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

Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model

In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are therefore necessary. To handle the resulting nonlinear monotone quasilinear elliptic equation, we use linearization based on the Picard iteration. We consider two types of basis functions, offline and online basis functions, following the general framework of GMsFEM. The offline basis functions depend nonlinearly on the solution. Thus, we design an indicator function and we will recompute the offline basis functions when the indicator function predicts that the material property has significant change during the iterations. On the other hand, we will use the residual based online basis functions to reduce the error substantially when updating basis functions is necessary. Our numerical results show that the above combination of offline and online basis functions is able to give accurate solutions with only a few basis functions per each coarse region and updating basis functions in selected iterations.Comment: 19 pages, 2 figures, submitted to Journal of Computational and Applied Mathematic

Adaptive Generalized Multiscale Model Reduction Techniques for Problems in Perforated Domains

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

Adaptive Generalized Multiscale Model Reduction Techniques for Problems in Perforated Domains

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