3,156 research outputs found
Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems
In this paper we propose a general framework for the uncertainty
quantification of quantities of interest for high-contrast single-phase flow
problems. It is based on the generalized multiscale finite element method
(GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a
hierarchy of approximations of different resolution, whereas the latter gives
an efficient way to estimate quantities of interest using samples on different
levels. The number of basis functions in the online GMsFEM stage can be varied
to determine the solution resolution and the computational cost, and to
efficiently generate samples at different levels. In particular, it is cheap to
generate samples on coarse grids but with low resolution, and it is expensive
to generate samples on fine grids with high accuracy. By suitably choosing the
number of samples at different levels, one can leverage the expensive
computation in larger fine-grid spaces toward smaller coarse-grid spaces, while
retaining the accuracy of the final Monte Carlo estimate. Further, we describe
a multilevel Markov chain Monte Carlo method, which sequentially screens the
proposal with different levels of approximations and reduces the number of
evaluations required on fine grids, while combining the samples at different
levels to arrive at an accurate estimate. The framework seamlessly integrates
the multiscale features of the GMsFEM with the multilevel feature of the MLMC
methods following the work in \cite{ketelson2013}, and our numerical
experiments illustrate its efficiency and accuracy in comparison with standard
Monte Carlo estimates.Comment: 29 pages, 6 figure
Enhanced multiscale restriction-smoothed basis (MsRSB) preconditioning with applications to porous media flow and geomechanics
A novel method to enable application of the Multiscale Restricted Smoothed
Basis (MsRSB) method to non M-matrices is presented. The original MsRSB method
is enhanced with a filtering strategy enforcing M-matrix properties to enable
the robust application of MsRSB as a preconditioner. Through applications to
porous media flow and linear elastic geomechanics, the method is proven to be
effective for scalar and vector problems with multipoint finite volume (FV) and
finite element (FE) discretization schemes, respectively. Realistic complex
(un)structured two- and three-dimensional test cases are considered to
illustrate the method's performance
Generalized multiscale finite element methods (GMsFEM)
In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method
Expanded mixed multiscale finite element methods and their applications for flows in porous media
We develop a family of expanded mixed Multiscale Finite Element Methods
(MsFEMs) and their hybridizations for second-order elliptic equations. This
formulation expands the standard mixed Multiscale Finite Element formulation in
the sense that four unknowns (hybrid formulation) are solved simultaneously:
pressure, gradient of pressure, velocity and Lagrange multipliers. We use
multiscale basis functions for the both velocity and gradient of pressure. In
the expanded mixed MsFEM framework, we consider both cases of separable-scale
and non-separable spatial scales. We specifically analyze the methods in three
categories: periodic separable scales, - convergence separable scales, and
continuum scales. When there is no scale separation, using some global
information can improve accuracy for the expanded mixed MsFEMs. We present
rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes
both conforming and nonconforming expanded mixed MsFEM. Numerical results are
presented for various multiscale models and flows in porous media with shales
to illustrate the efficiency of the expanded mixed MsFEMs.Comment: 33 page
Multiscale Solution Techniques For High-Contrast Anisotropic Problems
Anisotropy occurs in a wide range of applications. Examples include porous media, composite materials, heat transfer, and other fields in science and engineering. Due to the anisotropy, the physical property could vary significantly only in certain directions. As such, the discrete problem will have a very large condition number for traditional numerical methods. In addition, many anisotropic materials contain multiple scales and their physical properties could vary in orders of magnitude. These large variations bring an additional small-scale parameter into the problem. Thus, a proper treatment of the anisotropy not only helps to design robust iterative methods, but also provides accurate approximations of the problem.
Various well-developed techniques have been used to address anisotropic problems, such as multigrid methods, adaptive methods, and domain decomposition techniques. More recently, a large class of accurate reduced-order methods have been introduced and applied to many applications. These include multiscale finite element, multiscale finite volume, and mixed multiscale finite element methods.
The primary focus of this dissertation is to study a multiscale finite element method for the approximation of heterogeneous problems involving high-anisotropy, high-contrast, parameter dependency. First, we design robust two-level domain decomposition preconditioners using multiscale coarse spaces. Next, a general formulation of heterogeneous problem is investigated using this multiscale finite element method. Then, a multilevel multiscale finite element method is proposed and analyzed to reduce the computational cost. Last, this multiscale finite element method is extended to a convection-diffusion problem
Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT
We introduce a new geometric approach for the homogenization and inverse
homogenization of the divergence form elliptic operator with rough conductivity
coefficients in dimension two. We show that conductivity
coefficients are in one-to-one correspondence with divergence-free matrices and
convex functions over the domain . Although homogenization is a
non-linear and non-injective operator when applied directly to conductivity
coefficients, homogenization becomes a linear interpolation operator over
triangulations of when re-expressed using convex functions, and is a
volume averaging operator when re-expressed with divergence-free matrices.
Using optimal weighted Delaunay triangulations for linearly interpolating
convex functions, we obtain an optimally robust homogenization algorithm for
arbitrary rough coefficients. Next, we consider inverse homogenization and show
how to decompose it into a linear ill-posed problem and a well-posed non-linear
problem. We apply this new geometric approach to Electrical Impedance
Tomography (EIT). It is known that the EIT problem admits at most one isotropic
solution. If an isotropic solution exists, we show how to compute it from any
conductivity having the same boundary Dirichlet-to-Neumann map. It is known
that the EIT problem admits a unique (stable with respect to -convergence)
solution in the space of divergence-free matrices. As such we suggest that the
space of convex functions is the natural space in which to parameterize
solutions of the EIT problem
Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities
This paper presents a homogenisation-based constitutive model to describe the
effective tran- sient diffusion behaviour in heterogeneous media in which there
is a large contrast between the phase diffusivities. In this case mobile
species can diffuse over long distances through the fast phase in the time
scale of diffusion in the slow phase. At macroscopic scale, contrasted phase
diffusivities lead to a memory effect that cannot be properly described by
classical Fick's second law. Here we obtain effective governing equations
through a two-scale approach for composite materials consisting of a fast
matrix and slow inclusions. The micro-macro transition is similar to
first-order computational homogenisation, and involves the solution of a
transient diffusion boundary-value problem in a Representative Volume Element
of the microstructure. Different from computational homogenisation, we propose
a semi-analytical mean-field estimate of the composite response based on the
exact solution for a single inclusion developed in our previous work [Brassart,
L., Stainier, L., 2018. Effective transient behaviour of inclusions in
diffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of the
model is that the macroscopic concentration is not one-to-one related to the
macroscopic chemical potential, but obeys a local kinetic equation associated
with diffusion in the slow phase. The history-dependent macroscopic response
admits a representation based on internal variables, enabling efficient time
integration. We show that the local chemical kinetics can result in non-Fickian
behaviour in macroscale boundary-value problems.Comment: 36 pages, 14 figure
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