A topology optimization method in rarefied gas flow problems using the Boltzmann equation

This paper presents a topology optimization method in rarefied gas flow problems to obtain the optimal structure of a flow channel as a configuration of gas and solid domains. In this paper, the kinetic equation, the governing equation of rarefied gas flows, is extended over the entire design domain including solid domains assuming the solid as an imaginary gas for implicitly handling the gas-solid interfaces in the optimization process. Based on the extended equation, a 2D flow channel design problem is formulated, and the design sensitivity is obtained based on the Lagrange multiplier method and adjoint variable method. Both the rarefied gas flow and the adjoint flow are computed by a deterministic method based on a finite discretization of the molecular velocity space, rather than the DSMC method. The validity and effectiveness of our proposed method are confirmed through several numerical examples

An efficient topology optimization method for steady gas flows in all flow regimes

An efficient topology optimization method applicable to both continuum and rarefied gas flows is proposed in the framework of gas-kinetic theory. The areas of gas and solid are marked by the material density, based on which a fictitious porosity model is used to reflect the effect of the solid on the gas and mimic the diffuse boundary condition on the gas-solid interface. The formula of this fictitious porosity model is modified to make the model work well in all flow regimes, i.e., from the continuum to free-molecular flow regimes. To find the optimized material density, a gradient-based optimizer is adopted and the design sensitivity is obtained by the continuous adjoint method. To solve the primal kinetic equation and the corresponding adjoint equation, the numerical schemes efficient and accurate in all flow regimes are constructed. Several airfoil optimization problems are solved to demonstrate the good performance and high efficiency of the present topology optimization method, covering the flow conditions from continuum to rarefied, and from subsonic to supersonic

A Multiscale Kinetic-Fluid Solver with Dynamic Localization of Kinetic Effects

This paper collects the efforts done in our previous works [P. Degond, S. Jin, L. Mieussens, A Smooth Transition Between Kinetic and Hydrodynamic Equations, J. Comp. Phys., 209 (2005) 665--694.],[P.Degond, G. Dimarco, L. Mieussens, A Moving Interface Method for Dynamic Kinetic-fluid Coupling, J. Comp. Phys., Vol. 227, pp. 1176-1208, (2007).],[P. Degond, J.G. Liu, L. Mieussens, Macroscopic Fluid Model with Localized Kinetic Upscaling Effects, SIAM Multi. Model. Sim. 5(3), 940--979 (2006)] to build a robust multiscale kinetic-fluid solver. Our scope is to efficiently solve fluid dynamic problems which present non equilibrium localized regions that can move, merge, appear or disappear in time. The main ingredients of the present work are the followings ones: a fluid model is solved in the whole domain together with a localized kinetic upscaling term that corrects the fluid model wherever it is necessary; this multiscale description of the flow is obtained by using a micro-macro decomposition of the distribution function [P. Degond, J.G. Liu, L. Mieussens, Macroscopic Fluid Model with Localized Kinetic Upscaling Effects, SIAM Multi. Model. Sim. 5(3), 940--979 (2006)]; the dynamic transition between fluid and kinetic descriptions is obtained by using a time and space dependent transition function; to efficiently define the breakdown conditions of fluid models we propose a new criterion based on the distribution function itself. Several numerical examples are presented to validate the method and measure its computational efficiency.Comment: 34 page

Moment Closure - A Brief Review

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in mathematics, physics, chemistry and quantitative biolog

Application of general semi-infinite Programming to Lapidary Cutting Problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented

Porting of DSMC to multi-GPUs using OpenACC

The Direct Simulation Monte Carlo has become the method of choice for studying gas flows characterized by variable rarefaction and non-equilibrium effects, rising interest in industry for simulating flows in micro-, and nano-electromechanical systems. However, rarefied gas dynamics represents an open research challenge from the computer science perspective, due to its computational expense compared to continuum computational fluid dynamics methods. Fortunately, over the last decade, high-performance computing has seen an exponential growth of performance. Actually, with the breakthrough of General-Purpose GPU computing, heterogeneous systems have become widely used for scientific computing, especially in large-scale clusters and supercomputers. Nonetheless, developing efficient, maintainable and portable applications for hybrid systems is, in general, a non-trivial task. Among the possible approaches, directive-based programming models, such as OpenACC, are considered the most promising for porting scientific codes to hybrid CPU/GPU systems, both for their simplicity and portability. This work is an attempt to port a simplified version of the fm dsmc code developed at FLOW Matters Consultancy B.V., a start-up company supporting this project, on a multi-GPU distributed hybrid system, such as Marconi100 hosted at CINECA, using OpenACC. Finally, we perform a detailed performance analysis of our DSMC application on Volta (NVIDIA V100 GPU) architecture based computing platform as well as a comparison with previous results obtained with x64 86 (Intel Xeon CPU) and ppc64le (IBM Power9 CPU) architectures

Quantitative uniform in time chaos propagation for Boltzmann collision processes

This paper is devoted to the study of mean-field limit for systems of indistinguables particles undergoing collision processes. As formulated by Kac \cite{Kac1956} this limit is based on the {\em chaos propagation}, and we (1) prove and quantify this property for Boltzmann collision processes with unbounded collision rates (hard spheres or long-range interactions), (2) prove and quantify this property \emph{uniformly in time}. This yields the first chaos propagation result for the spatially homogeneous Boltzmann equation for true (without cut-off) Maxwell molecules whose "Master equation" shares similarities with the one of a L\'evy process and the first {\em quantitative} chaos propagation result for the spatially homogeneous Boltzmann equation for hard spheres (improvement of the %non-contructive convergence result of Sznitman \cite{S1}). Moreover our chaos propagation results are the first uniform in time ones for Boltzmann collision processes (to our knowledge), which partly answers the important question raised by Kac of relating the long-time behavior of a particle system with the one of its mean-field limit, and we provide as a surprising application a new proof of the well-known result of gaussian limit of rescaled marginals of uniform measure on the $N$-dimensional sphere as $N$ goes to infinity (more applications will be provided in a forthcoming work). Our results are based on a new method which reduces the question of chaos propagation to the one of proving a purely functional estimate on some generator operators ({\em consistency estimate}) together with fine stability estimates on the flow of the limiting non-linear equation ({\em stability estimates})