The overall objectives of the NACHOS project-team are the design, mathematical analysis and actual leveraging of numerical methods for the solution of first order linear systems of partial differential equations (PDEs) with variable coefficients modeling wave propagation problems. The two main physical contexts considered by the team are electrodynamics and elastodynamics. The corresponding applications lead to the simulation of electromagnetic or seismic wave interaction with media exhibiting space and time heterogeneities. Moreover, in most of the situations of practical relevance, the propagation settings involve structures or/and material interfaces with complex shapes. Both the heterogeneity of the media and the complex geometrical features of the propagation domains motivate the use of numerical methods that can deal with non-uniform discretization meshes. In this context, the research efforts of the team concentrate on numerical methods formulated on unstructured or hybrid structured/unstructured meshes for the solution of the systems of PDEs of electrodynamics and elastodynamics. Our activities include the implementation of these numerical methods in advanced 3D simulation software that efficiently exploit the capabilities of modern high performance computing platforms. In this respect, our research efforts are also concerned with algorithmic issues related to the design of numerical algorithms that perfectly fit to the hardware characteristics of petascale class supercomputers.

In the case of electrodynamics, the mathematical model of interest is the full system of unsteady Maxwell equations which is a first-order hyperbolic linear system of PDEs (if the underlying propagation media is assumed to be linear). This system can be numerically solved using so-called time-domain methods among which the Finite Difference Time-Domain (FDTD) method introduced by K.S. Yee in 1996 is the most popular and which often serves as a reference method for the works of the team. For certain types of problems, a time-harmonic evolution can be assumed leading to the formulation of the frequency-domain Maxwell equations whose numerical resolution requires the solution of a linear system of equations (i.e in that case, the numerical method is naturally implicit). Heterogeneity of the propagation media is taken into account in the Maxwell equations through the electrical permittivity, the magnetic permeability and the electric conductivity coefficients. In the general case, the electrical permittivity and the magnetic permeability are tensors whose entries depend on space (i.e heterogeneity in space) and frequency. In the latter case, the time-domain numerical modeling of such materials requires specific techniques in order to switch from the frequency evolution of the electromagnetic coefficients to a time dependency. Moreover, there exist several mathematical models for the frequency evolution of these coefficients (Debye model, Drude model, Drude-Lorentz model, etc.).

In the case of elastodynamics, the mathematical model of interest is the system of elastodynamic equations for which several formulations can be considered such as the velocity-stress system. For this system, as with Yee's scheme for time-domain electromagnetics, one of the most popular numerical method is the finite difference method proposed by J. Virieux in 1986. Heterogeneity of the propagation media is taken into account in the elastodynamic equations through the Lamé and mass density coefficients. A frequency dependence of the Lamé coefficients allows to take into account physical attenuation of the wave fields and characterizes a viscoelastic material. Again, several mathematical models are available for expressing the frequency evolution of the Lamé coefficients.

The research activities undertaken by the team aim at developing innovative numerical methodologies putting the emphasis on several features:

**Accuracy**. The foreseen numerical methods should rely on
discretization techniques that best fit to the geometrical
characteristics of the problems at hand. Methods based on
unstructured, locally refined, even non-conforming, simplicial
meshes are particularly attractive in this regard. In addition, the
proposed numerical methods should also be capable to accurately
describe the underlying physical phenomena that may involve highly
variable space and time scales. Both objectives are generally
addressed by studying so-called

**Numerical efficiency**. The simulation of unsteady problems
most often relies on explicit time integration schemes. Such
schemes are constrained by a stability criterion, linking some space
and time discretization parameters, that can be very restrictive
when the underlying mesh is highly non-uniform (especially for
locally refined meshes). For realistic 3D problems, this can
represent a severe limitation with regards to the overall computing
time. One possible overcoming solution consists in resorting to an
implicit time scheme in regions of the computational domain where
the underlying mesh size is very small, while an explicit time
scheme is applied elsewhere in the computational domain. The
resulting hybrid explicit-implicit time integration strategy raises
several challenging questions concerning both the mathematical
analysis (stability and accuracy, especially for what concern
numerical dispersion), and the computer implementation on modern
high performance systems (data structures, parallel computing
aspects). A second, often considered approach is to devise a local
time stepping strategy. Beside, when considering time-harmonic
(frequency-domain) wave propagation problems, numerical efficiency
is mainly linked to the solution of the system of algebraic
equations resulting from the discretization in space of the
underlying PDE model. Various strategies exist ranging from the
more robust and efficient sparse direct solvers to the more flexible
and cheaper (in terms of memory resources) iterative methods.
Current trends tend to show that the ideal candidate will be a
judicious mix of both approaches by relying on domain decomposition
principles.

**Computational efficiency**. Realistic 3D wave propagation
problems involve the processing of very large volumes of data. The
latter results from two combined parameters: the size of the mesh
i.e the number of mesh elements, and the number of degrees of
freedom per mesh element which is itself linked to the degree of
interpolation and to the number of physical variables (for systems
of partial differential equations). Hence, numerical methods must
be adapted to the characteristics of modern parallel computing
platforms taking into account their hierarchical nature (e.g
multiple processors and multiple core systems with complex cache and
memory hierarchies). In addition, appropriate parallelization
strategies need to be designed that combine SIMD and MIMD
programming paradigms.

From the methodological point of view, the research activities of the team are concerned with four main topics: (1) high order finite element type methods on unstructured or hybrid structured/unstructured meshes for the discretization of the considered systems of PDEs, (2) efficient time integration strategies for dealing with grid induced stiffness when using non-uniform (locally refined) meshes, (3) numerical treatment of complex propagation media models (e.g. physical dispersion models), (4) algorithmic adaptation to modern high performance computing platforms.

The Discontinuous Galerkin method (DG) was introduced in 1973 by Reed and Hill to solve the neutron transport equation. From this time to the 90's a review on the DG methods would likely fit into one page. In the meantime, the Finite Volume approach (FV) has been widely adopted by computational fluid dynamics scientists and has now nearly supplanted classical finite difference and finite element methods in solving problems of non-linear convection and conservation law systems. The success of the FV method is due to its ability to capture discontinuous solutions which may occur when solving non-linear equations or more simply, when convecting discontinuous initial data in the linear case. Let us first remark that DG methods share with FV methods this property since a first order FV scheme may be viewed as a 0th order DG scheme. However a DG method may also be considered as a Finite Element (FE) one where the continuity constraint at an element interface is released. While keeping almost all the advantages of the FE method (large spectrum of applications, complex geometries, etc.), the DG method has other nice properties which explain the renewed interest it gains in various domains in scientific computing as witnessed by books or special issues of journals dedicated to this method - - - :

It is naturally adapted to a high order approximation of the unknown field. Moreover, one may increase the degree of the approximation in the whole mesh as easily as for spectral methods but, with a DG method, this can also be done very locally. In most cases, the approximation relies on a polynomial interpolation method but the DG method also offers the flexibility of applying local approximation strategies that best fit to the intrinsic features of the modeled physical phenomena.

When the space discretization is coupled to an explicit time integration scheme, the DG method leads to a block diagonal mass matrix whatever the form of the local approximation (e.g. the type of polynomial interpolation). This is a striking difference with classical, continuous FE formulations. Moreover, the mass matrix may be diagonal if the basis functions are orthogonal.

It easily handles complex meshes. The grid may be a classical
conforming FE mesh, a non-conforming one or even a hybrid mesh made
of various elements (tetrahedra, prisms, hexahedra, etc.). The DG
method has been proven to work well with highly locally refined
meshes. This property makes the DG method more suitable (and
flexible) to the design of some

It is also flexible with regards to the choice of the time stepping scheme. One may combine the DG spatial discretization with any global or local explicit time integration scheme, or even implicit, provided the resulting scheme is stable.

It is naturally adapted to parallel computing. As long as an explicit time integration scheme is used, the DG method is easily parallelized. Moreover, the compact nature of DG discretization schemes is in favor of high computation to communication ratio especially when the interpolation order is increased.

As with standard FE methods, a DG method relies on a variational formulation of the continuous problem at hand. However, due to the discontinuity of the global approximation, this variational formulation has to be defined locally, at the element level. Then, a degree of freedom in the design of a DG method stems from the approximation of the boundary integral term resulting from the application of an integration by parts to the element-wise variational form. In the spirit of FV methods, the approximation of this boundary integral term calls for a numerical flux function which can be based on either a centered scheme or an upwind scheme, or a blending between these two schemes.

DG methods are at the heart of the activities of the team regarding the development of high order discretization schemes for the PDE systems modeling electromagnetic and elatsodynamic wave propagation.

**Nodal DG methods for time-domain problems**. For the
numerical solution of the time-domain Maxwell equations, we have
first proposed a non-dissipative high order DGTD (Discontinuous
Galerkin Time-Domain) method working on unstructured conforming
simplicial meshes . This DG method
combines a central numerical flux function for the approximation of
the integral term at the interface of two neighboring elements with
a second order leap-frog time integration scheme. Moreover, the
local approximation of the electromagnetic field relies on a nodal
(Lagrange type) polynomial interpolation method. Recent achievements
by the team deal with the extension of these methods towards
non-conforming unstructured
- and hybrid
structured/unstructured meshes , their
coupling with hybrid explicit/implicit time integration schemes in
order to improve their efficiency in the context of locally refined
meshes
--.
A high order DG method has also been proposed for the numerical
resolution of the elastodynamic equations modeling the propagation
of seismic waves .

**Hybridizable DG (HDG) method for time-domain and
time-harmonic problems**. For the numerical treatment of the
time-harmonic Maxwell equations, nodal DG methods can also be
considered . However, such DG
formulations are highly expensive, especially for the discretization
of 3D problems, because they lead to a large sparse and undefinite
linear system of equations coupling all the degrees of freedom of
the unknown physical fields. Different attempts have been made in
the recent past to improve this situation and one promising strategy
has been recently proposed by Cockburn *et al.* in the form of so-called hybridizable
DG formulations. The distinctive feature of these methods is that
the only globally coupled degrees of freedom are those of an
approximation of the solution defined only on the boundaries of the
elements. This work is concerned with the study of such
Hybridizable Discontinuous Galerkin (HDG) methods for the solution
of the system of Maxwell equations in the time-domain when the time
integration relies on an implicit scheme, or in the
frequency-domain. The team has been a precursor in the development
of HDG methods for the frequency-domain Maxwell
equations.

**Multiscale DG methods for time-domain problems**. More
recently, in collaboration with LNCC in Petropolis (Frédéric
Valentin) the framework of the HOMAR assoacite team, we are
investigating a family of methods specifically designed for an
accurate and efficient numerical treatment of multiscale wave
propagation problems. These methods, referred to as Multiscale
Hybrid Mixed (MHM) methods, are currently studied in the team for
both time-domain electromagnetic and elastodynamic PDE models. They
consist in reformulating the mixed variational form of each system
into a global (arbitrarily coarse) problem related to a weak
formulation of the boundary condition (carried by a Lagrange
multiplier that represents e.g. the normal stress tensor in
elastodynamic sytems), and a series of small, element-wise, fully
decoupled problems resembling to the initial one and related to some
well chosen partition of the solution variables on each element. By
construction, that methodology is fully parallelizable and
recursivity may be used in each local problem as well, making MHM
methods belonging to multi-level highly parallelizable methods. Each
local problem may be solved using DG or classical Galerkin FE
approximations combined with some appropriate time integration
scheme (

The use of unstructured meshes (based on triangles in two space dimensions and tetrahedra in three space dimensions) is an important feature of the DGTD methods developed in the team which can thus easily deal with complex geometries and heterogeneous propagation media. Moreover, DG discretization methods are naturally adapted to local, conforming as well as non-conforming, refinement of the underlying mesh. Most of the existing DGTD methods rely on explicit time integration schemes and lead to block diagonal mass matrices which is often recognized as one of the main advantages with regards to continuous finite element methods. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. There are basically three strategies that can be considered to cure this computational efficiency problem. The first approach is to use an unconditionally stable implicit time integration scheme to overcome the restrictive constraint on the time step for locally refined meshes. In a second approach, a local time stepping strategy is combined with an explicit time integration scheme. In the third approach, the time step size restriction is overcome by using a hybrid explicit-implicit procedure. In this case, one blends a time implicit and a time explicit schemes where only the solution variables defined on the smallest elements are treated implicitly. The first and third options are considered in the team in the framework of DG -- and HDG discretization methods.

Towards the general aim of being able to consider concrete physical
situations, we are interested in taking into account in the numerical
methodologies that we study, a better description of the propagation
of waves in realistic media. In the case of electromagnetics, a
typical physical phenomenon that one has to consider is *dispersion*. It is present in almost all media and expresses the way
the material reacts to an electromagnetic field. In the presence of
an electric field a medium does not react instantaneously and thus
presents an electric polarization of the molecules or electrons that
itself influences the electric displacement. In the case of a linear
homogeneous isotropic media, there is a linear relation between the
applied electric field and the polarization. However, above some
range of frequencies (depending on the considered material), the
dispersion phenomenon cannot be neglected and the relation between the
polarization and the applied electric field becomes complex. This is
rendered via a frequency-dependent complex permittivity. Several
models of complex permittivity exist. Concerning biological media,
the Debye model is commonly adopted in the presence of water,
biological tissues and polymers, so that it already covers a wide
range of applications . In the context
of nanoplasmonics, one is interested in modeling the dispersion
effects on metals on the nanometer scale and at optical
frequencies. In this case, the Drude or the Drude-Lorentz models are
generally chosen . In the context of
seismic wave propagation, we are interested by the intrinsic
attenuation of the medium . In realistic
configurations, for instance in sedimentary basins where the waves are
trapped, we can observe site effects due to local geological and
geotechnical conditions which result in a strong increase in
amplification and duration of the ground motion at some particular
locations. During the wave propagation in such media, a part of the
seismic energy is dissipated because of anelastic losses relied to the
internal friction of the medium. For these reasons, numerical
simulations based on the basic assumption of linear elasticity are no
more valid since this assumption results in a severe overestimation of
amplitude and duration of the ground motion, even when we are not in
presence of a site effect, since intrinsic attenuation is not taken
into account.

Beside basic research activities related to the design of numerical methods and resolution algorithms for the wave propagation models at hand, the team is also committed to demonstrate the benefits of the proposed numerical methodologies in the simulation of challenging three-dimensional problems pertaining to computational electromagnetics and computational geoseismics. For such applications, parallel computing is a mandatory path. Nowadays, modern parallel computers most often take the form of clusters of heterogeneous multiprocessor systems, combining multiple core CPUs with accelerator cards (e.g Graphical Processing Units - GPUs), with complex hierarchical distributed-shared memory systems. Developing numerical algorithms that efficiently exploit such high performance computing architectures raises several challenges, especially in the context of a massive parallelism. In this context, current efforts of the team are towards the exploitation of multiple levels of parallelism (computing systems combining CPUs and GPUs) through the study of hierarchical SPMD (Single Program Multiple Data) strategies for the parallelization of unstructured mesh based solvers.

Electromagnetic devices are ubiquitous in present day technology. Indeed, electromagnetism has found and continues to find applications in a wide array of areas, encompassing both industrial and societal purposes. Applications of current interest include (among others) those related to communications (e.g transmission through optical fiber lines), to biomedical devices (e.g microwave imaging, micro-antenna design for telemedecine, etc.), to circuit or magnetic storage design (electromagnetic compatibility, hard disc operation), to geophysical prospecting, and to non-destructive evaluation (e.g crack detection), to name but just a few. Equally notable and motivating are applications in defence which include the design of military hardware with decreased signatures, automatic target recognition (e.g bunkers, mines and buried ordnance, etc.) propagation effects on communication and radar systems, etc. Although the principles of electromagnetics are well understood, their application to practical configurations of current interest, such as those that arise in connection with the examples above, is significantly complicated and far beyond manual calculation in all but the simplest cases. These complications typically arise from the geometrical characteristics of the propagation medium (irregular shapes, geometrical singularities), the physical characteristics of the propagation medium (heterogeneity, physical dispersion and dissipation) and the characteristics of the sources (wires, etc.).

Although many of the above-mentioned application contexts can potentially benefit from numerical modeling studies, the team currently concentrates its efforts on two physical situations.

Two main reasons motivate our commitment to consider this type of problem for the application of the numerical methodologies developed in the NACHOS project-team:

First, from the numerical modeling point of view, the interaction between electromagnetic waves and biological tissues exhibit the three sources of complexity identified previously and are thus particularly challenging for pushing one step forward the state-of-the art of numerical methods for computational electromagnetics. The propagation media is strongly heterogeneous and the electromagnetic characteristics of the tissues are frequency dependent. Interfaces between tissues have rather complicated shapes that cannot be accurately discretized using cartesian meshes. Finally, the source of the signal often takes the form of a complicated device (e.g a mobile phone or an antenna array).

Second, the study of the interaction between electromagnetic waves and living tissues is of interest to several applications of societal relevance such as the assessment of potential adverse effects of electromagnetic fields or the utilization of electromagnetic waves for therapeutic or diagnostic purposes. It is widely recognized nowadays that numerical modeling and computer simulation of electromagnetic wave propagation in biological tissues is a mandatory path for improving the scientific knowledge of the complex physical mechanisms that characterize these applications.

Despite the high complexity both in terms of heterogeneity and
geometrical features of tissues, the great majority of numerical
studies so far have been conducted using variants of the widely known
FDTD method due to Yee . In this method, the whole
computational domain is discretized using a structured (cartesian)
grid. Due to the possible straightforward implementation of the
algorithm and the availability of computational power, FDTD is
currently the leading method for numerical assessment of human
exposure to electromagnetic waves. However, limitations are still
seen, due to the rather difficult departure from the commonly used
rectilinear grid and cell size limitations regarding very detailed
structures of human tissues. In this context, the general objective
of the contributions of the NACHOS project-team is to demonstrate the
benefits of high order unstructured mesh based Maxwell solvers for a
realistic numerical modeling of the interaction of electromagnetic
waves and biological tissues with emphasis on applications related to
numerical dosimetry. Since the creation of the team, our works on
this topic have mainly been focussed on the study of the exposure of
humans to radiations from mobile phones or wireless communication
systems (see Fig. ). This activity has been
conducted in close collaboration with the team of Joe Wiart at Orange
Labs/Whist Laboratory
(http://

Nanostructuring of materials has opened up a number of new possibilities for manipulating and enhancing light-matter interactions, thereby improving fundamental device properties. Low-dimensional semiconductors, like quantum dots, enable one to catch the electrons and control the electronic properties of a material, while photonic crystal structures allow to synthesize the electromagnetic properties. These technologies may, e.g., be employed to make smaller and better lasers, sources that generate only one photon at a time, for applications in quantum information technology, or miniature sensors with high sensitivity. The incorporation of metallic structures into the medium add further possibilities for manipulating the propagation of electromagnetic waves. In particular, this allows subwavelength localisation of the electromagnetic field and, by subwavelength structuring of the material, novel effects like negative refraction, e.g. enabling super lenses, may be realized. Nanophotonics is the recently emerged, but already well defined, field of science and technology aimed at establishing and using the peculiar properties of light and light-matter interaction in various nanostructures. Nanophotonics includes all the phenomena that are used in optical sciences for the development of optical devices. Therefore, nanophotonics finds numerous applications such as in optical microscopy, the design of optical switches and electromagnetic chips circuits, transistor filaments, etc. Because of its numerous scientific and technological applications (e.g. in relation to telecommunication, energy production and biomedicine), nanophotonics represents an active field of research increasingly relying on numerical modeling beside experimental studies.

Plasmonics is a related field to nanophotonics. Metallic nanostructures whose optical scattering is dominated by the response of the conduction electrons are considered as plasmomic media. If the structure presents an interface with e.g. a dielectric with a positive permittivity, collective oscillations of surface electrons create surface-plasmons-polaritons (SPPs) that propagate along the interface. SPPs are guided along metal-dielectric interfaces much in the same way light can be guided by an optical fiber, with the unique characteristic of subwavelength-scale confinement perpendicular to the interface. Nanofabricated systems that exploit SPPs offer fascinating opportunities for crafting and controlling the propagation of light in matter. In particular, SPPs can be used to channel light efficiently into nanometer-scale volumes, leading to direct modification of mode dispersion properties (substantially shrinking the wavelength of light and the speed of light pulses for example), as well as huge field enhancements suitable for enabling strong interactions with non-linear materials. The resulting enhanced sensitivity of light to external parameters (for example, an applied electric field or the dielectric constant of an adsorbed molecular layer) shows great promise for applications in sensing and switching. In particular, very promising applications are foreseen in the medical domain - .

Numerical modeling of electromagnetic wave propagation in interaction with metallic nanostructures at optical frequencies requires to solve the system of Maxwell equations coupled to appropriate models of physical dispersion in the metal, such as the Drude and Drude-Lorentz models. Here again, the FDTD method is a widely used approach for solving the resulting system of PDEs . However, for nanophotonic applications, the space and time scales, in addition to the geometrical characteristics of the considered nanostructures (or structured layouts of the latter), are particularly challenging for an accurate and efficient application of the FDTD method. Recently, unstructured mesh based methods have been developed and have demonstrated their potentialities for being considered as viable alternatives to the FDTD method - - . Since the end of 2012, nanophotonics/plasmonics is increasingly becoming a focused application domain in the research activities of the team in close collaboration with physicists from CNRS laboratories, and also with researchers from international institutions.

Elastic wave propagation in interaction with solids are encountered in a lot of scientific and engineering contexts. One typical example is geoseismic wave propagation for earthquake dynamics or resource prospection.

To understand the basic science of earthquakes and to help engineers better prepare for such an event, scientists want to identify which regions are likely to experience the most intense shaking, particularly in populated sediment-filled basins. This understanding can be used to improve buildings in high hazard areas and to help engineers design safer structures, potentially saving lives and property. In the absence of deterministic earthquake prediction, forecasting of earthquake ground motion based on simulation of scenarios is one of the most promising tools to mitigate earthquake related hazard. This requires intense modeling that meets the spatial and temporal resolution scales of the continuously increasing density and resolution of the seismic instrumentation, which record dynamic shaking at the surface, as well as of the basin models. Another important issue is to improve the physical understanding of the earthquake rupture processes and seismic wave propagation. Large-scale simulations of earthquake rupture dynamics and wave propagation are currently the only means to investigate these multiscale physics together with data assimilation and inversion. High resolution models are also required to develop and assess fast operational analysis tools for real time seismology and early warning systems.

Numerical methods for the propagation of seismic waves have been
studied for many years. Most of existing numerical software rely on
finite difference type methods. Among the most popular schemes, one
can cite the staggered grid finite difference scheme proposed by
Virieux and based on the first order
velocity-stress hyperbolic system of elastic waves equations, which is
an extension of the scheme derived by Yee for the
solution of the Maxwell equations. Many improvements of this method
have been proposed, in particular, higher order schemes in space or
rotated staggered-grids allowing strong fluctuations of the elastic
parameters. Despite these improvements, the use of cartesian grids is
a limitation for such numerical methods especially when it is
necessary to incorporate surface topography or curved interface.
Moreover, in presence of a non planar topography, the free surface
condition needs very fine grids (about 60 points by minimal Rayleigh
wavelength) to be approximated. In this context, our objective is to
develop high order unstructured mesh based methods for the numerical
solution of the system of elastodynamic equations for elastic media in
a first step, and then to extend these methods to a more accurate
treatment of the heterogeneities of the medium or to more complex
propagation materials such as viscoelastic media which take into
account the intrinsic attenuation. Initially, the team has considered
in detail the necessary methodological developments for the
large-scale simulation of earthquake dynamics
. More recently, the team has
collaborated with CETE Méditerranée which is a regional technical
and engineering centre whose activities are concerned with seismic
hazard assessment studies, and IFSTTAR
(https://

This application topic is considered in close collaboration with the
MAGIQUE-3D project-team at Inria Bordeaux - Sud-Ouest which is
coordinating the Depth Imaging Partnership (DIP
-http://

*DIscOntinuous GalErkin Nanoscale Solvers*

Keywords: High-Performance Computing - Computational electromagnetics - Discontinuous Galerkin - Computational nanophotonics

Functional Description: The DIOGENeS software suite provides several tools and solvers for the numerical resolution of light-matter interactions at nanometer scales. A choice can be made between time-domain (DGTD solver) and frequency-domain (HDGFD solver) depending on the problem. The available sources, material laws and observables are very well suited to nano-optics and nano-plasmonics (interaction with metals). A parallel implementation allows to consider large problems on dedicated cluster-like architectures.

Authors: Stéphane Lanteri, Nikolai Schmitt, Alexis Gobé and Jonathan Viquerat

Contact: Stéphane Lanteri

*discontinuous GalERkin Solver for microWave INteraction with biological tissues*

Keywords: High-Performance Computing - Computational electromagnetics - Discontinuous Galerkin - Computational bioelectromagnetics

Functional Description: GERShWIN is based on a high order DG method formulated on unstructured tetrahedral meshes for solving the 3D system of time-domain Maxwell equations coupled to a Debye dispersion model.

Contact: Stéphane Lanteri

URL: http://

*High Order solver for Radar cross Section Evaluation*

Keywords: High-Performance Computing - Computational electromagnetics - Discontinuous Galerkin

Functional Description: HORSE is based on a high order HDG (Hybridizable Discontinuous Galerkin) method formulated on unstructured tetrahedral and hybrid structured/unstructured (cubic/tetrahedral) meshes for the discretization of the 3D system of frequency-domain Maxwell equations, coupled to domain decomposition solvers.

Authors: Ludovic Moya and Alexis Gobé

Contact: Stéphane Lanteri

URL: http://

This study is concerned with reduced-order modeling for time-domain electromagnetics and nanophotonics. More precisely, we consider the applicability of the proper orthogonal decomposition (POD) technique for the system of 3D time-domain Maxwell equations, possibly coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. We introduce a discontinuous Galerkin (DG) approach for the discretization of the problem in space based on an unstructured tetrahedral mesh. A reduced subspace with a significantly smaller dimension is constructed by a set of POD basis vectors extracted offline from snapshots that are obtained by the global DGTD scheme with a second order leap-frog method for time integration at a number of time levels. POD-based ROM is established by projecting (Galerkin projection) the global semi-discrete DG scheme onto the low-dimensional space. The stability of the POD-based ROM equipped with the second order leap-frog time scheme has been analysed through an energy method. Numerical experiments have allowed to verify the accuracy, and demonstrate the capabilities of the POD-based ROM. These very promising preliminary results are currently consolidated by assessing the efficiency of the proposed POD-based ROM when applied to the simulation of 3D nanophotonic problems.

In this work, we focus on the development of the use of periodic boundary conditions with sources at oblique incidence in a DGTD framework. Whereas in the context of the Finite Difference Time Domain (FDTD) methods, an abundant literature can be found, for DGTD, the amount of contributions reporting on such methods is remarkably low. In this work, we supplement the existing references using the field transform technique with an analysis of the continuous system using the method of characteristics and provide an energy estimate. Furthermore, we also study the numerical stability of the resulting DGTD scheme. After numerical validations, two realistic test problems have been considered in the context of nanophotonics with our DIOGENeS DGTD solver. This work has been accepted for publication in 2019.

We go a step further toward a better understanding of the fundamental properties of the linearized hydrodynamical model studied in the PhD of Nikolai Schmitt . This model is especially relevant for small nanoplasmonic structures (below 10nm). Using a hydrodynamical description of the electron cloud, both retardation effects and non local spatial response are taken into account. This results in a coupled PDE system for which we study the linear response. In (submitted, under revision), we concentrate on establishing well posedness results combined to a theoretical and numerical stability analysis. We especially prove polynomial stability and provide optimal energy decay rate. Finally, we investigate the question of numerical stability of several explicit time integration strategies combined to a Discontinuous Galerkin spatial discretization.

Although losses in metal is viewed as a serious drawback in many plasmonics experiments, thermoplasmonics is the field of physics that tries to take advantage of the latter. Indeed, the strong field enhancement obtained in nanometallic structures lead to a localized raise of the temperature in its vicinity leading to interesting photothermal effects. Therefore, metallic nanoparticles may be used as heat sources that can be easily integrated in various environments. This is especially appealing in the field of nanomedecine and can for example be used for diagnosis purposes or nanosurgery to cite but just a few. This year, we initiated a preliminary work towards this new field in collaboration with Y. D'Angelo (Université Côte d'Azur) and G. Baffou (Fresnel Institute, Marseille) who is an expert in this field. Due to the various scales and phenomena that come into play, the numerical modeling present great challenges. The laser illumination first excite a plasmon oscillation (reaction of the electrons of the metal) that relaxes in a thermal equilibrium and in turn excite the metal lattice (phonons). The latter is then responsible for heating the environment. A relevant modeling approach thus consists in describing the electron-phonon coupling through the evolution of their respective temperature. Maxwell's equations is then coupled to a set of coupled nonlinear hyperbolic equations describing the evolution of the temperatures of electrons, phonons and environment. The nonlinearities and the different time scales at which each thermalization occurs make the numerical approximation of these equations quite challenging.

In this work, we study nanoplasmonic structures with corners (typically a diedral/triangular structure); a situation that raises a lot of issues. We focus on a lossles Drude dispersion model and propose to investigate the range of validity of the amplitude limit principle. The latter predicts the asymptotic harmonic regime of a structure that is monochromatically illuminated, which makes a frequency domain approach relevant. However, in frequency domain, several well posedness problems arise due to the presence of corners (addressed in the PhD thesis of Camille Carvalho). This should impact the validity of the limit amplitude principle and has not yet been addressed in the literature in this precise setting. Here, we combine frequency-domain and time-domain viewpoints to give a numerical answer to this question in two dimensions. We show that the limit amplitude principle does not hold for whole interval of frequencies, that are explicited using the well-posedness analysis. This work is now being finalized.

Although the DGTD method has already been successfully applied to complex electromagnetic wave propagation problems, its accuracy may seriously deteriorate on coarse meshes when the solution presents multiscale or high contrast features. In other physical contexts, such an issue has led to the concept of multiscale basis functions as a way to overcome such a drawback and allow numerical methods to be accurate on coarse meshes. The present work, which is conducted in the context of the HOMAR Associate Team, is concerned with the study of a particular family of multiscale methods, named Multiscale Hybrid-Mixed (MHM) methods. Initially proposed for fluid flow problems, MHM methods are a consequence of a hybridization procedure which caracterize the unknowns as a direct sum of a coarse (global) solution and the solutions to (local) problems with Neumann boundary conditions driven by the purposely introduced hybrid (dual) variable. As a result, the MHM method becomes a strategy that naturally incorporates multiple scales while providing solutions with high order accuracy for the primal and dual variables. The completely independent local problems are embedded in the upscaling procedure, and computational approximations may be naturally obtained in a parallel computing environment. In this study, a family of MHM methods is proposed for the solution of the time-domain Maxwell equations where the local problems are discretized either with a continuous FE method or a DG method (that can be viewed as a multiscale DGTD method). Preliminary results have been obtained in the two-dimensional case.

Hybridizable discontinuous Galerkin (HDG) methods have been
investigated in the team since 2012. This family of method employs
face-based degrees of freedom that can be viewed as a fine grain
domain decomposition technique. We originally focused on
frequency-domain applications, for which HDG methods enable the use of
static condensation, leading to drastic reduction in computational
time and memory consumption. More recently, we have investigated the
use of HDG discretization to solve time-dependent problems.
Specifically, in the context of the PhD thesis of Georges Nehmetallah,
we focused on two particular aspects. On the one hand, HDG methods
exhibit a superconvergence property that allows, by means of local
postprocessing, to obtain new improved approximations of the unknowns.
Our first contribution is to apply this methodology to time-dependent
Maxwell's equations, where the post-processed approximation converges
with order

Another interesting aspect of the HDG method is that it can be conveniently employed to blend different time-integration schemes in different regions of the mesh. This is especially useful to efficiently handle locally refined space grids, that are required to take into account geometrical details. These ideas have already been explored for standard discontinuous Galerkin discretization in the context of the PhD thesis of Ludovic Moya , . Here, we focused on HDG methods and we introduced a family of coupled implicit-explicit (IMEX) time integration methods for solving time-dependent Maxwell's equations. We established stability conditions that are independent of the size of the small elements in the mesh, and are only constrained by the coarse part. Numerical experiments on two-dimensional benchmarks illustrate the theory and the usefulness of the approach.

The development of *a posteriori* error estimators and
is a new topic of interest for the team. Concerning *a posteriori*
estimators, a collaboration with
the SERENA project-team has been initiated. We mainly focus on a
technique called equilibrated fluxes, which has the advantage to
produce

Together with the development of a posteriori estimators, a novel
activity in the team is the design of efficient

The design and analysis of multiscale methods for wave propagation is an important research line for team. The team actually mainly specializes in one family of multiscale methods, called multiscale hybrid-mixed (MHM). These developments started thanks to the close collaboration with Frédéric Valentin, who has recently been awarded an Inria international chair. Previous investigations in the context of this collaboration focused on time-dependent Maxwell's equations . Recent efforts have been guided towards the realization of a MHM method for time-harmonic Maxwell's equations. We first focused on the Helmholtz equation, that modelizes the particular case of polarized waves. Our first results include the implementation of the method for two-dimensional problems as well as rigorous, frequency-explicit, stability and convergence analysis. These findings have recently been accepted for publication . In the context of the internship of Zakaria Kassali, the method has been further adapted for the propagation of polarized waves in solar cells. Specifically, it is required in this case to take into account “quasi-periodic” boundary conditions that deserve a special treatment. We are currently undertaking further developments guided toward full three-dimensional Maxwell's equations with the PhD of Zakaria Kassali, which started in November 2019.

This work is undertaken in the context of PRACE 6IP project and aims
at the development of scalable frequency-domain electromagnetic wave
propagation solvers, in the framework of the DIOGENeS software suite.
This solver is based on a high order HDG scheme formulated on an
unstructured tetrahedral grid for the discretization of the system of
three-dimensional Maxwell equations in dispersive media, leading to
the formulation of large sparse undefinite linear system for the
hybrid variable unknowns. This system is solved with domain
decomposition strategies that can be either a purely algebraic
algorithm working at the matrix operator level (i.e. a black-box
solver), or a tailored algorithm designed at the continuous PDE level
(i.e. a PDE-based solver). In the former case, we collaborate with
the HIEPACS project-team at Inria Bordeaux - Sud-Ouest in view of
adapting and exploiting the MaPHyS (Massively Parallel Hybrid Solver -
https://

Metasurfaces are flat optical nanocomponents, that are the basis of
several more complicated optical devices. The optimization of their
performance is thus a crucial concern, as they impact a wide range of
applications. Yet, current design techniques are mostly based on *engineering knowledge*, and may potentially be improved by a
rigorous analysis based on accurate simulation of Maxwell's equations.
The goal of this study is to optimize phase gradient metasurfaces by
taking advantage of our fullwave high order Discontinuous Galerkin
time-Domain solver implemented in DIOGENeS, coupled with two advanced
optimization techniques based on statistical learning and evolutionary
strategies. Our key findings are novel designs for Gan semiconductor
phase gradient metasurfaces operating at visible wavelengths. Our
numerical results reveal that rectangular and cylindrical nanopillar
arrays can achieve more than respectively 88% and 85% of diffraction
efficiency for TM polarization and both TM and TE polarization
respectively, using only 150 fullwave simulations. To the best of our
knowledge, this is the highest blazed diffraction efficiency reported
so far at visible wavelength using such metasurface architectures.
Fig. depicts the superiority of the proposed
statistical learning approaches over standard gradient-based
optimization strategies. This work has been recently published
.

There is significant recent interest in designing ultrathin crystalline silicon solar cells with active layer thickness of a few micrometers. Efficient light absorption in such thin films requires both broadband antireflection coatings and effective light trapping techniques, which often have different design considerations. In collaboration with physicists from the Sunlit team at C2N-CNRS, we conduct a numerical study of solar cells based on nanocone gratings. Indeed, it has been previously shown that by employing a double-sided grating design, one can separately optimize the geometries for antireflection and light trapping purposes to achieve broadband light absorption enhancement . In the present study, we adopt the nanocone grating considered in . This structure contains a crystalline silicon thin film with nanocone gratings also made of silicon. The circular nanocones form two-dimensional square lattices on both the front and the back surfaces. The film is placed on a perfect electric conductor (PEC) mirror. The ultimate objective of this study is to devise a numerical optimization strategy to infer optimal values of the geometrical characteristics of the nanocone grating on each side of the crystalline silicon thin film. Absorption characteristics are here evaluated using the high order DGTD solver from the DIOGENeS software suite. We use two efficient global optimization techniques based on statistical learning to adpat the geometrical characteristics of the nanocones in order the maximize the light absorption properties of this type of solr cells.

Recent experiments have shown that spatial dispersion may have a conspicuous impact on the response of plasmonic structures. This suggests that in some cases the Drude model should be replaced by more advanced descriptions that take spatial dispersion into account, like the hydrodynamic model. Here we show that nonlocality in the metallic response affects surface plasmons propagating at the interface between a metal and a dielectric with high permittivity. As a direct consequence, any nanoparticle with a radius larger than 20 nm can be expected to be sensitive to spatial dispersion whatever its size. The same behavior is expected for a simple metallic grating allowing the excitation of surface plasmons, just as in Woods famous experiment. Finally, we carefully set up a procedure to measure the signature of spatial dispersion precisely, leading the way for future experiments. Importantly, our work suggests that for any plasmonic structure in a high permittivity dielectric, nonlocality should be taken into account.

The design of intrinsically flat two-dimensional optical components, i.e., metasurfaces, generally requires an extensive parameter search to target the appropriate scattering properties of their constituting building blocks. Such design methodologies neglect important near-field interaction effects, playing an essential role in limiting the device performance. Optimization of transmission, phase-addressing and broadband performances of metasurfaces require new numerical tools. Additionally, uncertainties and systematic fabrication errors should be analysed. These estimations, of critical importance in the case of large production of metaoptics components, are useful to further project their deployment in industrial applications. Here, we report on a computational methodology to optimize metasurface designs. We complement this computational methodology by quantifying the impact of fabrication uncertainties on the experimentally characterized components. This analysis provides general perspectives on the overall metaoptics performances, giving an idea of the expected average behavior of a large number of devices.

This contract with TOTAL CSE (Computational Science and Engineering) division in Houston, Texas, is concerned with the development of a DGTD solver for applications in geoseismics. The R&D division of the EP (Oil, Gas Exploration & Production) branch of TOTAL has been interested in DG type methods since many years. It acquired a know-how on these methods and developed internally software tools integrating DG methods as solvers of the direct problem (forward propagators) in different seismic imaging processes (RTM - Reverse Time Migration, and FWI - Full Waveform Inversion). These solvers are concerned with the numerical resolution of PDE systems of acoustics and elastodynamics. TOTAL is now interested in having a similar DGTD solver for the numerical resolution of the system of time-domain Maxwell equations, in view of the development of an electromagnetic imaging process to identify conductivity of a medium. This electromagnetic imaging process would then be coupled to the existing seismic imaging ones.

Type: ANR ASTRID Maturation

See also: http://

Duration: Avril 2019 - March 2022

Coordinator: Inria

Partner: CRHEA laboratory in Sophia Antipolis and NAPA Technologies in Archamps

Inria contact: Stéphane Lanteri

Abstract: In the OPERA project, we are investigating and optimizing the properties of planar photonic devices based on metasurfaces using numerical modelling. The scientific and technical activities that constitute the project work programme are organized around 4 main workpackages. The numerical characterization of the optical properties of planar devices based on metasurfaces, as well as their optimization are at the heart of the activities and objectives of two horizontal (transversal) workpackages. These numerical methodologies will be integrated into the DIOGENeS software framework that will eventually integrates (1) discontinuous Galerkin-type methods that have been tested over the past 10 years for the discretization of Maxwell equations in time and frequency regimes, mainly for applications in the microwave band, (2) parallel resolution algorithms for sparse linear systems based on the latest developments in numerical linear algebra, (3) modern optimization techniques based on learning and metamodeling methods and (4) software components adapted to modern high performance computing architectures. Two vertical workpackages complete this program. One of them aims to demonstrate the contributions of methodological developments and numerical tools resulting from transversal workpackages through their application to diffusion/radiation control by passive planar devices. The other, more prospective, concerns the study of basic building blocks for the realization of adaptive planar devices.

Title: PRACE Sixth Implementation Phase (PRACE-6IP) project

Duration: May 2019 - December 2021

Partners: see https://

Inria contact: Luc Giraud

PRACE, the Partnership for Advanced Computing is the permanent pan-European High Performance Computing service providing world-class systems for world-class science. Systems at the highest performance level (Tier-0) are deployed by Germany, France, Italy, Spain and Switzerland, providing researchers with more than 17 billion core hours of compute time. HPC experts from 25 member states enabled users from academia and industry to ascertain leadership and remain competitive in the Global Race. Currently PRACE is finalizing the transition to PRACE 2, the successor of the initial five year period. The objectives of PRACE-6IP are to build on and seamlessly continue the successes of PRACE and start new innovative and collaborative activities proposed by the consortium. These include: assisting the development of PRACE 2; strengthening the internationally recognised PRACE brand; continuing and extend advanced training which so far provided more than 36 400 person·training days; preparing strategies and best practices towards Exascale computing, work on forward-looking SW solutions; coordinating and enhancing the operation of the multi-tier HPC systems and services; and supporting users to exploit massively parallel systems and novel architectures. A high level Service Catalogue is provided. The proven project structure will be used to achieve each of the objectives in 7 dedicated work packages. The activities are designed to increase Europe's research and innovation potential especially through: seamless and efficient Tier-0 services and a pan-European HPC ecosystem including national capabilities; promoting take-up by industry and new communities and special offers to SMEs; assistance to PRACE 2 development; proposing strategies for deployment of leadership systems; collaborating with the ETP4HPC, CoEs and other European and international organisations on future architectures, training, application support and policies. This will be monitored through a set of KPIs.

Title: European joint effort toward a highly productive programming environment for heterogeneous exascale computing

Program: H2020

See also: https://

Duration: October 2018 - September 2021

Coordinator: Barcelona Supercomputing Center

Partner: Barcelona Supercomputing Center (Spain)

Coordinator: CEA

Partners:

Fraunhofer–Gesellschaft (Germany)

CINECA (Italy)

IMEC (Blegium)

INESC ID (Portugal)

Appentra Solutions (Spain)

Eta Scale (Sweden)

Uppsala University (Sweden)

Inria (France)

Cerfacs (France)

Inria contact: Stéphane Lanteri

EPEEC’s main goal is to develop and deploy a production-ready parallel programming environment that turns upcoming overwhelmingly-heterogeneous exascale supercomputers into manageable platforms for domain application developers. The consortium will significantly advance and integrate existing state-of-the-art components based on European technology (programming models, runtime systems, and tools) with key features enabling 3 overarching objectives: high coding productivity, high performance, and energy awareness. An automatic generator of compiler directives will provide outstanding coding productivity from the very beginning of the application developing/porting process. Developers will be able to leverage either shared memory or distributed-shared memory programming flavours, and code in their preferred language: C, Fortran, or C++. EPEEC will ensure the composability and interoperability of its programming models and runtimes, which will incorporate specific features to handle data-intensive and extreme-data applications. Enhanced leading-edge performance tools will offer integral profiling, performance prediction, and visualisation of traces. Five applications representative of different relevant scientific domains will serve as part of a strong inter-disciplinary co-design approach and as technology demonstrators. EPEEC exploits results from past FET projects that led to the cutting-edge software components it builds upon, and pursues influencing the most relevant parallel programming standardisation bodies.

**PHOTOM**

Title: PHOTOvoltaic solar devices in Multiscale computational simulations

International Partners:

Center for Research in Mathematical Engineering, Universidad de Concepcion (Chile), Rodolfo Araya

Laboratório Nacional de Computação Científica (Brazil), Frédéric Valentin

Instito de Matemáticas, PUCV (Chile), Diego Paredes

Duration: 2018 - 2020

Start year: 2018

See also: http:////

The work consists of devising, analyzing and implementing new multiscale finite element methods, called Multiscale Hybrid-Mixed (MHM) method, for the Helmholtz and the Maxwell equations in the frequency domain. The physical coefficients involved in the models contain highly heterogeneous and/or high contrast features. The goal is to propose numerical algorithms to simulate wave propagation in complex geometries as found in photovoltaic devices, which are naturally prompt to be used in massively parallel computers. We demonstrate the well-posedness and establish the optimal convergence of the MHM methods. Also, the MHM methods are shown to induce a new face-based a posteriori error estimator to drive space adaptivity. An efficient parallel implementation of the new multiscale algorithm assesses theoretical results and is shown to scale on a petaflop parallel computer through academic and realistic two and three-dimensional solar cells problems.

Prof. Kurt Busch, Humboldt-Universität zu Berlin, Institut für Physik, Theoretical Optics & Photonics

** IIC VALENTIN Frédéric**

Title: Innovative multiscale numerical algorithms for wave-matter interaction models at the nanoscale

International Partner (Institution - Laboratory - Researcher):

Laboratório Nacional de Computação Científica (Brazil), Frédéric Valentin

Duration: 2018 - 2022

Start year: 2018

See also: https://

The project addresses complex three-dimensional nanoscale wave-matter interaction models, which are relevant to the nanophotonics and nanophononics fields, and aims at devising innovative multiscale numerical methods, named Multiscale Hybrid-Mixed methods (MHM for short), to solve them with high accuracy and high performance.

David Pardo (Basque Center for Applied Mathematics, Spain) at Inria, France, April 2-5, 2019.

Christophe Geuzaine (University of Liège, Belgium) at Inria, France, April 29-30, 2019.

Jean-Francois Remacle (Ecole Polytechnique de Louvain, Belgium) at Inria, France, April 29-30, 2019.

Jay Gopalakrishnan (University of Portland, USA) at Inria, France, June 4-5, 2019.

Stéphane Lanteri has chaired the second workshop of the PHOTOM (PHOTOvoltaic solar devices in Multiscale computational simulations) project that took place at Inria Sophia Antipolis-Méditerranée, France, Jan 28-Feb 01, 2019.

Stéphane Lanteri has chaired the third workshop of the CLIPhTON (advanCed numericaL modelIng for multiscale and multiphysics nanoPhoTONics) network that took place ati Inria Sophia Antipolis-Méditerranée, France, October 24-25, 2019.

Théophile Chaumont-Frelet: ESAIM Math. Model. Numer. Anal., NUMA, NME

Yves D'Angelo: Nanotechnology, Journal of Geophysical & Astrophysical Fluid Dynamics

Claire Scheid: SIAM J. Numer. Anal., SIAM J. Sci. Comput.

Stéphane Lanteri: J. Comput. Phys., Comp. Meth. Appl. Mech. Engrg.

Théophile Chaumont-Frelet, Journées Ondes Sud-Ouest, MIOS, France , January 2019.

Théophile Chaumont-Frelet, Séminaire LJAD, Nice, France , November 2019.

Théophile Chaumont-Frelet at University of Basel, Switzerland, December 11-13, 2019.

Théophile Chaumont-Frelet at University of Bath, UK, November 11-15, 2019.

Théophile Chaumont-Frelet at the Basque Center for Applied Mathematics, Spain, May 1-3, 2019.

Stéphane Lanteri is a member of the Scientific Committee of CERFACS.

Yves D'Angelo is the head of the “Laboratoire J.A. Dieudonné” (LJAD, UMR 7351).

Stéphane Descombes is the head of the “Maison de la Modélisation, de la Simulation et des Interactions” (MSI) of Université Côte d'Azur

Stéphane Lanteri is a member of the Project-team Committee's Bureau of the Inria Sophia Antipolis-Méditerranée research center.

License: Yves D'Angelo, *Analyse des Séries de Fourier* , 30 h, L3, Univ. Côte d'Azur

Lisence: Claire Scheid, *Fondements 2* , 24 h, L1, Univ. Côte d'Azur

Master: Yves D'Angelo, *Modélisation et Simulation Numérique* , 48 h, M1, Univ. Côte d'Azur

Master: Yves D'Angelo, *Modélisation de la Turbulence fluid* , 30 h, M2, Univ. Côte d'Azur

Master: Claire Scheid, *Analyse, Lecture and practical works* , 27 h, M2, Univ. Côte d'Azur

Master: Claire Scheid, *Méthodes numériques en EDP, Lectures and practical works* , 63 h, M1, Univ. Côte d'Azur

Master: Claire Scheid, *Option Modélisation, Lectures and practical works* , 48 h, M2, Univ. Côte d'Azur

Master: Claire Scheid, *Soutien Analyse Fonctionnelle et Esepacde de Hilbert* , 18 h, M1, Univ. Côte d'Azur

Master: Stéphane Descombes, *Introduction aux EDP* , 30 h, M1, Univ. Côte d'Azur

Master: Stéphane Descombes, *ACP et reconnaissance de caractères* , 9 h, M2, Univ. Côte d'Azur

License: Stéphane Descombes, *Travaux dirigés de mathématiques pour l'économie*, 18 h, L1, Univ. Côte d'Azur

Engineering: Stéphane Lanteri, *High performance scientific computing*, , 24 h, MAM5, Polytech Nice Sophia

PhD in progress: Alexis Gobé, Multiscale hybrid-mixed methods for time-domain nanophotonics, vNovember 2016, Stéphane Lanteri

PhD in progress: Georges Nehmetallah, Efficient finite element type solvers for the numerical modeling of light transmission in nanostructured waveguides and cavities, November 2017, Stéphane Descombes and Stéphane Lanteri

PhD in progress: Zakaria Kassali, Multiscale finite element simulations applied to the design of photovoltaic cells, November 2019, Théophile Chaumont-Frelet and Stéphane Lanteri

PhD in progress: Massimiliano Montone, High order finite element type solvers for the coupled Maxwell-semiconductor equations in the time-domain, December 2019, Stéphane Lanteri and Claire Scheid

Yves D'Angelo: Basile Radisson, IRPHE Marseille, France, Avril 2019, Rapporteur.

Claire Scheid: Pierre Mennuni, Lille, France, November 2019, Examinatrice.

Claire Scheid: Weslley Da Silva Peireira, LNCC, Petropolis, Brazil, September 2019, Examinatrice.

Stéphane Lanteri: Aurélien Citrain, Inria Pau, December 2019, Rapporteur.

Stéphane Lanteri: Nicolas Lebbe, CEA LETI, Grenoble, Novembre 2019, Examinateur.

Stéphane Lanteri: Matthieu Patrizio, ISAE, Toulouse, Mai 2019: Examinateur.