639 research outputs found
A Generalized Multiscale Finite Element Method for the Brinkman Equation
In this paper we consider the numerical upscaling of the Brinkman equation in
the presence of high-contrast permeability fields. We develop and analyze a
robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for
the Brinkman model. In the fine grid, we use mixed finite element method with
the velocity and pressure being continuous piecewise quadratic and piecewise
constant finite element spaces, respectively. Using the GMsFEM framework we
construct suitable coarse-scale spaces for the velocity and pressure that yield
a robust mixed GMsFEM. We develop a novel approach to construct a coarse
approximation for the velocity snapshot space and a robust small offline space
for the velocity space. The stability of the mixed GMsFEM and a priori error
estimates are derived. A variety of two-dimensional numerical examples are
presented to illustrate the effectiveness of the algorithm.Comment: 22 page
Recommended from our members
Mini-Workshop: Numerical Upscaling for Media with Deterministic and Stochastic Heterogeneity
This minisymposium was third in series of similar events, after two very successful meetings in 2005 and 2009. The aim was to provide a forum for an extensive discussion on the theoretical aspects and on the areas of application and validity of numerical upscaling approaches for heterogeneous problems with deterministic and stochastic coefficients. The intensive discussions during the meeting contributed to a better understanding of upscaling approaches for multiscale problems with stochastic coefficients, and for synergy between scientists coming to this topic from the area of deterministic multiscale problems on one hand, and those coming from the area of SPDE on the other hand. Recent advanced results on upscaling approaches for deterministic multiscale problems were presented, well mixed with strong presentations on SDE and SPDE. The open problems in these areas were discussed, with emphasis on the case of stochastic coefficients brainstorming numerous numerical upscaling approaches. A number of young researchers, very actively working in these areas, were involved in the workshop discussing the links between scales., thus ensuring the continuity between the generations of researchers
Homogenized lattice Boltzmann model for simulating multi-phase flows in heterogeneous porous media
A homogenization approach for the simulation of multi-phase flows in heterogeneous porous media is presented. It is based on the lattice Boltzmann method and combines the grayscale with the multi-component ShanâChen method. Thus, it mimics fluidâfluid and solidâfluid interactions also within pores that are smaller than the numerical discretization. The model is successfully tested for a broad variety of single- and two-phase flow problems. Additionally, its application to multi-scale and multi-phase flow problems in porous media is demonstrated using the electrolyte filling process of realistic 3D lithium-ion battery electrode microstructures as an example. The approach presented here shows advantages over comparable methods from literature. The interfacial tension and wetting conditions are independent and not affected by the homogenization. Moreover, all physical properties studied here are continuous even across interfaces of porous media. The method is consistent with the original multi-component ShanâChen method (MCSC). It is as stable as the MCSC, easy to implement, and can be applied to many research fields, especially where multi-phase fluid flow occurs in heterogeneous and multi-scale porous media
Homogenized lattice Boltzmann methods for fluid flow through porous media -- part I: kinetic model derivation
In this series of studies, we establish homogenized lattice Boltzmann methods
(HLBM) for simulating fluid flow through porous media. Our contributions in
part I are twofold. First, we assemble the targeted partial differential
equation system by formally unifying the governing equations for nonstationary
fluid flow in porous media. A matrix of regularly arranged, equally sized
obstacles is placed into the domain to model fluid flow through porous
structures governed by the incompressible nonstationary Navier--Stokes
equations (NSE). Depending on the ratio of geometric parameters in the matrix
arrangement, several homogenized equations are obtained. We review existing
methods for homogenizing the nonstationary NSE for specific porosities and
discuss the applicability of the resulting model equations. Consequently, the
homogenized NSE are expressed as targeted partial differential equations that
jointly incorporate the derived aspects. Second, we propose a kinetic model,
the homogenized Bhatnagar--Gross--Krook Boltzmann equation, which approximates
the homogenized nonstationary NSE. We formally prove that the zeroth and first
order moments of the kinetic model provide solutions to the mass and momentum
balance variables of the macrocopic model up to specific orders in the scaling
parameter. Based on the present contributions, in the sequel (part II), the
homogenized NSE are consistently approximated by deriving a limit consistent
HLBM discretization of the homogenized Bhatnagar--Gross--Krook Boltzmann
equation
Recommended from our members
Mini-Workshop: Numerical Upscaling for Flow Problems: Theory and Applications
The objective of this workshop was to bring together researchers working in multiscale simulations with emphasis on multigrid methods and multiscale ïŹnite element methods, aiming at chieving of better understanding and synergy between these methods. The goal of multiscale ïŹnite element methods, as upscaling methods, is to compute coarse scale solutions of the underlying equations as accurately as possible. On the other hand, multigrid methods attempt to solve ïŹne-scale equations rapidly using a hierarchy of coarse spaces. Multigrid methods need âgoodâ coarse scale spaces for their eïŹciency. The discussions of this workshop partly focused on approximation properties of coarse scale spaces and multigrid convergence. Some other presentations were on upscaling, domain decomposition methods and nonlinear multiscale methods. Some researchers discussed applications of these methods to reservoir simulations, as well as to simulations of ïŹltration, insulating materials, and turbulence
Asymptotic expansions for high-contrast elliptic equations
In this paper, we present a high-order expansion for elliptic equations in
high-contrast media. The background conductivity is taken to be one and we
assume the medium contains high (or low) conductivity inclusions. We derive an
asymptotic expansion with respect to the contrast and provide a procedure to
compute the terms in the expansion. The computation of the expansion does not
depend on the contrast which is important for simulations. The latter allows
avoiding increased mesh resolution around high conductivity features. This work
is partly motivated by our earlier work in \cite{ge09_1} where we design
efficient numerical procedures for solving high-contrast problems. These
multiscale approaches require local solutions and our proposed high-order
expansion can be used to approximate these local solutions inexpensively. In
the case of a large-number of inclusions, the proposed analysis can help to
design localization techniques for computing the terms in the expansion. In the
paper, we present a rigorous analysis of the proposed high-order expansion and
estimate the remainder of it. We consider both high and low conductivity
inclusions
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
- âŠ