The overall objectives of the NACHOS project-team are the formulation, analysis and evaluation of numerical methods and high performance algorithms for the solution of first order linear systems of partial differential equations (PDEs) with variable coefficients pertaining to electrodynamics and elastodynamics. In both domains, the applications targeted by the team involve the interaction of the underlying physical fields with media exhibiting space and time heterogeneities. Moreover, in most of the situations of practical relevance, the computational domain is irregularly shaped or/and it includes geometrical singularities. Both the heterogeneity and the complex geometrical features of the underlying media motivate the use of numerical methods working on non-uniform discretizations of the computational domain. In this context, the research efforts of the team aim at the development of unstructured (or hybrid structured/unstructured) mesh based methods with activities ranging from the mathematical analysis of numerical methods for the solution of the systems of PDEs of electrodynamics and elastodynamics, to the development of prototype 3d simulation software that efficiently exploits the capabilities of modern high performance computing platforms.

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. In the vast majority of existing time-domain methods, time advancing relies on an explicit time scheme. 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 teams focuses on physical applications dealing with electromagnetic or elastodynamic wave propagation in interaction with heterogeneous media and irregularly shaped structures. The underlying wave propagation phenomena can be purely unsteady or they can be periodic (because the imposed source term follows a time-harmonic evolution). In this context, 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 strategy in the context of a fully explicit time integration
scheme. Beside, when considering time-harmonic 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 nonlinear 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 nonlinear 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 et the heart of the activities of the pteam 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 meshes and

**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
consiered
-. 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 the framework of a collaboration with LNCC in
Petropolis (Frédéric Valentin), we have started to investigate 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 traduces the way
the material reacts to the presence of electromagnetic waves. 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 traduced by a frequency-dependent complex
permittivity. Several such models for the characterization of the
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
. If one is interested in
modeling the dispersion effects on metals on the nanometer scale and
at optical frequencies, which are the conditions that one has to deal
with in the context of nanoplasmonics, then the Drude or the
Drude-Lorentz models are generally adopted
. 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 result 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 computation 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 (Finite Difference Time Domain) 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. Mettalic 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 nonlinear 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 the Drude and Drude-Lorentz models. Her 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/plamonics 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, in particular in the context of 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 initiated a
close collaboration with CETE Méditerranée
http://

This application topic has been considered recently by the NACHOS
project-team and this is done in close collaboration with the
MAGIQUE-3D project-team at Inria Bordeaux - Sud-Ouest which is
coordinating the Depth Imaging Partnership (DIP)
http://

MAXW-DGTD is a software suite for the simulation of time domain electromagnetic wave propagation. It implements a solution method for the Maxwell equations in the time-domain. MAXW-DGTD is based on a discontinuous Galerkin method formulated on unstructured triangular (2d case) or tetrahedral (3d case) meshes . Within each element of the mesh, the components of the electromagnetic field are approximated by a arbitrary high order nodal polynomial interpolation method. This discontinuous Galerkin method combines a centered scheme for the evaluation of numerical fluxes at a face shared by two neighboring elements, with an explicit Leap-Frog time scheme. The software and the underlying algorithms are adapted to distributed memory parallel computing platforms thanks to a parallelization strategy that combines a partitioning of the computational domain with message passing programming using the MPI standard. Besides, a peripheral version of the software has been recently developed which is able to exploit the processing capabilities of a hybrid parallel computing system comprising muticore CPU and GPU nodes.

AMS: AMS 35L50, AMS 35Q60, AMS 35Q61, AMS 65N08, AMS 65N30, AMS 65M60

Keywords: Computational electromagnetics, Maxwell equations, discontinuous Galerkin, tetrahedral mesh.

OS/Middelware: Linux

Required library or software: MPI (Message Passing Interface), CUDA

Programming language: Fortran 77/95

MAXW-DGFD is a software suite for the simulation of time-harmonic electromagnetic wave propagation. It implements a solution method for the Maxwell equations in the frequency domain. MAXW-DGFD is based on a discontinuous Galerkin method formulated on unstructured triangular (2d case) or tetrahedral (3d case) meshes. Within each element of the mesh, the components of the electromagnetic field are approximated by a arbitrary high order nodal polynomial interpolation method. The resolution of the sparse, complex coefficients, linear systems resulting from the discontinuous Galerkin formulation is performed by a hybrid iterative/direct solver whose design is based on domain decomposition principles. The software and the underlying algorithms are adapted to distributed memory parallel computing platforms thanks to a paralleization strategy that combines a partitioning of the computational domain with a message passing programming using the MPI standard. Some recent achievements have been the implementation of non-uniform order DG method in the 2d case and of a new hybridizable discontinuous Galerkin (HDG) formulation also in the 2d and 3d cases.

AMS: AMS 35L50, AMS 35Q60, AMS 35Q61, AMS 65N08, AMS 65N30, AMS 65M60

Keywords: Computational electromagnetics, Maxwell equations, discontinuous Galerkin, tetrahedral mesh.

OS/Middelware: Linux

Required library or software: MPI (Message Passing Interface)

Programming language: Fortran 77/95

SISMO-DGTD is a software for the simulation of time-domain seismic wave propagation. It implements a solution method for the velocity-stress equations in the time-domain. SISMO-DGTD is based on a discontinuous Galerkin method formulated on unstructured triangular (2d case) or tetrahedral (3d case) meshes . Within each element of the mesh, the components of the electromagnetic field are approximated by a arbitrary high order nodal polynomial interpolation method. This discontinuous Galerkin method combines a centered scheme for the evaluation of numerical fluxes at a face shared by two neighboring elements, with an explicit Leap-Frog time scheme. The software and the underlying algorithms are adapted to distributed memory parallel computing platforms thanks to a paralleization strategy that combines a partitioning of the computational domain with a message passing programming using the MPI standard.

AMS: AMS 35L50, AMS 35Q74, AMS 35Q86, AMS 65N08, AMS 65N30, AMS 65M60

Keywords: Computational geoseismics, elastodynamic equations, discontinuous Galerkin, tetrahedral mesh.

OS/Middelware: Linux

Required library or software: MPI (Message Passing Interface)

Programming language: Fortran 77/95

The system of Maxwell equations describes the evolution of the
interaction of an electromagnetic field with a propagation medium.
The different properties of the medium, such as isotropy, homogeneity,
linearity, among others, are introduced through *constitutive
laws* linking fields and inductions. In the present study, we focus
on nonlinear effects and address nonlinear Kerr materials
specifically. In this model, any dielectric may become nonlinear
provided the electric field in the material is strong enough. As a
first setp, we consider the one-dimensional case and study the
numerical solution of the nonlinear Maxwell equations thanks to DG
methods. In particular, we make use of an upwind scheme and
limitation techniques because they have a proven ability to capture
shocks and other kinds of singularities in the fluid dynamics
framework. The numerical results obtained in this preliminary study
gives us confidence towards extending this work to higher spatial
dimensions.

Usually, unstructured mesh based methods rely on tessellations
composed of straight-edged elements mapped linearly from a reference
element, on domains which physical boundaries are indifferently
straight or curved. Such meshes represent a serious hindrance for
high order finite element (FE) methods since they limit the accuracy
to second order in the spatial discretization. Thus, exploiting an
enhanced representation of physical geometries is in agreement with
the natural procedure of high order FE methods, such as the DG method.
There are several ways to account for curved geometries. One could
choose to incorporate the knowledge coming from CAD in the method to
design the geometry and the approximation. These methods are called
*isogeometric*, and have received a lot of attention
recently. This naturally implies to have access to CAD models of the
geometry. On the other hand, *isoparametric* usually rely on a
polynomial approximation of both the boundary and the solution. This
can be added fairly easily on top of existing implementations. In the
present study we focus on the latter type of method, since our goal is
first to envisage the benefit of curvilinear meshes for light/matter
interaction with nanoscale structures.

When metallic nanostructures have sub-wavelength sizes and the illuminating frequencies are in the regime of metal's plasma frequency, electron interaction with the exciting fields have to be taken into account. Due to these interactions, plasmonic surface waves can be excited and cause extreme local field enhancements (surface plasmon polariton electromagnetic waves). Exploiting such field enhancements in applications of interest requires a detailed knowledge about the occurring fields which can generally not be obtained analytically. For the numerical modeling of light/matter interaction on the nanoscale, the choice of an appropriate model is a crucial point. Approaches that are adopted in a first instance are based on local (no interaction between electrons) dispersive models e.g. Drude or Drude-Lorentz. From the mathematical point of view, these models lead to an additional ordinary differential equation in time that is coupled to Maxwell's equations. When it comes to very small structures in a regime of 2 nm to 25 nm, non-local effects due to electron collisions have to be taken into account. Non-locality leads to additional, in general non-linear, partial differential equations and is significantly more difficult to treat, though. In this work, we study a DGTD method able to solve the system of Maxwell equations coupled to a linearized non-local dispersion model relevant to nanoplasmonics. While the method is presented in the general 3d case, in this preliminary stdudy, numerical results are given for 2d simulation settings.

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 has been initiated in the context of the visit of Frédéric Valentin in the 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 2d case for models problems.

This study is concerned with the development of accurate and efficient solution strategies for the system of 3d time-domain Maxwell equations coupled to local dispersion models (e.g. Debye, Drude or Drude-Lorentz models) in the presence of locally refined meshes. Such meshes impose a constraint on the allowable time step for explicit time integration schemes that can be very restrictive for the simulation of 3d problems. We consider here the possibility of using an unconditionally stable implicit time integration scheme combined to a HDG discretization method. As a first step, we extend our former study in which was dealing with the 2d time-domain Maxwell equations for non-dispersive media.

In the context of the ANR TECSER project, we continue our efforts
towards the development of scalable high order HDG methods for the
solution of the system of 3d frequency-domain Maxwell equations. We
aim at fully exploiting the flexibiity of the HDG discretization
framework with regards to the adaptation of the interpolation order
(

This work is concerned with the development of high order DGTD methods formulated on unstructured simplicial meshes for the numerical solution of the system of time-domain elastodynamic equations. These methods share some ingredients of the DGTD methods developed by the team for the time-domain Maxwell equations among which, the use of nodal polynomial (Lagrange type) basis functions, a second order leap-frog time integration scheme and a centered scheme for the evaluation of the numerical flux at the interface between neighboring elements. A recent novel contribution is the numerical treatment of viscoelastic attenuation. For this, the velocity-stress first order hyperbolic system is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. In practice, we need solving 3L additional equations in 2d (and 6L in 3d), where L is the number of relaxation mechanisms of the generalized Maxwell body. This method has been implemented in 2d and 3d.

We have recently devised an extension of the DGTD method for elastic wave propagation in arbitrary heterogeneous media. In realistic geological media (sedimentary basins for example), one has to include strong variations in the material properties. Then, the classical hypothesis that these properties are constant within each element of the mesh can be a severe limitation of the method, since we need to discretize the medium with very fine meshes resulting in very small time steps. For these reasons, we propose an improvement of the DGTD method allowing non-constant material properties within the mesh elements. A change of variables on the stress components allows writing the elastodynamic system in a pseudo-conservative form. Then, the introduction of non-constant material properties inside an element is simply treated by the calculation, via convenient quadrature formulae, of a modified local mass matrix depending on these properties. This new extension has been validated for a smoothly varying medium or a strong jump between two media, which can be accurately approximated by the method, independently of the mesh.

One of the most used seismic imaging methods is the full waveform
inversion (FWI) method which is an iterative procedure whose algorithm
is the following. Starting from an initial velocity model, (1) compute
the solution of the wave equation for the

In the context of the visit of Frédéric Valentin in the team, we have initiated a study aiming at the design of novel multiscale methods for the solution of the time-domain elastodynamic equations, in the spirit of MHM (Multiscale Hybrid-Mixed) methods previously proposed for fluid flow problems. Motivation in that direction naturally came when dealing with non homogeneous anisotropic elastic media as those encountered in geodynamics related applications, since multiple scales are naturally present when high contrast elasticity parameters define the propagation medium. Instead of solving the usual system expressed in terms of displacement or displacement velocity, and stress tensor variables, a hybrid mixed-form is derived in which an additional variable, the Lagrange multiplier, is sought as representing the (opposite) of the surface tension defined at each face of the elements of a given discretization mesh. We consider the velocity/stress formulation of the elastodynamic equations, and study a MHM method defined for a heterogeneous medium where each elastic material is considered as isotropic to begin with. If the source term (the applied given force on the medium) is time independent, and if we are given a arbitrarily coarse conforming mesh (triangulation in 2d, tetrahedrization in 3d), the proposed MHM method consists in first solving a series of fully decoupled (therefore parallelizable) local (element-wise) problems defining parts of the full solution variables which are directly related to the source term, followed by the solution of a global (coarse) problem, which yields the degrees of freedom of both the Lagrange multiplier dependent part of the full solution variables and the Lagrange multiplier itself. Finally, the updating of the full solution variables is obtained by adding each splitted solution variables, before going on the next time step of a leap-frog time integration scheme. Theoretical analysis and implementation of this MHM method where the local problems are discretized with a DG method, are underway.

Since January 2013, the team is coordinating the C2S@Exa
http://

Type: ANR ASTRID

Duration: May 2014 - April 2017

Coordinator: Inria

Partner: Airbus Group Innovations, Inria, Nuclétudes

Inria contact: Stéphane Lanteri

Abstract: the objective of the TECSER projet is to develop an innovative high performance numerical methodology for frequency-domain electromagnetics with applications to RCS (Radar Cross Section) calculation of complicated structures. This numerical methodology combines a high order hybridized DG method for the discretization of the frequency-domain Maxwell in heterogeneous media with a BEM (Boundary Element Method) discretization of an integral representation of Maxwell's equations in order to obtain the most accurate treatment of boundary truncation in the case of theoretically unbounded propagation domain. Beside, scalable hybrid iterative/direct domain decomposition based algorithms are used for the solution of the resulting algebraic system of equations.

Type: FP7

Defi: Special action

Instrument: Integrated Project

Objectif: Exascale computing platforms, software and applications

Duration: October 2013 - September 2016

Coordinator: Forschungszentrum Juelich Gmbh (Germany)

Partner: Intel Gmbh (Germany), Bayerische Akademie der Wissenschaften (Germany), Ruprecht-Karls-Universitaet Heidelberg (Germany), Universitaet Regensburg (Germany), Fraunhofer-Gesellschaft zur Foerderung der Angewandten Forschung E.V (Germany), Eurotech Spa (Italy), Consorzio Interuniversitario Cineca (Italy), Barcelona Supercomputing Center - Centro Nacional de Supercomputacion (Spain), Xyratex Technology Limited (United Kingdom), Katholieke Universiteit Leuven (Belgium), Stichting Astronomisch Onderzoek in Nederland (The Netherlands) and Inria (France).

Inria contact: Stéphane Lanteri

Abstract: the DEEP-ER project aims at extending the Cluster-Booster Architecture that has been developed within the DEEP project with a highly scalable, efficient, easy-to-use parallel I/O system and resiliency mechanisms. A Prototype will be constructed leveraging advances in hardware components and integrate new storage technologies. They will be the basis to develop a highly scalable, efficient and user-friendly parallel I/O system tailored to HPC applications. Building on this I/O functionality a unified user-level checkpointing system with reduced overhead will be developed, exploiting multiple levels of storage. The DEEP programming model will be extended to introduce easy-to-use annotations to control checkpointing, and to combine automatic re-execution of failed tasks and recovery of long-running tasks from multi-level checkpoint. The requirements of HPC codes with regards to I/O and resiliency will guide the design of the DEEP-ER hardware and software components. Seven applications will be optimised for the DEEP-ER Prototype to demonstrate and validate the benefits of the DEEP-ER extensions to the Cluster-Booster Architecture.

Dr. Maciej Klemm: University of Bristol, Communication Systems & Networks Laboratory, Centre for Communications Research (United Kingdom)

Since July 2012, the team is coordinating the HOSCAR
http://

The general objective of the project is to setup a multidisciplinary Brazil-France collaborative effort for taking full benefits of future high-performance massively parallel architectures. The targets are the very large-scale datasets and numerical simulations relevant to a selected set of applications in natural sciences: (i) resource prospection, (ii) reservoir simulation, (iii) ecological modeling, (iv) astronomy data management, and (v) simulation data management. The project involves computer scientists and numerical mathematicians divided in 3 fundamental research groups: (i) numerical schemes for PDE models (Group 1), (ii) scientific data management (Group 2), and (iii) high-performance software systems (Group 3). Several Brazilian institutions are participating to the project among which: LNCC (Laboratório Nacional de Computaçäo Científica), COPPE/UFRJ (Instituto Alberto Luiz Coimbra de Pós-Graduaçäo e Pesquisa de Engenharia/Alberto Luiz Coimbra Institute for Grad<uate Studies and Research in Engineering, Universidade Federal do Rio de Janeiro), INF/UFRGS (Instituto de Informática, Universidade Federal do Rio Grande do Sul) and LIA/UFC (Laboratórios de Pesquisa em Ciência da Computaçäo Departamento de Computaçäo, Universidade Federal do Ceará). The French partners are research teams from several Inria research centers.

Liang Li, UESTC, China, July 15-August 8

Jay Gopalakrishnan, Portland University, USA, December 8-11

Maciej Klemm, University of Bristol, UK, July 29-August 2

Stéphane Lanteri, *Computational electromagnetics*, MAM5,
20 h, Polytech Nice.

Claire Scheid, *Practical works on ordinary differential equations*,
36 h, L3, University of Nice-Sophia Antipolis.

Claire Scheid, *Lectures and practical works in Numerical Analysis*,
36 h, M1, Mathematics engineering, University of Nice-Sophia Antipolis.

Stéphane Descombes, *Analyse numérique et applications
en finances*, M2, 30 h, University of Nice-Sophia Antipolis.

PhD defended in December 2014 : Caroline Girard, *Numerical
modeling of the electromagnetic susceptibility of innovative planar
circuits*, October 2011, Stéphane Lanteri, Ronan Perrussel and
Nathalie Raveu (Laplace Laboratory, INP/ENSEEIHT/UPS, Toulouse).

PhD in progress : Fabien Peyrusse, *Numerical simulation of
strong earthquakes by a discontinuous Galerkin method*, University
of Nice-Sophia Antipolis, October 2010, Nathalie Glinsky and
Stéphane Lanteri.

PhD in progress : Marie Bonnasse-Gahot, *Numerical simulation of
frequency domain elastic and viscoelastic wave propagation using
discontinuous Galerkin methods*, University of Nice-Sophia
Antipolis, October 2012, Julien Diaz (MAGIQUE3D project-team, Inria
Bordeaux - Sud-Ouest) and Stéphane Lanteri.

PhD in progress : Jonathan Viquerat, *Discontinuous Galerkin
Time-Domain methods for nanophotonics applications*, October 2012,
Stéphane Lanteri and Claire Scheid.

PhD in progress : Colin Vo Cing Tri, *Numerical modeling of
non-local dispersion for plasmonic nanostructures*, November 2014,
Stéphane Lanteri and Claire Scheid.