Hierarchical Upscaling and Model Reduction Techniques for Multiscale Dual-continuum Systems
Abstract
Simulation in media with multiple interacting continua is often challenging due to
distinct properties of the continua, multiple scales and high contrast. Thus, some type of
model reduction is required. One of the approaches is a multi-continuum technique, where
every process in each continuum is modeled separately and an interaction term is added.
Direct numerical simulation in multiscale multi-continuum media is very expensive as
it requires a large number of degrees of freedom to completely resolve the micro-scale
variation. In this work, we present efficient upscaling and model reduction methods for
multiscale dual-continuum systems.
We first consider the numerical homogenization of a multiscale dual-continuum system
where the interaction terms between the continua are scaled as O(1/ε²) where ε is the
microscopic scale. Computing the effective coefficients of the homogenized equations
can be expensive because one needs to solve local cell problems for a large number of
macroscopic points. We develop a hierarchical approach for solving these cell problems
at a dense network of macroscopic points with an essentially optimal computation cost.
The method employs the fact that neighboring representative volume elements (RVEs)
share similar features; and effective properties of the neighboring RVEs are close to each
other. The hierarchical approach reduces computation cost by using different levels of
resolution for cell problems at different macroscopic points. Solutions of the cell problems
which are solved with a higher level of resolution are employed to correct the solutions at
neighboring macroscopic points that are computed by approximation spaces with a lower
level of resolution.
We then consider the case where the interaction terms of the dual-continuum system
are scaled as O(1/ε). We derive the homogenized problem that is a dual-continuum system which contains features that are not in the original two scale problem. In particular,
the homogenized dual-continuum system contains extra convection terms and negative
interaction coefficients while the interaction coefficient between the continua in the original
two scale system obtains both positive and negative values. We prove rigorously
the homogenization convergence and homogenization convergence rate. Homogenization
of dual-continuum system of this type has not been considered before. We present the
numerical examples for computing effective coefficients using hierarchical finite element
methods.
We assume the above mentioned homogenized equation still possess some degree of
multiscale and high contrast features caused by channels in the media. This motivates us
to develop the generalized multiscale finite element method (GMsFEM) for an upscaled
multiscale dual-continuum equations with general convection and interaction terms. GMs-
FEM systematically generates either uncoupled or coupled multiscale basis, via establishing
local snapshots and spectral decomposition in the snapshot space. Then the global
problem is solved in the constructed multiscale space with a reduced dimensional structure.
Convergence analysis of the proposed GMsFEM is accompanied with the numerical
results, which support the theoretical results.
Citation
Park, Jun Sur (2020). Hierarchical Upscaling and Model Reduction Techniques for Multiscale Dual-continuum Systems. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /192448.