Publications

2025

• Méthode hybride de simulation de champs ultrasonores dans une grande structure stratifiée avec des objets au contact
  
  • Kubecki Romain
  • Ducasse Eric
  • Bonnet Marc
  • Deschamps Marc
  , 2025. Ce travail a pour objectif de simuler la propagation d'ultrasons dans une structure stratifiée de grande taille comportant des objets au contact (de type traducteur, raidisseur, ou autre), dans un contexte de contrôle non destructif. La taille modérée des objets permet leur simulation par éléments finis, qui est par contre prohibitive pour la structure stratifiée de base. Si cette dernière est de géométrie canonique (plane ou tubulaire à symétrie de révolution), les champs peuvent en revanche être calculés par une méthode semi-analytique rapide utilisant des transformées de Laplace en temps et de Fourier par rapport aux coordonnées « longitudinales » (plan de la plaque ou positions axiale et azimutale dans le tube). En effet, dans le domaine <latex>(k,r,s)</latex> (<latex>k~vecteur</latex> d'onde, <latex>r~position</latex> dans l'épaisseur, <latex>s~variable</latex> de Laplace), le problème de propagation peut être résolu de manière exacte, et massivement parallélisable. Pour exploiter les atouts des deux méthodes, nous proposons une approche itérative de couplage par <i>décomposition de domaine</i> (DDM), reposant sur une suite de problèmes de propagation dans chaque sous-domaine comprenant sur leur interface commune des conditions aux limites dépendant des solutions de l'itération précédente. La littérature montre que le choix de conditions de Robin (de type impédance) entre deux domaines couplés garantit dans beaucoup de situations la convergence des itérations de couplage. Nous prouvons que cette convergence a bien lieu pour notre contexte particulier et présentons une validation numérique préliminaire en configuration 2D. Le caractère spatialement non-local du traitement semi-analytique de la structure stratifiée nous conduit ensuite à construire des fonctions de base négligeables en-dehors d'un voisinage de l'interface et à développer un protocole spécifique pour leur couplage avec les éléments finis. Ces deux aspects constituent les principaux ingrédients de la méthode hybride proposée ici. <latex>\medskip\hspace20mm</latex><i>Ce travail est financé par la DGA-AID et le CEA-List.</i>
• Roadmap on metamaterial theory, modelling and design
  
  • Davies Bryn
  • Szyniszewski Stefan
  • Dias Marcelo
  • de Waal Leo
  • Kisil Anastasia
  • P Smyshlyaev Valery
  • Cooper Shane
  • Kamotski Ilia
  • Touboul Marie
  • Craster Richard
  • Capers James
  • Horsley Simon
  • Hewson Robert
  • Santer Matthew
  • Murphy Ryan
  • Thillaithevan Dilaksan
  • Berry Simon
  • Conduit Gareth
  • Earnshaw Jacob
  • Syrotiuk Nicholas
  • Duncan Oliver
  • Kaczmarczyk Łukasz
  • Scarpa Fabrizio
  • Pendry John
  • Martí-Sabaté Marc
  • Guenneau Sébastien
  • Torrent Daniel
  • Cherkaev Elena
  • Wellander Niklas
  • Alù Andrea
  • Madine Katie
  • Colquitt Daniel
  • Sheng Ping
  • Bennetts Luke
  • Krushynska Anastasiia
  • Zhang Zhaohang
  • Mirzaali Mohammad
  • Zadpoor Amir
  Journal of Physics D: Applied Physics, IOP Publishing, 2025, 58 (20), pp.203002. This Roadmap surveys the diversity of different approaches for characterising, modelling and designing metamaterials. It contains articles covering the wide range of physical settings in which metamaterials have been realised, from acoustics and electromagnetics to water waves and mechanical systems. In doing so, we highlight synergies between the many different physical domains and identify commonality between the main challenges. The articles also survey a variety of different strategies and philosophies, from analytic methods such as classical homogenisation to numerical optimisation and data-driven approaches. We highlight how the challenging and many-degree-of-freedom nature of metamaterial design problems call for techniques to be used in partnership, such that physical modelling and intuition can be combined with the computational might of modern optimisation and machine learning to facilitate future breakthroughs in the field. (10.1088/1361-6463/adc271)
  DOI : 10.1088/1361-6463/adc271
• A Rellich-type theorem for the Helmholtz equation in a junction of stratified media
  
  • Al Humaikani Sarah
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Hazard Christophe
  , 2025. <div><p>We prove that there are no non-zero square-integrable solutions to a two-dimensional Helmholtz equation in some unbounded inhomogeneous domains which represent junctions of stratified media. More precisely, we consider domains that are unions of three half-planes, where each half-plane is stratified in the direction orthogonal to its boundary. As for the well-known Rellich uniqueness theorem for a homogeneous exterior domain, our result does not require any boundary condition. Our proof is based on half-plane representations of the solution which are derived through a generalization of the Fourier transform adapted to stratified media. A byproduct of our result is the absence of trapped modes at the junction of open waveguides as soon as the angles between branches are greater than π/2.</p></div>
• An inverse problem related to an elasto-plastic beam
  
  • Bourgeois Laurent
  • Mercier Jean-François
  Inverse Problems, IOP Publishing, 2025, 41 (10). We consider an elasto-plastic beam and address the following inverse problem: external forces have created some plastic strains in this beam, which therefore shows a residual observable deformation once the structure is load-free. Can we retrieve the loading history from this observation, or at least the plastic strains ? After proving the well-posedness of the forward problem, we show that the solution can be described in a semi-explicit way in the pure bending case, so that the forward problem amounts to a one dimensional non linear problem. Such problem is smooth enough for us to solve the inverse problem by using a classical least square method, which is illustrated with the help of some numerical examples. (10.1088/1361-6420/ae0e49)
  DOI : 10.1088/1361-6420/ae0e49
• Explicit T-coercivity for the Stokes problem: a coercive finite element discretization
  
  • Ciarlet Patrick
  • Jamelot Erell
  , 2024, pp.137-159. Using the T -coercivity theory as advocated in Chesnel-Ciarlet [Numer. Math., 2013], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when the viscosity is small (10.1016/j.camwa.2025.03.028)
  DOI : 10.1016/j.camwa.2025.03.028
• Stability of time stepping methods for discontinuous Galerkin discretizations of Friedrichs' systems
  
  • Imperiale Sébastien
  • Joly Patrick
  • Rodríguez Jerónimo
  , 2025. In this work we study new various energy-based theoretical results on the stability of s-stages, s-th order explicit Runge-Kutta integrators as well as a modified leap-frog scheme applied to discontinuous Galerkin discretizations of transient linear symmetric hyperbolic Friedrichs' systems. We restrict the present study to conservative systems and Cauchy problems.
• A complex-scaled boundary integral equation for the embedded eigenvalues and complex resonances of the Neumann-Poincaré operator on domains with corners
  
  • Maltez Faria Luiz
  • Monteghetti Florian
  , 2025. The adjoint of the harmonic double-layer operator, also known as the Neumann-Poincaré (NP) operator, is a boundary integral operator whose real eigenvalues are associated with surface modes that find applications in e.g. photonics. On 2D domains with corners, the NP operator looses its compactness, as each corner induces a bounded interval of essential spectrum, and can exhibit both embedded eigenvalues and complex resonances. This work introduces a non-self-adjoint boundary integral operator whose discrete spectrum contains both embedded eigenvalues and complex resonances of the NP operator. This operator is obtained using a Green's function that is complex-scaled at each corner of the boundary. Numerical experiments using a Nyström discretization on a graded mesh demonstrates the accuracy of the method and its advantage over a 2D finite element discretization implementing the same complex scaling technique. (10.1016/j.camwa.2025.08.012)
  DOI : 10.1016/j.camwa.2025.08.012
• Mathematical and numerical analysis of the modes of a heterogeneous electromagnetic waveguide.
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2025. In the homogeneous case, i.e. with constant epsilon and mu, the modes (E_n, H_n, \beta_n) are easily obtained by solving scalar problems in the section S of the guide and are pairwise orthogonal in L^2(S). They are either propagating (\beta in R) or purely evanescent (\beta in iR) and they have phase and group velocities of the same sign. For heterogeneous guides, i.e. with varying epsilon and mu in the section, these properties are generally not true and the mathematical analysis of the modes is much more delicate. In this talk, we present different formulations to study them and discuss their respective advantages. For strong variations of epsilon and/or mu, we show numerically that inverse modes, with group and phase velocities of opposite sign, can exist. Such cases for which PMLs fail to capture the outgoing solution are one of the reasons why we develop modal transparent conditions.
• Adaptive mesh refinement on Cartesian meshes applied to the mixed finite element discretization of the multigroup neutron diffusion equations
  
  • Ciarlet Patrick
  • Do Minh-Hieu
  • Madiot François
  , 2025. <div><p>The multigroup neutron diffusion equations are often used to model the neutron density at the nuclear reactor core scale. Classically, these equations can be recast in a mixed variational form.This chapter presents an adaptive mesh refinement approach based on a posteriori estimators. We focus on refinement strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications.</p><p>This preprint corresponds to the Chapter 19 of volume 60 in AAMS, Advances in Applied Mechanics (to appear).</p></div>
• Solving numerically the two-dimensional time harmonic Maxwell problem with sign-changing coefficients
  
  • Chaaban Farah
  • Ciarlet Patrick
  • Rihani Mahran
  , 2025. We are investigating the numerical solution to the 2D time-harmonic Maxwell equations in the presence of a classical medium and a metamaterial, that is with sign-changing coefficients. As soon as the problem has a unique solution, we are able to build a converging numerical approximation based on the finite element method, for which there is no constraint on the meshes related to the sign-changing behavior. Numerical examples illustrate the theory.
• Energy stable and linearly well-balanced numerical schemes for the non-linear Shallow Water equations with Coriolis force
  
  • Audusse Emmanuel
  • Dubos Virgile
  • Gaveau Noémie
  • Penel Yohan
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2025, 47 (01), pp.A1-A23. We analyse a class of energy-stable and linearly well-balanced numerical schemes dedicated to the nonlinear Shallow Water equations with Coriolis force. The proposed algorithms rely on colocated finite-difference approx- imations formulated on cartesian geometries. They involve appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic equilibrium. We show that the resulting methods ensure semi-discrete energy estimates. Among the proposed algorithms a colocated finite-volume scheme is described. Numerical results show a very clear improvement around the nonlinear geostrophic equilibrium when compared to those of classic Godunov-type schemes. (10.1137/22M1515707)
  DOI : 10.1137/22M1515707
• Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach
  
  • Baudet Cédric
  Asymptotic Analysis, IOS Press, 2025. We consider the Helmholtz equation in an angular sector partially covered by a homogeneous layer of small thickness, denoted ε. We propose in this work an asymptotic expansion of the solution with respect to ε at any order. This is done using matched asymptotic expansion, which consists here in introducing different asymptotic expansions of the solution in three subdomains: the vicinity of the corner, the layer and the rest of the domain. These expansions are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have singular behaviors. Our approach is to reformulate these matching conditions purely algebraically by writing all asymptotic expansions as formal series. By using algebraic calculus we reduce the matching conditions to scalar relations linking the singular behaviors of the fields. These relations have a convolutive structure and involve some coefficients that can be computed analytically. Our asymptotic expansion is justified rigorously with error estimates.
• Shape optimization of slip-driven axisymmetric microswimmers
  
  • Liu Ruowen
  • Zhu Hai
  • Guo Hanliang
  • Bonnet Marc
  • Veerapaneni Shravan
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2025, 47 (2), pp.A1065-A1090. In this work, we develop a computational framework that aims at simultaneously optimizing the shape and the slip velocity of an axisymmetric microswimmer suspended in a viscous fluid. We consider shapes of a given reduced volume that maximize the swimming efficiency, i.e., the (size-independent) ratio of the power loss arising from towing the rigid body of the same shape and size at the same translation velocity to the actual power loss incurred by swimming via the slip velocity. The optimal slip and efficiency (with shape fixed) are here given in terms of two Stokes flow solutions, and we then establish shape sensitivity formulas of adjoint-solution that provide objective function derivatives with respect to any set of shape parameters on the sole basis of the above two flow solutions. Our computational treatment relies on a fast and accurate boundary integral solver for solving all Stokes flow problems. We validate our analytic shape derivative formulas via comparisons against finite-difference gradient evaluations, and present several shape optimization examples. (10.1137/24M1659649)
  DOI : 10.1137/24M1659649
• Maxwell's equations with hypersingularities at a negative index material conical tip
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Rihani Mahran
  Pure and Applied Analysis, Mathematical Sciences Publishers, 2025, 7 (1), pp.127–169. We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity ε and the permeability µ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in ε and in µ, the corresponding scalar operators are not of Fredholm type in the usual H^1 spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical L^2 framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the T-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.
• Scattering of transient waves by an interface with time-modulated jump conditions
  
  • Michaël Darche
  • Assier Raphaël
  • Guenneau S
  • Lombard Bruno
  • Touboul Marie
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2025, 335, pp.923-951. Time modulation of the physical parameters offers interesting new possibilities for wave control. Examples include amplification of waves, harmonic generation and non-reciprocity, without resorting to non-linear mechanisms. Most of the recent studies focus on the time-modulation of the bulk physical properties. However, as the temporal modulation of these properties is difficult to achieve experimentally, we will concentrate here on the special case of an interface with time-varying jump conditions, which is simpler to implement. This work is focused on wave propagation in a one-dimensional medium containing one modulated interface. Properties of the scattered waves are investigated theoretically: energy balance, generation of harmonics, impedance matching and non-reciprocity. A fourth-order numerical method is also developed to simulate transient scattering. Numerical experiments are conducted to validate the numerical scheme and to illustrate the theoretical findings.
• Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (II) A modal approach
  
  • Cassier Maxence
  • Joly Patrick
  • Martínez Luis Alejandro Rosas
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2025. This work concerns the analysis of electromagnetic dispersive media modelled by generalized Lorentz models. More precisely, this paper is the second of two articles dedicated to the long time behaviour of solutions of Maxwell's equations in dissipative Lorentz media, via the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. In opposition to the frequency dependent Lyapunov functions approach used in [Cassier, Joly, Rosas Martínez, Z. Angew. Math. Phys. 74 (2023), 115], we develop a method based on the spectral analysis of the underlying non-self-adjoint operator of the model. Although more involved, this approach is closer to physics, as it uses the dispersion relation of the model, and has the advantage to provide more precise and more optimal results, leading to distinguish the notion of weak and strong dissipation. (10.48550/arXiv.2312.12231)
  DOI : 10.48550/arXiv.2312.12231
• Early-Reverberation Imaging Functions for Bounded Elastic Domains
  
  • Ducasse Eric
  • Rodriguez Samuel
  • Bonnet Marc
  Acta Acustica, EDP Sciences, 2025. For the ultrasonic inspection of bounded elastic structures, finite-duration imaging functions are derived in the Fourier-Laplace domain.The signals involved are exponentially windowed, so that early reflections are taken into account more strongly than later ones in the imaging methodology.Applying classical approaches to the general case of anisotropic elasticity, we express the Fréchet derivatives of the relevant data-misfit functional with respect to arbitrary perturbations of the mass density and stiffnesses in terms of forward and adjoint solutions.Their definitions incorporate the exponentially decaying weighting. The proposed finite-duration imaging functions are then defined on that basis.As some areas of the structure are less insonified than others, it is necessary to define normalized imaging functions to compensate for these variations.Our approach in particular aims to overcome the difficulty of dealing with bounded domains containing defects not located in direct line of sight from the transducers and measured signals of long duration.For this initiation work, we demonstate the potential of the proposed method on a two-dimensional test case featuring the imaging of mass and elastic stiffness variations in a region of a bounded isotropic medium that is not directly visible from the transducers. (10.2139/ssrn.4854889)
  DOI : 10.2139/ssrn.4854889
• An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 2: transmission problems.
  
  • Cassier Maxence
  • Joly Patrick
  , 2025. In this second chapter, we analyse transmission problems between a dielectric and a dispersive negative material. In the first part, we consider a transmission problem between two half-spaces, filled respectively by the vacuum and a Drude material, and separated by a planar interface. In this setting, we answer the following question: does this medium satisfy a limiting amplitude principle? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. In the second part, we consider the transmission problem of an infinite strip made of a Drude material embedded in the vacuum and analyse the existence and dispersive properties of guided waves. In both problems, our spectral analysis elucidates new and unusual physical phenomena for the considered transmission problems due to the presence of the dispersive negative material. In particular, we prove the existence of an interface resonance in the first part and the existence of slow light phenomena for guiding waves in the second part.
• On the constitutive behavior of linear viscoelastic solids under the plane stress condition
  
  • Guzina Bojan B
  • Bonnet Marc
  Journal of Elasticity, Springer Verlag, 2025, 157, pp.45. Motivated by the recent experimental and analytical developments enabling high-fidelity material characterization of (heterogeneous) sheet-like solid specimens, we seek to elucidate the constitutive behavior of linear viscoelastic solids under the plane stress condition. More specifically, our goal is to expose the relationship between the plane-stress viscoelastic constitutive parameters and their (native) ``bulk'' counterparts. To facilitate the sought reduction of the three-dimensional (3D) constitutive behavior, we deploy the concept of projection operators and focus on the frequency-domain behavior by resorting to the Fourier transform and the mathematical framework of tempered distributions, which extends the Fourier analysis to functions (common in linear viscoelasticity) for which Fourier integrals are not convergent. In the analysis, our primary focus is the on class of linear viscoelastic solids whose 3D rheological behavior is described by a set of constant-coefficient ordinary differential equations, each corresponding to a generic arrangement of ``springs'' and ``dashpots''. On reducing the general formulation to the isotropic case, we proceed with an in-depth investigation of viscoelastic solids whose bulk and shear modulus each derive from a suite of classical ``spring and dashpot'' configurations. To enable faithful reconstruction of the 3D constitutive parameters of natural and engineered solids via (i) thin-sheet testing and (ii) applications of the error-in-constitutive-relation approach to the inversion of (kinematic) sensory data, we also examine the reduction of thermodynamic potentials describing linear viscoelasticity under the plane stress condition. The analysis is complemented by a set of analytical and numerical examples, illustrating the effect on the plane stress condition on the behavior of isotropic and anisotropic viscoelastic solids. (10.1007/s10659-025-10136-6)
  DOI : 10.1007/s10659-025-10136-6
• Efficient boundary integral method to evaluate the acoustic scattering from coupled fluid-fluid problems excited by multiple sources
  
  • Pacaut Louise
  • Chaillat Stéphanie
  • Mercier Jean-François
  • Serre Gilles
  Journal of Computational Physics, Elsevier, 2025, 524, pp.113736. In the naval industry many applications require to study the behavior of a penetrable obstacle embedded in water, notably in presence of a turbulent flow. Such configuration is encountered in particular when noise is scattered by two-phase fluids, e.g., turbulent flows with air bubbles. Fast and efficient numerical methods are required to compute this scattering in the presence of realistic 3D geometries, such as bubble curtains. In [1], we have developed a very efficient approach in the case of a rigid obstacle of arbitrary shape, excited by a turbulent flow. It is based on the numerical evaluation of tailored Green's functions. Here we extend this fast method to the case of a penetrable obstacle. It is not a straightforward extension and we propose two main contributions. First, tailored Green's functions for a fluid-fluid coupled problem are derived theoretically and determined numerically. Second, we show the need of a regularized Boundary Integral formulation to obtain these Green's functions accurately in all configurations. Finally, we illustrate the efficiency of the method on various applications related to the scattering by multiple bubbles. (10.1016/j.jcp.2025.113736)
  DOI : 10.1016/j.jcp.2025.113736
• A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic acoustic problems in heterogeneous media
  
  • Pescuma Simone
  • Gabard Gwenael
  • Chaumont-Frelet Théophile
  • Modave Axel
  Journal of Computational Physics, Elsevier, 2025, 534, pp.114009. We consider the finite element solution of time-harmonic wave propagation problems in heterogeneous media with hybridizable discontinuous Galerkin (HDG) methods. In the case of homogeneous media, it has been observed that the iterative solution of the linear system can be accelerated by hybridizing with transmission variables instead of numerical traces, as performed in standard approaches. In this work, we extend the HDG method with transmission variables, which is called the CHDG method, to the heterogeneous case with piecewise constant physical coefficients. In particular, we consider formulations with standard upwind and general symmetric fluxes. The CHDG hybridized system can be written as a fixed-point problem, which can be solved with stationary iterative schemes for a class of symmetric fluxes. The standard HDG and CHDG methods are systematically studied with the different numerical fluxes by considering a series of 2D numerical benchmarks. The convergence of standard iterative schemes is always faster with the extended CHDG method than with the standard HDG methods, with upwind and scalar symmetric fluxes. (10.1016/j.jcp.2025.114009)
  DOI : 10.1016/j.jcp.2025.114009
• An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 1: general results.
  
  • Cassier Maxence
  • Joly Patrick
  , 2025. In this chapter, we investigate the mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media for which the dielectric permittivity $\varepsilon$ and magnetic permeability $\mu$ depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to the existence of Herglotz functions that determine the dispersion of the material. We consider successively the cases of the general passive media and the so-called local media for which $\varepsilon$ and $\mu$ are rational functions of the frequency. This leads us to analyse the important classes of non-dissipative and dissipative generalized Lorentz models. In particular, we discuss the connection between mathematical and physical properties of models through the notions of stability, energy conservation, dispersion and modal analyses, group and phase velocities and energy decay in dissipative systems.
• Analysis of time-harmonic electromagnetic problems with elliptic material coefficients
  
  • Ciarlet Patrick
  • Modave Axel
  Mathematical Methods in the Applied Sciences, Wiley, 2025. We consider time-harmonic electromagnetic problems with material coefficients represented by elliptic fields, covering a wide range of complex and anisotropic material media. The properties of elliptic fields are analyzed, with particular emphasis on scalar fields and normal tensor fields. Time-harmonic electromagnetic problems with general elliptic material fields are then studied. Well-posedness results for classical variational formulations with different boundary conditions are reviewed, and hypotheses for the coercivity of the corresponding sesquilinear forms are investigated. Finally, the proposed framework is applied to examples of media used in the literature: isotropic lossy media, geometric media, and gyrotropic media. (10.1002/mma.70318)
  DOI : 10.1002/mma.70318