Multigrid and M-Matrices in the Finite Pointset Method

We consider the Finite Pointset Method (FPM) for incompressible flows. In the classical FPM derivatives are approximated by a least squares approximation. In general this approach yields stencils with both positive and negative entries. We present how optimization routines can force the stencils to have only positive weights aside from the central point. This approach yields an M-matrix structure, which is of interest for various linear solvers. We investigate algebraic multigrid to solve the arising linear systems. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From the BGK model to the Navier-Stokes equations

We give here a complete derivation of the Navier-Stokes-Fourier equations from a model collisional kinetic equation, the BGK model. Though physically unrealistic, this model shares some common features with more classical models such as the Boltzmann equation.Then the program developed by Bardos, Golse and Levermore [Fluid dynamic limits of kinetic equations II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46 (5) (1993) 667-753] to study hydrodynamic limits of the steady Boltzmann equation, and extended by Lions and Masmoudi [From Boltzmann equations to Navier-Stokes equations I, Archive Rat. Mech. Anal. 158 (2001) 173-193] in the time-dependent case, can be adapted here, and gives the expected convergence result provided that the particle density f satisfies some integrability assumption.The originality of the present work is to remove this assumption by establishing refined a priori estimates. The crucial idea is to decompose f as (f−M_f)+M_f where M_f is the local Maxwellian associated with f. The first term is then estimated by means of the entropy dissipation, while the other is smooth in v. A mixing property of the operator (ε∂_t+v.∇_x) allows to transfer some of this extra-integrability on the variable x.     

Kinetic decomposition for kinetic models of BGK type

In this paper, we consider kinetic models of BGK type which describe the scalar conservation law at the microscopic scale. We use new technique developed in Comm. Partial Differential Equations 27 (2002) 1229 in order to get the convergence. First, we obtain the approximate transport equation for the given kinetic models of BGK type. Then using the averaging lemma, we obtain the convergence. This paper shows how to relate the given kinetic model with the averaging lemma to get the convergence.     

Numerical properties of high order discrete velocity solutions to the BGK kinetic equation

A high order numerical method for the solution of model kinetic equations is proposed. The new method employs discontinuous Galerkin (DG) discretizations in the spatial and velocity variables and Runge-Kutta discretizations in the temporal variable. The method is implemented for the one-dimensional Bhatnagar-Gross-Krook equation. Convergence of the numerical solution and accuracy of the evaluation of macroparameters are studied for different orders of velocity discretization. Synthetic model problems are proposed and implemented to test accuracy of discretizations in the free molecular regime. The method is applied to the solution of the normal shock wave problem and the one-dimensional heat transfer problem.     

Application of the Finite Cell Method to patient-specific femur simulations

Standard methods for predicting the mechanical response of a human femur bone from quantitative computer-tomography (qCT) scans are classically based on the h-version of the finite element method. These methods are often limited in accuracy and efficiency due to the need for segmentation and the slow convergence rate. With the Finite Cell Method (FCM) a high-order fictitious domain method has been developed that overcomes the aforementioned problems and provides accurate results when compared to high-order finite element methods and experimental results. Herein the FCM applied to the analysis of a patient-specific femur is presented. The femur model is determined based on qCT-scans and the elastic response under compression is presented in terms of strains and displacements. The results are compared with a p-FE analysis and validated by results from an in-vitro test of the modeled femur. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rarefied gas channel flow using the BGK model

Zusammenfassung Das Problem der Strömungsbildung in Couetteströmung suspendierter Partikel in Flüssigkeiten wird mit Hilfe der Theorie der Flüssigkeiten mit Mikrostruktur entwickelt. Die Reaktion der Mikrostruktur auf das entwickelte Geschwindigkeitsfeld, die Grenzschicht und die resultierende Schubspannung werden untersucht.     

聚合物时温等效模型有限元应用研究

为更好地描述聚合物材料力学性能的温度相关性问题,对目前广泛应用的WLF模型进行改进研究,并引入“零时间”因子提高了粘弹性材料变温松弛模量的获取精度.在此基础上基于ABAQUS用户材料子程序UTRS将时温等效模型应用到数值计算中.根据不同温度水平下的应力松弛实验获得模型参数,并通过等速拉伸实验与数值结果的对比验证了该模型及其有限元方法的可行性及正确性.结果表明:引入“零时间”因子的变温松弛模量精度更高;改进WLF模型对复合推进剂具有更好的适用性和更高的精确度.     

Long-time behaviour of a one-dimensional BGK model: convergence to macroscopic rarefaction waves

For one-dimensional kinetic BGK models, regarded as relaxation models for scalar conservation laws with genuinely nonlinear fluxes, we prove that the macroscopic density converges to the rarefaction wave solution of the corresponding scalar conservation law in the long time limit, and that the phase space density approaches an equilibrium distribution with the rarefaction wave as macroscopic density. The proof requires a smallness assumption on the amplitude of the rarefaction waves and uses a micro-macro decomposition of the perturbation equation.     

Cauchy problem for the ellipsoidal BGK model for polyatomic particles

We establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal BGK model, which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level.     

Long-time behaviour of a one-dimensional BGK model: convergence to macroscopic rarefaction waves

For one-dimensional kinetic BGK models, regarded as relaxation models for scalar conservation laws with genuinely nonlinear fluxes, we prove that the macroscopic density converges to the rarefaction wave solution of the corresponding scalar conservation law in the long time limit, and that the phase space density approaches an equilibrium distribution with the rarefaction wave as macroscopic density. The proof requires a smallness assumption on the amplitude of the rarefaction waves and uses a micro-macro decomposition of the perturbation equation.     

Global weak solutions to the relativistic BGK equation

Abstract

In this paper, the global existence of weak solutions to the relativistic BGK model for the relativistic Boltzmann equation is analyzed. The proof relies on the strong compactness of the density, velocity, and temperature under minimal assumptions on the control of some moments of the initial condition together with the initial entropy.     

Modeling of two-phase flows with surface tension by finite pointset method (FPM)

A meshfree method for two-phase immiscible incompressible flows including surface tension is presented. The continuum surface force (CSF) model is used to include the surface tension force. The incompressible Navier–Stokes equation is considered as the mathematical model. Application of implicit projection method results in linear second-order partial differential equations for velocities and pressure. These equations are then solved by the finite pointset method (FPM), which is a meshfree and Lagrangian method. The fluid is represented as finite number of particles and the immiscible fluids are distinguished by the color of each particle. The interface is tracked automatically by advecting the color functions for each particle. Two test cases, Laplace's law and the Rayleigh–Taylor instability in 2D have been presented. The results are found to be consistent with the theoretical results.     

Finite time ruin problems for the Erlang(2) risk model

We consider the Erlang(2) risk model and derive expressions for the density of the time to ruin and the joint density of the time to ruin and the deficit at ruin when the individual claim amount distribution is (i) an exponential distribution and (ii) an Erlang(2) distribution. We also consider the special case when the initial surplus is zero.     

Convergence of a continuous BGK model for conservation laws on the quarter plane

We consider a scalar conservation law in the quarter plan. This equation is approximated in a kinetic BGK model with infinite set of velocities. The convergence is established in the general BV framework, without special restrictions on the flux nor on the equilibrium problem's data.     

BGK方程矩的传播

对Boltzman方程的BGK模型,我们证明了,对任意s>2,如果初始值的s阶矩有限,则其分布解的s阶矩在任何时间区间[0,T]上保持有界.