Pierre Charrier

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.

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.

Rarefied gas flow over an in-line array of circular cylinders

A steady rarefied gas flow through periodic porous media kept at a uniform temperature is considered on the basis of the Bhatnagar–Gross–Krook equation and the diffuse reflection condition on the solid boundary. Under the assumption that the period is much smaller than the length scale of variation of the global pressure distribution, a macroscopic fluid model describing the pressure distribution and the mass flux of the gas in the medium is derived by the homogenization previously proposed by Charrier and Dubroca [Multiscale Model. Simul. 2, 124 (2003)]. The effective diffusion coefficient contained in the model is constructed numerically as a function of the Knudsen number, in the case of the medium consisting of an in-line array of circular cylinders, with the help of the numerical analysis of a rarefied gas flow in an infinite expanse of the cylinder array driven by a uniform small pressure gradient. An application of the model to an isothermal flow in a porous slab induced by a pressure difference is...

Discrete-Velocity Models for Numerical Simulations in Transitional Regime for Rarefied Flows and Radiative Transfer

In this paper we propose a deterministic eulerian approach for kinetic relaxation models. This approach is based on adefinition of the discrete equilibrium using a discrete equivalent of the minimum entropy principle. This leads to discrete models which are entropic and conservative. Two applications to gas dynamics and to radiative transfer are presented. Numerical experiments illustrate the performance of numerical codes based on this approach.

Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation

It is well known that the roughness of the wall has an effect on microscale gas flows. This effect can be shown for large Knudsen numbers by using a numerical solution of the Boltzmann equation. However, when the wall is rough at a nanometric scale, it is necessary to use a very small mesh size which is much too expansive. An alternative approach is to incorporate the roughness effect in the scattering kernel of the boundary condition, such as the Maxwell-like kernel introduced by the authors in a previous paper. Here, we explain how this boundary condition can be implemented in a Discrete Velocity approximation of the Boltzmann equation. Moreover, the influence of the roughness is shown by computing the structure scattering pattern of mono-energetic beams of the incident gas molecules. The effect of the angle of incidence of these molecules, of their mass, and of the morphology of the wall is investigated and discussed in a simplified two-dimensional configuration. The effect of the azimuthal angle of the incident beams is shown for a three-dimensional configuration. Finally, the case of non-elastic scattering is considered. All these results suggest that our approach is a promising way to incorporate enough physics of gas-surface interaction, at a reasonable computing cost, to improve kinetic simulations of micro and nano-flows.

Boundary conditions for the Boltzmann equation for rough walls

In some applications, rarefied gases have to considered in a domain whose boundary presents some nanoscale roughness. That is why, we have considered (Brull,2014) a new derivation of boundary conditions for the Boltzmann equation, where the wall present some nanoscale roughness. In this paper, the interaction between the gas and the wall is represented by a kinetic equation defined in a surface layer at the scale of the nanometer close to the wall. The boundary conditions are obtained from a formal asymptotic expansion and are describded by a scattering kernel satisfying classical properties (non-negativeness, normalization, reciprocity). Finally, we present some numerical simulations of scattering diagrams showing the importance of the consideration of roughness for small scales in the model.

Partitioning and mapping for parallel nested dissection on distributed memory architectures

In this paper, we consider the parallel implementation of a block Cholesky factorization based on a nested dissection ordering for unstructured problems. We focus on loosely coupled networks of many processors with local memory and message passing mechanism. More precisely, we study a parallel block solver associated with refined partitions from the separator partition; the aim is to find the partition corresponding to the correct granularity leading to a high quality mapping (in terms of load balancing for the processors, of average length for the routing paths, and of average edge contention on the network). Then, we propose a refinement algorithm leading to this good granularity, and we provide some numerical measurements using the mapping tool included in the ADAM environment.

‘Pointwise Degeneracy’ for Delay Evolutionary Equations

This chapter describes the pointwise degeneracy for delay evolutionary equations. A well-known property of linear differential systems in ℝ n is that the set of values of all solutions at a given time t is ℝ n itself. This property does not extend to delay differential systems. There exist systems with lag for which the set of values of all solutions is a proper subspace of ℝ n (for some time t). They are called pointwise degenerate. The chapter reviews this property in infinite-dimensional spaces. As in finite dimensions, it is easy to see that if an equation is pointwise degenerate at time t 1, it remains pointwise degenerate at every time t ≤ t 1 . It presents a condition of degeneracy as a consequence of the controllability properties of infinite-dimensional control system. It also presents an assumption in which H is a separable Hilbert space, T ( t ) is a bounded, strongly continuous semigroup on H , and A is its infinitesimal generator, with domain D ( A ). A condition of degeneracy as a consequence of the controllability properties of this infinite-dimensional control system is obtained. It is found that if the semigroup T ( t ) is holomorphic, it can be proven that the minimum time of degeneracy is kh , where k is an integer greater than or equal to 2. Similar work can be done for systems with two lags and for systems including a lag in a derivative.

Implementation of a symmetric boundary element method on distributed memory computers for 3D maxwell equations

Publisher Summary The severe computational requirements for the numerical simulation of high frequency electromagnetism problems led to a strong interest for parallel processing in this domain. This chapter focuses on the design and analysis of a parallel implementation for finite element approximations of such problems on multiple instructions, multiple data (MIMD) distributed-memory computers. Single process, multiple data (SPMD) programming model was used with a message passing mechanism for communications. The algorithm works in two steps. During the first step, the assembly constructs the linear system associated with the approximation method; and during the second step, it solves the system created. These two steps are sequentially performed and use the same distributed data structure to store the matrix of the linear system. The parallelisms induced by each of these two steps are discussed in the chapter. The choice of a symmetric variational formulation allows one to reduce the total amount of elementary computation and the storage. However, it is more difficult to achieve a good parallel efficiency in this context of symmetry for 3D problems; this chapter mainly focuses on this difficulty and in particular optimizes the first step (load balancing and communication) with respect to a well suited distribution of data for the linear solver chosen in the second step.