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We are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [ 4 ]. We consider a conservative underlying kinetic model, the Vlasov-BGK-Poisson system. We perform a scaling on this system in order to obtain its hydrodynamic limit. We present a deterministic numerical method to approximate this kinetic system. The method is shown to be Asymptotic-Preserving in the hydrodynamic limit, which means that any stability condition of the method is independant of any parameter \begin{document}$ \varepsilon $\end{document} , with \begin{document}$ \varepsilon \rightarrow 0 $\end{document} . We prove that the method is, under appropriate choices, consistant with the solution for bi-temperature Euler. Finally, our method is compared to methods for the fluid model (HLL, Suliciu).

This article presents the use of artificial neural networks (ANN) to predict nonlocal heat flux transport within hydrodynamic simulations. Several cases of laser driven ablation of a plastic target are considered. The database for the ANN training phase is built using the transport module of the hydrodynamic code CHIC. It covers a range of parameters characteristic of laser experiments in the context of high-energy-density physics. Results show that an ANN can efficiently replace a module of nonlocal transport in one- and two-dimensional hydrodynamic simulations, with an error less than 3% in a radius of 0.5μm and an average computation gain of a factor 433 in two dimensions.

This work consists of the validation of a new Grid Based Boltzmann Solver (GBBS) conceived for the description of the transport and energy deposition by energetic particles for radiotherapy purposes. The entropic closure and a compact mathematical formulation allow our code (M1) to calculate the delivered dose with an accuracy comparable to the Monte-Carlo (MC) codes with a computational time that is reduced to the order of few minutes without any special processing power requirement. A validation protocol with heterogeneity inserts has been defined for different photon sources. The comparison with the MC calculated depth-dose curves and transverse profiles of the beam at different depths shows an excellent accuracy of the M1 model.

One of the big challenges of the emerging MRI-guided radiotherapy is the prediction of an external magnetic field effect on the deposited dose induced by a beam of charged particles. In this paper, we present the results of the implementation of the Lorentz force in the deterministic M1 model. The validation of our code is performed by comparisons with the Monte-Carlo code FLUKA. The relevant examples show a significant modification of the shape of dose deposition volume induced by the external magnetic field in presence of heterogeneities. A gamma-index analysis 3%/3 mm shows a good agreement of our model with FLUKA simulations.

In the present work the $M_1$ angular moments model in a velocity-adaptive frame is presented for rarefied gas dynamics applications. First of all, the derivation of the angular $M_1$ moments model in the particle mean velocity frame is introduced. The choice of the mean velocity framework in order to enforce the Galilean invariance property of the model is highlighted. In addition, it is shown that the model rewritten in terms of the entropic variables is Friedrichs-symmetric. Also, the derivation of the associated conservation laws and the zero mean velocity condition are detailed. Second, a suitable numerical scheme, preserving the realizable requirement of the numerical solution for the angular $M_1$ moments model in the mean velocity frame, is proposed. Third, some numerical results obtained considering several test cases in different collisional regimes are displayed.

The present paper concerns the study of the nonconservative bitem-perature Euler system with transverse magnetic field. We firstly introduce an underlying conservative kinetic model coupled to Maxwell equations. The nonconservative bitemperature Euler system with transverse magnetic field is then established from this kinetic model by hydrodynamic limit. Next we present the derivation of a finite volume method to approximate weak solutions. It is obtained by solving a relaxation system of Suliciu type, and is similar to HLLC type solvers. The solver is shown in particular to preserve positivity of density and internal energies. Moreover we use a local minimum entropy principle to prove discrete entropy inequalities, ensuring the robustness of the scheme.

Particle transport in radiation therapy can be modelled by a kinetic equation which must be solved numerically. Unfortunately, the numerical solution of such equations is generally too expensive for applications in medical centers. Moment methods provide a hierarchy of models used to reduce the numerical cost of these simulations while preserving basic properties of the solutions. Moment models require a closure because they have more unknowns than equations. The entropy-based closure is based on the physical description of the particle interactions and provides desirable properties. However, computing this closure is expensive. We propose an approximation of the closure for the first two models in the hierarchy, the $$M_1$$M1 and $$M_2$$M2 models valid in one, two or three dimensions of space. Compared to other approximate closures, our method works in multiple dimensions. We obtain the approximation by a careful study of the domain of realizability and by invariance properties of the entropy minimizer. The $$M_2$$M2 model is shown to provide significantly better accuracy than the $$M_1$$M1 model for the numerical simulation of a dose computation in radiotherapy. We propose a numerical solver using those approximated closures. Numerical experiments in dose computation test cases show that the new method is more efficient compared to numerical solution of the minimum entropy problem using standard software tools.

The transport of photons and electrons is studied in the field of radiotherapy to compute the dose, that is, the quantity of energy transferred to the medium by a beam of particles at each position. A kinetic model is proposed, and to decrease the computation times, it is reduced through a moment extraction. Entropy-based angular moment models of order up to two (M1 and M2 models) are shown to provide accurate results compared to a reference code with much lower computational costs.

The electronic M 1 model is widely used for electron transport studies in a hot collisional plasma. However, the moment extraction of the electron–electron collision operator from the kinetic collision operator, for this angular moments model, is challenging and some approximations are required. In this work a characterisation of the electron–electron and electron–ion collision operators is given and the electron plasma transport coefficients are derived. It is shown that in the high Z limit the electronic M 1 model and the Fokker–Planck–Landau equation coincide in the case of near equilibrium. Also, in general, the electron–electron collision operator proposed for the electronic M 1 model recovers accurate electron transport plasma coefficients.

This work deals with the numerical resolution of the M1-Maxwell system in the quasi-neutral regime. In this regime the stiffness of the stability constraints of classical schemes causes huge calculation times. That is why we introduce a new stable numerical scheme consistent with the transitional and limit models. Such schemes are called Asymptotic-Preserving (AP) schemes in literature. This new scheme is able to handle the quasi-neutrality limit regime without any restrictions on time and space steps. This approach can be easily applied to angular moment models by using a moments extraction. Finally, two physically relevant numerical test cases are presented for the Asymptotic-Preserving scheme in different regimes. The first one corresponds to a regime where electromagnetic effects are predominant. The second one on the contrary shows the efficiency of the Asymptotic-Preserving scheme in the quasi-neutral regime. In the latter case the illustrative simulations are compared with kinetic and hydrodynamic numerical results.

A new deterministic method for calculating the dose distribution in the electron radiotherapy field is presented. The aim of this work was to validate our model by comparing it with the Monte Carlo simulation toolkit, GEANT4. A comparison of the longitudinal and transverse dose deposition profiles and electron distributions in homogeneous water phantoms showed a good accuracy of our model for electron transport, while reducing the calculation time by a factor of 50. Although the Bremsstrahlung effect is not yet implemented in our model, we propose here a method that solves the Boltzmann kinetic equation and provides a viable and efficient alternative to the expensive Monte Carlo modeling.

International audience; This work is devoted to the derivation of an admissible asymptotic-preserving scheme for the electronic M1 model in the diffusive regime. A numerical scheme is proposed in order to deal with the mixed derivatives which arise in the diffusive limit leading to an anisotropic diffusion. The derived numerical scheme preserves the realisability domain and enjoys asymptotic-preserving properties correctly handling the diffusive limit recovering the relevant limit equation. In addition, the cases of non constants electric field and collisional parameter are naturally taken into account with the present approach. Numerical test cases validate the considered scheme in the non-collisional and diffusive limits.

International audience; This paper is devoted to the study of the nonconservative bitemperature Euler system. We firstly introduce an underlying two species kinetic model coupled with the Poisson equation. The bitemperature Euler system is then established from this kinetic model according to an hydrodynamic limit. A dissipative entropy is proved to exist and a solution is defined to be admissible if it satisfies the related dissipation property. Next, four different numerical methods are presented. Firstly, the kinetic model gives rise to kinetic schemes for the fluid system. The second approach belongs to the family of the discrete BGK schemes introduced by Aregba-Driollet and Natalini. Finally, a quasi-linear relaxation approach and a Lagrange-remap scheme are considered. 1991 Mathematics Subject Classification. 65M08, 35L60, 35L65. Secondary: 82D10, 76X05. The dates will be set by the publisher.

Typical external radiotherapy treatments consist in emitting beams of energetic photons targeting the tumor cells. Those photons are transported through the medium and interact with it. Such interactions affect the motion of the photons but they are typically weakly deflected which is not well modeled by standard numerical methods. The present work deals with the transport of photons in water. The motion of those particles is modeled by an entropy-based moment model, i.e. the M 1 model. The main difficulty when constructing numerical approaches for photon beam modelling emerges from the significant difference of magnitude between the diffusion effects in the forward and transverse directions. A numerical method for the M 1 equations is proposed with a special focus on the numerical diffusion effects.

In plasma physics domain, the electrons transport can be des cribed from kinetic and hydrodynamical models. Both methods prese nt disadvantages and thus cannot be considered in practical computations for Inertial Confinement Fusion (ICF). That is why we propose in this paper a new mo del which is intermediate between these two descriptions. More precise ly, the derivation of such models is based on an angular closure in the phase space a nd retains only the energy of particles as a kinetic variable. The closure of the moment system is obtained from a minimum entropy principle. The resulting continuous model is proved to satisfy fundamental properties. Moreover the m odel is discretized w.r.t the energy variable and the semi-discretized scheme i s shown to satisfy conservation properties and entropy decay.

International audience

Introduction The majority of patients affected by cancer are nowadays treated by radiotherapy, which consists in delivering a homogeneous dose with energetic particles. Magnetic Resonance Imaging (MRI)-guided radiotherapy shows precisely the tumor position in real time and allows to direct the photon beams to the tumor while sparing organs at risks. Our work consists in the validation of a new model developped to simulate the energy deposition of the particles used in radiotherapy (electrons, photons and protons) in the presence of strong magnetic field, within human tissues, and its adaptation to this new technology. Material and methods This model is based on a kinetic entropic closure of the linearized Boltzmann equation [1] , which describes the transport of energetic particles in the matter. This equation takes a lot of computation time to be solved due to the high number of variables. To simplify this, we replace fluences by angular moments, which allows getting rid of the angular variables and improve the calculation time. We obtain a set of angular moments equations, and we close this set using the Boltzmann’s principle of entropy maximization on the two first equations of the set. This model has an accuracy comparable to references Monte-Carlo (MC) codes [2] (Geant4, MCNPx, Penelope), and is less time-consuming than these ones. We added the Lorentz force into our code to study the influence of a magnetic field on the dose deposition, and compared it to the reference full MC code FLUKA. To further validate our model, we have carried out an experimental campaign in which we irradiated a composition of materials of different mass densities, with 6 and 18 MV photon beams of a Varian Clinac 21Ex accelerator, without and with the influence of a 0.87 T magnetic field of amplitude induced by a permanent magnet. Results We show a good agreement between our model and the FLUKA code ( Fig. 1 ). We highlight the effects that occur on the propagation of particles in the matter in presence of a magnetic field of a few Tesla. Indeed, it modifies the dose deposition by shifting the deposition of energy, leading to an increased diffusion depending on the orientation of the magnetic field. Moreover, it also leads to an increase or decrease of the dose deposited at the interfaces between materials depending on their difference between mass densities. The results of our experiments lead to the same conclusions. These effects have to be taken into account in order to prevent creation of hot spots or a spread of energy distribution in a human body, within computation times compatible with the clinical environment. Conclusion Our model is a good candidate for future Treatment Planning Systems since it allows a faster and efficient way to plan a viable treatment for a patient in presence of strong magnetic fields. A magnetic field modifies the profile of the dose deposition such as it has to be taken into account in calculations to prevent the dramatic changes which occur at interfaces between media of different density. This work takes place in the framework of POPRA (Programme Optique Physique et Radiotherapie en Aquitaine), which involves several laboratories around problematics on the topic of cancer treatment.

Introduction This work proposes a completely new Grid-Based Boltzmann Solver (GBBS) developed for the transport and energy deposition by energetic particles. Its entropic closure and mathematical formulation allow our entropic model to calculate the delivered dose with an accuracy comparable to Monte Carlo (MC) codes with a computational time that is reduced to the order of few seconds without any special processing power requirement. Methods In contrast to discrete ordinates angular discretization methods, such as Acuros®, our method is based on a reduced number of moment equations closed with Boltzmann’s H-theorem. Keeping a good accuracy of calculations, the algorithm can simulate different treatment techniques such as the external radiotherapy even in presence of magnetic field [1] , [2] (e.g., MRI-guided radiotherapy), brachytherapy or intra-operative radiation therapy. The first validation step consists in simulating dose distributions in complex numerical phantoms including a large number of heterogeneity shapes such as bone, lung and air. For both brachytherapy and external beam radiotherapy, simulations based on CT scan, using the real phase-space of the source, have been performed. The entropic model is validated by a direct comparison with the reference MC code PENELOPE. Results The code is capable of calculating 3D dose distributions with 1 mm3voxels without statistical uncertainties in few seconds instead of several minutes like PENELOPE. In brachytherapy applications the dose distributions significantly differ from those calculated with the TG-43 approximations, thanks to its capability to account for inhomogeneities and strong density gradients. Moreover, for both applications the code shows an excellent agreement with PENELOPE within the 1%/1 mm gamma-index criterion. Conclusions In the comparison with the MC results the excellent accuracy of the model is demonstrated. Thanks to its reduced computational time and its accuracy, this model is a promising candidate to become a real-time dose calculation algorithm.

Because of stability constraints, most numerical schemes applied to hyperbolic systems of equations turn out to be costly when the flux term is multiplied by some very large scalar. This problem emerges with the M1 system of equations in the field of radiotherapy when considering heterogeneous media with very disparate densities. Additionally, the flux term of the M1 system is non-linear, and in order for the model to be well-posed the numerical solution needs to fulfill conditions called realizability. In this paper, we propose a numerical method that overcomes the stability constraint and preserves the realizability property. For this purpose, we relax the M1 system to obtain a linear flux term. Then we extend the stencil of the difference quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation example.

An accurate and rapid kinetic model describing the collisional transport of particles is presented. It is derived from the Fokker-Planck equation and involves an angular closure in the phase space leading to a set of hyperbolic equations for the moments of the distribution function evolving in time, space and energy. This method provides an alternative to the prohibitive cost of a direct solution to the full kinetic equation. Moreover, it is exact for the cases of collimated beams and the quasi-isotropic distributions. It can be approximated with the usual numerical schemes of the non-linear hyperbolic analysis. This model has features that are required to simulate the electron or ion transport for the inertial confinement fusion and the dose deposition for radiation therapy of cancers.

A reduced mathematical model for the transport of high current relativistic electron beams in a dense collisional plasma is developed. Based on the hypothesis that the density of relativistic electrons is much less than the plasma density and their energy is much higher than the plasma temperature, a model with two energy scales is proposed, where the beam and plasma electrons are considered as two coupled sub-systems, which exchange the energy and particles due to collisions. The process of energy exchange is described in the Fokker-Planck approximation, where the pitch angle electron-ion and electron-electron collisions dominate. The process of particle exchange between populations, leading to the production of secondary energetic electrons, is described with a Boltzmann term. The electron-electron collisions with small impact parameters make an important contribution in the overall dynamics of the beam electrons.

A new relaxation scheme is exhibited to approximate the weak solutions of the M 1 model to simulate radiative transfer. This numerical method uses better wavespeed approximations than the current schemes. In addition, it is proved to satisfy all the required stability properties and the asymptotic preserving property. To cite this article: C. Berthon et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

International audience; The present paper concerns the derivation of finite volume methods to approximate weak solutions of Ten-Moments equations with source terms. These equations model compressible anisotropic flows. A relaxation-type scheme is proposed to approximate such flows. Both robustness and stability conditions of the suggested finite volume methods are established. To prove discrete entropy inequalities, we derive a new strategy based on local minimum entropy principle and never use some approximate PDE's auxiliary model as usually recommended. Moreover, numerical simulations in 1D and in 2D illustrate our approach.

Hydrodynamic simulations of high-energy-density plasmas require a detailed description of energy fluxes. For low and intermediate atomic number materials, the leading mechanism is the electron transport, which may be a nonlocal phenomenon requiring a kinetic modeling. In this paper, we present and test the results of a nonlocal model based on the first angular moments of a simplified Fokker-Planck equation. This multidimensional model is closed thanks to an entropic relation (the Boltzman H-theorem). It provides a better description of the electron distribution function, thus enabling studies of small scale kinetic effects within the hydrodynamic framework. Examples of instabilities of electron plasma and ion-acoustic waves, driven by the heat flux, are presented and compared with the classical formula.

We extend the half moment entropy closure for the radiative heat transfer equations presented in Dubroca and Klar [B. Dubroca, A. Klar, Half moment closure for radiative transfer equations, J. Comput. Phys. 180 (2002) 584-596] and Turpault et al. [R. Turpault, M. Frank, B. Dubroca, A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys. 198 (2004) 363-371] to multi-D. To that end, we consider a partial moment system with general partitions of the unit sphere closed by an entropy minimization principle. We give physical and mathematical reasons for this choice of model and study its properties. Several numerical examples in different physical regimes are presented.

The M 1 radiative transfer model is considered in the present work in order to simulate the radiative fields and their interactions with the matter. The model is governed by an hyperbolic system of conservation laws supplemented by relaxation source terms. Several difficulties arise when approximating the solutions of the model; namely the positiveness of the energy, the flux limitation and and the limit diffusion behavior have to be satisfied. An HLLC scheme is exhibited and it is shown to satisfy all the required properties. A particular attention is payed concerning the approximate extreme waves. These approximations are crucial to obtain an accurate scheme. The extension to the full 2D problem is proposed. It satisfies, once again, all the expected properties. Numerical experiments are proposed. They show that the considered scheme is actually less diffusive than the currently used numerical methods.

Generation of magnetic fields around the laser speckles in underdense plasma due to an anisotropic laser heating is shown to be an important effect under the conditions previewed for the inertial confinement fusion. The anisotropy of electron distribution persists much longer than the electron collision time and creates a steady source for the quasistationary magnetic field. The structure of magnetic field around the laser beam is studied along with the effects of plasma dynamics, speckle intensity profile, and the electron heat transport.

The half space moment approximation to the radiative heat transfer equations developed in [J. Comp. Phys. 180 (2002) 584] for frequency-independent absorption and scattering is extended to frequency-dependent coefficients using averaging over frequency groups. We compare numerical results obtained with this new model to known approximations in different physical regimes.

We propose in this paper an approach for deriving in a rigorous way a family of models of mass and heat transfer in reactive porous media. At a microscopic level we propose a model coupling the Boltzmann equation in the gas phase, the heat equation on the solid phase, and appropriate interface conditions, including adsorption-desorption reactions. Several scalings are proposed, each one corresponding to a particular regime. Then an asymptotic expansion mixing homogenization and fluid limit leads to a system of coupled diffusion equations where the effective diffusion tensors are defined from the microscopic geometry of the material through auxiliary problems. Finally, we prove that the diffusion operator is elliptic, and we give algebraic and geometric conditions of degeneracy.

In plasma physics domain, the electron transport is described with the Fokker-Planck-Landau equation. The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. That is why we propose in this paper a new model whose derivation is based on an angular closure in the phase space and retains only the energy of particles as kinetic dimension. To find a solution compatible with physics conditions, the closure of the moment system is obtained under a minimum entropy principle. This model is proved to satisfy the fundamental properties like a H theorem. Moreover an entropic discretization in the velocity variable is proposed on the semi-discrete model. Finally, we validate on numerical test cases the fundamental properties of the full discrete model.

Resume Nous appliquons ici le principe de minimisation d'entropie a l'equation de transfert radiatif. Comme dans le cadre des equations cinetiques classiques, nous obtenons a l'aide de ce principe une hierarchie de systemes aux moments fermes. Chaque systeme de la hierarchie est hyperbolique, possede un flux limite par la vitesse de la lumiere et dissipe localement l'entropie. Nous analysons plus particulierement les proprietes du premier systeme de la hierarchie.

Resume On demontre le principe d'entropie minimum sur un ensemble de vitesses discret. On utilise ce resultat pour definir des modeles a vitesses discretes de l'equation BGK et des systemes aux moments de Levermore. Enfin, on demontre quelques proprietes essentielles de ces modeles.

Summary form only given. One major issue to address in Inertial Confinement Fusion (ICF) is the detailed description of the kinetic transport of laser generated fast electrons in the time and space scales of the hydrodynamic evolution of the imploded target. We have developed, at CELIA, a fast reduced kinetic model for relativistic electrons transport based on the angular moments of the relativistic Fokker-Planck equation, the M1 model1. This model takes into account the slowing down of fast electrons through collisions with plasma electrons (free and bounded), plasmons and the elastic scattering of fast electrons on plasma ions and electrons. The self-consistent magnetic and electric fields are computed thanks to a generalized Ohm law. This module has been implemented into the 2D radiation hydrodynamic code CHIC2. The M1 model is used as well as for the Fast Ignition (FI) than for the Shock Ignition (SI) schemes. A recent experiment of relativistic electrons transport through Aluminum foils is analyzed thanks to this multi-scales tool. Because of its computing speed, various initial configurations have been tested to reproduce experimental data. In addition, due to its structure, the effects of electric and magnetic fields can easily be highlighted and so the resistive fast electrons losses are directly compared to the collisional losses. Concerning Shock Ignition scheme, it is shown that the energy transfer by fast electrons from the corona to the compressed shell is a important mechanism in the creation of ablation pressure. A 30 keV energy electron beam of 2 - 5 PW/cm2 energy flux may create a pressure amplitude of more than 300 Mbar within few tens of ps in a precompressed solid material3. The dynamics of the ablation layer and the shock evolution are also presented in realistic configurations.

International audience; The present work deals with the establishment of stability conditions of finite volume methods to approximate weak solutions of the general Euler equations to simulate compressible flows. In oder to ensure discrete entropy inequalities, we derive a new technique based on a local minimum principle to be satisfied by the specific entropy. Sufficient conditions are exhibited to satisfy the required local minimum entropy principle. Arguing these conditions, a class of entropy preserving schemes is thus derived.

Treatment with high energy ionizing radiation is one of the main methods in modern cancer therapy that is in clinical use. During the last decades, two main approaches to dose calculation were used, Monte Carlo simulations and semi-empirical models based on Fermi-Eyges theory. A third way to dose calculation has only recently attracted attention in the medical physics community. This approach is based on the deterministic kinetic equations of radiative transfer. Starting from these, we derive a macroscopic partial differential equation model for electron transport in tissue. This model involves an angular closure in the phase space. It is exact for the free-streaming and the isotropic regime. We solve it numerically by a newly developed HLLC scheme based on [BerCharDub], that exactly preserves key properties of the analytical solution on the discrete level. Several numerical results for test cases from the medical physics literature are presented., Comment: 20 pages, 7 figures

Treatment with high energy ionizing radiation is one of the main methods in modern cancer therapy that is in clinical use. During the last decades, two main approaches to dose calculation were used, Monte Carlo simulations and semiempirical models based on Fermi-Eyges theory. A third way to dose calculation has only recently attracted attention in the medical physics community. This approach is based on the deterministic kinetic equations of radiative transfer. In this work, we study a full discretization of the transport equation, whose solution is supposed to serve as a benchmark for simplified methods. The computational challenge is that scattering is forward-peaked, which makes a fine resolution and thus a very large linear system of equations necessary. Traditional methods like source iteration are inefficient or fail in this case. Therefore we propose a new method which combines an incomplete factorization of the scattering matrix and several iterative steps to obtain a fast and accurate solution. Numerical examples are given.

International audience; A high order, deterministic direct numerical method is proposed for the nonrelativistic $2D_{\bf x} \times 3D_{\bf v}$ Vlasov-Maxwell system, coupled with Fokker-Planck-Landau type operators. Such a system is devoted to the modelling of electronic transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It describes the kinetics of plasma physics in the nonlocal thermodynamic equilibrium regime. Strong numerical constraints lead us to develop specific methods and approaches for validation, that might be used in other fields where couplings between equations, multiscale physics, and high dimensionality are involved. Parallelisation (MPI communication standard) and fast algorithms such as the multigrid method are employed, that make this direct approach be computationally affordable for simulations of hundreds of picoseconds, when dealing with configurations that present five dimensions in phase space.

International audience; Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper presents a numerical method based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described by a digitized 3D image. The homogenization procedure -- neglecting the effect of gas density gradients on heat transfer through the solid -- leads to closure problems in R^6 for the obtention of effective properties ; they are then simplified using a Galerkin method based on a 21-element basis set. The kinetic equations are then discretized in R^3 space with a finite-volume scheme. The method is validated against experimental data in the case of a closed test tube. It shows to be coherent with past approaches of thermal transpiration. Then, it is applied to several 3D images of increasing complexity. Another validation is brought by comparison with other distinct numerical approaches for the evaluation of the Darcian permeability tensor and of the Knudsen diffusion tensor. Results show that thermal transpiration has to be described by an effective transport tensor which is distinct from the other tensors.

In many cases, ceramic materials undergoing an ablative process (i.e. erosion, dissolution, sublimation, oxidation, etching) display a characteristic "scalloped" morphology. The length scales are not related to the ceramic microstructure, and cannot always be related to some flow pattern above the surface. We propose a simple isothermal model, where diffusion perpendicularly to the average surface and mass loss are in competition. A Hamilton-Jacobi equation is set up, and a steady, non-trivial, analytical solution consists of intersecting circle arcs in 2D. The solution is found to be stable; the curvature radius is given by the diffusion-to-reaction ratio. A 3-dimensional isothermal simulation confirms the results and the validity of the vertical flux approximation. Such a model could help to identify intrinsic mass-loss rate constants from morphology examination and the knowledge of gas diffusion coefficients.

On considere le probleme mixte pour un systeme de n lois de conservation a une dimension d'espace avec une condition aux limites en x=0, qui s'ecrit: (1.1) u t +f(u) x =0 x>0, t>0; (1.2) u(0,x)=u 0 (x) x>0; (1.3) u(t,0)=a 0 (t) t>0

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper develops a model based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described e. g. by a digitized 3D image. The homogenization procedure -- neglecting the effect of gas density gradients on heat transfer through the solid -- leads to macroscopic transfer relations, and to closure problems in R^6 for the obtention of effective properties. Coherence of the approach with previous literature on the subject is discussed. The asymptotic limits of the model (rarefied and continuum regimes) are also studied. One of the main results is that the effect of the geometry on thermal transpiration has to be described by a tensor which is distinct from the permeability and Knudsen diffusion tensors.

We present a consistent numerical model for coupling radiation to hydrodynamics at low Mach number. The hydrodynamical model is based on a low-Mach asymptotic in the compressible flow that removes acoustic wave propagation while retaining the compressibility effects resulting from combustion. Radiative transfer is modelled by the M"1 entropy equations that can be viewed as a moment method. The radiation model possesses the capability to accurately approximate solution of radiative transfer at low computational cost while retaining the main physical properties of radiative energy. Consistent numerical approaches are developed for space and time discretizations in both hydrodynamics and radiation. A modified projection method is used for hydrodynamics, whereas an HLL-type discretization is implemented for radiation transport. The combined methods permit time steps that are controlled by the advective time scales resulting in a substantial improvement in computational efficiency compared to a compressible formulation. Numerical results are presented for the natural convection in a squared cavity with large temperature difference and also for a diffusion methane/air flame with four-step reduced chemical reactions in non-gray participating media. The present approach has been found to be feasible and satisfactory.

International audience; Angular moments closures are widely used in numerical solutions of kinetic equations. While in the strongly collisional limit they provide a good approximation of the full kinetic equation, their validity domain in the weakly collisional limit is unknown. This work is devoted to defining the validity domain of the M1 model and its extensions, the two populations M1 and the M2 angular moments models for the collisionless kinetic physics applications. Three typical kinetic plasma effects are considered, which are the charged particle beams interaction, the Landau damping and the electromagnetic wave absorption in an overdense semi-infinite plasma. For each case, a perturbative analysis is performed and the dispersion relation is established using the moments models. These relations are compared with those computed by considering the Vlasov equation. The validity limits of each model are demonstrated.

This document presents a general model for Thermal Protection Systems ablation. This method is based on the wall sites concept, extended to heterogeneous and phase change reactions. It is demonstrated that relations exist between homogeneous and heterogeneous reactions and a general derivation of these relations is described. Ablation of carbon is treated to illustrate the influence of various parameters (sites, species, kinetics). This model is then applied to a realistic description of a re-entry vehicle for which the coupling with the external flow is significant and taken into account using a fully coupled code. The second part of the study compares this exact calculation to various approximations. The first one assumes a steady-state behaviour in the material. The second one is the asymptotic case of local thermodynamic equilibrium, which takes into account the species issued from the wall. Nomenclature

a significant overheating. The composites materials undergo an ablative process that consumes the heat flux by physico-chemical reactions. The composite disappears gradually by sublimation, oxidation, pyrolysis. In the case of polycristalline graphite, plasma jet experiments show a structural surface roughness like a scalloped pattern when the flow is turbulent. The main idea of this paper is to propose a reaction-diusion model in two and three dimensions in order to reproduce this shape and an experimental method to evaluate the sticking coecient for sublimation. Firstly we use an analytical expression of the recession velocity with an explicit non trivial solution (circle arcs in 2D). This velocity is evaluated by a vertical mass flux and the gaseous phase is considered isothermal and isobaric. Secondly a model is developed with a pure diusion of the carbon sublimation with a resolution in three dimensions of the mass conservation law. The simulations are performed and confirm the results of the analytical velocity and the vertical flux hypothesis.