The project aims at studying mathematical models issued from environmental and energy management questions. We consider systems of PDEs of hydrodynamic type or hybrid fluid/kinetic systems. The problems we have in mind involve unusual coupling, which in turn leads to challenging difficulties for mathematical analysis and the need of original numerical solutions. By nature many different scales arise in the problems, which allows to seek hierarchies of reduced models based on asymptotic arguments. The topics require a deep understanding of the modeling issues and, as far as possible boosted by the mathematical analysis of the equations and the identification of key structure properties, we wish to propose innovative and performing numerical schemes. To this end, the development of innovative Finite Volumes schemes with unstructured meshes on complex geometries will be a leading topic of the team activity.

Mathematical modeling and computer simulation are among the main research tools for environmental management, risks evaluation and sustainable development policy. Many aspects of the computer codes as well as the PDEs systems on which these codes are based can be considered as questionable regarding the established standards of applied mathematical modeling and numerical analysis. This is due to the intricate multiscale nature and tremendous complexity of those phenomena that require to set up new and appropriate tools. Our research group aims to contribute to bridging the gap by developing advanced abstract mathematical models as well as related computational techniques.

The scientific basis of the proposal is two–fold. On the one hand, the project is “technically–driven”: it has a strong content of mathematical analysis and design of general methodology tools. On the other hand, the project is also “application–driven”: we have identified a set of relevant problems motivated by environmental issues, which share, sometimes in a unexpected fashion, many common features. The proposal is precisely based on the conviction that these subjects can mutually cross-fertilize and that they will both be a source of general technical developments, and a relevant way to demonstrate the skills of the methods we wish to design.

To be more specific:

We can distinguish the following fields of expertise

Our research focuses on the numerical modeling of multiphase porous media flows accounting for complex geology and for nonlinear and multi-physics couplings. It is applied to various problems in the field of energy such as the simulation of geothermal systems in collaboration with BRGM, of nuclear waste repositories in collaboration with Andra, and of oil and gas recovery in collaboration with Total. We are starting a new program through the Inria-IFPEN initiative. Our research directions include the development of advanced numerical schemes adapted to polyhedral meshes and highly heterogeneous media in order to represent more accurately complex geologies. A special focus is made on the modeling of multiphase flows in network of faults or fractures represented as interfaces of co-dimension one coupled to the surrounding matrix. We also investigate nonlinear solvers adapted to the nonlinear couplings between gravity, capillary and viscous forces in highly heterogeneous porous media. In the same line, we study new domain decomposition algorithms to couple non-isothermal compositional liquid gas flows in a porous medium with free gas flows occurring at the interface between the ventilation gallery and the nuclear waste repository or between a geothermal reservoir and the atmosphere. We are exploring the coupling between the multiphase flow in the porous matrix and the solid mechanics involved in opening fractures.

We investigate fluid mechanics models referred to as “multi–fluids” flows. A large part of our activity is more specifically concerned with the case where a disperse phase interacts with a dense phase. Such flows arise in numerous applications, like for pollutant transport and dispersion, the combustion of fuel particles in air, the modelling of fluidized beds, the dynamic of sprays and in particular biosprays with medical applications, engine fine particles emission... There are many possible modelings of such flows: microscopic models where the two phases occupy distinct domains and where the coupling arises through intricate interface conditions; macroscopic models which are of hydrodynamic (multiphase) type, involving non standard state laws, possibly with non conservative terms, and the so–called mesoscopic models. The latter are based on Eulerian–Lagrangian description where the disperse phase is described by a particle distribution function in phase space. Following this path we are led to a Vlasov-like equation coupled to a system describing the evolution of the dense phase that is either the Euler or the Navier-Stokes equations. It turns out that the leading effect in such models is the drag force. However, the role of other terms, of more or less phenomenological nature, deserves to be discussed (close packing terms, lift term, Basset force...). Of course the fluid/kinetic model is interesting in itself and needs further analysis and dedicated numerical schemes. In particular, in collaboration with the Atomic Energy Commission (CEA), we have proposed a semi-Lagrangian scheme for the simulation of particulate flows, extending the framework established in plasma physics to such flows.

We also think it is worthwhile to identify hydrodynamic regimes: it leads to discuss hierarchies of coupled hydrodynamic systems, the nature of which could be quite intriguing and original, while they share some common features of the porous media problems. We are particularly interested in revisiting the modeling of mixture flows through the viewpoint of kinetic models and hydrodynamic regimes. We propose to revisit the derivation of new mixture models, generalizing Kazhikov-Smagulov equations, through hydrodynamic asymptotics. The model is of “hybrid” type in the sense that the constraint reduces to the standard incompressibility condition when the disperse phase is absent, while it involves derivatives of the particle volume fraction when the disperse phase is present.

The simulation of motions of solid bodies in a fluid, as well as fracturation, fissuration phenomena leads to numerical difficulties: they can undergo deformations, fragmentation and contact which deform dramatically the fluid domain, rendering remeshing techniques less effective. The numerical suite Mka2d/Mka3d/Celia2d/Celia3d/Precis addresses this issue. On the solid side, the adopted discrete element discretization works on general polyhedral meshes, and again uses different types of degree of freedom, stored at the cell and face centers, with suitable reconstruction procedures in order to guaranty the conservation properties. On the fluid side, we use a cut-cell approach designed in order to preserve exactly the discrete mass and energy conservations for general Finite Volume methods. This approach is an alternative to ALE methods which would require costly remeshing and can induce severe stability conditions due to mesh deformations. It is particularly adapted to manage fragmentation events, which lead to changes in the topology of the fluid domain. The delicate geometrical issues are handled by using robust, efficient and fast geometric intersection procedures. Such issues are also investigated for the modeling of urban floods; our methodologies on finite volume schemes on complex geometries and domain decomposition methods are reinvested on such problems through the ANR project Top-up (exploiting formal analogies between Shallow-Water equations and Richard's equation in the regimes of interest).

The NeuroMod Institute for Modeling in Neuroscience and Cognition aims at promoting modeling as an approach for integrating brain mechanisms and cognitive functions. It has selected the project proposed by C. Guerrier and S. Krell about the modeling and simulation of the variations of the electric field in the dendritic tree of neurons as well as the electrodiffusion of ions in the neuronal cytoplasm, considered as an electrolyte (P. Paragot's PhD thesis). Thus, the model is based on the Nernst-Planck equation coupled to the Poisson equation; it has many similarities with convection-diffusion models arising for flows in porous media. Difficulties arise from the multi-scale configuration, and the presence of boundary layers between the cytoplasm, the membrane, and the external neuron’s environment. This leads to new developments for the DDFV framework and its application with domain decomposition approaches.

Members of the team have started an original research program
devoted to fungal network growth. We started working on this subject
through a collaboration with biologists and physicists at LIED
(Université Paris Diderot) and probabilists in CMAP (Ecole
Polytechnique) and Université Paris Sud, involving Rémi
Catellier and Yves D'Angelo (team Atlantis). The motivation is to
understand branching networks as an efficient space exploration
strategy, with fungus Podospora Anserina being the biological
model considered. This research is submitted as an ANR-project and
has been supported by various local fundings.

We have developed a size and space structured model describing interaction of tumor cells with immune cells based on a system of partial differential equations. This model is intended to describe the earliest stages of this interaction and takes into account the migration of the tumor antigen-specific cytotoxic effectors cells towards the tumor microenvironment by a chemotactic mechanism. This study reveals cancer persistent equilibrium states as expected by biologists, as well as escape phases when protumoral immune responses are activated. This effect which leads to persistent tumors at a controlled level was inferred from clinical observations and demonstrations using mouse model. Therefore, the maintenance of cancer in a viable equilibrium state represents a relevant goal of cancer immunotherapy. The mathematical interpretation of the equilibrium state by means of eigenvalue problems and constrained equations, has permitted us to develop new numerical algorithms in order to predict at low numerical cost the main features of the equilibrium and to discriminate, in biologically relevant cases, the parameters that are the most influential on the equilibrium.

This topic is addressed mainly with Paulo Amorim (Univ. Federal Rio de Janeiro) and Fernando Peruani (Lab. de Physique Théorique et Modélisation, Cergy Paris Université).

We are interested in the mathematical modeling of physico-biological phenomena that drive towards a self-organization of a population of individuals reacting to external signals. It might lead to the formation of remarkable patterns or the following of traveling external signal. We develop microscopic and hydrodynamic models for such phenomena, with a specific interest in the modeling of ant foraging.

Our main software achievement is the code ComPASS. It is developped since 2013
through several collaborations, by means of PhD and postdocs,
with BRGM, ANDRA, MdS, Storengy, LJLL and has benefitted from the support of Carnot Institute and ANR through the project Charms. (The project has not been fortunate enough to receive an engineer support from Inria, though.)
The objective is to propose an alternative to commercial codes, like Tough2, which faces limitations, at least for some
specific situations. The code
is an open source parallel code,
it works on complex geometry, with complex unstructured meshes, it accounts for faults, fractures and deals with polyphasic and compositional flows.
It applies in particular to geothermal flows.

Since 2021, both ANDRA and BRGM are committed to the partnership and the development of ComPASS.
The management of the code is shared with S. Lopez and L. Beaude from BRGM.
The common
ANDRA/BRGM
roadmap
explicitly refers to ComPASS in the simulation strategy of these
organisms.
The
code is distributed from the Inria Gitlab platform under the opensource license CeCILL2.1/GPLv3.
It has been effectively used for several user-cases:

Further information available on the Compass website.

The numerical suite Mka2d/Mka3d/Celia2d/Celia3d/Precis simulates an elastic solid by discretizing the solid into rigid particles in 2d or 3d configurations. An adequate choice of forces and torques between particles allows to recover the equations of elastodynamics. The code Celia2d-3d is devoted to fluid-structure interactions. The code Precis is a more mature version of these softwares, with further visualization procedures. The codes are on a GitLab plateform, with the objective of a diffusion beyond our circle of close collaborators. Moreover, a part of our methodologies aim at being reinvested in industrial collaborations (in discussion).

Further information available on the website: Mka3d and Celia3d

New results are concerned with

The research of the team is regularly supported by several contracts
with industrial partners: BRGM, Storengy, IFP-EN, on scientific computing issues in geosciences.
This has permitted to welcome many PhD and postdocs.
The transfer strategy is built on the development of the opensource code ComPASS, specifically
oriented towards the simulation of mass and heat transfers in fractured media.
The code is identified by the consortium Andra/BRGM
as an alternative of the commercial code Tough2 for security simulations of transient hydraulic-gas flows.

We are currently in touch with the company Altair for the development of simulation tools fro fluid structure ineractions problems.

The on-going project, supported by ANR, about the modeling of urban floods, involves Frédéric Valentin from LNCC in Brazil who benefits from a INRIA International Chair at the Inria research centre Université Côte d'Azur.

The collaboration with J. Droniou is supported by the Inria associated team HDTHM; it has also been supported by Univ. Côte d'Azur chairs (chaire Montel). J. Droniou visits the team for a couple of month in Fall 2022.

We have benefitted from several visits of M. Gander from Geneva Univ.

LJAD-Inria and BRGM are the French partners of the Norvegian-German-French project In-
SPiRE (for International Open Source Simulation Software Partnership in Research and
Education) supported by the Research Council of Norway with the code
ComPASS as one of the softwares of this project together with Dune, Dumux and OPM.

We are member of the European Research Consortium Eurad, the European Joint Program on Radioactive Waste Management. It gathers 51 organisations across 23 EU Member States having received a mandate by their official National Programme owner for radioactive waste management and nationally funded Research Entities. ComPASS is part of the program Development and Improvement Of Numerical methods and Tools for modelling coupled processes (Donut), see for instance on a mixed-dimensional 3D-2D-1D applied to the simulation of the desaturation of the fractured excavated damaged zone at the interface with ventilated tunnels in Andra nuclear waste storage facilities. We are also member of the European Consortium Nupus (Interdisciplinary Union of Porous Media Research) led by the University of Stuttgart.

GdR MathGeoPhy

The research group MathGeoPhy has activities centered around scientific computing, design of new numerical schemes and mathematical modelling (upscaling, homogenization, sensitivity studies, inverse problems,...). Its goal is to coordinate research in this area, as well as to promote the emergence of focused groups around specific projects

GdR Mamovi

The team is involved in the activities of the research group dedicated to applications to life sciences.

T. Goudon is founding editor and Co-Editor in chief of SMAI Journal of Computational Mathematics SMAI-JCM

T. Goudon is academic editor at PlosOne.

We are regularly committed to review research papers for journals in applied mathematical analysis (ARMA, SIMA, Ann. PDE, ANN. ENS...), numerical analysis (SINUM, Numerische, JCP...), scientific computing (SISC), geosciences (SIAM Geosc.), life sciences (JTB, NHM,...)...

T. Goudon is Scientific officer for Mathematics at the General Directorate for Research and Innovation of the Ministry of Higher Education and Research. As such, he is member of the Board of IHP, CIMPA, CIRM, IHES and gives advices on any subject that concerns research in mathematics. He is responsible of the follow-up of mathematics in initiatives with national funding (Labex, Equipex, PEPR, IA national plan), and of the “Assises des Mathématiques”.

T. Goudon is regularly expert for funding agencies (France, Switzerland, Belgium, Chile, Hong-Kong, Slovakia...). He serves as expert for the Marie Skłodowska-Curie call.

Beyond mentoring PhD thesis and postdoctoral fellowships, we also supervise masters internships, and apprenticeship with companies.

Team members are regularly invited to serve as reviewers of PhD thesis and habilitation to conduct research, in France, abroad and also on connected disciplines.

Stella Krell has participated to the creation and organisation of “mathematical rallyes” devoted to K-grades students. The objective is to develop the pleasure to practice mathematics, around clever games, see the website of the Académie de Nice.

Laurent Monasse participated to the program “Regards de géomètre” with speeches to K-grades in high-schools about mathematics, origamis and illusions. He participated to the “Cordées de la réussite” with a project on origamis for dress-production.

The team also regularly welcomes students from 4ème/3ème for short internships.