of variational formulations of mixed linear problems is achieved through the inf-sup condition on the constraint. In this note, we propose an alternative framework to study such problems by using the T-coercivity approach to derive a global inf-sup condition. Generally speaking, this is a constructive approach that, in addition, drives the design of suitable approximations. As a matter of fact, the derivation of the uniform discrete inf-sup condition for the approximate problems follows easily from the study of the original problem. To support our view, we solve a series of classical mixed problems with the T-coercivity approach. Among others, the celebrated Fortin Lemma appears naturally in the numerical analysis of the approximate problems. (10.5802/crmath.590)
  DOI : 10.5802/crmath.590
• Study of a degenerate non-elliptic equation to model plasma heating
  
  • Ciarlet Patrick
  • Kachanovska Maryna
  • Peillon Etienne
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2024. In this manuscript, we study solutions to resonant Maxwell's equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div (α∇u) − ω 2 u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [13] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series a numerical tests, comparing the numerical results of Nicolopoulos et al to those obtained with our numerical method, for which we observe better convergence.
• Coupling of discontinuous Galerkin and pseudo-spectral methods for time-dependent acoustic problems
  
  • Meyer Rose-Cloé
  • Bériot Hadrien
  • Gabard Gwenael
  • Modave Axel
  Journal of Theoretical and Computational Acoustics, World Scientific, 2024, 32 (4), pp.2450017. Many realistic problems in computational acoustics involve complex geometries and sound propagation over large domains, which requires accurate and efficient numerical schemes. It is difficult to meet these requirements with a single numerical method. Pseudo-spectral (PS) methods are very efficient, but are limited to rectangular shaped domains. In contrast, the nodal discontinuous Galerkin (DG) method can be easily applied to complex geometries, but can become expensive for large problems. In this paper, we study a coupling strategy between the PS and DG methods to efficiently solve time-domain acoustic wave problems. The idea is to combine the strengths of these two methods: the PS method is used on the part of the domain without geometric constraints, while the DG method is used around the PS region to accurately represent the geometry. This combination allows for the rapid and accurate simulations of large-scale acoustic problems with complex geometries, but the coupling and the parameter selection require great care. The coupling is achieved by introducing an overlap between the PS and DG regions. The solutions are interpolated on the overlaps, which allows the use of unstructured finite element meshes. A standard explicit Runge-Kutta time-stepping scheme is used with the DG scheme, while implicit schemes can be used with the PS scheme due to the peculiar structure of this scheme. We present one-and two-dimensional results to validate the coupling technique. To guide future implementations of this method, we extensively study the influence of different numerical parameters on the accuracy of the schemes and the coupling strategy. (10.1142/S2591728524500178)
  DOI : 10.1142/S2591728524500178
• Adaptive solution of the domain decomposition+ $L^2$ -jumps method applied to the neutron diffusion equation on structured meshes
  
  • Gervais Mario
  • Madiot François
  • Do Minh-Hieu
  • Ciarlet Patrick
  EPJ Web of Conferences, EDP Sciences, 2024, 302, pp.02011. At the core scale, neutron deterministic calculations are usually based on the neutron diffusion equation. Classically, this equation can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. The goal is to extend the Adaptive Mesh Refinement (AMR) strategy previously proposed in [1] to the Domain Decomposition+ $L^2$ jumps which allows non conformity at the interface between subdomains. We are able to refine each subdomain independently, which eventually leads to a more optimal refinement. We numerically investigate the improvements made to the AMR strategy. (10.1051/epjconf/202430202011)
  DOI : 10.1051/epjconf/202430202011
• Active Design of Diffuse Acoustic Fields in Enclosures
  
  • Aquino Wilkins
  • Rouse Jerry
  • Bonnet Marc
  Journal of the Acoustical Society of America, Acoustical Society of America, 2024, 155, pp.1297-1307. This paper presents a numerical framework for designing diffuse fields in rooms of any shape and size, driven at arbitrary frequencies. That is, we aim at overcoming the Schroeder frequency limit for generating diffuse fields in an enclosed space. We formulate the problem as a Tikhonov regularized inverse problem and propose a lowrank approximation of the spatial correlation that results in significant computational gains. Our approximation is applicable to arbitrary sets of target points and allows us to produce an optimal design at a computational cost that grows only linearly with the (potentially large) number of target points. We demonstrate the feasibility of our approach through numerical examples where we approximate diffuse fields at frequencies well below the Schroeder limit. (10.1121/10.0024770)
  DOI : 10.1121/10.0024770
• Construction of polynomial particular solutions of linear constant-coefficient partial differential equations
  
  • Anderson Thomas G.
  • Bonnet Marc
  • Faria Luiz
  • Pérez-Arancibia Carlos
  Computers & Mathematics with Applications, Elsevier, 2024, 162C, pp.94-103. This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and also have potential applications in certain kinds of Trefftz finite element methods. The equations covered in this work include the isotropic and anisotropic Poisson, Helmholtz, Stokes, linearized Navier-Stokes, stationary advection-diffusion, elastostatic equations, as well as the time-harmonic elastodynamic and Maxwell equations. Several solutions to complex PDE systems are obtained by a potential representation and rely on the Helmholtz or Poisson solvers. Some of the cases addressed, namely Stokes flow, Maxwell’s equations and linearized Navier-Stokes equations, naturally incorporate divergence constraints on the solution. This article provides a generic pattern whereby solutions are constructed by leveraging solutions of the lowest-order part of the partial differential operator (PDO). With the exception of anisotropic material tensors, no matrix inversion or linear system solution is required to compute the solutions. This work is accompanied by a freely-available Julia library, ElementaryPDESolutions.jl, which implements the proposed methodology in an efficient and user-friendly format. (10.1016/j.camwa.2024.02.045)
  DOI : 10.1016/j.camwa.2024.02.045
• Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation
  
  • Anderson Thomas G.
  • Bonnet Marc
  • Faria Luiz
  • Pérez‐Arancibia Carlos
  Journal of Computational Physics, Elsevier, 2024, 511, pp.113091. This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented. (10.1016/j.jcp.2024.113091)
  DOI : 10.1016/j.jcp.2024.113091