Efficient particle methods for solving the Boltzmann equation

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2007.Includes bibliographical references (leaves 85-86).A new particle simulation method for solving the Boltzmann equation is presented and tested. This method holds a significant computational efficiency advantage for low-signal flows compared to traditional particle methods such as the Direct Simulation Monte Carlo (DSMC). More specifically, the proposed algorithm can efficiently simulate arbitrarily small deviations from equilibrium (e.g. low speed flows) at a computational cost that does not scale with the deviation from equilibrium, while maintaining the basic algorithmic structure of DSIMC. This is achieved by incorporating the variance reduction ideas presented in [L. L. Baker and N. G. Hadjiconstantinou, Physics of Fluids, vol 17, art. no 051703, 2005] within a collision integral formulation; the latter ensures that the deviation from equilibrium remains finite and thus the calculation remains stable for collision dominated flows, in contrast to previous attempts. The formulation, developed within this thesis, is described in detail. The resulting scheme is validated for a wide range of Knudsen numbers (ratio of molecular mean free path to characteristic flow lengthscale) -- ranging from collision-dominated flow -- to collisionless flow- and a wide range of deviations from equilibrium. Excellent agreement is found with DSMC solutions for linear and weakly non-linear flows.by Thomas Homolle.S.M

An explicit time-integrator for non-smooth dynamics

National audienc

Benchmark cases for robust explicit time integrators in non-smooth transient dynamics

Abstract This article introduces benchmark cases for time integrators devoted to non-smooth impact dynamics. It focuses on numerical properties of explicit integrators. Each case tests one necessary numerical property in computational impact dynamics: energy behaviour at impact, angular momentum conservation, non-linear behaviour. The cases are easy to implement and analyse, providing a benchmark well-suited to first numerical studies. We rewrite explicit schemes for non-smooth impact dynamics with unified notations, and analyse them with the benchmark cases

Shape sensitivity analysis of an elastic contact problem: convergence of the Nitsche based finite element approximation

In a recent work, we introduced a finite element approximation for the shape optimization of an elastic structure in sliding contact with a rigid foundation where the contact condition (Signorini's condition) is approximated by Nitsche's method and the shape gradient is obtained via the adjoint state method. The motivation of this work is to propose an a priori convergence analysis of the numerical approximation of the variables of the shape gradient (displacement and adjoint state) and to show some numerical results in agreement with the theoretical ones. The main difficulty comes from the non-differentiability of the contact condition in the classical sense which requires the notion of conical differentiability

An explicit time-integrator for non-smooth dynamics

National audienc

Shape sensitivity analysis of an elastic contact problem: convergence of the Nitsche based finite element approximation

In a recent work, we introduced a finite element approximation for the shape optimization of an elastic structure in sliding contact with a rigid foundation where the contact condition (Signorini's condition) is approximated by Nitsche's method and the shape gradient is obtained via the adjoint state method. The motivation of this work is to propose an a priori convergence analysis of the numerical approximation of the variables of the shape gradient (displacement and adjoint state) and to show some numerical results in agreement with the theoretical ones. The main difficulty comes from the non-differentiability of the contact condition in the classical sense which requires the notion of conical differentiability

Homogénéisation numérique pour un couche renforcée par des fibres en grandes déformations élastiques en utilisant une méthode itérative découplée

We propose a procedure to approximate the large elastic deformations of a fiber reinforced layer by a two-scale decoupled homogenization numerical procedure. The nonlinear micro and macroscopic scales are strongly coupled in most homogenization methods which is very costly. Our method consists in decoupling the micro and macro scales by considering separate boundary value problems and an intermediate anisotropic constitutive law optimised over a training set. We propose an iterative procedure based on this method which allows to improve the quality of the approximation to get closer to the coupled homogenization and keeping a reasonable computational cost. We perform representative numerical studies for a layer with heterogeneous hyperelastic material in order to demonstrate the capability and reliability of the proposed method and test several intermediate constitutive laws

A multi-scale patch approximation for Poisson problems with a small inhomogeneous inclusion

The paper deals with the multi-scale approximation of the influence of a small inhomogeneity of arbitrary shape in an elastic medium. A new multi-scale patch method is introduced, whose caracteristic is to deal with a large scale problem without inclusion, a small-scale problem on a patch surrounding the inclusion defining a corrector and an iterative procedure between these two problems. Theoretical results of convergence of the iterations, a posteriori error estimate and comparison of the corrector with the asymptotic expansion are provided. The finite element approximation is also addressed together with some numerical tests