A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling
In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and mechanics in heterogenous media with nonlinear coupling. The c...
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| Format: | Article |
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Elsevier
2015
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| Online Access: | https://eprints.nottingham.ac.uk/31219/ |
| _version_ | 1848794153237872640 |
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| author | Brown, Donald Vasilyeva, Maria |
| author_facet | Brown, Donald Vasilyeva, Maria |
| author_sort | Brown, Donald |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and mechanics in heterogenous media with nonlinear coupling. The central issue being how to handle the nonlinearities and the multiscale scale nature of the media. To compute an efficient numerical solution we develop and implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves nonlinear problems on a coarse grid by constructing local multiscale basis functions and treating part of the nonlinearity locally as a parametric value. After linearization with a Picard Iteration, the procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block by treating the staggered nonlinearity as a parametric value. Using a snapshot space and local spectral problems, we construct an offline basis of reduced dimension. From here an online, parametric dependent, space is constructed. Finally, after multiplying by a multiscale partitions of unity, the multiscale basis is constructed and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions. We implement this algorithm on a geometry with a linear and nonlinear pressure dependent permeability field and compute error between the multiscale solution with the fine-scale solutions. |
| first_indexed | 2025-11-14T19:11:40Z |
| format | Article |
| id | nottingham-31219 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:11:40Z |
| publishDate | 2015 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-312192020-05-04T17:21:26Z https://eprints.nottingham.ac.uk/31219/ A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling Brown, Donald Vasilyeva, Maria In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and mechanics in heterogenous media with nonlinear coupling. The central issue being how to handle the nonlinearities and the multiscale scale nature of the media. To compute an efficient numerical solution we develop and implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves nonlinear problems on a coarse grid by constructing local multiscale basis functions and treating part of the nonlinearity locally as a parametric value. After linearization with a Picard Iteration, the procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block by treating the staggered nonlinearity as a parametric value. Using a snapshot space and local spectral problems, we construct an offline basis of reduced dimension. From here an online, parametric dependent, space is constructed. Finally, after multiplying by a multiscale partitions of unity, the multiscale basis is constructed and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions. We implement this algorithm on a geometry with a linear and nonlinear pressure dependent permeability field and compute error between the multiscale solution with the fine-scale solutions. Elsevier 2015-11-27 Article PeerReviewed Brown, Donald and Vasilyeva, Maria (2015) A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling. Journal of Computational and Applied Mathematics, 297 . pp. 132-146. ISSN 1879-1778 Multiscale Geomechanics Finite Elements http://www.sciencedirect.com/science/article/pii/S037704271500552X doi:10.1016/j.cam.2015.11.007 doi:10.1016/j.cam.2015.11.007 |
| spellingShingle | Multiscale Geomechanics Finite Elements Brown, Donald Vasilyeva, Maria A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling |
| title | A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling |
| title_full | A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling |
| title_fullStr | A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling |
| title_full_unstemmed | A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling |
| title_short | A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling |
| title_sort | generalized multiscale finite element method for poroelasticity problems ii: nonlinear coupling |
| topic | Multiscale Geomechanics Finite Elements |
| url | https://eprints.nottingham.ac.uk/31219/ https://eprints.nottingham.ac.uk/31219/ https://eprints.nottingham.ac.uk/31219/ |