INRIA, Evaluation of Theme 1

Modeling, simulation and numerical analysis

Project-team NACHOS

March 17-19, 2009

Project-team title : Numerical modeling and high performAnce computingfor evolution problems in Complex domains and HeterogeneOuS media

Scienti�c leader : Stéphane Lanteri

Research center : Sophia Antipolis-Méditerranée

Common project-team with : J.A. Dieudonné Mathematics Laboratory, UMRCNRS 6621, University of Nice-Sophia Antipolis

The NACHOS project-team has started its research activities in July 2006 and has been o�ciallycreated in July 2007 as a follow-up of the CAIMAN project-team.

1 Personnel

Personnel (March 17-19, 2009)

Misc. INRIA CNRS University Total

DR / Professors 2 2

CR / Assistant Professor 1 2 3

Permanent engineer 0

Temporary engineer 1 1

PhD student 2 2

Postdoc 1 1

Total 1 6 0 2 9

External Collaborators 2 2 6 10

Visitors (> 1 month) 0

(1) �Senior Research Scientist (Directeur de Recherche)�(2) �Junior Research Scientist (Chargé de Recherche)�(3) �Civil servant (CNRS, INRIA, ...)�(4) �Associated with a contract (Ingénieur Expert or Ingénieur Associé)�

Changes in sta�

DR / Professors Misc. INRIA CNRS University totalCR / Assistant Professors

Arrival

Leaving

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Current composition of the project-team (March 17-19, 2009)

• INRIA sta�

� Montserrat Argente [TR, project-team assistant, 30%]

� Loula Fezoui [DR2]

� Stéphane Lanteri [DR2], scienti�c leader

• University of Nice-Sophia Antipolis

J.A. Dieudonné Mathematics Laboratory sta�

� Victorita Dolean [Assistant Professor]

� Francesca Rapetti [Assistant Professor, HDR]

• Ecole des Ponts ParisTech, CERMICS 1 sta�

� Nathalie Glinsky [CR]

• PhD student

� Joseph Charles [INRIA grant]

� Mohamed El Bouajaji [INRIA grant]

• Temporary engineer

� Christian Konrad [INRIA grant]

• Postdoc

� Siham Layouni [INRIA grant]

• External Collaborators

� Academic

∗ Victor Manuel Cruz-Atienza (Assistant Professor, Instituto de Geo�sica, Depar-tamento de Sismologia, Universidad Nacional Autonoma de Mexico)

∗ Sarah Delcourte (Assistant Professor, Claude Bernard University - Lyon 1)

∗ Martin Gander (Professor, Mathematics Department, University of Geneva)

∗ Luc Giraud (Professor, Parallel Algorithms and Optimization Group, LIMA-IRIT, ENSEEIHT, Toulouse)

∗ Stéphane Operto (CR CNRS, Géosciences Azur Laboratory, Villefranche surMer)

∗ Ronan Perrussel (CR CNRS, Ampère Laboratory, Ecole Centrale de Lyon)

∗ Serge Moto Mpong (Assistant Professor, University of Yaoundé 1, Cameroon)

∗ Jean Virieux (Professor, Joseph Fourier University and LGIT Laboratory)

� Industrial

∗ Muriel Sesques (Research engineer, CEA DAM, CESTA Center, Bordeaux)

∗ Joe Wiart (Research engineer, Orange Labs, Issy-les-Moulineaux center)

1Mathematics and scienti�c computing teaching and research centre

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Current position of former project-team membersduring the July 2006 - March 2009 period

• Former PhD students

� Mondher Benjemaa (PhD defended in November 2007), Postdoc (since January 2008)at Ecole des Mines ParisTech (Géophysique/Centre de Géosciences), Fontainebleau

� Antoine Bouquet (PhD defended in December 2007), Research engineer, DCNS Gr-oup, Saint-Tropez

� Adrien Catella (PhD defended in December 2008), student in specialized Masterprogram of the CERAM

� Hassan Fahs (PhD defended in December 2008), Postdoc (since February 2009) atIFP, Rueil-Malmaison

� Hugo Fol (PhD defended in December 2006), Research engineer (since April 2007),Société Générale, Paris

• Former Postdocs

� Sarah Delcourte (from October 2007 to September 2008), Assistant Professor, ClaudeBernard University - Lyon 1

� Ronan Perrussel (from November 2005 in the CAIMAN project-team to October2006), CR CNRS, Ampère Laboratory, Ecole Centrale de Lyon

Last INRIA enlistments

• NA

Other comments

2 Work progress

2.1 Keywords

• Electromagnetic wave propagation

• Seismic wave propagation

• Time domain, frequency domain

• Computational electromagnetics and bioelectromagnetics

• Computational geoseismics

• Finite volume method

• Discontinuous Galerkin method

• Simplicial mesh

• Arbitrary high order approximation

• Non-conforming discretization

• hp-adaptivity

• Hybrid explicit-implicit time integration

• Domain decomposition method

• High performance computing

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2.2 Context and overall goal of the project

The research activities of the NACHOS project-team aim at the formulation, analysis andevaluation of numerical methods and high performance resolution algorithms for the computersimulation of evolution problems in complex domains and heterogeneous media. Although wedo not forbid ourselves to study physical problems involving non-linear phenomena, we beforeall consider mathematical models that rely on �rst-order hyperbolic linear systems of PDEswith variable coe�cients and, in practice, we focus on mathematical models pertaining to elec-trodynamics [Jackson, 1999] and elastodynamics [Landau et al., 1990] with applications thatare concerned with computational electromagnetics and computational geoseismics. In gen-eral, realistic applications in these �elds involve the interaction of the underlying physical �eldswith media exhibiting space and time heterogeneities such as when studying the propagationof electromagnetic waves in biological tissues or the propagation of seismic waves in complexgeological media. Moreover, in most of the situations of practical relevance, the computationaldomain is irregularly shaped or/and it includes geometrical singularities. Both the heterogene-ity and the complex geometrical features of the underlying media motivate the use of numericalmethods working on non-uniform discretizations of the computational domain. In this context,our research e�orts are turned towards the development of unstructured (or hybrid unstruc-tured/structured) mesh based methods, capitalizing on previous e�orts undertaken in formerlocal scienti�c computing teams (i.e the CAIMAN and SINUS project-teams). We undertakeactivities ranging from the mathematical analysis of numerical methods for the discretizationof systems of PDEs of electrodynamics and elastodynamics, to the development of prototypesimulation software that e�ciently exploit the capabilities of modern high performance com-puting platforms. From the point of view of applications, our objective is to demonstrate thecapabilities of the proposed numerical methodologies and associated high performance simula-tion software through the realization of realistic, large-scale, three-dimensional (3D) numericalsimulations that are relevant to computational electromagnetics and computational geoseismics.Three physical situations currently attract our attention which are discussed below. In eachcase, it is worth noting that the team is committed to setup active collaborations with physi-cists or engineers that are directly concerned with the considered applications. Some of thesecollaborations are already in progress, while others are planned or will be sought for.

2.2.1 Application domain 1: computational electromagnetics

Electromagnetic devices are ubiquitous in present day technology. Indeed, electromagnetism hasfound and continues to �nd applications in a wide array of areas, encompassing both industrialand societal purposes. Applications of current interest include (among others) those related tocommunications (e.g transmission through optical �ber lines), to biomedical devices (e.g mi-crowave imaging, micro-antenna design for telemedecine, etc.), to circuit or magnetic storagedesign (electromagnetic compatibility, hard disc operation), to geophysical prospecting, and tonon-destructive evaluation (e.g crack detection), to name but just a few. Equally notable andmotivating are applications in defense which include the design of military hardware with de-creased signatures, automatic target recognition (e.g bunkers, mines and buried ordnance, etc.)propagation e�ects on communication and radar systems, etc. Although the principles of electro-magnetics are well understood, their application to practical con�gurations of current interest,such as those that arise in connection with the examples above, is signi�cantly complicated andfar beyond manual calculation in all but the simplest cases. These complications typically arisefrom the geometrical characteristics of the propagation medium (irregular shapes, geometricalsingularities), the physical characteristics of the propagation medium (heterogeneity, physicaldispersion and dissipation) and the characteristics of the sources (wires, etc.).

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Part of the activities of the NACHOS project-team aim at the development of high performance,high order, unstructured mesh based solvers for the full system of Maxwell equations, in the timedomain and frequency domain regimes. Although many of the above-mentioned electromagneticwave propagation problems can potentially bene�t from the proposed numerical methodologies,the team concentrates its e�orts on the following two situations.

Interaction of electromagnetic waves with biological tissues. Two main reasons mo-tivate our commitment to consider this type of problem for the application of the numericalmethodologies developed in the NACHOS project-team:

• �rst, from the numerical modeling point of view, the interaction between electromagneticwaves and biological tissues exhibit the three sources of complexity identi�ed previouslyand are thus particularly challenging for pushing one step forward the state-of-the art ofnumerical methods for computational electromagnetics. The propagation media is stronglyheterogeneous and the electromagnetic characteristics of the tissues are frequency depen-dent. Interfaces between tissues have rather complicated shapes that cannot be accuratelydiscretized using cartesian meshes. Finally, the source of the signal often takes the formof a complicated device (e.g a mobile phone or an antenna array).

• second, the study of the interaction between electromagnetic waves and living tissues �ndsapplications of societal relevance such as the assessment of potential adverse e�ects ofelectromagnetic �elds [Bernardi et al., 2000] or the utilization of electromagnetic waves fortherapeutic or diagnostic purposes [Rosen et al., 2002]. It is widely recognized nowadaysthat numerical modeling and computer simulation of electromagnetic wave propagationin biological tissues is a mandatory path for improving the scienti�c knowledge of thecomplex 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 widelyknown FDTD (Finite Di�erence Time Domain) method due to Yee [Yee, 1966]. In this method,the whole computational domain is discretized using a structured (cartesian) grid. Due to thepossible straightforward implementation of the algorithm and the availability of computationalpower, FDTD is currently the leading method for numerical assessment of human exposure toelectromagnetic waves. However, limitations are still seen, due to the rather di�cult depar-ture from the commonly used rectilinear grid and cell size limitations regarding very detailedstructures of human tissues. In this context, the general objective of the contributions of theNACHOS project-team is to demonstrate the bene�ts of high order unstructured mesh basedMaxwell solvers for a realistic numerical modeling of the interaction of electromagnetic wavesand biological tissues with emphasis on applications related to the numerical dosimetry of expo-sure to radiations from mobile phones or wireless communication systems [Scarella et al., 2006].This activity is conducted in close collaboration with the team of Joe Wiart at Orange Labs(formerly, France Telecom Research & Development) in Issy-les-Moulineaux.

Interaction of electromagnetic waves with charged particle beams. Physical phenom-ena involving charged particles take place in various physical and technological situations suchas in plasmas, semiconductor devices, hyper-frequency devices, charged particle beams and moregenerally, in electromagnetic wave propagation problems including the interaction with chargedparticles by taking into account self consistent �elds. The numerical simulation of the evolutionof charged particles under their self-consistent or applied electromagnetic �elds can be modeledby the three dimensional Vlasov-Maxwell equations. The Vlasov equation describes the transport

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in phase space of charged particles submitted to external as well as self-consistent electromag-netic �elds. It is coupled non-linearly to the Maxwell equations which describe the evolution ofthe self-consistent electromagnetic �elds. The numerical method which is mostly used for thesolution of these equations is the Particle-In-Cell (PIC) method [Birdsall and Langdon, 1985]-[Hockney and Eastwood, 1981]. Its basic idea is to discretize the distribution function f ofmacro-particles which is the solution of the Vlasov equation, by a particle method, which con-sists in representing f by a �nite number of macro-particles and advancing those using theLorentz equations of motion. On the other hand, Maxwell equations are solved on a computa-tional mesh of the physical space. The coupling is done by gathering the charge and currentdensities from the particles on the mesh to get the sources for the Maxwell equations, and byinterpolating the �eld data on the particles when advancing them.

Such Vlasov-Maxwell PIC solvers have become a major research tool in di�erent areas of physicsinvolving self-consistent interaction of charged particles, in particular in plasma and beamphysics. Two-dimensional simulations have now become very reliable and can be used as well forqualitative as for quantitative results that can be compared to experiments with good accuracy.As the power of supercomputers was increasing three dimensional codes have been developed inthe recent years. However, even in order to just make qualitative 3D simulations, an enormouscomputing power is required. Today's and future massively parallel supercomputers allow toenvision the simulation of realistic problems involving complex geometries and multiple scales.In order to achieve this e�ciently, new numerical methods need to be designed. This includesthe investigation of high order Maxwell solvers, the use of hybrid grids with several homoge-neous zones having their own structured or unstructured mesh type and size, and a �ne analysisof load balancing issues. These issues are considered in the NACHOS project-team throughthe development of high order unstructured mesh based Vlasov-Maxwell PIC parallel solvers.This activity is conducted in close collaboration with the CESTA Center of the CEA DAM inBordeaux and also in the framework of a multi-partner project funded by the ANR (see thedescription of the HOUPIC project in section 5).

2.2.2 Application domain 2: computational geoseismics

Computational challenges in geoseismics span a wide range of disciplines and have signi�cantscienti�c and societal implications. Two important topics are mitigation of seismic hazardsand discovery of economically recoverable petroleum resources. The research activities of theNACHOS project-team in this domain have before all focused on the development of numericalmethodologies and simulation tools for seismic risk assessment, while the involvement on thesecond topic has been rather minimal so far.

Seismic risk assessment. To understand the basic science of earthquakes and to help en-gineers better prepare for such an event, scientists want to identify which regions are likely toexperience the most intense shaking, particularly in populated sediment-�lled basins. This un-derstanding can be used to improve building codes in high risk areas and to help engineers designsafer structures, potentially saving lives and property. In the absence of deterministic earthquakeprediction, forecasting of earthquake ground motion based on simulation of scenarios is one themost promising tools to mitigate earthquake related hazard. This requires intense modelingthat meets the spatial and temporal resolution scales of the continuously increasing density andresolution of the seismic instrumentation, which record dynamic shaking at the surface, as wellas of the basin models. Another important issue is to improve our physical understanding ofthe earthquake rupture processes and seismicity. Large scale simulations of earthquake rupturedynamics, and of fault interactions, are currently the only means to investigate these multi-scale

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physics together with data assimilation and inversion. High resolution models are also requiredto develop and assess fast operational analysis tools for real time seismology and early warningsystems. Modeling and forecasting earthquake ground motion in large basins is a challenging andcomplex task. The complexity arises from several sources. First, multiple scales characterize theearthquake source and basin response: the shortest wavelengths are measured in tens of meters,whereas the longest measure in kilometers; basin dimensions are on the order of tens of kilome-ters, and earthquake sources up to hundreds of kilometers. Second, temporal scales vary fromthe hundredths of a second necessary to resolve the highest frequencies of the earthquake sourceup to as much as several minutes of shaking within the basin. Third, many basins have a highlyirregular geometry. Fourth, the soil's material properties are highly heterogeneous. And �fth,geology and source parameters are observable only indirectly and thus introduce uncertaintyin the modeling process. Because of its modeling and computational complexity, earthquakesimulation is currently recognized as a grand challenge problem.

Numerical methods for the propagation of seismic waves have been studied for many years.Most of existing numerical software rely on �nite di�erence type methods. Among the mostpopular schemes, one can cite the staggered grid �nite di�erence scheme proposed by Virieux[Virieux, 1986] and based on the �rst order velocity-stress hyperbolic system of elastic wavesequations, which is an extension of the scheme derived by Yee [Yee, 1966] for the solution ofthe Maxwell equations. The use of cartesian meshes is a limitation for such numerical meth-ods especially when it is necessary to incorporate surface topography or curved interface. Inthis context, our objective is to develop high order unstructured mesh based methods for thenumerical solution of the system of elastodynamic equations for elastic media in a �rst step,and then to extend these methods to the treatment of more complex propagation materials suchas viscoelastic media. Additionally, the team considers in detail the necessary mathematicaland numerical developments for the large-scale simulation of earthquake dynamics. This ac-tivity is conducted in close collaboration with Jean Virieux (Waves and Internal Structure ofthe Earth team of the LGIT Laboratory in Grenoble) and Victor Manuel Cruz-Atienza (Insti-tuto de Geo�sica, Departamento de Sismologia, Universidad Nacional Autonoma de Mexico)[Ben Jemaa et al., 2007]-[Ben Jemaa et al., 2009].

Seismic exploration. The project-team is an associate partner of the Seiscope consortium(seismic imaging of complex structures from multicomponent global o�set data by full waveforminversion), which is coordinated by the Géosciences Azur Laboratory in Sophia Antipolis. Inthis context, the team collaborates with Stéphane Operto and Vincent Etienne (PhD studentat the Géosciences Azur Laboratory) on the development of a numerical methodology for three-dimensional elastic wave inversion.

2.2.3 Research directions

The applications in computational electromagnetics and computational geoseismics discussedpreviously lead to the numerical simulation of wave propagation in heterogeneous media andoften involve complex shape objects or domains including geometrical details or singularities.The underlying wave propagation phenomena can be purely unsteady or they can be periodic(because the imposed source term follows a time harmonic evolution). Although time domainformulations of wave propagation problems correspond to the most general situation, frequencydomain problems are interesting for at least two reasons: they lead to more challenging math-ematical problems (de�ne on complex valued quantities) and thus motivate the research ofpowerful numerical methods (especially in the case of highly heterogeneous media) and, theirsolutions can be used to validate that of equivalent time domain problems. When a time har-

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monic behavior applies, numerical methods can take into account this fact as soon as the initialphase of their design. The overall objective of the NACHOS project-team is to develop unstruc-tured (or hybrid structured/unstructured) mesh based numerical methods for solving systemsof PDEs modeling the propagation of electromagnetic and geoseismic waves, with a particularattention to several distinguishing features that are discussed below.

Accuracy. On one hand, we consider numerical methods relying on discretization techniquesthat best �t to the geometrical characteristics of the problems at hand, more particularly, meth-ods working on unstructured, locally re�ned, even non-conforming, simplicial meshes. On theother hand, these methods should also be capable to accurately describe the underlying physicalphenomena that may involve highly variable space and time scales. With reference to this char-acteristic, two main strategies are possible: adaptive local re�nement/coarsening of the mesh (i.eh-adaptivity) and adaptive local variation of the interpolation order (i.e p-adaptivity). Ideally,these two strategies are combined leading to the so-called hp-adaptive methods. Indeed, devel-oping such hp-adaptive discretization methods represents an ultimate objective of the researchactivities of the NACHOS project-team and for that purpose, we currently concentrate oure�orts on a family of discontinuous �nite element methods.

Stability. In most of the cases, the targeted applications yield numerical simulations thatmust be carried out over a long physical duration. It is then mandatory that numerical methodsare able to reproduce the physical phenomena free of any arti�cial di�usion and in a stableway. As a general rule, for time domain problems, it is highly desirable to design numericalmethods that conserve a certain discrete energy (or, more generally, numerical methods thatverify a discrete form of an energy conservation principle such as Poynting's theorem in the caseof electromagnetic wave propagation).

Numerical e�ciency. The numerical simulation of unsteady problems most often rely on ex-plicit time integration schemes. Such schemes are subjected to stability criteria, linking the spaceand time discretization parameters, that can be very restrictive when the underlying mesh ishighly non-uniform (especially for locally re�ned meshes). For realistic three-dimensional prob-lems, this simply translates into unfeasible computing times. In order to improve this situation,one possible approach consists in applying an implicit time scheme in regions of the computa-tional domain where the underlying mesh is highly re�ned. The resulting hybrid explicit-implicit(or locally implicit) time integration strategy raises several challenges both from the mathemati-cal analysis viewpoint (stability and accuracy, especially for what concern numerical dispersion)and from the computer implementation viewpoint (data structures, parallel computing aspects).For implicit time integration schemes on one hand, and for the numerical treatment of frequencydomain problems on the other hand, numerical e�ciency also refers to a foreseen property oflinear system solvers.

Computational e�ciency. Despite the ever increasing performances of microprocessors, thenumerical simulation of realistic three-dimensional problems is hardly performed on a high-endworkstation and parallel computing is a mandatory path. This is all the more true that we aimat demonstrating the bene�ts of the numerical methods that we will propose through the simu-lation of large-scale wave propagation problems leading to the processing of very large volumesof data. The latter results from two combined parameters: the size of the mesh (measured bythe total number of elements) and the number of degrees of freedom per mesh element whichis itself linked to the degree of interpolation and to the number of physical variables (since we

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solve systems of PDEs). Hence, numerical methods must be adapted to the characteristics ofmodern parallel computing platforms taking into account their hierarchical nature (e.g multipleprocessors and multiple core systems with complex cache and memory hierarchies). Appropriateparallelization strategies need to be designed that combine distributed memory and shared mem-ory programming paradigms. Moreover, maximizing the e�ective �oating point performanceswill require the design of numerical algorithms that can bene�t from the optimized BLAS linearalgebra kernels.

Taking the above objectives into consideration, the research activities of the NACHOS project-team are organized along four directions:

• arbitrary high order, hp-adaptive, discontinuous Galerkin (DG) methods designed on un-structured simplicial meshes, or hybrid structured/unstructured meshes, for the discretiza-tion of the systems of PDEs of electrodynamics and elastodynamics, including the numer-ical treatment of physically relevant propagation media models.

• Hybrid explicit-implicit time integration schemes for the treatment of grid-induced sti�nessin the case of the numerical simulation of time domain electromagnetic and elastodynamicwave propagation problems on highly non-uniform (i.e locally re�ned) meshes.

• Domain decomposition solution methods for wave propagation problems modeled by thesystems of PDEs of electrodynamics and elastodynamics in the time domain and frequencydomain regimes.

• High performance numerical algorithms adapted to architectural characteristics of modernparallel computing platforms in view of performing large-scale, three-dimensional, numer-ical simulation for applications relevant to computational electromagnetics and computa-tional geoseismics.

2.3 Objectives for the evaluation period

The midterm (4 years) objectives that were given at the creation of the NACHOS project-teamare the following:

1. arbitrary high order DG methods based on nodal interpolation methods (referred as DG-Pp

in the following) designed on unstructured conforming simplicial meshes for the discretiza-tion of the systems of PDEs of electrodynamics and elastodynamics, in the 2D and 3Dcases, for time domain and frequency domain formulations of these systems.

2. Arbitrary high order DG-Pp methods designed on unstructured non-conforming (locallyre�ned) simplicial meshes for the discretization of the systems of PDEs of electrodynamicsand elastodynamics, in the 2D and 3D cases, with emphasis on time domain formulationsof these systems.

3. hybrid explicit-implicit time integration strategies for arbitrary high order DG-Pp methodsdesigned on unstructured conforming simplicial meshes for the solution of the system oftime domain Maxwell equations in the 2D and 3D cases.

4. domain decomposition methods coupled to arbitrary high order DG-Pp methods designedon unstructured conforming simplicial meshes for the solution of the system of Maxwellequations, in the 2D and 3D cases, for time domain and frequency domain formulations ofthis system.

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When considering the systems of PDEs of electrodynamics and elastodynamics, only simplepropagation media models where planned to be taken into account i.e:

• linear, heterogeneous, isotropic and non-dispersive media in the case of electromagneticwave propagation,

• linear, heterogeneous, isotropic and elastic media in the case of seismic wave propagation.

The above-mentioned methodological developments have to be considered as �rst steps towardsaccurate and e�cient numerical methodologies for addressing the physical situations and relatedapplications discussed in subsections 2.2.1 and 2.2.2 which lead to the numerical modeling ofproblems involving the interaction of:

• electromagnetic waves with dispersive media (i.e biological tissues),

• electromagnetic waves with charged particles beams,

• seismic waves with viscoelastic geological media.

Achievements regarding these objectives are described in more details in subsections 2.4 to 2.6below. In doing so, we have concentrated of methodological aspects while speci�c milestonesregarding applications have been discussed in subsection 2.4 with reference to collaborations withengineers and physicists that are directly concerned with the considered applications. Besides,designing methods, algorithms and software that are well adapted to modern high performancecomputing platforms has to be considered as a transversal objective to our research activities andis thus not addressed separately. For the sake of completeness, we mention that our simulationsare performed on a local platform (for simulations conducted on several tens to a few hundredsof CPUs) and on tera�opic platforms (for large-scale simulations) which are made available tothe French academic community under the auspices of GENCI.

2.4 High order discontinuous Galerkin methods on simplicial meshes

The discontinuous Galerkin method was introduced in 1973 by Reed and Hill to solve the neu-tron transport equation. From this time to the 90's a review of the DG methods would likely�t into one page. In the meantime, the �nite volume approach has been widely adopted bycomputational �uid dynamics scientists and has now nearly supplanted classical �nite di�erenceand �nite element methods in solving problems of non-linear convection. The success of the�nite volume method is due to its ability to capture discontinuous solutions which may occurwhen solving non-linear equations or more simply, when convecting discontinuous initial datain the linear case. Let us �rst remark that DG methods share with �nite volumes this propertysince a �rst order �nite volume scheme can be viewed as a 0th order DG scheme. Howevera DG method may be also considered as a �nite element one where the continuity constraintat an element interface is released. While it keeps almost all the advantages of the �nite ele-ment method (large spectrum of applications, complex geometries, etc.), the DG method hasother nice properties which explain the renewed interest it gains in various domains in scien-ti�c computing as witnessed by books or special issues of journals dedicated to this method[Cockburn et al., 2000]-[Cockburn and Shu, 2005]-[Dawson, 2006]:

• it is naturally adapted to a high order approximation of the unknown �eld. Moreover, onemay increase the degree of the approximation in the whole mesh as easily as for spectralmethods but, with a DG method, this can also be done very locally. In most cases,

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the approximation relies on a polynomial interpolation method but the DG method alsoo�ers the �exibility of applying local approximation strategies that best �t to the intrinsicfeatures of the modeled physical phenomena.

• When the discretization in space is coupled to an explicit time integration method, theDG method leads to a block diagonal mass matrix independently of the form of the localapproximation (e.g the type of polynomial interpolation). This is striking di�erence withclassical, continuous �nite element formulations. Moreover, the mass matrix is diagonal ifan orthogonal basis is chosen.

• It easy handles complex meshes. The grid may be a classical conforming �nite elementmesh, a non-conforming one or even a hybrid mesh made of various elements (tetrahedron,prism, hexahedron, etc.). The DG method has been proved to work well with highly locallyre�ned meshes. This property makes the DG method more suitable to the design of a hp-adaptive solution strategy (i.e where the characteristic mesh size h and the interpolationdegree p changes locally wherever it is needed).

• It is �exible with regards to the choice of the time stepping scheme. One may combinethe DG spatial discretization with any explicit or even implicit time integration schemeprovided that stability is ensured,

• it is naturally adapted to parallel computing. As long as an explicit time integrationscheme is used, the DG method is easily parallelized. Moreover, the compact nature of DGdiscretization schemes is in favor of high computation to communication ratio especiallywhen the interpolation order is increased.

As with standard �nite element methods, a DG method relies on a variational formulation ofthe continuous problem at hand. However, due to the discontinuity of the global approximation,this variational formulation has to be de�ned at the element level. Then, a degree of freedomin the design of a DG method stems from the approximation of the boundary integral termcoming from the application of an integration by parts to the element-wise variational form. Inthe spirit of �nite volume methods, the approximation of this boundary integral term calls for anumerical �ux function which can be based on either a centered scheme or an upwind scheme,or a blending between these two schemes.

For the solution of the time domain Maxwell equations, we have �rst proposed (in the context ofthe CAIMAN project-team) a DGTD (Discontinuous Galerkin Time Domain) method workingon unstructured conforming simplicial meshes [Fezoui et al., 2005]. This DGTD method com-bines a central numerical �ux function with a second order leap-frog time integration scheme,while the local approximation of the electromagnetic �eld within a simplex relies on a nodal(Lagrange type) polynomial interpolation method. This DGTD-Pp method is non-dissipativeand stable under a CFL-like condition. The initial implementation of this DGTD-Pp method[Bernacki et al., 2006b]-[Bernacki et al., 2006a] has been limited to a linear interpolation me-thod. Since the creation of the NACHOS project-team, our research activities have beenconcerned with the extension of this method towards higher accuracy and increased e�ciencyin line with the planned midterm objectives listed in subsection 2.3. Moreover, we have alsostudied the development of such methods for the frequency domain Maxwell equations, and thetime domain elastodynamic equations modeling the propagation of seismic waves. Finally, anactivity that has been launched more recently aims at designing a high order DGTD-Pp coupledto a PIC method for the solution of the system of Vlasov-Maxwell equations.

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2.4.1 Personnel

• For the time domain Maxwell equations

� Permanent researchers: Victorita Dolean, Loula Fezoui, Stéphane Lanteri andFrancesca Rapetti

� PhD students: Hassan Fahs and Joseph Charles

� Postdocs: Siham Layouni

• For the frequency domain Maxwell equations

� Permanent researchers: Victorita Dolean and Stéphane Lanteri

� PhD students: Mohamed El Bouajaji and Hugo Fol

� Postdocs: Ronan Perrussel

� External collaborators: Ronan Perrussel (since November 2007)

• For the time domain elastodynamic equations

� Permanent researchers: Loula Fezoui and Nathalie Glinsky

� PhD students: Mondher Benjemaa

� Postdocs: Sarah Delcourte

� External collaborators: Victor Manuel Cruz-Atienza, Sarah Delcourte (since October2008), Serge Moto Mpong, Stéphane Operto and Jean Virieux

2.4.2 Project-team positioning

Originally devised for dealing with hyperbolic problems and largely adopted by the computa-tional �uid dynamics community, the DG method is now applied to a variety of problems ofpractical interest. Compressible �uid mechanics whose mathematical modeling calls for non-linear hyperbolic and mixed hyperbolic-parabolic systems of PDEs is certainly the domain thathas been the most in�uential to the development of DG methods, from both the theoreticaland application points of view. Two major contributors to this picture are Bernardo Cockburnat University of Minnesota and Chi-Wang Shu at Brown University. In addition to numerousresults for convection-di�usion problems and systems of conservation laws, Bernardo Cockburn,Chi-Wang Shu and colleagues have also studied the applicability of DG methods for the nu-merical resolution of the time domain Maxwell equations, see [Cockburn et al., 2004] (locallydivergence-free DG methods) and [Chen et al., 2005] (high order Runge-Kutta DG methods).

Jan S. Hesthaven at Brown University and Tim Warburton at Rice University are two otherresearchers that have extensively studied theoretical and practical issues of DG methods for solv-ing the time domain [Hesthaven and Warburton, 2002]-[Hesthaven and Warburton, 2004a] orfrequency domain [Hesthaven and Warburton, 2004b]-[Warburton and Embree, 2006] Maxwellequations. One of their important achievements is the development of high order Rung-KuttaDG (RKDG) methods based on nodal interpolation on simplicial meshes. Jan S. Hesthavenhas visited the NACHOS team for a few days in July 2007 and will visit us again in July2008, with the common objective of setting up a collaboration on the development of high orderDGTD-Pp-PIC methods for the solution of the coupled system of Vlasov-Maxwell equations.

RKDG methods have also been studied by Jaap van der Vegt and colleagues at the University ofTwente [Sarmany et al., 2007]. More generally, the latter group and the NACHOS team clearly

12

share some common research concerns regarding the study of hp-adaptive methods for the solu-tion of the time domain and frequency domain Maxwell equations [Harutyunyan et al., 2008a]-[Harutyunyan et al., 2008b], and in particular on the development of a (object oriented) softwareframework for hp-adaptive DG methods. DG methods applied to the frequency domain Maxwellequations have also been studied theoretically by several other authors (stability, convergence,presence of spurious modes for the Maxwell eigenvalue problem, see [Houston et al., 2005b]-[Houston et al., 2005a]-[Bu�a and Perugia, 2006]-[Bu�a et al., 2007] among other publicationsof these authors): Annalisa Bu�a (Istituto di Matematica Applicata e Tecnologie Informaticheof CNR, Pavia, Italy), Paul Houston (University of Nottingham), Ilaria Perugia (University ofPavia), Dominik Schotzau (University of British Columbia).

At the national level, ONERA is actively studying the application of high order DG meth-ods on hexahedral meshes for the numerical solution of the time domain Maxwell equations[Cohen et al., 2006]-[Montseny et al., 2008]-[Pernet and Ferrieres, 2007] in collaboration withthe POEMS project-team at INRIA Paris-Rocquencourt and the CERFACS (Electro-Magnetism- EMC group). There is clearly a room for a strong collaboration between ONERA and the NA-CHOS project-team in particular, and that could also involved the CERFACS (EMC group)and the POEMS project-team, on the development of high order DG methods for the solutionof the time domain Maxwell equations. One possible topic that could be at the heart of such acollaboration and that would largely bene�t from our respective expertise, is the study of DGformulations on hybrid meshes combining tetrahedra and hexahedra.

Finally, we note that a general theory for DG methods applied to Friedrichs systems has recentlybeen proposed by Alexandre Ern (Ecole des Ponts ParisTech, Cermics) and Jean-Luc Germond(Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur)[Ern and Guermond, 2006a]-[Ern and Guermond, 2006b].

The development of DG methods for the numerical solution of the elastodynamic equationsand their applications to seismic waves propagation is still in its infancy. Recent achievementsin this direction have been obtained by Martin Käser (Department of Earth and Environmen-tal Sciences, Geophysics, University of Munich) and Michael Dumbser (Institut für Aerody-namik und Gasdynamik, University of Stuttgart) in the framework of ADER2-DG methods[Käser and Dumbser, 2006]-[Dumbser and Käser, 2006]-[Käser et al., 2007]-[Dumbser et al., 2007]-[Käser et al., 2008].

Within INRIA, DG methods for seismic wave propagation are also studied in the MAGIQUE-3D project-team at the Bordeaux - Sud-Ouest research center. The MAGIQUE-3D and NA-CHOS project-teams plan to collaborate on this topic in a very near future and in the frameworkof the Depth Imaging Partnership strategic action between the TOTAL group and INRIA thathas been launched in 2008.

2.4.3 Scienti�c achievements

Time domain Maxwell equations. The previously developed discontinuous Galerkin me-thod [Fezoui et al., 2005] for the solution of the time domain Maxwell equations on unstructuredsimplicial meshes has been extended to arbitrary high order in space and time. The resultingDGTD-Pp method relies on the following ingredients: a central numerical �ux function for theapproximation of the integral term at an interface between two neighboring elements, a highorder nodal (Lagrange type) polynomial interpolation method for the approximation of the elec-tromagnetic �eld components within a simplex element and a high order leap-frog scheme for

2Arbitrary accuracy DErivative Riemann problem

13

time integration. Two important features of discontinuous Galerkin methods are their �exibil-ity with regards to the local approximation of the �eld quantities and their natural ability todeal with non-conforming meshes. Whereas conforming discontinuous Galerkin time domainmethods have been developed rather readily, the development of non-conforming discontinu-ous Galerkin methods has been less impressive. The non-conformity can result from a localre�nement of the mesh (h-adaptivity), or of the approximation order (p-adaptivity) or of bothof them (hp-adaptivity). In the context of the PhD thesis of Hassan Fahs (defended in De-cember 2008) [Fahs, 2008], we have studied non-dissipative discontinuous Galerkin methods forsolving the 2D time domain Maxwell equations on non-conforming, locally re�ned, triangularmeshes. In particular, a hp-like method which allows for both a local non-conforming re�nementof the mesh and a locally de�ned approximation order has been designed, analyzed (stabilitystudy [Fahs et al., 2008], hp a priory convergence study [Fahs, 2008]) and evaluated throughits application to several 2D propagation problems in homogeneous and heterogeneous media[Fahs, 2009]. The DGTD-Pp methods discussed here are implemented in the MAXW2D-DGTDand MAXW3D-DGTD software (see section 4.2).

Frequency domain Maxwell equations. A large number of electromagnetic wave propaga-tion problems can be modeled by assuming a time harmonic behavior of the electromagnetic �eldand thus considering the numerical solution of the frequency domain Maxwell equations. Our�rst achievements in this direction have taken place in the framework of the PhD thesis of HugoFol (defended in December 2006) [Fol, 2006] with the development of DGFD (DiscontinuousGalerkin Frequency Domain) method on tetrahedral meshes for the solution of the 3D frequencydomain Maxwell equations. As for the time domain Maxwell equations, a nodal (Lagrange type)polynomial interpolation method has been adopted for the approximation of the electromagnetic�eld components within a simplex element The initial implementation of this DGFD-Pp methodhas been limited to a linear interpolation method. Preliminary investigations towards higheraccuracy have been conducted in the 2D case [Dolean et al., 2008a]. It is worth to note thatcontrary to the time domain case, the use of a centered numerical �ux for the approximationof the integral term at an interface between two neighboring elements is questionable in thecontext of time harmonic problems. In particular, we have observed numerically that the use ofan upwind scheme yields an optimally convergent DGFD-Pp method. These DGFD-Pp methodshave been implemented in the MAXW2D-DGFD and MAXW3D-DGFD software (see section4.2).

Time domain elastodynamic equations. Following our achievements for the time domainMaxwell equations, we have developed a high order non-dissipative DGTD-Pp method on un-structured simplicial meshes for the solution of the �rst order hyperbolic linear system of elas-todynamic equations in the 2D and 3D cases [Delcourte et al., 2009]. This method is essentiallybased on the same ingredients than those adopted for the time domain Maxwell equations amongwhich, the use of nodal polynomial (Lagrange type) basis functions, a second order leap-frogtime integration scheme and a centered scheme for the evaluation of the numerical �ux at theinterface between neighboring elements. This DGTD-Pp method has been validated and assessedin detail in the context of propagation problems in both homogeneous and heterogeneous me-dia including problems for which analytical solutions can be computed. Finally, the impact ofhigh order time stepping on the overall accuracy and e�ciency of the proposed non-dissipativeDGTD-Pp has recently been investigated through the adoption of a family of high order leap-frogschemes 3. The DGTD-Pp methods discussed here are implemented in the SISMO2D-DGTD

3S. Moto Mpong and N. Glinsky, Arbitrary high order non-dissipative discontinuous Galerkin method for

elastic wave propagation, in preparation.

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and SISMO3D-DGTD software (see section 4.2).

2.4.4 Collaborations

For the main part, ongoing collaborations in relation with our research activities on high orderdiscontinuous Galerkin methods are oriented towards the adaptation and application of theproposed discretization methods to the study of the wave propagation problems associated withthe applications described in subsections 2.2.1 and 2.2.2.

Human exposure to electromagnetic �elds. In collaboration with Joe Wiart (OrangeLabs, Issy-les-Moulineaux center), the DGTD-Pp methods developed in the team for the solu-tion of the time domain Maxwell equations are applied to the realistic numerical modeling of theinteraction of electromagnetic �elds with biological tissues. The objective is to obtain high reso-lution distributions of the electromagnetic �eld in the tissues in order to assess whether localizede�ects (so called, hot spots) appear that are not correctly modeled by the FDTD method whichis currently the most commonly used method for numerical dosimetry studies of human exposureto the radiation of wireless communication systems. So far, we have focused on the case of mobilephone radiation. In particular, we aim at applying our DGTD-Pp methods on locally re�ned,possibly in a non-conforming way, tetrahedral meshes of head tissues which include an accuratediscretization of interfaces between tissues. Such meshes have been constructed thanks to a col-laboration with the GEOMETRICA project-team (Pierre Alliez and Mariette Yvinec) at INRIASophia Antipolis - Méditerranée research center [Scarella et al., 2006]-[Clatz et al., 2006a].

Interaction of electromagnetic waves with charged particle beams. We collaboratewith Muriel Sesques (CEA DAM, CESTA Center, Bordeaux) on the development of a coupledVlasov-Maxwell solver combining the high order DGTD-Pp method on tetrahedral meshes devel-oped in the team and a Particle-In-Cell method. The resulting DGTD-PIC solver will be usedfor electrical vulnerability studies of the experimental chamber of the Laser Mégajoule system.This topic is also considered in the HOUPIC project funded by ANR (see subsection 5) in thecontext of which we collaborate with Eric Sonnendrücker and other researchers of the CALVIat INRIA Nancy - Grand Est research center.

Seismic wave propagation and dynamic fault modeling. We are interested in the nu-merical simulation of seismic activity, including wave propagation and dynamic fault modeling,more precisely, the simulation of a fault whose location is prescribed and which undergoes atransient behavior. Then, the numerical modeling of a dynamic fault rupture is realized by tak-ing into account a time dependent boundary condition on the fault geometry, and requires theadaptation of the DGTD-Pp methods developed for the solution of the system of elastodynamicsequations. Our �rst contributions on this topic have been obtained in the context of a �nitevolume method [Ben Jemaa et al., 2007]-[Ben Jemaa et al., 2009], in the context of the PhDthesis of Mondher Benjemaa, in collaboration with Victor Manuel Cruz-Atienza (Instituto deGeo�sica, Departamento de Sismologia, Universidad Nacional Autonoma de Mexico) and JeanVirieux (LGIT Laboratory in Grenoble and Géosciences Azur Laboratory in Sophia Antipolis).Beside, for what concern the development and analysis of high order DGTD-Pp methods for thesolution of the system of elastodynamic equations, we collaborate with Sarah Delcourte (ClaudeBernard University - Lyon 1, former postdoc of the team) and Serge Moto Mpong (Universityof Yaoundé 1, Cameroon).

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2.4.5 External support

• Research grant with France Télécom R&D/Orange Labs, Issy-les-Moulineaux center (highorder DGTD-Pp methods on non-conforming simplicial meshes for time domain electro-magnetic wave propagation).

• Research grant with CEA DAM, CESTA Center, Bordeaux and HOUPIC ANR project(high order DGTD-Pp/PIC solver for the Vlasov-Maxwell equations).

• MAXWELL (high order DGFD-Pp methods on simplicial meshes for frequency domainelectromagnetic wave propagation) ANR project.

• QSHA (�nite volume method on simplicial meshes for time domain seismic wave propaga-tion) ANR project.

2.4.6 Self assessment

In the high order DG methods developed by the team so far, for electromagnetic or elastody-namic wave propagation problems, the local approximation of the physical �eld relies on a nodal(Lagrange type) polynomial interpolation method. However, it is clear that other polynomialinterpolation methods could be adopted as well. The choice of an interpolation method and as-sociated set of basis functions should ideally take into account several criteria among which, themodal or nodal nature of the functions, the orthogonality of the basis functions, the hierarchicalstructure of the basis functions, the conditioning of the elemental matrices to be inverted (e.gthe mass matrix in explicit DGTD-Pp methods), the spectral properties of the interpolation andthe programming simplicity. A dedicated study on this question is desirable, especially in viewof the development of p-adaptive DG methods. Such a study is at the heart of the PhD thesisof Joseph Charles (started in October 2008), and is conducted in the context of the solution ofthe time domain Maxwell equations.

Whereas we have devoted most of our e�orts so far towards the design of arbitrary high orderin space DGTD-Pp methods, little attention has been given to time accuracy. A �rst step inthis direction has been realized in the PhD thesis of Hassan Fahs (defended in December 2008)and during the visit of Serge Moto Mpong (from November 2007 to August 2008) with thestudy of a family of high order leap-frog schemes in conjunction with DGTD-Pp methods forthe systems of Maxwell and elastodynamic equations respectively. Other time stepping methodssuch as Runge-Kutta methods could be considered as well and the �nal choice should take intoconsideration accuracy, stability and computational e�ciency (CPU cost, memory overhead)issues. Such a study is also planned to be undertaken in the PhD thesis of Joseph Charles.

For what concern the design of DG methods for frequency domain electromagnetic wave propa-gation problems (i.e DGFD-Pp methods), exploiting the locality of the approximation (i.e localde�nition of the approximation order) and of the discretization (i.e local re�nement of the mesh)and ultimately, of hp-adaptivity, will be mandatory to obtain a competitive method with re-gards to more conventional, continuous �nite element (edge element) methods. A �rst step inthis direction is considered in the context of the PhD thesis of Mohamed El Bouajaji (started inOctober 2008) through the development of a p-adaptive DGFD-Pp method for the 2D and 3Dfrequency domain Maxwell equations.

2.5 Time integration strategies for grid-induced sti�ness in DGTD methods

The use of unstructured meshes (based on triangles in two space dimensions and tetrahedra inthree space dimensions) is an important feature of the DGTD methods developed in the NA-

16

CHOS project-team which can thus easily deal with complex geometries and heterogeneouspropagation media. Moreover, discontinuous Galerkin discretization methods are naturallyadapted to local, conforming as well as non-conforming, re�nement of the underlying mesh,Most of the existing DGTD methods rely on explicit time integration schemes and lead to blockdiagonal mass matrices which is often recognized as one of the main advantages with regards tocontinuous �nite element methods. However, explicit DGTD methods are also constrained bya stability condition that can be very restrictive on highly re�ned meshes and when the localapproximation relies on high order polynomial interpolation. This is indeed the case for theDGTD-Pp methods discussed in subsection 2.4. There are basically two strategies that can beconsidered to cure this computational e�ciency problem. The �rst approach consists in using alocal time stepping strategy combined with an explicit time integration scheme. In the secondapproach, which is the one actually considered in the NACHOS project-team, the objective isto design a DGTD-Pp method with an enlarged stability domain.

Although the adoption of an unconditionally stable implicit time integration scheme allows toovercome the restrictive constraint on the time step for locally re�ned meshes, it is not clearwhether the resulting numerical methodology will demonstrate a superiority in terms of accu-racy and overall e�ciency over the original methodology based on an explicit time integrationscheme. On one hand, the numerical dispersion error of an implicit time integration scheme de-�nes a limit on the time step used in practice. On the other hand, the computational e�ciency isdirectly impacted by the fact that at each time step, an implicit time integration scheme leads tothe inversion of a large sparse linear system. For the linear Maxwell equations, the matrix of thissystem is constant as far as the time step is �xed during the simulation and non-dispersive prop-agation media are considered. The linear system solver can certainly exploit this fact but thiswill probably not translate into a drastic reduction of the cost of a single time step. Taking intoaccount all these issues, a locally implicit time integration scheme could be the best compromisewhen solving an unsteady wave propagation problem on a locally re�ned mesh. Such a hybridexplicit-implicit time integration strategy raises several challenges both from the mathematicalanalysis viewpoint (stability and accuracy, especially for what concern numerical dispersion)and from the computer implementation viewpoint (data structures, parallel computing aspects,especially load balancing issues).

Since the creation of the NACHOS project-team, we have concentrated our e�orts on thestudy of such implicit or hybrid explicit-implicit time integration strategies for dealing withgrid-induced sti�ness in DGTD-Pp methods in the context of the numerical solution of the timedomain Maxwell equations.

2.5.1 Personnel

• Permanent researchers: Victorita Dolean and Stéphane Lanteri

• PhD students: Adrien Catella

2.5.2 Project-team positioning

As mentioned above, the treatment of grid-induced sti�ness in unstructured mesh based solversfor time domain wave propagation problems can be addressed either through the design of a localtime stepping strategy in conjunction with an explicit time integration scheme, or by adoptingan implicit (or a hybrid explicit-implicit) time integration scheme. It is noteworthy that thegreat majority of recent or ongoing research works on this topic concentrate on the �rst of theseapproaches.

17

Local time stepping schemes have been studied or applied by several authors in the recentyears. Fumeaux et al. at ETH Zurich [Fumeaux et al., 2004] have designed such a strategyfor a dissipative FVTD method on non-uniform tetrahedral meshes. A multi-class local timestepping scheme has been proposed by Piperno in [Piperno, 2006b] in the context of a non-dissipative leap-frog based DGTD method. A two-class strategy is also considered by Cohenal. [Cohen et al., 2006] in the framework of the non-dissipative DGTD method on hexahedralmeshes developed at ONERA. This strategy makes use of interpolations to approximate unknown�elds when updating the cells values within a given class, however interpolations are recognizedto be too expensive in terms of computational time further motivating the development ofmulti-class local time stepping strategies. Such a scheme is then developed by essentially thesame authors in [Montseny et al., 2008] but this time in the context of a dissipative versionof the DGTD method considered in [Cohen et al., 2006]. The bene�ts of the resulting localtime stepping DGTD method are demonstrated for the numerical solution of the 3D Maxwell'sequations. A remark that applies to all the multi-class local time stepping strategies discussed sofar is that a stability criterion is di�cult to obtain and in some cases a reduction on the smallesttime step is necessary for long time stability. More recently, a promising local time steppingstrategy of arbitrary high order has been proposed by Diaz and Grote [Diaz and Grote, 2007] fora second order scalar wave equation discretized in space by either a continuous or a discontinuous�nite element method. The proposed strategy is based on the second order leap-frog schemewhich is extended to arbitrarily high order by a modi�ed equation approach. This local timestepping method is proved to conserve a discrete energy and explicit CFL conditions are exhibitedfor the second order and fourth order accurate in time methods. According to the authors,this strategy extends easily to the Maxwell equations in second order form and for variousdiscretization in space methods but this has not been demonstrated so far.

A few implicit variants of Yee's FDTD method have been developed among which, the alternat-ing direction implicit �nite di�erence time domain (ADI-FDTD) method [Namiki, 2000] which isa non-dissipative implicit FDTD method. The ADI-FDTD method o�ers unconditional stabilitywith modest computational overhead despite its implicit formulation, because it relies on a fac-torization of the implicit matrix operator leading to the inversion of tri-diagonal linear systems.Explicit-implicit methods for the solution of the system of Maxwell equations have been studiedby several authors with the shared goal of designing numerical methodologies able to deal withhybrid structured-unstructured meshes. For example, a stable hybrid FDTD-FETD method isconsidered by Rylander and Bondeson in [Rylander and Bondeson, 2002], while Degerfeldt andRylander [Degerfeldt and Rylander, 2006] propose a FETD method with stable hybrid explicit-implicit time-stepping working on brick-tetrahedral meshes that do not require an intermediatelayer of pyramidal elements. On the other hand, Kanevsky et al. [Kanevsky et al., 2007] studythe application of explicit-implicit Runge-Kutta (so-called IMEX-RK) methods in conjunctionwith high order discontinuous Galerkin discretizations on unstructured triangular meshes, inthe framework of unsteady compressible �ow problems (i.e the numerical solution of the sys-tems of Euler or Navier-Stokes equations). Originally developed to solve the sti� operator ofconvection-di�usion-reaction models, IMEX-RK methods are in this work used for separatingthe time integration of sti� and non-sti� portions of the computational domain with regards togrid-induced sti�ness.

2.5.3 Scienti�c achievements

In the context of the PhD thesis of Adrien Catella [Catella, 2008] (defended in December 2008),we have �rst studied the applicability of an implicit time integration scheme in conjunction withthe high order DGTD-Pp discretization method discussed in subsection 2.4 for the solution of

18

the time domain Maxwell equations. We have proposed to use the Crank-Nicholson scheme inplace of the explicit leap-frog scheme initially adopted in this method. As a result, we obtainan unconditionally stable, non-dissipative, implicit DGTD-Pp method, at the expense of theinversion of a global linear system. For two-dimensional problems, a sparse direct (LU factor-ization) method can be used for solving this linear system. Moreover, the factorization stepcan be performed once for all prior to entering the time stepping loop since the implicit matrixis constant in time, and the solution of the linear system at each time iteration amounts to aforward-backward solve. Despite the computational overhead of the solution of a linear system,the resulting implicit DGTD-Pp method allows for a noticeable reduction of the overall comput-ing time as compared to its explicit counterpart based on a leap-frog time integration scheme, asfar as the underlying triangular mesh is non-uniform (e.g locally re�ned) [Catella et al., 2008]-[Catella et al., 2009]. However, in the three-dimensional case, a globally implicit method basedon a sparse direct solver su�ers from large memory overheads and is therefore not a feasiblesolution strategy.

A more viable approach for 3D simulations consists in applying an implicit time integrationscheme locally i.e in the re�ned regions of the mesh, while preserving an explicit time scheme inthe complementary part, yielding an hybrid explicit-implicit (or locally implicit) time integrationscheme. Several strategies can be considered to achieve this goal and, as a preliminary feasibilitystudy in this direction, we have studied a hybrid explicit-implicit DGTD-Pp method initiallyintroduced by Piperno in [Piperno, 2006b]. In this method, the set of elements of the mesh isassumed to be partitioned into two subsets, one made of the smallest elements and the otherone gathering the remaining elements. The distinction between the two subsets can be doneaccording to a geometrical threshold, or/and a physical criterion as well. Then, the smallelements are handled using a Crank-Nicolson scheme while the other elements are time advancedusing a variant of the classical leap-frog scheme known as the Verlet method. This hybridexplicit-implicit DGTD-Pp method has been applied to the solution of the 2D and 3D timedomain Maxwell equations. For practical 3D simulations on locally re�ned tetrahedral meshesand as soon as a few percents (typically less than 1%) of the elements are treated implicitly), thesimulation time can be reduced by a factor between 5 to 10 [Catella et al., 2008]. This study asdemonstrated that a hybrid explicit-implicit DGTD-Pp can be a viable strategy for large-scaletime domain electromagnetic wave simulations using highly re�ned meshes.

2.5.4 Collaborations

The study of globally implicit and hybrid explicit-implicit DGTD-Pp methods has not been thesubject of an external collaboration during the evaluation period. However, this will be the casefor the short-term sequel of this study (this is discussed in section 6).

2.5.5 External support

• Research grant with CEA DAM, CESTA Center, Bordeaux and HOUPIC ANR project(high order DGTD-Pp/PIC solver for the Vlasov-Maxwell equations).

2.5.6 Self assessment

It is noteworthy that, in [Piperno, 2006b], the proposed hybrid explicit-implicit DGTD-Pp me-thod has not been assessed numerically neither studied theoretically apart from an energy con-servation analysis which however does not state the stability limit of the method. We haverecently completed this stability analysis of the method showing that a su�cient condition onthe global time step for having a stable hybrid explicit-implicit DGTD-Pp method is that the

19

explicit method based on the Verlet time scheme alone is stable 4. The next step is to studythe convergence of the method to have a complete picture of its theoretical properties, althoughwe cannot expect more that second order accuracy due to the building blocks of the method.Moreover, the computer implementation of this method relies on the de�nition of a separationcriterion for the de�nition of the sets of explicit and implicit elements. In practice, the choiceof the maximum value of the threshold cg (and thus, of the global time step) yielding an errorlevel not exceeding that of the fully explicit DGTD-Pp method is not a trivial task. It requiresa detailed analysis of the numerical dispersion error of the hybrid explicit-implicit DGTD-Pp

method and the de�nition of an error estimator allowing the development of an auto-adaptivenumerical methodology. These tasks have not been considered so far and are postponed for afuture study.

Besides, it is clear that other directions can be considered in order to design hybrid explicit-implicit time integration schemes for the treatment of grid-induced sti�ness, especially in viewof obtaining high order (i.e. more than second order) accurate methods. In particular, methodsand results coming from the ODE community should be very helpful in this respect.

2.6 Domain decomposition methods for wave propagation models

Domain Decomposition (DD) methods are �exible and powerful techniques for the parallel nu-merical solution of systems of PDEs. As clearly described in [Smith et al., 1996], they can beused as a process of distributing a computational domain among a set of interconnected proces-sors or, for the coupling of di�erent physical models applied in di�erent regions of a computa-tional domain (together with the numerical methods best adapted to each model) and, �nallyas a process of subdividing the solution of a large linear system associated to the discretizationof a system of PDEs into smaller problems whose solutions can be used to devise a parallelpreconditioner or a parallel solver. In all cases, DD methods (1) rely on a partitioning of thecomputational domain into subdomains, (2) solve in parallel the local problems using a direct oriterative solver and, (3) call for an iterative procedure to combine the local solutions to obtainthe solution of the global (original) problem. Subdomain solutions are connected by means ofsuitable transmission conditions at the arti�cial interfaces between the subdomains. The choiceof these transmission conditions greatly in�uences the convergence rate of the DD method. Onegenerally distinguish three kinds of DD methods:

• overlapping methods use a decomposition of the computational domain in overlappingpieces. The so-called Schwarz method belongs to this class. Schwarz initially introducedthis method for proving the existence of a solution to a Poisson problem. In the Schwarzmethod applied to the numerical resolution of elliptic PDEs, the transmission conditionsat arti�cial subdomain boundaries are simple Dirichlet conditions. Depending on the waythe solution procedure is performed, the iterative process is called a Schwarz multiplicativemethod (the subdomains are treated in sequence) or an additive method (the subdomainsare treated in parallel).

• non-overlapping methods are variants of the original Schwarz DD methods with no overlapbetween neighboring subdomains. In order to ensure convergence of the iterative processin this case, the transmission conditions are not trivial and are generally obtained througha detailed inspection of the mathematical properties of the underlying PDE or system ofPDEs.

4V. Dolean, A. Catella, H. Fahs and S. Lanteri, A hybrid explicit-implicit discontinuous Galerkin method for

time domain electromagnetics, in preparation.

20

• substructuring methods rely on a non-overlapping partition of the computational domain.They assume a separation of the problem unknowns in purely internal unknowns and in-terface ones. Then, the internal unknowns are eliminated thanks to a Schur complementtechnique yielding to the formulation of a problem of smaller size whose iterative reso-lution is generally easier. Nevertheless, each iteration of the interface solver requires therealization of a matrix/vector product with the Schur complement operator which in turnamounts to the concurrent solution of local subproblems.

Schwarz algorithms have enjoyed a second youth over the last decades, as parallel computersbecame more and more powerful and available. Fundamental convergence results for the classicalSchwarz methods were derived for many partial di�erential equations, and can now be found inseveral books [Smith et al., 1996]-[Quarteroni and Valli, 1999]-[Toselli and Widlund, 2004]. Ev-en if they have been introduced for the �rst time two centuries ago, over the last two decades,classical Schwarz methods have regained a lot of popularity with the development of the par-allel computers. First developed for the elliptic problems, they have been recently extendedto systems of hyperbolic partial di�erential equations, and it was observed that the classicalSchwarz method can be convergent even without overlap in certain cases. This is in strongcontrast to the behavior of classical Schwarz methods applied to elliptic problems, for whichoverlap is essential for convergence. Over the last decade, optimized versions of Schwarz meth-ods have been developed for elliptic partial di�erential equations. These methods use moree�ective transmission conditions between subdomains, and are also convergent without overlapfor elliptic problems. The extension of such methods to systems of equations and more preciselyto Maxwell's system (time harmonic and time discretized equations) has been done recentlyin [Dolean and Gander, 2007]-[Dolean et al., 2009b]. These new transmission conditions wereoriginally proposed for three di�erent reasons: �rst, to obtain Schwarz algorithms that are con-vergent without overlap; secondly, to obtain a convergent Schwarz method for the Helmholtzequation, where the classical Schwarz algorithm is not convergent, even with overlap; and third,to accelerate the convergence of classical Schwarz algorithms. Several studies towards the devel-opment of optimized Schwarz methods for the frequency domain Maxwell equations have beenconducted this last decade, most often in combination with conforming edge element approxi-mations. Optimized Schwarz algorithms can involve transmission conditions that are based onhigh order derivatives of the interface variables. However, the e�ectiveness of the new optimizedinterface conditions has been proved so far only for simple geometries and applications.

The research activities of the NACHOS project-team on this topic aim at the formulation,analysis and evaluation of classical and optimized Schwarz type domain decomposition methodsin conjunction with discontinuous Galerkin approximation methods on unstructured simplicialmeshes for the solution of time domain and frequency domain wave propagation problems. Ongo-ing works in this direction are concerned with the design of non-overlapping Schwarz algorithmsfor the solution of the frequency domain Maxwell equations.

2.6.1 Personnel

• Permanent researchers: Victorita Dolean and Stéphane Lanteri

• PhD students: Mohamed El Bouajaji and Hugo Fol

• Postdocs: Ronan Perrussel

• External collaborators: Martin Gander and Ronan Perrussel (since November 2007)

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2.6.2 Project-team positioning

Domain decomposition methods are currently studied by a lot of researchers worldwide. Thereis a well known international conference dedicated to these methods, as well as a special pur-pose Web site. In general, researchers working on domain decomposition methods do not limitthemselves to the study of a speci�c PDE or system of PDEs. In some cases, a particular ap-proach is devised, analyzed and applied to several mathematical models. A popular example isthe FETI (Finite Element Tearing and Interconnecting) method extensively studied by CharbelFarhat (Stanford University) and François-Xavier Roux (Laboratoire Jacques-Louis Lions, Uni-versité Pierre et Marie Curie, Paris) for structural mechanics problems [Farhat and Roux, 1994]and then adapted to other systems of PDEs among which Maxwell equations discretized byedge elements [Rapetti and Toselli, 2001]. Schwarz type domain decomposition methods basedon optimized interface conditions have been introduced more recently, with pioneering con-tributions from Frédéric Nataf (Laboratoire Jacques-Louis Lions, Université Pierre et MarieCurie, Paris) for advection/di�usion problems. Optimized Schwarz methods for wave propaga-tion models are currently investigated by several authors among which Ana Alonso Rodriguez(University of Trento), Martin J. Gander (University of Geneva), Luca Gerardo-Giorda (Uni-versity of Trento) Yvon Maday (Laboratoire Jacques-Louis Lions, Université Pierre et MarieCurie, Paris) and Frédéric Magoules (Laboratoire Mathématiques Appliquées aux Systèmes,Ecole Centrale de Paris). Finally, we note that for the system of Maxwell equations, mostof the existing results are concerned with non-overlapping domain decompositions methods inconjunction with conforming �nite element methods (see [Hut and Zou, 2004]-[Lee et al., 2005]-[Alonso-Rodriguez and Gerardo-Giorda, 2006] for recent examples).

2.6.3 Scienti�c achievements

In the context of the PhD thesis of Hugo Fol [Fol, 2006] (defended in December 2006) we have �rststudied classical overlapping and non-overlapping Schwarz algorithms for the solution of 2D and3D frequency domain Maxwell equations. These algorithms share the fact that a Després typecondition [Després et al., 1992] is imposed at the interfaces between neighboring subdomains.For the system of Maxwell equations, this interface condition is equivalent to a Dirichlet conditionfor characteristic variables associated to incoming waves. For this reason, this condition is oftenreferred as a natural interface condition [Dolean and Gander, 2008]. The convergence propertiesof overlapping and non-overlapping Schwarz algorithms based on this interface condition havebeen assessed theoretically for a two-subdomain decomposition. For that purpose, a Fourieranalysis is applied to the formulation of the Schwarz algorithms in the continuous case, andallows to obtain and analytical expression of the convergence rate of the algorithms.

At the discrete level and in the context of the DGFD-Pp methods discussed in subsection 2.4, thenatural interface condition is taken into account in the weak formulation through a boundaryintegral term associated to subdomain interfaces (arti�cial boundaries). This boundary integralterm is approximated using a �ux splitting scheme yielding a natural and intuitive treatment ofthe interface condition at the discrete level. Moreover, the Schwarz algorithm can be applied asa global solver or it can be reformulated as a Richardson iterative method acting on an interfacesystem. In the latter case, the resolution of the interface system can be accelerated using a Krylovmethod. In our implementation, a multi-frontal sparse direct solver [Amestoy et al., 2000] isused for solving the subdomain problems and the resulting domain decomposition solvers can beviewed as hybrid iterative-direct solution methods for the large, sparse and complex coe�cientsalgebraic system coming from the discretization of the frequency domain Maxwell equations by adiscontinuous Galerkin method. These algorithms have been implemented in the context of loworder discontinuous Galerkin methods (�nite volume method and discontinuous Galerkin method

22

based on linear interpolation) [Dolean et al., 2006b]-[Dolean et al., 2007]-[Dolean et al., 2007].

Preliminary investigations of optimized Schwarz algorithms combined to high order discontin-uous Galerkin frequency domain methods on triangular meshes have been conducted in thecontext of the solution of the 2D frequency domain Maxwell equations [Dolean et al., 2008b].Starting from the optimized interface conditions proposed for the continuous Maxwell equationsin [Dolean et al., 2009b], our �rst objective has been to study the formulation of discrete vari-ants of the associated optimized Schwarz algorithms in the context of the high order DGFD-Pp

methods discussed in subsection 2.4. This work continues in the context of the PhD thesis ofMohamed El Bouajaji (started in October 2008).

2.6.4 Collaborations

The team is actively collaborating with Martin Gander (Mathematics Department, Universityof Geneva) on the design of optimized interface conditions for the systems of time domain andfrequency domain Maxwell equations [Dolean et al., 2009b], and with Ronan Perrussel (Am-père Laboratory, Ecole Centrale de Lyon), who is a former postdoctoral fellow of the NA-CHOS project-team, on the development of optimized Schwarz algorithms in conjunction withdiscontinuous Galerkin discretization methods [Dolean et al., 2007]-[Dolean et al., 2008b].

Finally, since 2007, we collaborate with Luc Giraud (Parallel Algorithms and OptimizationGroup, LIMA-IRIT, ENSEEIHT, Toulouse) on the design of algebraic preconditioning tech-niques for the interface systems characterizing the Schwarz algorithms.

2.6.5 External support

• Research grant with France Télécom R&D/Orange Labs, La Turbie center (domain de-composition solvers for the frequency domain Maxwell equations discretized by low orderdiscontinuous Galerkin methods).

• MAXWELL (high order DGFD-Pp methods on simplicial meshes for frequency domainelectromagnetic wave propagation) ANR project.

2.6.6 Self assessment

One di�culty linked to the use of a sparse direct method for solving the subdomain problems isthat the �ll-in of the local L and U factors is generally not well balanced (except for relativelysimple problems (simply shaped domain, uniform mesh and homogeneous media). We believethat this drawback is recurrent to almost all similar implementations of domain decompositionalgorithms (i.e based on exact factorization methods for the subdomain solves). This problemcould be �gured out by resorting to constrained level of �ll-in subdomain solvers or/and byimproving the quality of the mesh partitions (with regards to the resulting �ll-in unbalance). Itis noteworthy that the partitioning of a mesh using a dedicated tool such as MeTiS, is dictatedby two main criteria, namely the minimization of the subdomains separator and the achievementof a well balanced computational load, while a balance of the �ll-in is rarely an objective.

Besides, it is clear that the overall performances of a hybrid iterative-direct domain decomposi-tion based solver such as those that we are studying are notably impacted by the e�ciency ofthe method used for solving the subdomain problems. The adoption of a sparse direct methodcertainly is one of the most e�cient options however it also induces a memory consumptionthat can be a serious obstacle to the solution of very large 3D problems. Alternative solutionstrategies have to be investigated that o�er a good compromise between numerical e�ciency and

23

memory overhead. This topic is actually considered in the framework of the PhyLeaS associatedteam (see also subsection 5) to which the NACHOS project-team participate.

The optimized interface conditions proposed in [Dolean et al., 2009b] have been designed for non-dissipative propagation media. Clearly, these conditions have to be extended if more complexpropagation media such as biological tissues have to be considered.

3 Bibliography of the project-team

Doctoral dissertations and Habilitation theses

[Benjemaa, 2007] Benjemaa, M. (December 2007). Etude et simulation numérique de la rupturedynamique des séismes par des méthodes d'éléments �nis discontinus. Doctoral thesis inmathematics, Nice-Sophia Antipolis University.

[Bouquet, 2007] Bouquet, A. (December 2007). Caractérisation de structures rayonnantes parune méthode Galerkin discontinue associée à une technique de domaines �ctifs. Doctoral thesisin mathematics, Nice-Sophia Antipolis University.

[Catella, 2008] Catella, A. (December 2008). Schémas d'intégration en temps e�caces pour larésolution numérique des équations de Maxwell instationnaires par des méthodes Galerkindiscontinues d'ordre élevé en maillages non-structurés. Doctoral thesis in mathematics, Nice-Sophia Antipolis University.

[Fahs, 2008] Fahs, H. (December 2008). Méthodes de type Galerkin discontinu d'ordre élevé pourla résolution numérique des équations de Maxwell instationnaires sur des maillages simplexesnon-conformes. Doctoral thesis in mathematics, Nice-Sophia Antipolis University.

[Fol, 2006] Fol, H. (December 2006). Méthodes de type Galerkin discontinu pour la résolutionnumérique des équations de Maxwell 3D en régime harmonique. Doctoral thesis in mathemat-ics, Nice-Sophia Antipolis University.

[Rapetti, 2008] Rapetti, F. (June 2008). Discrétisation variationnelle d'ordre élevé sur sim-plexes: applications à l'électromagnétisme numérique. Habilitation thesis, Nice-Sophia An-tipolis University.

Articles in referred journals and book chapters

[Ben Jemaa et al., 2007] Ben Jemaa, M., Glinsky-Olivier, N., Cruz-Atienza, V., Piperno, S., andVirieux, J. (2007). Dynamic non-planar crack rupture by a �nite volume method. Geophys.J. Int., 171:271�285.

[Ben Jemaa et al., 2009] Ben Jemaa, M., Glinsky-Olivier, N., Cruz-Atienza, V., and Virieux(2009). 3D dynamic rupture simulations by a �nite volume method. Geophys. J. Int. Toappear.

[Bernacki et al., 2006a] Bernacki, M., Fezoui, L., Lanteri, S., and Piperno, S. (2006a). Parallelunstructured mesh solvers for heterogeneous wave propagation problems. Appl. Math. Model.,30(8):744�763.

[Bernacki et al., 2006b] Bernacki, M., Lanteri, S., and Piperno, S. (2006b). Time-domain par-allel simulation of heterogeneous wave propagation on unstructured grids using explicit non-di�usive, discontinuous Galerkin methods. J. Comp. Acoustics, 14(1):57�81.

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[Bolze et al., 2006] Bolze, R., Cappello, F., Caron, E., Daydé, M., Desprez, F., Jeannot, E.,Jégou, Y., Lanteri, S., Leduc, J., Melab, N., Mornet, G., Namyst, R., Primet, P., Quetier, B.,Richard, O., Talbi, E., and Touche, I. (2006). Grid'5000: a large scale and highly recon�g-urable experimental Grid testbed. Int. J. High Perf. Comput. App., 20(4):481�494.

[Catella et al., 2008] Catella, A., Dolean, V., and Lanteri, S. (2008). An unconditionally sta-ble discontinuous Galerkin method for solving the 2D time-domain Maxwell equations onunstructured triangular meshes. IEEE Trans. Magn., 44(6):1250�1253.

[Catella et al., 2009] Catella, A., Dolean, V., and Lanteri, S. (2009). An implicit discontin-uous Galerkin time-domain method for two-dimensional electromagnetic wave propagation.COMPEL. to appear.

[Delcourte et al., 2009] Delcourte, S., Fezoui, L., and Glinsky-Olivier, N. (2009). A high-orderdiscontinuous Galerkin method for the seismic wave propagation. ESAIM: proceedings. toappear.

[Dolean et al., 2008a] Dolean, V., Fol, H., Lanteri, S., and Perrussel, R. (2008a). Solution ofthe time-harmonic Maxwell equations using discontinuous Galerkin methods. J. Comp. Appl.Math., 218(2):435�445.

[Dolean et al., 2009a] Dolean, V., Fol, H., Lanteri, S., and Perrussel, R. (2009a). Schwarz algo-rithms combined to discontinuous Galerkin formulations for time-harmonic Maxwell's equa-tions. In Magoules, F., editor, Schwarz algorithms and domain decomposition methods. Saxe-Coburg Publications. to appear.

[Dolean et al., 2009b] Dolean, V., Gander, M., and Gerardo-Giorda, L. (2009b). OptimizedSchwarz methods for Maxwell equations. SIAM J. Scient. Comp. to appear.

[Dolean et al., 2007] Dolean, V., Lanteri, S., and Perrussel, R. (2007). A domain decompositionmethod for solving the three-dimensional time-harmonic Maxwell equations discretized bydiscontinuous Galerkin methods. J. Comput. Phys., 227(3):2044�2072.

[Dolean et al., 2008b] Dolean, V., Lanteri, S., and Perrussel, R. (2008b). Optimized Schwarzalgorithms for solving time-harmonic Maxwell's equations discretized by a discontinuousGalerkin method. IEEE Trans. Magn., 44(6):954�957.

[Fahs, 2009] Fahs, H. (2009). Development of a hp-like discontinuous Galerkin time-domainmethod on non-conforming simplicial meshes for electromagnetic wave propagation. Int. J.Numer. Anal. Mod., 6(2):193�216.

[Fahs et al., 2008] Fahs, H., Fezoui, L., Lanteri, S., and Rapetti, F. (2008). Preliminary in-vestigation of a non-conforming discontinuous Galerkin method for solving the time-domainMaxwell equations. IEEE Trans. Magn., 44(6):1254�1257.

[Pasquetti and Rapetti, 2006] Pasquetti, R. and Rapetti, F. (2006). Spectral element methodson unstructured meshes: comparisons and recent advances. J. Sci. Comput., 27(1�3):377�387.

[Rapetti, 2007] Rapetti, F. (2007). High order edge elements on simplicial meshes. ESAIM:Math. Model. Num. Anal. (M2AN), 41(6):1001�1020.

[Scarella et al., 2006] Scarella, G., Clatz, O., Lanteri, S., Beaume, G., Oudot, S., Pons, J.-P.,Piperno, S., Joly, P., and Wiart, J. (2006). Realistic numerical modelling of human headtissue exposure to electromagnetic waves from cellular phones. Comptes Rendus Physique,7(5):501�508.

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Publications in Conferences and Workshops

[Ben Jemaa et al., 2006a] Ben Jemaa, M., Glinsky-Olivier, N., Cruz-Atienza, V., Virieux, J.,Piperno, S., and Lanteri, S. (2006a). 2D and 3D non-planar dynamic rupture by a �nitevolume method. In AGU (American Geophysical Union) Fall Meeting, San Francisco. Posterpresentation.

[Ben Jemaa et al., 2006b] Ben Jemaa, M., Glinsky-Olivier, N., Virieux, J., and Piperno, S.(2006b). Dynamic non-planar crack rupture by a �nite volume method. In 68th EAGE (Eu-ropean Association of Geoscientists and Engineers) Conference & Exhibition, Vienna, Austria.

[Ben Jemaa et al., 2006c] Ben Jemaa, M., Glinsky-Olivier, N., Virieux, J., and Piperno, S.(2006c). Dynamic non-planar crack rupture by a �nite volume method. In EGU (EuropeanGeosciences Union) General Assembly 2006, Vienna, Austria.

[Benjemaa et al., 2007] Benjemaa, M., Glinsky-Olivier, N., Cruz-Atienza, V., and Virieux, J.(2007). 3D dynamic crack rupture by a �nite volume method. In AGU (American GeophysicalUnion) Fall Meeting, San Francisco. Poster presentation.

[Catella et al., 2008] Catella, A., Dolean, V., Fezoui, L., and Lanteri, S. (2008). E�cient timeintegration strategies for high-order discontinuous Galerkin time-domain methods. In 24thInternational Review of Progress in Applied Computational Electromagnetics (ACES 2008),Niagara Falls, Canada.

[Catella et al., 2007] Catella, A., Dolean, V., and Lanteri, S. (2007). Investigation of implicittime stepping for grid-induced sti�ness in discontinuous Galerkin time-domain methods onunstructured triangular meshes. In 8th International Conference on Mathematical and Nu-merical Aspects of Waves (Waves 2007), pages 257�259, University of Reading, UK.

[Clatz et al., 2006a] Clatz, O., Lanteri, S., Oudot, S., and Piperno, S. (2006a). Unstructuredmesh solvers for the simulation of electromagnetic wave propagation and induced temper-ature elevation in living tissues. In 7th International Symposium on Computer Methods inBiomechanics and Biomedical Engineering, Antibes, France.

[Clatz et al., 2006b] Clatz, O., Lanteri, S., Oudot, S., Pons, J.-P., Piperno, S., Scarella, G., andWiart, J. (2006b). Modélisation numérique réaliste de l'exposition des tissus de la tête àun champ électromagnétique issu d'un téléphone mobile. In 13ème Colloque International etExposition sur la Compatibilité Electromagnétique, pages 377�397, Saint Malo, France.

[Dolean et al., 2009] Dolean, V., El Bouajaji, M., Lanteri, S., and Perrussel, R. (2009). Domaindecomposition solvers for discontinuous Galerkin discretization of the time-harmonic Maxwellequations. In SIAM Conference on Computational Science & Engineering (CSE09), Miami,Florida.

[Dolean et al., 2006a] Dolean, V., Fezoui, L., Lanteri, S., and Rapetti, F. (2006a). Discontinu-ous galerkin methods on unstructured meshes for the numerical resolution of the time-domainMaxwell equations. In 5ème Conférence Européenne sur les Méthodes Numériques en Elec-tromagnétisme (NUMELEC 2006), Lille, France.

[Dolean et al., 2006b] Dolean, V., Fol, H., Lanteri, S., and Perrussel, R. (2006b). Resolutionof the time-harmonic Maxwell equations using discontinuous galerkin methods and domaindecomposition algorithms. In 20th International Congress on Computational and AppliedMathematics (ICCAM06), Leuven, Belgium. http://hal.archives-ouvertes.fr/hal-00106201.

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[Dolean et al., 2007] Dolean, V., Fol, H., Lanteri, S., and Perrussel, R. (2007). Hybrid iter-ative/direct strategies for solving the three-dimensional time-harmonic Maxwell equationsdiscretized by discontinuous Galerkin methods. In 2007 International Conference On Precon-ditioning Techniques For Large Sparse Matrix Problems In Scienti�c And Industrial Applica-tions, pages 69�71, Météopole, Toulouse, France.

[Dolean and Gander, 2007] Dolean, V. and Gander, M. (2007). The relation between optimizedSchwarz methods for scalar and systems of partial di�erential equations. In 6th InternationalCongress on Industrial and Applied Mathematics (ICIAM07), ETH Zurich, Switzerland.

[Dolean and Gander, 2008] Dolean, V. and Gander, M. (2008). Why classical Schwarz methodsapplied to hyperbolic systems can converge even without overlap. In 17th International Con-ference on Domain Decomposition Methods in Science and Engineering, volume 60 of LectureNotes in Computational Science and Engineering (LNCSE), pages 467�475, St. Wolfgang-Strobl, Austria. Springer Verlag.

[Fahs, 2007] Fahs, H. (2007). Une méthode Galerkin discontinu pour la résolution numérique deséquations de Maxwell 2D en domaine temporel sur des maillages triangulaires localement raf-�nés non-conformes. In 3ème Congrès National de Mathématiques Appliquées et Industrielles(SMAI 2007), VVF l'Alisier, Praz sur Arly.

[Fahs, 2008] Fahs, H. (2008). A non-conforming discontinuous Galerkin method for solvingMaxwell's equations. In 6ème Conférence Européenne sur les Méthodes Numériques en Elec-tromagnétisme (NUMELEC 2008), Liège, Belgium.

[Fahs and Lanteri, 2008] Fahs, H. and Lanteri, S. (2008). A high-order non-conforming discon-tinuous Galerkin method for time-domain electromagnetics. In 13th International Congresson Computational and Applied Mathematics (ICCAM08), Ghent, Belgium.

[Fahs et al., 2008] Fahs, H., Lanteri, S., and Rapetti, F. (2008). Development of a non-conforming discontinuous Gakerkin method on simplex meshes for electromagnetic wave prop-agation. In 4th International Conference on Advanced COmputational Methods in ENgineering(ACOMEN'08), Liège, Belgium.

[Glinsky-Olivier et al., 2006a] Glinsky-Olivier, N., Ben Jemaa, M., Virieux, J., and Piperno, S.(2006a). 2D seismic wave propagation by a �nite volume method. In 68th EAGE (EuropeanAssociation of Geoscientists and Engineers) Conference & Exhibition, Vienna, Austria.

[Glinsky-Olivier et al., 2006b] Glinsky-Olivier, N., Ben Jemaa, M., Virieux, J., and Piperno, S.(2006b). A �nite volume method for the 2d seismic wave propagation. In EGU (EuropeanGeosciences Union) General Assembly 2006, Vienna, Austria.

Internal reports

[Ben Jemaa et al., 2006] Ben Jemaa, M., Glinsky-Olivier, N., and Piperno, S. (2006). Etude destabilité d'un schéma volumes �nis pour les équations de l'élasto-dynamique en maillages nonstructurés. Technical Report RR-5817, INRIA.

[Catella et al., 2007] Catella, A., Dolean, V., and Lanteri, S. (2007). An implicit DGTD methodfor solving the two-dimensional Maxwell equations on unstructured triangular meshes. Tech-nical Report RR-6110, INRIA.

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[Dolean et al., 2006] Dolean, V., Fol, H., Lanteri, S., and Piperno, S. (2006). Méthodes detype Galerkin discontinu pour la résolution numérique des équations de Maxwell en régimefréquentiel. Technical Report RR-5904, INRIA.

[Dolean et al., 2009] Dolean, V., Lanteri, S., and Perrussel, R. (2009). Méthodes de typeGalerkin discontinu pour les équations de Maxwell en régime harmonique: �ux numériques etalgorithmes multigrille. Technical Report RR-6805, INRIA.

[Fahs, 2007] Fahs, H. (2007). Numerical evaluation of a non-conforming discontinuous Galerkinmethod on triangular meshes for solving the time-domain Maxwell equations. Technical Re-port RR-6311, INRIA.

[Fahs et al., 2007] Fahs, H., Fezoui, L., Lanteri, S., and Rapetti, F. (2007). A hp-like discontinu-ous Galerkin method for solving the 2D time-domain Maxwell's equations on non-conforminglocally re�ned triangular meshes. Technical Report RR-6162, INRIA.

[Fahs and Lanteri, 2008] Fahs, H. and Lanteri, S. (2008). Convergence and stability of a high-order leap-frog based discontinuous Galerkin method for the Maxwell equations on non-conforming meshes. Technical Report RR-6699, INRIA.

[Fahs et al., 2006] Fahs, H., Lanteri, S., and Rapetti, F. (2006). Etude de stabilité d'une méth-ode galerkin discontinu pour la résolution numérique des équations de maxwell 2d en domainetemporel sur des maillages triangulaires non-conformes. Technical Report RR-6023, INRIA.

[Konrad, 2008] Konrad, C. (2008). On domain decomposition with space �lling curves for theparallel solution of the coupled Maxwell/Vlasov equations. Technical Report RR-6693, INRIA.

4 Knowledge dissemination

4.1 Publications

We recall at this point that the NACHOS project-team has started its research activities in July2006 and has been o�cially created in July 2007. For this reason, we have decided to includeseparate entries for piblications accepted in 2008 and that will appear in 2009.

2006 2007 2008 2009

PhD thesis 1 2 2

H.D.R (*) 1

Journal 5 3 4 5

Conference proceedings (**) 9 5 5 1

Book chapter 1

Book (written)

Book (edited)

Patent

Technical report 3 3 2 1

Deliverable(*) HDR Habilitation à diriger des Recherches(**) Conference with a program committee

Indicate the major journals in the �eld and, for each, indicate the number of papers coau-thored by members of the project-team that have been accepted during the evaluation period.

28

1. Concerning applied mathematics and scienti�c computing

• Journal of Computational Physics, 1 paper

• SIAM Journal Scienti�c Computing, 1 paper

• Journal of Computational and Applied Mathematics, 1 paper

• International Journal of Numerical Analysis and Modeling, 1 paper

• ESAIM: Mathematical Modelling and Numerical Analysis (M2AN), 1 paper

• Journal of Scienti�c Computing, 1 paper

2. Concerning computational electromagnetics

• The International Journal for Computation and Mathematics in Electrical and Elec-tronic Engineering (COMPEL), 1 paper

• IEEE Transactions on Magnetics, 3 papers

3. Concerning computational geoseismics

• Geophysical Journal International, 2 papers

Indicate the major conferences in the �eld and, for each, indicate the number of paperscoauthored by members of the project-team that have been accepted during the evaluationperiod.

1. In relation to applied mathematics and scienti�c computing

• International Conference on Mathematical and Numerical Aspects ofWaves (WAVES), 1 paper

• International Congress on Computational and AppliedMathematics (ICCAM), 2 papers

• International Conference on Domain Decomposition Methods in Science and Engi-neering (DDM), 1 paper

• SIAM Conference on Computational Science & Engineering (CSE), 1 paper

2. In relation to computational electromagnetics

• International Review of Progress in Applied ComputationalElectromagnetics (ACES), 1 paper

• International Conference on the Computation ofElectromagnetic Fields (COMPUMAG), 3 papers

3. In relation to computational geoseismics

• European Geosciences Union General Assembly, 2 papers

• European Association of Geoscientists and Engineers Conference & Exhibition, 2papers

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4.2 Software

MAXW2D-DGTD and MAXW3D-DGTD: these software implement the high order DGTD-Pp methods on simplicial meshes discussed in subsection 2.4 for the solution of the 2Dand 3D time domain Maxwell equations. Existing competitors are Sledge++ co-developedby Jan Hesthaven (Division of Applied Mathematics at Brown University) Tim Warbur-ton (Department of Computational and Applied Mathematics at Rice University) andcolleagues, hpGEM developed in the Numerical Analysis and Computational MechanicsGroup at Twente University by Jaap van der Vegt and colleagues and the DGTD softwaredeveloped at ONERA by Xavier Ferrieres [Cohen et al., 2006]-[Montseny et al., 2008] incollaboration with Garry Cohen (POEMS project-team at INRIA Paris-Rocquencourt),Sébastien Pernet (Electro-Magnetism - EMC group at CERFACS) and other colleagues.

MAXW2D-DGFD and MAXW3D-DGFD: these software implement the high order DGFD-Pp

methods on simplicial meshes discussed in subsection 2.4 for the solution of the 2D and3D time domain Maxwell equations. These software also implement the hybrid iterative-direct domain decomposition based solvers discussed in subsection 2.6. From the pointof view of discontinuous Galerkin treatment of the frequency domain Maxwell equations,a known competitor is Sledge++ co-developed by Jan Hesthaven (Division of AppliedMathematics at Brown University) Tim Warburton (Department of Computational andApplied Mathematics at Rice University) and colleagues but this software does not seemto rely on a domain decomposition method of any kind for the solution of the associatedsparse linear systems.

SISMO2D-DGTD and SISMO3D-DGTD: these software implement the high order DGTD-Pp

methods on simplicial meshes discussed in subsection 2.4 for the solution of the 2D and3D time domain elastodynamic equations. The SEISSOL software developed by MichaelDumbser (Institut für Aerodynamik und Gasdynamik at University of Stuttgart) and Mar-tin Kaeser (Department of Earth and Environmental Sciences Geophysics at Universityof Munich) seems to be the only competitor. This software is based on the ADER-DG method described in a series of papers of the authors [Käser and Dumbser, 2006]-[Dumbser and Käser, 2006]-[Käser et al., 2007]- [Dumbser et al., 2007]-[Käser et al., 2008].

So far (i.e since the creation of the NACHOS project-team), the above software have not beendistributed under any form of license.

4.3 Valorization and technology transfer

In the context of our collaboration with CEA DAM, CESTA research center in Bordeaux (seesubsection 5), the high order DGTD-Pp method developed in the team for the solution of thetime domain Maxwell equations is adapted and integrated by our partner in the in-house sim-ulation platform EMMA which implements a coupled Vlasov-Maxwell solver on unstructuredtetrahedral meshes. For hardening studies of electromagnetic devices, the CESTA routinely ex-ploits a FDTD-PIC solver which is the result of several years of development. However, with theincrease of complexity of the studied electromagnetic devices, this cartesian mesh based solver isshowing its modeling limitations especially with regards to the treatment of the underlying geo-metrical features, thus motivating the development of an unstructured mesh based solver. Ourcollaboration with the CESTA materializes by the fact that the numerical ingredients relatedto the solution of the time domain Maxwell equations in the EMMA software are exactly thosethat have been considered in the MAXW3D-DGTD software developed in the team. Besides,

30

the coupling with a PIC method also raises new research issues both on the numerical analysisside (assessment of the charge conservation property and development of correction techniques,design of assignment schemes for particles induced current and charge densities to the physicalmesh which are compatible with the DG method features) and on high performance computingaspects (load balancing issues).

As mentioned in subsection 2.2.1, the FDTD method is still the prevailing method in numericaldosimetry studies of human exposure to electromagnetic �elds. It is clear that the simplicity ofthe FDTD method as well as its ability to easily model the radiation sources, together with thecomputational e�ciency of the method, are the main reasons of this success. It is also noteworthythat most if not all of these studies are conducted by researchers and engineers coming fromthe electrical engineering domain, the biomedical domain or the telecommunication industry.For these groups, the FDTD method is in general seen as the on the shelf method that can bestraightforwardly adapted to the particular modeling situations underlying numerical dosimetrystudies. Moreover, for what concern exposure limits, the established international norms arede�ned in terms of certain average values of the SAR (Speci�c Absorption Rate) which is abasic quantity of interest in microwave numerical dosimetry studies. Thus the stair-casing e�ectresulting from the use of cartesian meshes in the FDTD method is somewhat reduced and indeed,localized features of the electromagnetic �eld are rarely assessed. NACHOS is one of the veryfew teams developing unstructured mesh based numerical dosimetry methodologies of humanexposure to electromagnetic �elds. Our objective here is not to develop a methodology that willsupersede the FDTD method but rather to propose a complementary method that will be usedfor two main purposes:

• �rst, to obtain high resolution distributions of the local SAR induced by mobile phoneradiation using the high order DGTD-Pp method developed in the team applied on locallyre�ned tetrahedral meshes of head tissues;

• Second, to perform parametric studies at low cost, using coarse albeit realistic geometricalmodels of human tissues (the head or the full body as well) in view of uncertainty analysiswith respect to the values of the electromagnetic parameters (electrical permittivity andconductivity) or the morphology of the tissues (in particular for taking into account theage incidence on the morphology).

Both of these topics are at the heart of our collaboration with the WAVE (Interaction of electro-magnetic wave with humans) group at Orange Labs, Issy-les-Moulineaux center. In this context,we ultimately plan to transfer to this group a dedicated version of the MAXW3D-DGTD soft-ware developed in the NACHOS project-team. Besides, or recent involvement in the COST BM0704 action (see subsection 4.5) should also give us the opportunity to introduce our numericalmodeling methodology to a wider scienti�c community committed to the study of biologicale�ects of electromagnetic �elds.

4.4 Teaching

Victorita Dolean, Université de Nice Sophia-Antipolis (Faculté de Sciences and Ecole Polytech-nique Universitaire) (192h/year), 2006-2009 period

• Finite elements (lecture and exercice classes), Master degree.

• Numerical analysis using C language (exercice classes), Master degree.

• Numerical analysis (lecture and exercice classes), 1st year of engineering school.

31

• Numerical methods for PDEs (lecture classes), Master degree.

• Dynamical systems (exercice classes), 3rd year.

Francesca Rapetti Université de Nice Sophia-Antipolis (Faculté de Sciences and Ecole Poly-technique Universitaire) (192h/year)

• 2006-2007

� Numerical analysis (exercice classes), Master degree, 20 h

• 2007-2008

� Numerical analysis (exercice classes), Master degree, 40 h

� Numerical analysis (lecture classes), Master degree, 30 h

• 2007-2008

� Numerical analysis (exercice classes), Master degree, 63 h

� Numerical analysis (lecture classes), Master degree, 36 h

� Scienti�c computing, �nite elements (lecture classes), Master degree, 45 h

� Algebra (exercice classes), License degree, 36 h

4.5 Visibility

Co-organization (Stéphane Lanteri) of the CEA-EDF-INRIA school �High performance scien-ti�c computing: algorithms, software tools and applications�, November 6-9 2006, INRIARocquencourt, France.

Co-organization (Stéphane Lanteri, Victorita Dolean) of the CEA-EDF-INRIA school �Ro-bust methods and algorithms for solving large algebraic systems on modern high perfor-mance computing systems�, March 30-April 3 2009, INRIA Sophia Antipolis-Méditerranée,France.

Organization (Stéphane Lanteri) of a minisymposium �High order methods for the solution ofwave propagation PDE models with applications to electromagnetics and geoseismics�atthe 9th International Conference on Mathematical and Numerical Aspects of Waves, Waves2009, June 15-19 2009, Pau, France.

Co-organization (Stéphane Lanteri) of minisymposium �Toward robust hybrid parallel sparsesolvers for large scale applications�at the SIAM Conference on Computational Science andEngineering CSE09, March 2-6 2009, Miami, Florida, USA.

Co-coordination (Victorita Dolean) of the MathMods - Erasmus Mundus MSc Course (par-ticipating institutions: Università degli Studi dell'Aquila - Italy, Universitat Autònomade Barcelona - Spain, Gdansk University of Technology - Poland, Universität Hamburg -Germany and University of de Nice - Sophia Antipolis - France).

Participation to the working group on �Electromagnetic �eld computational dosimetry� of theCOST BM 0704 action. The main objective of this action is to create a structure in whichresearchers in the �eld of electromagnetic �elds (EMF) and health can share knowledgeand information on: (1) how existing EMF technologies change either in their operatingcharacteristics or in novel ways and applications in which they are used, (2) identifyingwhat entirely new EMF technologies are introduced and on what timescale, (3) what novelemission and operating characteristics might result and what impact these would have onthe device-speci�c and overall EMF exposure of people, (4) what possible health e�ects

32

could consequently arise and the scienti�c evidence for health concerns if any, (5) how suchconcerns should be addressed through the use of evidence-based information and (6) whattools are e�ective in communicating and managing such risks and perceived risks.

Participation to the Euroseistest Numerical Benchmark initiative. The objective of this bench-mark, organized in the framework of the Cashima project (CEA Cadarache, the LGITin Grenoble and Aristotle University of Thessaloniki), is to perform simulations of realevents on the Volvi area (a well documented region near Thessaloniki) including complexcharacteristics of the medium.

5 External Funding

We recall at this point that the NACHOS project-team has started its research activities in July2006 and has been o�cially created in July 2007. For this reason, we have decided to includebudget information for 2009.

(k euros) 2006 2007 2008 2009

INRIA Research Initiatives

ARC†LSIA‡National initiatives

ANR DiscoGrid 5 56 15

ANR HOUPIC 19 88

ANR MAXWELL 2 2

European projects

Associated teams

PhyLeaS 2 3

Industrial contracts

CEA 17 17 17

FTR&D #1 23 37

FTR&D #2 28 42

CNRS Géosciences 5 5 10

Scholarships

PhD (Mondher Benjeema - INRIA grant) 17 14

PhD (Hassan Fahs - MESR grant) 31 31 22

PhD (Charles Joseph - INRIA CORDIS ) 8 34

Post Doc (Ronan Perrussel - INRIA grant) 31

Post Doc (Sarah Delcourte - INRIA grant) 7 31

AI+

ODL#

Other funding

INRIA 23 40 15 6

Total 147 156 210 190

† INRIA Cooperative Research Initiatives‡ Large-scale Initiative Actions

33

∗ other than those supported by one of the above projects+ junior engineer supported by INRIA# engineer supported by INRIA

ARCs

NA

National initiatives

QSHA: �Quantitative Seismic Hazard Assessment�(from January 2006 to September 2009).This project if funded by the ANR in the framework of the program Catastrophes Tel-luriques et Tsunami. The participants are: CNRS/Géosciences Azur, BRGM (Bureau deRecherches Géologiques et Minières, Service Aménagement et Risques Naturels, Orléans),CNRS/LGIT (Laboratoire de Géophysique Interne et Technophysique, Observatoire deGrenoble), CEA/DAM (Bruyères le Chatel), LCPC, INRIA Sophia Antipolis (NACHOSteam), ENPC (CERMICS), CEREGE (Centre europeen de Recherche et d'Enseignementdes Géosciences de l'Environnement, Aix en Provence), IRSN (Institut de Radioprotectionet de Surete Nucléaire), CETE Méditerranée (Nice), LAM (Laboratoire de Mécanique,Université de Marne la Vallée), LMS (Laboratoire de Mécanique des Solides, Ecole Poly-technique). The activities undertaken in the QSHA project aim at (1) obtaining a moreaccurate description of crustal structures for extracting rheological parameters for wavepropagation simulations, (2) improving the identi�cation of earthquake sources and thequanti�cation of their possible size, (3) improving the numerical simulation techniques forthe modeling of waves emitted by earthquakes, (4) improving empirical and semi-empiricaltechniques based on observed data and, (5) deriving a quantitative estimation of groundmotion. From the numerical modeling viewpoint, essentially all of the existing familiesof methods (boundary element method, �nite di�erence method, �nite volume method,spectral element method and discrete element method) are extended for the purpose ofthe QSHA objectives.

DiscoGrid: �Distributed objects and components for high performance scienti�c comput-ing�(from January 2006 to June 2009). The project-team is coordinating (StéphaneLanteri) the DiscoGrid project which is funded by ANR in the framework of Calcul Intensifet Grilles de Calcul program. The DiscoGrid project aims at studying and promoting anew paradigm for programming non-embarrassingly parallel scienti�c computing applica-tions on distributed, heterogeneous, computing platforms. The target applications requirethe numerical solution of systems of PDEs modeling electromagnetic wave propagationand �uid �ow problems. The partners of this project are: INRIA Sophia Antipolis -Méditerranée (NACHOS , OASIS and SMASH project-teams), INRIA Rennes - BretagneAtlantique (PARIS project-team), the ID-IMAG laboratory Grenoble (MOAIS team), theLABRi laboratory in Bordeaux (Distributed Sytems and Objects team) and EADS Inno-vation Works in Toulouse.

HOUPIC: �High order �nite element particle-in-cell solvers on unstructured grids�(from Jan-uary 2006 to December 2009). The project-team is a partner of the HOUPIC projectwhich is funded by ANR in the framework of Calcul Intensif et Simulations program. Thisproject is coordinated by Eric Sonnendrücker for the IRMA (Institut de Recherche Math-ématique Avancée) Laboratory in Strasbourg, and the other partners are the LSIIT (Lab-oratoire des Sciences de l'Image, de l'Informatique et de la Télédétection) in Strasbourg,the CEA/CESTA in Bordeaux, the PSI (Paul Scherrer Institut) in Villigen (Switzerland)

34

and the IAG (Institut für Aerodynamik und Gasdynamik) in Stuttgart (Germany). Themain objective of this project is to develop and compare Finite Element Time Domain(FETD) solvers based on high order Hcurl conforming elements and high order Discontin-uous Galerkin (DGTD) �nite elements and investigate their coupling to a PIC method.

MAXWELL: �Novel, ultra-wideband, bistatic, multipolarization, wide o�set, microwave dataacquisition, microwave imaging, and inversion for permittivity�(from January 2008 to De-cember 2010). The project-team is a partner of the MAXWELL project which is fundedby ANR under the non-thematic program. This project is coordinated by Christian Pichotfrom the LEAT (Laboratoire d'Electronique Antennes et Télécommunications) in SophiaAntipolis and the other partners are the Géosciences Azur Laboratory in Sophia Antipolisand the MIGP (Laboratoire de Modélisation et Imagerie en Géosciences de Pau) Labo-ratory. This project aims at the development of a complete microwave imaging system,with a frequency bandwidth of 1.87 GHz, ranging from 130 MHz to 2 GHz, using un-structured mesh solvers of the time harmonic Maxwell equations which drive a generalizedleast-squares inversion engine, whose output is a subsurface map of the relative permit-tivity. Subsidiary goals of the project are: (a) the construction and calibration of twoultra-wideband antennas, (b) the construction of two types of carriages for performingdata acquisition, (c) the acquisition of dense microwave data with very wide o�set for theentire bandwidth from 130 MHz to 2 GHz and for 2 orthogonal co-polarizations and onecross-polarization, (d) the reprocessing of data, including gain and kinematic inversion us-ing conventional seismic processing formulations and (e) the development of discontinuousGalerkin solvers on simplicial meshes for the numerical solution of the frequency domainMaxwell equations and their integration into an inversion system.

European projects

NA

Associated teams and other international projects

PhyLeaS: since January 2008, the NACHOS project-team is a partner of the PhyLeaS INRIAassociate team (Design and high performance implementation of parallel hybrid sparselinear solvers) which is coordinated by Jean Roman (SCALAPPLIX project-team, INRIABordeaux - Sud-Ouest Research Center) and associates the following partners: YousefSaad (Department of Computer Science and Engineering, University of Minnesota, USA),Matthias Bollhoefer (Institute of Computational Mathematics Department of Mathemat-ics and Computer Science, TU Brunswick, Germany), Luc Giraud (Parallel Algorithmsand Optimization Group, LIMA-IRIT UMR CNRS 5505, ENSEEIHT, Toulouse). Theresearch activities undertaken in the framework of the PhyLeaS associate team aim at thedesign and e�cient implementation of parallel hybrid linear system solvers which combinethe robustness of direct methods with the implementation �exibility of iterative schemes.These approaches are candidate to get scalable solvers on massively parallel computers.

Industrial contracts

France Télécom R&D/Orange Labs, La Turbie center, �DGFD methods for the frequencydomain Maxwell equations�(from July 2006 to November 29007). France Télécom R&Dis developing its own software (SR3D) for the solution of the frequency domain Maxwellequations using a Boundary Element Method (BEM) and is interested in coupling SR3Dwith a �nite element software able to deal with multi-material media for antenna design.

35

This research grant was a �rst step in this direction being concerned with the developmentof a DGFD-Pp method on unstructured tetrahedral meshes for the solution of the frequencydomain Maxwell equations. The PhD thesis of Hugo Fol has been partially funded by thisgrant.

CEA DAM, CESTA research center in Bordeaux, �High order DGTD-PIC solver for the Vlasov-Maxwell equations�(from October 2006 to September 2008 - a follow-up is under negotia-tion). The subject of this research grant is the development of a coupled Vlasov-Maxwellsolver combining the high order DGTD-Pp method on tetrahedral meshes developed inthe team and a Particle-In-Cell method. The resulting DGTD-PIC solver will be used forelectrical vulnerability studies of the experimental chamber of the Laser Mégajoule system.The PhD thesis of Adrien Catella has been partially funded by this grant.

France Télécom R&D/Orange Labs, Issy-les-Moulineaux center, �DGTD methods onnon-conforming simplicial meshes�(from October 2006 to September 2009). This researchgrant which aims at the development of high order DGTD-Pp methods on non-conformingsimplicial meshes for the numerical modeling of the interaction of electromagnetic waveswith biological tissues. The PhD thesis of Hassan Fahs has been partially funded by thisgrant.

Other funding,

NA

6 Objectives for the next four years

6.1 Scienti�c objectives

6.1.1 High order DG methods

For each of the considered systems of PDEs (Maxwell equations and elastodynamic equations)and propagation regimes (time domain and frequency domain problems), an important objectiveof our future activities will be to extend our current achievements on arbitrary high order DGmethods towards more adaptivity (i.e targeting hp-adaptive DG methods). This will requireconsidering several complementary aspects:

• formulation, analysis and computer implementation of non-conforming DG methods (com-bining local h-re�nement and p-enrichement) on simplicial meshes. The work undertakenin the context of the PhD thesis of Hassan Fahs for the 2D time domain Maxwell equa-tions was a �rst step in this direction. More recently, in the context of the PhD thesis ofMohamed El Bouajaji (started in October 2008), a similar study has been initiated for the2D and 3D frequency domain Maxwell equations.

• investigation of various polynomial interpolation methods. As mentioned in section 2.4, inthe high order DG methods developed by the team to date, the local approximation of thephysical �eld within a simplex relies on a nodal (Lagrange type) polynomial interpolationmethod. However, it is clear that other polynomial interpolation methods could be adoptedas well. The choice of an interpolation method and associated set of basis functionsshould ideally take into account several criteria among which, the modal or nodal natureof the functions, the orthogonality of the basis functions, the hierarchical structure of thebasis functions, the conditioning of the elemental matrices to be inverted (e.g the mass

36

matrix in explicit DGTD-Pp methods), the spectral properties of the interpolation and theprogramming simplicity. The development of expansion bases for high order interpolationon simplicial elements has been studied quite extensively in the last decade as witnessedby several books totally or partially dedicated to the subject (see [Solin et al., 2004] and[Karniadakis and Sherwin, 2005] among others). In this context, a reasonable preliminaryobjective would be to assess the properties of some of the existing candidate solutions inthe context of our DG formulations. It seems that a few works have been conducted so farwith the objective to study or design expansion bases in conjunction with DG methods forhyperbolic PDE models [Deng and Cai, 2005]-[Piperno, 2006a]-[Luoa et al., 2008]). Sucha study is at the heart of the PhD thesis of Joseph Charles (started in October 2008).

• investigation of a posteriori error estimators for hp-adaptive DG methods. This is a veryimportant topic which is currently the subject of numerous studies in various applicationcontexts. For the simulation of wave phenomena, most of the existing results have beenobtained for quasi-static or time harmonic PDE models (see [Harutyunyan et al., 2008a]-[Harutyunyan et al., 2008b] for the frequency domain Maxwell equations discretized by aedge element method and [Perugia and Schotzau, 2003] for the low frequency frequencydomain Maxwell equations discretized by a DG method among others). It is clear that thistopic has not been considered so far by any of the members of the NACHOS project-teamand for that reason, we do not intend to propose new foundation results on a posteriorierror estimation for the systems of PDEs and discretization methods that we consider. Ourobjective will rather be to build upon new collaborations with researchers that are activelyinvolved in the subject in order to extend and adapt existing results to our mathematicaland numerical modeling contexts for time domain and frequency domain wave propagationproblems. In particular, the investigation of a posteriori error estimators for hp-adaptiveDG methods for the solution of the elastodynamic equations will be conducted in closecollaboration with Sarah Delcourte (Claude Bernard University - Lyon 1).

6.1.2 Time integration schemes

We will devote some e�orts to time integration strategies for the high order DGTD-Pp methodsthat we develop for the solution of the time domain Maxwell and elastodynamic equations. Theobjective will before all be to improve the convergence properties of the explicit variants of thesemethods, but also to devise accurate and e�cient solution strategies for situations where thestability limit of a globally explicit method becomes a concern. On one hand, we intend tostudy various arbitrary high order explicit time stepping schemes, both in terms of theoreticalproperties (stability and a priori convergence in the fully discrete case) and computational issues.For the time domain Maxwell equations, this topic will be considered in the context of the PhDthesis of Joseph Charles (started in October 2008). On the other hand, building on the promisingresults obtained in the context of the PhD thesis of Adrien Catella (defended in December 2008),we will study further hybrid explicit-implicit time integration strategies for overcoming grid-induced sti�ness for simulations involving non-uniform meshes. On both of these subjects, wehave recently initiated a collaboration with Stéphane Descombes who is a professor in the J.A.Dieudonné Mathematics Laboratory at the University of Nice-Sophia Antipolis.

6.1.3 Numerical treatment of complex propagation media models

An important component of our planned activities for the coming years will aimed at the ex-tension of the proposed numerical methodologies towards more complex propagation media,more precisely, for taking into account physical dispersion and dissipation models such as those

37

characterizing dispersive media in the case of electromagnetic wave propagation and viscoelasticmedia in the case of seismic wave propagation. Up to know the media that we have consideredfor electromagnetic and elastodynamic wave propagation are linear, isotropic and non-dispersive.The latter terms means that waves travel at time- or frequency-independent velocity throughthe material. The DG methods we have developed so far can deal with heterogeneous in spacemedia without modifying the original design, however this will no longer be true in presence ofphysical dispersion since not only a dispersion model has to be chosen according to the selectedmaterials and/or the selected frequency spectrum, but also an appropriate numerical treatmentof this model has to be de�ned. Despite the increasing amount of works dealing with DG meth-ods for computational electromagnetics and computational geoseismics, there are presently onlya very few of them that consider complex materials models [Lin et al., 2005]-[Käser et al., 2007].

6.1.4 Domain decomposition methods for wave propagation problems

We will carry on our e�orts for the development of scalable and robust domain decompositionalgorithms for wave propagation problems in the time harmonic regime with emphasis on thesolution of the frequency domain Maxwell equations discretized by high order DGFD-Pp methodsformulated on unstructured simplicial meshes. This activity will actually span several topics:

• optimized interface conditions for Schwarz algorithms applied to the system of Maxwellequations including physical dissipation (in collaboration with Martin Gander from theMathematics Department of University of Geneva);

• discrete variants of optimized Schwarz algorithms in the framework of high order DGFD-Pp methods (in collaboration with Ronan Perrussel from the Ampère Laboratory at EcoleCentrale de Lyon);

• e�cient algebraic solution strategies for the sparse, complex coe�cients, linear systemscoming from the discretization of the frequency domain Maxwell equations by high orderDGFD-Pp methods, to be used as subdomain solvers in domain decomposition algorithms(this activity will be conducted in the context of the PhyLeaS associated team).

6.1.5 Software developments

Although the applications described in subsection 2.2.1 and 2.2.2 are very di�erent from the pointof view of the underlying physics, the resulting simulation software will share a large commonpart especially if the same numerical approach (i.e the DG-Pp methods that we develop) is usedtogether with unstructured meshes. This common part consists of geometrical and topologicaldata and functions such as volumes, normals, number of vertices and/or element neighbors,etc. and interpolation data such as degrees of freedom per element, local mass and/or sti�nessmatrices, etc. All these data which are usually computed during the simulation (for memorystorage considerations) remain constant and are not a�ected by the time stepping scheme orother peculiarities of the problem under consideration. Thus, aiming at a single software platformthat makes all the mesh and interpolation data and functions independent from the rest of thesolver, while remaining �exible (to incorporate new element types for example) and easy to use,becomes very attractive if not necessary if we want to save time in programming and maintainingcomputer codes, and also in view of capitalizing our research e�orts in a software platform thatcould be distributed to the academic community. It is clear that such a software platform shouldrely on concepts of object-oriented programming. Although Fortran 90x may not be consideredas an OOP language, we plan to develop our library in Fortran to make use of large parts ofexisting codes in one part and to avoid a break in our development works and our ongoing

38

contractual duties (either industrial or academic) for which Fortran 77 or 90 has been adopted.This choice is also motivated by the fact that a major called, Fortran 2003, not only introducesall the features to make it a true OOP language but also enhances the interfacing with C whichis known as a drawback when dealing with allocatable arrays especially when used as argumentsof functions or subroutines. In the coming years, we plan to initiate the development of an OOPsoftware platform in Fortran that will integrate the achievements materialized in the softwaredescribed in subsection 4.2. Beside, we will also devote some e�orts towards the adaptation ofthe numerical kernels of high order DG-Pp methods to modern computer architectures and inparticular, to hybrid parallel systems combining CPUs and GPUs.

6.2 Sta� evolution

An important evolution concerning the permanent members of the team will concern NathalieGlinsky. We recall that Nathalie Glinsky currently has a full-time researcher position at ENPC.She was a member of the former CAIMAN project-team and has been associated to the creationof the NACHOS project-team where she is responsible for the major part of our activitiesregarding computational geoseismics and in particular, of the development of high order DGTD-Pp methods for the solution of the system of elastodynamic equations. Early 2010, she will jointhe CETE Méditérannée) center in Nice (25 km from Sophia Antipols) where she will conductnumerical modeling studies on seismic risk mitigation at urban and regional scales. Althoughshe will remain a collaborator of the NACHOS project-team (we actually plan to setup a long-term collaboration with CETE Méditérannée), our planned objectives in relation to seismic wavepropagation would greatly bene�t from the reinforcement of a junior researcher or an assistantprofessor who will commit fully on this topic. Besides, such an enlistment will also allow us toactively contribute to the recently launched Depth Imaging Partnership strategic action betweenthe TOTAL group and INRIA. Worthwhile to note, the scienti�c program of this partnershipincludes a component dedicated to the study of high order methods for the discretization of timedomain seismic wave propagation models on unstructured simplicial meshes.

39

Project-team members's bibliography (before 2006)

[Fezoui et al., 2005] Fezoui, L., Lanteri, S., Lohrengel, S., and Piperno, S. (2005). Convergence andstability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equationson unstructured meshes. ESAIM: Math. Model. Num. Anal., 39(6):1149�1176.

[Rapetti and Toselli, 2001] Rapetti, F. and Toselli, A. (2001). A FETI preconditioner for two dimen-sional edge element approximations of Maxwell's equations on nonmatching grids. SIAM J. Sci.Comput., 23(1):92�108.

General bibliography

[Alonso-Rodriguez and Gerardo-Giorda, 2006] Alonso-Rodriguez, A. and Gerardo-Giorda, L. (2006).New nonoverlapping domain decomposition methods for the harmonic Maxwell system. SIAM J.Sci. Comput., 28(1):102�122.

[Amestoy et al., 2000] Amestoy, P., Du�, I., and L'Excellent, J.-Y. (2000). Multifrontal parallel dis-tributed symmetric and unsymmetric solvers. Comput. Meth. App. Mech. Engng., 184:501�520.

[Bernardi et al., 2000] Bernardi, P., Cavagnaro, M., Pisa, S., and Piuzzi, E. (2000). Speci�c absorptionrate and temperature increases in the head of a cellular phone user. IEEE Trans. Microwave TheoryTech., 48(7):1118�1126.

[Birdsall and Langdon, 1985] Birdsall, C. and Langdon, A. (1985). Plasma physics via computer simu-lation. Mc Graw-Hill.

[Bu�a et al., 2007] Bu�a, A., Houston, P., and Perugia, I. (2007). Discontinuous Galerkin computationof the Maxwell eigenvalues on simplicial meshes. J. Comput. Appl. Math., 204(2):317�333.

[Bu�a and Perugia, 2006] Bu�a, A. and Perugia, I. (2006). Discontinuous Galerkin approximation ofthe Maxwell eigenproblem. SIAM J. Num. Anal., 44(5):2198�2226.

[Chen et al., 2005] Chen, M.-H., Cockburn, B., and Reitich, F. (2005). High-order RKDG methods forcomputational electromagnetics. J. Sci. Comput., 22-23:205�226.

[Cockburn et al., 2000] Cockburn, B., Karniadakis, G., and Shu, C., editors (2000). DiscontinuousGalerkin methods. Theory, computation and applications, volume 11 of Lecture Notes in ComputationalScience and Engineering. Springer-Verlag.

[Cockburn et al., 2004] Cockburn, B., Li, F., and Shu, C.-W. (2004). Locally divergence-free discontin-uous Galerkin methods for the Maxwell equations. J. Comp. Phys., 194(2):588�610.

[Cockburn and Shu, 2005] Cockburn, B. and Shu, C., editors (2005). Special issue on discontinuousGalerkin methods, volume 22-23 of J. Sci. Comput. Springer.

[Cohen et al., 2006] Cohen, G., Ferrieres, X., and Pernet, S. (2006). A spatial high spatial order hexa-hedral discontinuous galerkin method to solve Maxwell's equations in time domain. J. Comp. Phys.,217(2):340�363.

[Dawson, 2006] Dawson, C., editor (2006). Special issue on discontinuous Galerkin methods, volume 195of Comput. Meth. App. Mech. Engng. Elsevier.

[Degerfeldt and Rylander, 2006] Degerfeldt, D. and Rylander, T. (2006). A brick-tetrahedron �nite-element interface with stable hybrid explicit-implicit time-stepping for Maxwell's equations. J. Com-put. Phys., 220(1):383�393.

[Deng and Cai, 2005] Deng, S. and Cai, W. (2005). Analysis and application of an orthogonal nodal basison triangles for discontinuous spectral element methods. Appl. Num. Anal. Comp. Math., 2(3):326�345.

[Després et al., 1992] Després, B., Joly, P., and Roberts, J. (1992). A domain decomposition method forthe harmonic Maxwell equations. In Iterative methods in linear algebra, pages 475�484, Amsterdam.North-Holland.

40

[Diaz and Grote, 2007] Diaz, J. and Grote, M. (2007). Energy conserving explicit local time-steppingfor second-order wave equations. Technical Report 2007-02, Department of Mathematics, Universityof Basel.

[Dumbser and Käser, 2006] Dumbser, M. and Käser, M. (2006). An arbitrary high order discontinuousGalerkin method for elastic waves on unstructured meshes ii: the three-dimensional isotropic case.Geophys. J. Int., 167(1):319�336.

[Dumbser et al., 2007] Dumbser, M., Käser, M., and Toro, E. (2007). An arbitrary high-order discon-tinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping andp-adaptivity. Geophys. J. Int., 171(2):695�717.

[Ern and Guermond, 2006a] Ern, A. and Guermond, J.-L. (2006a). Discontinuous Galerkin methods forFriedrichs systems I. General theory. SIAM J. Numer. Anal., 44(2):753�778.

[Ern and Guermond, 2006b] Ern, A. and Guermond, J.-L. (2006b). Discontinuous Galerkin methods forFriedrichs systems II. Second-order elliptic PDE's. SIAM J. Numer. Anal., 44(6):2363�2388.

[Farhat and Roux, 1994] Farhat, C. and Roux, F. (1994). Implicit parallel processing in structural me-chanics, volume 2. North Holland.

[Fumeaux et al., 2004] Fumeaux, C., Baumann, D., Leuchtmann, P., and Vahldieck, R. (2004). A gener-alized local time-step scheme for e�cient FVTD simulations in strongly inhomogeneous meshes. IEEETrans. Microwave Theory Tech., 52(3):1067�1076.

[Harutyunyan et al., 2008a] Harutyunyan, D., Izsak, F., van der Vegt, J., and Botchev, M. (2008a).Adaptive �nite element techniques for the Maxwell equations using implicit a posteriori error estimates.Comput. Meth. App. Mech. Engng., 197(17�18):1620�1638.

[Harutyunyan et al., 2008b] Harutyunyan, D., Izsak, F., van der Vegt, J., and Botchev, M. (2008b).Implicit a posteriori error estimates for the Maxwell equations. Math. Comp., 77:1355�1386.

[Hesthaven and Warburton, 2002] Hesthaven, J. and Warburton, T. (2002). Nodal high-order methodson unstructured grids. I. Time-domain solution of Maxwell's equations. J. Comput. Phys., 181(1):186�221.

[Hesthaven and Warburton, 2004a] Hesthaven, J. and Warburton, T. (2004a). High-order accuratemethods for time-domain electromagnetics. Comp. Mod. Engin. Sci, 5(5):395�408.

[Hesthaven and Warburton, 2004b] Hesthaven, J. and Warburton, T. (2004b). High-order nodal discon-tinuous Galerkin methods for the Maxwell's eigenvalue problem. Royal Soc. London Ser. A, 362:493�524.

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