1 Research

1.1 History

After aeronautics engineering studies, I did my PhD under the supervision of Pierre-Alain Mazet at the French aerospace research agency (ONERA) in Toulouse. I worked on numerical simulations of RADAR cross sections. I developed an original parallel method for coupling boundary integral equations and a Discontinuous Galerkin (DG) method.

In 1994 I moved to a permanent Assistant Professor position in the University of Toulon, at the engineering Institute (ISITV: “Institut des Sciences de l’Ingénieur de Toulon et du Var” now called “SeaTech” since 2014). In Toulon, my main research subject was the mathematical and numerical modeling of compressible multiphase flows. With several collaborators I developed new finite volume methods based on entropy optimization principles. Those methods were implemented in parallel software and applied to flows with phase transitions, wave breaking simulations, internal ballistics of guns. As I was in Toulon, I also worked on theoretical aspects of the Navier-Stokes equations. I also made a series of papers on inverse problems in ocean geoacoustics.

I passed my habilitation thesis in 2005 and was hired in 2006 on a full professor position in the University of Strasbourg. I continue to work on mathematical and numerical modeling of multiphase flows. I also tackled new subjects concerning plasma physics modeling and software implementation on new computer architectures with hybrid CPU/GPU computing.

1.2 Main topics

In this paragraph I present shortly a few of my favorite former works.

1.2.1 Gas-liquid flow modelling

In this work [8] with Thomas Barberon and Sandra Rouy, we studied a compressible gas-liquid flow occurring in a submarine missile ejection device. We applied a fully Eulerian finite volume method able to track naturally the liquid-gas interface. For obtaining correct results it is necessary to adapt carefully several techniques: contact preserving schemes, time-dependent boundary condition that change of type, local time stepping. We were then able to reproduce precisely actual experiments. On Figure 1 the liquid-gas interface can be seen at different times.

I used the same kind of approaches for computing wave breaking. See http://www-irma.u-strasbg.fr/~helluy/soliton.htm.

In order to improve the modeling, we extended the approach in [9] in order to take into account the vaporization (cavitation process) that arises in the liquid due to violent pressure drop. This improved method is based on a finite volume method with a relaxation source term constructed from an entropy optimization principle. An interesting point of the paper is that applying directly the liquid-vapor pressure law is not a correct approach because there exists a continuous family of entropy solutions. Different solutions can be obtained by changing the CFL number of the simulation. With the entropy optimization approach we recover the physical solution, which has a maximal entropy dissipation rate.

I continue to work in Strasbourg on interface capturing method. In [10] I developed with Jonathan Jung a new fully conservative and stable finite volume approach for computing very stiff gas-liquid problems without pressure oscillations at the gas-liquid interface. To my knowledge, this is the only finite volume scheme that is both conservative and stable on these kinds of problems. It is based on a ALE approach with a random remap step.

1.2.2 Granular flows

In [28] we studied a compressible multiphase flow made of a solid and a gas phase. Each phase has its own velocity. The objective is to model the grain combustion inside a gun (internal ballistics). Some models suppose that the gas pressure p_g of the gas and the solid pressure p_s are linked by a pressure equilibrium relation of the form

where R > 0 is the granular stress. Generally, those models are not hyperbolic and thus unstable. Some authors have proposed models with pressure evolution equations for each phase and a relaxation source term in order to recover the pressure equilibrium.

In our work, we perform a rigorous analysis of this relaxation model, give an analytic form for the granular stress that ensures entropy dissipation. We also apply the model to an actual gun and compare it with another model.

1.2.3 Thermodynamics and (max,+) algebra

The Legendre transform is a theoretical tool that is used in many fields of mathematics and physics. For a convex function f the Legendre transform is defined by

There is a beautiful analogy between the Legendre transform and the Fourier transform in the theory of the (max,+) algebra. Indeed, if we consider the two following operations

it is possible to draw the following equivalence between classical analysis and (max,+) analysis.

classical analysis	(max,+) analysis
a ⋅ b	a ⊙ b = a + b
a + b	a ⊕ b = max(a,b) (a ⊕ a = a)
∫ Ωf(x)dx	f(x) = f(x)
characters: χ(s,x + y) = χ(s,x) ⋅ χ(s,y)	χ(s,x + y) = χ(s,x) ⊙ χ(s,y)
χ(s,x) = exp(−isx)	χ(s,x) = s ⋅ x
Fourier: (s) = ∫ f(x)exp(−isx)dx	Legendre: f∗(s) = f(x) ⊙ χ(s,x) = sx + f(x)
Convolution: (f ∗ g)(x) = ∫ yf(x − y)g(y)dy	Sup-convolutionf□g(x) = f(x − y) + g(y)
(f ∗ g)∧ = ⋅ĝ	(f□g)∗ = f∗⊙ g∗ = f∗ + g∗ (f,g concave usc)

A consequence of this analogy is that it is possible to construct a fast algorithm, similar to the fast Fourier transform, for computing Legendre transform and sup-convolution of sampled functions.

In [11] we apply the above theory to the thermodynamics of mixture. We consider a mixture of two components i = 1,2 characterized by their energy laws ε_i(ρ,σ), function of the density ρ and entropy σ. In the Legendre formalism the dual variables of ρ and σ are the chemical potential μ and the temperature θ. The pressure p_i(μ,θ) is then the Legendre transform of ε_i(ρ,σ). After a miscible mixture of the two components, the pressure and energy are given by

where □ denotes the sup-convolution operation. For an immiscible mixture the relations become

where co(f) denotes the convex envelope of f.

It is much easier to compute max and + operations than sup-convolutions or convex envelopes. Therefore, we propose an algorithm, based on the fast Legendre transform, in order to compute in an efficient way, the mixture equation of state from tabulated laws of each component. We apply the method to phase transition and to mixture of reactive gases.

1.2.4 GPU and hybrid computing

Since 2009 I generally implement my software using the OpenCL library. OpenCL is a programming framework, similar to CUDA in order to address GPU or multicore accelerator in a unified way.

In [23] with Anaïs Crestetto we have coupled a Particle-In-Cell (PIC) method and a Discontinuous Galerkin (DG) method for Vlasov-Maxwell simulations. The method is implemented on GPU with OpenCL. It is applied to the numerical simulation of a medical X-ray generator. With our software, we were awarded a prize at the AMD OpenCL innovation challenge in 2011 https://community.topcoder.com/amdapp/

With Jonathan Jung I also applied multi-GPU computing on the scheme that we developed in [39]. With OpenCL accelerations, we were able to compute test cases on very fine meshes (an example of a liquid-shock interaction is shown on Figure 2 )

For addressing more computational power it becomes almost mandatory to follow a task graph approach. The method consists in splitting the whole simulation into several elementary computational tasks with their dependencies. The tasks are then distributed automatically at runtime on the available resources for efficient parallel computations. This approach is described in [49], where we develop our own home-made runtime system, and apply it to electromagnetic simulations. More recently, we switched to a more general environment, developed by Inria specialists for more than ten years: StarPU. It is a runtime system for distributing the tasks on hybrid accelerators (CPU and GPU). The results are not yet published but are described in the thesis of Michel Massaro (Chapter 5):

https://tel.archives-ouvertes.fr/tel-01410049

1.2.5 Other works

Navier-Stokes theory In this paper [12] written with F. Golay, we prove a rigorous mathematical result of existence and uniqueness for weakly compressible Navier-Stokes equations. The proof is based on an abstract fixed point method in Sobolev spaces. The fixed point approach can also be applied numerically. The resulting scheme is not very efficient but I like it anyway because the fixed point algorithm requires to solving three different types of PDE with adapted Finite Element (FE) methods: a Laplace equation solved by standard finite elements, a transport equation, which we solved with the SUPG approach, and a Stokes problem that we solved with Crouzeix-Raviart elements. This was a good exercise for learning about the FE method.

In the end, we were able for instance to evaluate the contraction constant of the theoretical fixed point method (see Figure 3 ).

Inverse problems in ocean geoacoustics In [15] we use an optimal control technique for identifying the acoustic characteristics of the submarine ground from measurements. The method is applied to a popular reduced acoustic model in ocean engineering: the paraxial Tappert model, which has the same mathematical structure as the Schrödinger equation. See Figure 4 where an example of the reconstruction process is presented.

1.3 PhD supervisions

In this table I give the list of the thesis that I supervised. The indicated rate has no administrative meaning. It is an indication of my actual investment in the thesis supervision.

1.4 Prize

With Anaïs Crestetto I was awarded the fourth prize at the AMD OpenCL innovation challenge 2011: “Numerical simulation of a medical X-ray generator on GPU:

https://community.topcoder.com/amdapp/

1.5 Collaborations

Here is a non-exhaustive list of present and former collaborations (the PhD students are listed in the above table).

• Frédéric Golay (Toulon, specialist of mechanics): Navier-Stokes equations, wave breaking simulations.
• Sylvain Maire (Toulon, numerical analysis): quadrature methods for singular integral equations.
• Frédéric Coquel, Nicolas Seguin (University Paris VI, applied mathematics): entropy dissipative finite volume methods.
• Stephan Grilli (University of Rhode Island, USA, oceanography): wave breaking simulations.
• Mark Asch (Amiens, applied mathematics): inverse problems in acoustics.
• Thierry Gallouët (Marseille, applied mathematics): granular flow modeling. Thierry was my habilitation thesis advisor when I was in Toulon.
• Jean-Marc Hérard, Olivier Hurisse (Paris, senior research engineer at EDF, French electricity company): finite volume methods, multiphase flows, granular flows, well-balanced schemes.
• Siegfried Müller (Aachen, Germany, scientific computing): multiphase flows, finite volume methods with Arbitrary Mesh Refinement (AMR).
• Claus-Dieter Munz (Stuttgart, Germany, scientific computing): 3D-1D coupling in finite volume methods.
• Eric Sonnendrücker (Max Planck Institute for Plasma Physics, Garching, Germany, scientific computing): MHD, Vlasov, local time stepping in DG methods.
• Laurent Navoret (Strasbourg, applied mathematics): reduction methods for plasma physics, kinetic modeling for acoustics, implicit lattice-Boltzmann methods.
• Emmanuel Franck (Strasbourg, applied mathematics): implicit lattice-Boltzmann methods, physical preconditioning.
• Michel Mehrenberger (Strasbourg, applied mathematics): implicit lattice-Boltzmann methods, gyrokinetic plasma modeling.
• Christian Klingenberg (Würtzburg, Germany, applied mathematics): work in progress on implicit DG method for MHD.

1.6 Organizations of scientific events

I was co-organizer of several workshops and conferences. For instance:

• “Mathematical and Numerical Aspects of Low Mach Number flows”, Porquerolles, France, June 21-25 2003. Organizer of the numerical workshop on wave breaking: http://www-irma.u-strasbg.fr/~helluy/soliton.htm
• “Numerical Simulation of Complex and Multiphase Flows”, 18th - 22nd April 2005, Porquerolles, France. Special edition of flow turbulence and combustion: http://link.springer.com/journal/10494/76/4/page/1
• Third Workshop "Micro-Macro Modelling and Simulation of Liquid-Vapour Flows" IRMA, 23-25 Janvier 2008, http://www-irma.u-strasbg.fr/article565.html
• Fifth workshop "Micro-Macro Modelling and Simulation of Liquid-Vapour Flows" IRMA - April, 14-16, 2010, http://www-irma.u-strasbg.fr/article983.html
• CEMRACS 2011, summer research session “Multiscale Coupling of Complex Models” July 18th - August 26th, 2011 — Marseille, France. http://smai.emath.fr/cemracs/cemracs11/ . Special issue of ESAIM Procs: http://www.esaim-proc.org/articles/proc/abs/2012/04/contents/contents.html
• NUMKIN 2016 : International Workshop on Numerical Methods for Kinetic Equations IRMA Strasbourg, 17-21 October 2016. http://www-irma.u-strasbg.fr/article1573.html

I am regularly invited to conferences, workshops or summer schools. For instance:

• Applied Mathematics In Savoie 2012: http://www.lama.univ-savoie.fr/AMIS2012/Speakers.php
• FVCA 7 in Berlin, 2014: http://www.wias-berlin.de/workshops/fvca7/speakers.jsp
• NUMKIN 2015: https://www.ipp.mpg.de/3874756/NumKin2015
• “Ecole thématique Maths-Info-HPC”, 2016, Lyon (France): https://mathsinfohpc.sciencesconf.org
• etc.

1.7 Reviews

1.7.1 Journals

Over the years I made reviews for various journals. For instance: Math Reviews, M2AN, M3AS, Computers and Fluids, SIAM Journal on Numerical Analysis, Journal of Computational Physics, International Journal for Numerical Methods in Fluids, IJNMF, Journal of Mechanical Science and Technology, ESAIM, Oil & Gas Science and Technology, Numerical Methods for Partial Differential Equations, International Journal of Offshore and Polar Engineering (!), SIAM Journal on Applied Mathematics, CRAS, etc.

I am associate editor of the International Journal of Finite Volumes: http://www.i2m.univ-amu.fr/IJFV/

1.7.2 Other reviews

Regularly I write reviews on PhD or habilitation thesis or on for research project calls: CNRS calls on mathematics and physics, ANR (French research agency), US Army, French regions calls, etc.