Publications

2024

• Proximal Point Type Algorithms with Relaxed and Inertial Effects Beyond Convexity
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  Optimization, Taylor & Francis, 2024, pp.1-18. We show that the recent relaxed-inertial proximal point algorithm due to Attouch and Cabot remains convergent when the function to be minimized is not convex, being only endowed with certain generalized convexity properties. Numerical experiments showcase the improvements brought by the relaxation and inertia features to the standard proximal point method in this setting, too. (10.1080/02331934.2024.2329779)
  DOI : 10.1080/02331934.2024.2329779
• Cell seeding dynamics in a porous scaffold material designed for meniscus tissue regeneration
  
  • Jäger Henry
  • Grosjean Elise
  • Plunder Steffen
  • Redenbach Claudia
  • Keilmann Alex
  • Simeon Bernd
  • Surulescu Christina
  , 2024, 24 (2). We study the dynamics of a seeding experiment where a fibrous scaffold material is colonized by two types of cell populations. The specific application that we have in mind is related to the idea of meniscus tissue regeneration. In order to support the development of a promising replacement material, we discuss certain rate equations for the densities of human mesenchymal stem cells and chondrocytes and for the production of collagen‐containing extracellular matrix. For qualitative studies, we start with a system of ordinary differential equations and refine then the model to include spatial effects of the underlying nonwoven scaffold structure. Numerical experiments as well as a complete set of parameters for future benchmarking are provided. (10.1002/pamm.202400133)
  DOI : 10.1002/pamm.202400133
• An implicit–explicit time discretization for elastic wave propagation problems in plates
  
  • Methenni Hajer
  • Imperiale Alexandre
  • Imperiale Sébastien
  International Journal for Numerical Methods in Engineering, Wiley, 2024, 125, pp.e7393. We propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest. (10.1002/nme.7393)
  DOI : 10.1002/nme.7393
• Reconstruction of averaging indicators for highly heterogeneous media
  
  • Audibert Lorenzo
  • Haddar Houssem
  • Pourre Fabien
  Inverse Problems, IOP Publishing, 2024, 40 (4), pp.045028. We propose a new imaging algorithm capable of producing quantitative indicator functions for unknown and possibly highly oscillating media from multistatic far field measurements of scattered fields at a fixed frequency. The algorithm exploits the notion of modified transmission eigenvalues and their determination from measurements. We propose in particular the use of a new modified background obtained as the limit of a metamaterial background. It has the specificity of having a unique non trivial eigenvalue, which is particularly suited for the proposed imaging procedure. We show the efficiency of this new algorithm on some 2D experiments and emphasize its superiority with respect to some clasical approaches such as the Linear Sampling Method. (10.1088/1361-6420/ad2f64)
  DOI : 10.1088/1361-6420/ad2f64
• Modélisation hybride modale-éléments finis pour le contrôle ultrasonore d'une plaque élastique. Traitement des intégrales oscillantes de la méthode HSM
  
  • Allouko Amond
  , 2024. Cette thèse porte sur la méthode Half-Space Matching (HSM) pour la résolution de problèmes de diffraction dans une plaque élastique non-bornée, en vue de la simulation du contrôle non-destructif de plaques composites. La méthode HSM est une approche hybride qui couple un calcul éléments finis dans une boite contenant les défauts, avec des représentations semi-analytiques dans quatre demi-plaques qui recouvrent la partie saine de la plaque. Les représentations semi-analytiques de demi-plaques font intervenir des tenseurs de Green, exprimés à l'aide d'intégrales de Fourier et de séries modales. Or ces expressions peuvent être délicates à évaluer en pratique (coût et précision), rendant la méthode HSM inexploitable industriellement. Les difficultés sont d'abord analysées dans un cas scalaire bidimensionnel (acoustique). Deux méthodes sont proposées pour une évaluation efficace des intégrales de Fourier : la première exploite une approximation de type champ lointain et la seconde repose sur une déformation du chemin d'intégration dans le plan complexe (méthode de la complexification). Ces deux méthodes sont validées dans les cas scalaires isotrope et anisotrope où l'on dispose des valeurs exactes des intégrales de Fourier exprimées à l'aide de fonctions de Hankel. Elles sont ensuite généralisées au cas tridimensionnel de la plaque élastique. Dans ce cas, la formule de représentation est obtenue en faisant une transformée de Fourier suivant une direction parallèle à la plaque, puis, pour chaque valeur de la variable de Fourier ξ, une décomposition modale dans l'épaisseur. Les modes mis en jeu, appelés ξ-modes, sont étudiés en détail et comparés aux modes classiques (Lamb et SH dans le cas isotrope). Afin d'exploiter la bi-orthogonalité des ξ-modes, la formule de demi-plaque requiert la connaissance à la fois du déplacement et de la contrainte normale sur la frontière. Dans le cas isotrope, les propriétés d'analyticité des ξ-modes permettent de justifier et d'étendre la méthode de la complexification, y compris en présence de modes inverses. Ceci réduit les effets de couplage modal parasite induits par la discrétisation des intégrales de Fourier. La méthode de la complexification est ensuite utilisée pour le calcul des opérateurs intervenant dans la méthode HSM, qui dérivent tous de la formule de demiplaque. Différentes validations de la méthode HSM sont ainsi effectuées dans le cas isotrope. Des résultats préliminaires encourageants sont également obtenus pour une plaque orthotrope. Les améliorations réalisées ont permis à la fois de réduire significativement le temps de calcul et d'assurer une plus grande précision de la méthode HSM, permettant d'envisager son exploitation systématique dans un cadre de simulation industrielle.
• Introduction aux équations aux dérivées partielles hyperboliques et à leur approximation numérique
  
  • Fliss Sonia
  • Bonnet-Ben Dhia Anne-Sophie
  • Joly Patrick
  • Moireau Philippe
  , 2024.
• Flexible Optimization for Cyber-Physical and Human Systems
  
  • Simonetto Andrea
  IEEE Control Systems Letters, IEEE, 2024, 8, pp.1475-1480. Can we allow humans to pick among different, yet reasonably similar, decisions? Are we able to construct optimization problems whose outcome are sets of feasible, close-to-optimal decisions for human users to pick from, instead of a single, hardly explainable, do-as-I-say ``optimal'' directive? In this paper, we explore two complementary ways to render optimization problems stemming from cyber-physical applications flexible. In doing so, the optimization outcome is a trade off between engineering best and flexibility for the users to decide to do something slightly different. The first method is based on robust optimization and convex reformulations. The second method is stochastic and inspired from stochastic optimization with decision-dependent distributions. (10.1109/LCSYS.2024.3411931)
  DOI : 10.1109/LCSYS.2024.3411931
• Correlation Clustering Problem under Mediation
  
  • Alès Zacharie
  • Engelbeen Céline
  • Figueiredo Rosa
  INFORMS Journal on Computing, Institute for Operations Research and the Management Sciences (INFORMS), 2024, 36 (2), pp.672-689. In the context of community detection, correlation clustering (CC) provides a measure of balance for social networks as well as a tool to explore their structures. However, CC does not encompass features such as the mediation between the clusters, which could be all the more relevant with the recent rise of ideological polarization. In this work, we study correlation clustering under mediation (CCM), a new variant of CC in which a set of mediators is determined. This new signed graph clustering problem is proved to be NP-hard and formulated as an integer programming formulation. An extensive investigation of the mediation set structure leads to the development of two efficient exact enumeration algorithms for CCM. The first one exhaustively enumerates the maximal sets of mediators in order to provide several relevant solutions. The second algorithm implements a pruning mechanism, which drastically reduces the size of the exploration tree in order to return a single optimal solution. Computational experiments are presented on two sets of instances: signed networks representing voting activity in the European Parliament and random signed graphs. History: Accepted by Van Hentenryck, Area Editor for Pascal. Funding: This work was supported by Fondation Mathématique Jacques Hadamard [Grant P-2019-0031]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0129 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0129 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ . (10.1287/ijoc.2022.0129)
  DOI : 10.1287/ijoc.2022.0129
• DISCRETE HONEYCOMBS, RATIONAL EDGES AND EDGE STATES
  
  • Fefferman Charles Louis
  • Fliss Sonia
  • Weinstein Michael
  Communications on Pure and Applied Mathematics, Wiley, 2024, 77 (3), pp.1575-1634. Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of "zigzag type" and those of "armchair type", generalizing the classical zigzag and armchair edges. We prove that zero energy/flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most l. (10.1002/cpa.22141)
  DOI : 10.1002/cpa.22141
• Fast Imaging of Local Perturbations in a Unknown Bi-Periodic Layered Medium
  
  • Cakoni Fioralba
  • Haddar Houssem
  • Nguyen Thi-Phong
  Journal of Computational Physics, Elsevier, 2024, 501, pp.112773. We discuss a novel approach for imaging local faults inside an infinite bi-periodic layered medium in R 3 using acoustic measurements of scattered fields at the bottom or the top of the layer. The faulted area is represented by compactly supported perturbations with erroneous material properties. Our method reconstructs the support of perturbations without knowing or reconstructing the constitutive material parameters of healthy or faulty bi-period layer; only the size of the period is needed. This approach falls under the class of non-iterative imaging methods, known as the generalized linear sampling method with differential measurements, first introduced in [2] and adapted to periodic layers in [25]. The advantage of applying differential measurements to our inverse problem is that instead of comparing the measured data against measurements due to healthy structures, one makes use of periodicity of the layer where the data operator restricted to single Floquet-Bloch modes plays the role of the one corresponding to healthy material. This leads to a computationally efficient and mathematically rigorous reconstruction algorithm. We present numerical experiments that confirm the viability of the approach for various configurations of defects. (10.1016/j.jcp.2024.112773)
  DOI : 10.1016/j.jcp.2024.112773
• 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
• Design and Dimensioning of Natural Gas Pipelines with Hydrogen Injection
  
  • Jorquera-Bravo Natalia
  • Elloumi Sourour
  • Plateau Agnès
  , 2024. The global focus on reducing air pollution and dependence on fossil fuels has led to efforts to shift to renewable energy sources. Hydrogen is a promising alternative due to its high energy capacity and ability to regulate electricity production through electrolysis. In this context, the problem of designing and sizing natural gas pipelines with hydrogen injection is presented. The objective is to establish the network topology and diameter dimensions of each pipeline section for hydrogen distribution, in order to cover the demand at a minimum cost. To address the proposed problem, we consider the dimensioning as the selection of a diameter from a set of available measures, i.e., a discrete diameter approach, and we compare it with a continuous diameter approach from the literature, including a mixed integer nonlinear programming (MINLP) formulation of degree six. In our discrete diameter approach, we propose a non-convex quadratic (MIQLP) model, and we derive a mixed-integer quadratic convex relaxation (MIQCP). Finally, we adapt a Delta Change heuristic to this context. We implement several solution methods for a real case study in France. These include solving the dimensioning problem on a fixed Minimum Spanning Tree topology, considering both continuous and discrete diameters, employing the Delta Change heuristic for both cases, continuous and discrete, and solving the MIQCP relaxation problem. The strengths and weaknesses of each of these proposals are demonstrated through the study.
• Relaxed-Inertial Proximal Point Algorithms for Nonconvex Equilibrium Problems with Applications
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Tintaya Marcavillaca Raúl
  Journal of Optimization Theory and Applications, Springer Verlag, 2024, 203 (3), pp.2233-2262. We propose a relaxed-inertial proximal point algorithm for solving equilibrium problems involving bifunctions which satisfy in the second variable a generalized convexity notion called strong quasiconvexity, introduced by Polyak in 1966. The method is suitable for solving mixed variational inequalities and inverse mixed variational inequalities involving strongly quasiconvex functions, as these can be written as special cases of equilibrium problems. Numerical experiments where the performance of the proposed algorithm outperforms the one of the standard proximal point methods are provided, too. (10.1007/s10957-023-02375-1)
  DOI : 10.1007/s10957-023-02375-1
• Exact solution methods for large-scale discrete p-facility location problems
  
  • Durán Mateluna Cristian
  , 2024. This thesis focuses on the exact solution of the NP-hard problems p-median and p-center, combinatorial optimization problems that quickly become difficult to solve as the instance size increases. These discrete location problems involve opening a defined number p of facilities and then allocating to them a set of clients according to an objective function to be minimized.First, we study the p-median problem, which seeks to minimize the sum of distances between clients and the open facilities to which they are allocated. We develop an algorithm based on Benders decomposition that outperforms state-of-the-art exact methods. The algorithm considers a two-stage approach and an efficient algorithm for separating Benders cuts. The method has been evaluated on over 230 benchmark instances with up to 238025 clients and sites. Many instances are solved to optimality for the first time or have their best known solution improved.Secondly, we explore the p-center problem, which seeks to minimize the largest distance between a client and its nearest open facility. We first compare the five main MILP formulations in the literature. We study the Benders decomposition and also propose an exact algorithm based on a client clustering procedure based on the structure of the problem. All the proposed methods are compared with the state-of-the-art on benchmark instances. The results obtained are analyzed, highlighting the advantages and disadvantages of each method.Finally, we study a robust two-stage p-center problem with uncertainty on node demands and distances. We introduce the robust reformulation of the problem based on the five main deterministic MILP formulations in the literature. We prove that only a finite subset of scenarios from the infinite uncertainty set can be considered without losing optimality. We also propose a column and constraint generation algorithm and a branch-and-cut algorithm to efficiently solve this problem. We show how these algorithms can also be adapted to solve the robust single-stage problem. The different proposed formulations are tested on randomly generated instances and on a case study drawn from the literature.
• Commande des Systèmes
  
  • Jean Frédéric
  , 2024.
• Degenerate McKean-Vlasov equations with drift in anisotropic negative Besov spaces
  
  • Issoglio Elena
  • Pagliarani Stefano
  • Russo Francesco
  • Trevisani Davide
  , 2024. The paper is concerned with a McKean-Vlasov type SDE with drift in anisotropic Besov spaces with negative regularity and with degenerate diffusion matrix under the weak Hörmander condition. The main result is of existence and uniqueness of a solution in law for the McKean-Vlasov equation, which is formulated as a suitable martingale problem. All analytical tools needed are derived in the paper, such as the well-posedness of the Fokker-Planck and Kolmogorov PDEs with distributional drift, as well as continuity dependence on the coefficients. The solutions to these PDEs naturally live in anisotropic Besov spaces, for which we developed suitable analytical inequalities, such as Schauder estimates.
• 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
• Bi-objective finite horizon optimal control problems with Bolza and maximum running cost
  
  • Chorobura Ana Paula
  • Zidani Hasnaa
  , 2024. In this paper, we investigate optimal control problems with two objective functions of different nature that need to be minimized simultaneously. One objective is in the classical Bolza form and the other one is defined as a maximum running cost. Our approach is based on the Hamilton-Jacobi-Bellman framework. In the problem considered here the existence of Pareto solutions is not guaranteed. So first, we consider the bi-objective problem to be minimized over the convexified dynamical system. We show that if a vector is (weak) Pareto optimal solution for the convexified problem, then there exists an (weak) ε-Pareto optimal solution of the original problem that is in the neighborhood of this vector. After we define an auxiliary optimal control problem and show that the weak Pareto front of the convexified problem is a subset of the zero level set of the corresponding value function. Moreover, with a geometrical approach we establish a characterization of the Pareto front. It is also proved that the (weak) ε-Pareto front is contained in the negative level set of the auxiliary optimal control problem that is less or equal ε. Some numerical examples are considered to show the relevance of our approach.
• 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, 22 (1). 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. (10.1137/22M1495512)
  DOI : 10.1137/22M1495512
• Deterministic optimal control on Riemannian manifolds under probability knowledge of the initial condition
  
  • Jean Frédéric
  • Jerhaoui Othmane
  • Zidani Hasnaa
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2024, 56 (3), pp.3326-3356. In this article, we study an optimal control problem on a compact Riemannian manifold M with imperfect information on the initial state of the system. The lack of information is modelled by a Borel probability measure along which the initial state is distributed. The state space of this problem is the space of Borel probability measures over M. We define a notion of viscosity in this space by taking as test functions a subset of the set of functions that can be written as a difference of two semi-convex functions. With this choice of test functions, we extend the notion of viscosity solution to Hamilton-Jacobi-Bellman equations in Wasserstein space, we also establish that the value function of the control problem with imperfect information is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the space of Borel probability measures. (10.1137/23M1575251)
  DOI : 10.1137/23M1575251
• Combined field-only boundary integral equations for PEC electromagnetic scattering problem in spherical geometries
  
  • Faria Luiz
  • Pérez-Arancibia Carlos
  • Turc Catalin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2024, 84 (1). We analyze the well posedness of certain field-only boundary integral equations (BIE) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field $\mathbf{E}^s(\mathbf{x})$ and (2) scalar quantity $\mathbf{E}^s(\mathbf{x})\cdot\mathbf{x}$ are radiative solutions of the Helmholtz equation, novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIE. In this paper we use the Combined Field methodology of Burton and Miller within the field-only BIE approach and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are non zero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one -- a property similar to second kind integral equations. (10.1137/23M1561865)
  DOI : 10.1137/23M1561865
• Optimization of a domestic microgrid equipped with solar panel and battery: Model Predictive Control and Stochastic Dual Dynamic Programming approaches
  
  • Pacaud François
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  Energy Systems, Springer, 2024, 15 (1), pp.115-139. In this study, a microgrid with storage (battery, hot water tank) and solar panel is considered. We benchmark two algorithms, MPC and SDDP, that yield online policies to manage the microgrid, and compare them with a rule based policy. Model Predictive Control (MPC) is a well-known algorithm which models the future uncertainties with a deterministic forecast. By contrast, Stochastic Dual Dynamic Programming (SDDP) models the future uncertainties as stagewise independent random variables with known probability distributions. We present a scheme, based on out-of-sample validation, to fairly compare the two online policies yielded by MPC and SDDP. Our numerical studies put to light that MPC and SDDP achieve significant gains compared to the rule based policy, and that SDDP overperforms MPC not only on average but on most of the out-of-sample assessment scenarios. (10.1007/s12667-022-00522-7)
  DOI : 10.1007/s12667-022-00522-7
• 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, 2024, 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. (10.1137/23M1607866)
  DOI : 10.1137/23M1607866
• 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