Bruno Dubroca

An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions

The M1 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.

Partial moment entropy approximation to radiative heat transfer

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.

High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications

A high order, deterministic direct numerical method is proposed for the non-relativistic 2 D x i? 3 D v Vlasov-Maxwell system, coupled with Fokker-Planck-Landau collision operators. The magnetic field is perpendicular to the 2 D x plane surface of computation, whereas the electric fields occur in this plane. Such a system is devoted to modelling of electron transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It is able to describe the kinetics of the plasma electrons in the nonlocal equilibrium regime, and permits to consider a large anisotropy degree of the distribution function. We 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. Fast algorithms are employed, which makes this direct approach computationally affordable for simulations of hundreds of collisional times.

A deterministic partial differential equation model for dose calculation in electron radiotherapy

High-energy ionizing radiation is a prominent modality for the treatment of many cancers. The approaches to electron dose calculation can be categorized into semi-empirical models (e.g. Fermi-Eyges, convolution-superposition) and probabilistic methods (e.g.Monte Carlo). A third approach 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. 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 Berthon et al (2007 J. Sci. Comput. 31 347-89) that exactly preserves the key properties of the analytical solution on the discrete level. We discuss several test cases taken from the medical physics literature. A test case with an academic Henyey-Greenstein scattering kernel is considered. We compare our model to a benchmark discrete ordinate solution. A simplified model of electron interactions with tissue is employed to compute the dose of an electron beam in a water phantom, and a case of irradiation of the vertebral column. Here our model is compared to the PENELOPE Monte Carlo code. In the academic example, the fluences computed with the new model and a benchmark result differ by less than 1%. The depths at half maximum differ by less than 0.6%. In the two comparisons with Monte Carlo, our model gives qualitatively reasonable dose distributions. Due to the crude interaction model, these so far do not have the accuracy needed in clinical practice. However, the new model has a computational cost that is less than one-tenth of the cost of a Monte Carlo simulation. In addition, simulations can be set up in a similar way as a Monte Carlo simulation. If more detailed effects such as coupled electron-photon transport, bremsstrahlung, Compton scattering and the production of delta electrons are added to our model, the computation time will only slightly increase. Its margin of error, on the other hand, will decrease and should be within a few per cent of the actual dose. Therefore, the new model has the potential to become useful for dose calculations in clinical practice.

Magnetic field generation in plasmas due to anisotropic laser heating

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.

Multigroup model for radiating flows during atmospheric hypersonic re-entry

The system that has to be solved to compute radiation hydrodynamics is quite difficult from a numerical point of view. For most applications, the simulations are done thanks to uncoupled or eventually loosely coupled codes. However, in some hypersonic regimes, the effects of radiative transfer can drastically modify the hydrodynamics flow. For such applications, it is important to have a model that fully couples hydrodynamics and radiation in order to have a good behaviour of the solution. However, coupling with the full radiative transfer equation is usually very expensive hence it is not reasonable for multidimensionnal unsteady computations. Our choice is to use a moment model for the radiation part, which is way cheaper. This model uses an entropic closure that allows to be consistant with the fundamental physical properties such as energy conservation, entropy dissipation and flux-limitation. We also developed it to be multigroup in order to correctly predict the solution of strongly frequency-dependent problems.

Asymptotic Transport Models for Heat and Mass Transfer in Reactive Porous Media

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.

Deterministic model for the transport of energetic particles: Application in the electron radiotherapy

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.

Reduced entropic model for studies of multidimensional nonlocal transport in high-energy-density plasmas

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.

A Local Entropy Minimum Principle for Deriving Entropy Preserving Schemes

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 order 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.