Adaptive Generalized Multiscale Model Reduction Techniques for Problems in Perforated Domains
Abstract
Multiscale modeling of complex physical phenomena in many areas, including hydrogeology,
material science, chemistry and biology, consists of solving problems in highly
heterogeneous porous media. In many of these applications, differential equations are formulated
in perforated domains which can be considered as the region outside of inclusions
or connected bodies of various sizes. Due to complicated geometries of these inclusions,
solutions to these problems have multiscale features. Taking into account the uncertainties,
one needs to solve these problems extensively many times. Model reduction techniques
are significant for problems in perforated domains in order to improve the computational
efficiency.
There are some existing approaches for model reduction in perforated domains including
homogenization, heterogeneous multiscale methods and multiscale finite element
methods. These techniques typically consider the case when there is a scale separation or
the perforation distribution is periodic, and assume that the solution space can be approximated
by the solutions of directional cell problems and the effective equations contain a
limited number of effective parameters.
For more complicated problems where the effective properties may be richer, we are
interested in developing systematic local multiscale model reduction techniques to obtain
accurate macroscale representations of the underlying fine-scale problem in highly heterogeneous
perforated domains. In this dissertation, based on the framework of Generalized
Multiscale Finite Element Method, we develop novel methods and algorithms including
(1) development of systematic local model reduction techniques for computing multiscale
basis in perforated domains, (2) numerical analysis and exhaustive simulation utilizing the
proposed basis functions, (3) design of different applicable global coupling frameworks
and (4) applications to various problems with challenging engineering backgrounds. Our
proposed methods can significantly advance the computational efficiency and accuracy for
multiscale problems in perforated media.
Citation
Wang, Yating (2018). Adaptive Generalized Multiscale Model Reduction Techniques for Problems in Perforated Domains. Doctoral dissertation, Texas A & M University. Available electronically from https : / /hdl .handle .net /1969 .1 /173657.