37 research outputs found
Numerical extraction of a macroscopic pde and a lifting operator from a lattice Boltzmann model
Lifting operators play an important role in starting a lattice Boltzmann
model from a given initial density. The density, a macroscopic variable, needs
to be mapped to the distribution functions, mesoscopic variables, of the
lattice Boltzmann model. Several methods proposed as lifting operators have
been tested and discussed in the literature. The most famous methods are an
analytically found lifting operator, like the Chapman-Enskog expansion, and a
numerical method, like the Constrained Runs algorithm, to arrive at an implicit
expression for the unknown distribution functions with the help of the density.
This paper proposes a lifting operator that alleviates several drawbacks of
these existing methods. In particular, we focus on the computational expense
and the analytical work that needs to be done. The proposed lifting operator, a
numerical Chapman-Enskog expansion, obtains the coefficients of the
Chapman-Enskog expansion numerically. Another important feature of the use of
lifting operators is found in hybrid models. There the lattice Boltzmann model
is spatially coupled with a model based on a more macroscopic description, for
example an advection-diffusion-reaction equation. In one part of the domain,
the lattice Boltzmann model is used, while in another part, the more
macroscopic model. Such a hybrid coupling results in missing data at the
interfaces between the different models. A lifting operator is then an
important tool since the lattice Boltzmann model is typically described by more
variables than a model based on a macroscopic partial differential equation.Comment: submitted to SIAM MM
A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations
The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of C(1 + log(H/h))2 is obtained for the condition number of the preconditioned system where C is a constant independent of h or H or large jumps in the coefficient of the problem. Numerical simulations are presented which confirm the theoretical
results. A key component for the development and analysis of the BDDC algorithm is a novel perspective presenting the DG discretization as the sum of element-wise “local” bilinear forms. The element-wise perspective allows for a simple unified analysis of a variety of DG methods and leads naturally to the appropriate choice for the subdomain-wise local bilinear forms. Additionally, this new perspective enables a connection to be drawn between the DG discretization and a related continuous finite element discretization to simplify the analysis of the BDDC algorithm.Boeing CompanyMassachusetts Institute of Technology (Zakhartchenko Fellowship
The Moment Guided Monte Carlo method for the Boltzmann equation
In this work we propose a generalization of the Moment Guided Monte Carlo
method developed in [11]. This approach permits to reduce the variance of the
particle methods through a matching with a set of suitable macroscopic moment
equations. In order to guarantee that the moment equations provide the correct
solutions, they are coupled to the kinetic equation through a non equilibrium
term. Here, at the contrary to the previous work in which we considered the
simplified BGK operator, we deal with the full Boltzmann operator. Moreover, we
introduce an hybrid setting which permits to entirely remove the resolution of
the kinetic equation in the limit of infinite number of collisions and to
consider only the solution of the compressible Euler equation. This
modification additionally reduce the statistical error with respect to our
previous work and permits to perform simulations of non equilibrium gases using
only a few number of particles. We show at the end of the paper several
numerical tests which prove the efficiency and the low level of numerical noise
of the method.Comment: arXiv admin note: text overlap with arXiv:0908.026
Compactness of linearized kinetic operators
International audienceThis article reviews various results on the compactness of the linearized Boltzmann operator and of its generalization to mixtures of non-reactive monatomic gases
A comparison of homogenization and standard mechanics analyses for periodic porous composites
Composite material elastic behavior has been studied using many approaches, all of which are based on the concept of a Representative Volume Element (RVE). Most methods accurately estimate effective elastic properties when the ratio of the RVE size to the global structural dimensions, denoted here as ν, goes to zero. However, many composites are locally periodic with finite ν. The purpose of this paper was to compare homogenization and standard mechanics RVE based analyses for periodic porous composites with finite ν. Both methods were implemented using a displacement based finite element formulation. For one-dimensional analyses of composite bars the two methods were equivalent. Howver, for two- and three-dimensional analyses the methods were quite different due to the fact that the local RVE stress and strain state was not determined uniquely by the applied boundary conditions. For two-dimensional analyses of porous periodic composites the effective material properties predicted by standard mechanics approaches using multiple cell RVEs converged to the homogenization predictions using one cell. In addition, homogenization estimates of local strain energy density were within 30% of direct analyses while standard mechanics approaches generally differed from direct analyses by more than 70%. These results suggest that homogenization theory is preferable over standard mechanics of materials approaches for periodic composites even when the material is only locally periodic and ν is finite.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47812/1/466_2004_Article_BF00369853.pd
Calcul du buckling associe a une equation de transport de neutrons dans un milieu periodique
Available from CEN Saclay, Service de Documentation, 91191 Gif-sur-Yvette Cedex (France) / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc