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International Journal for Multiscale Computational Engineering
Fator do impacto: 1.016 FI de cinco anos: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Imprimir: 1543-1649
ISSN On-line: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2019031829
pages 529-550

SPACE-TIME NONLINEAR UPSCALING FRAMEWORK USING NONLOCAL MULTICONTINUUM APPROACH

Wing T. Leung
Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas, Austin, TX 78712, USA
Eric T. Chung
Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong SAR, China
Yalchin Efendiev
Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX 77840, USA; Multiscale Model Reduction Laboratory, North-Eastern Federal University, Yakutsk, Russia, 677980
Maria Vasilyeva
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX 77840, USA; Multiscale Model Reduction Laboratory, North Eastern Federal University, Yakutsk, Russia
Mary Wheeler
Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas, Austin, TX 78712, USA

RESUMO

In this paper, we develop a space-time upscaling framework that can be used for many challenging porous media applications without scale separation and high contrast. Our main focus is on nonlinear differential equations with multiscale coefficients. The framework is built on a nonlinear nonlocal multicontinuum upscaling concept and significantly extends the results of earlier work. Our approach starts with a coarse space-time partition and identifies test functions for each partition, which play the role of multicontinua. The test functions are defined via optimization and play a crucial role in nonlinear upscaling. In the second stage, we solve nonlinear local problems in oversampled regions with some constraints defined via test functions. These local solutions define a nonlinear map from macroscopic variables determined with the help of test functions to the fine-grid fields. This map can be thought as a downscaled map from macroscopic variables to the fine-grid solution. In the final stage, we seek macroscopic variables in the entire domain such that the downscaled field solves the global problem in a weak sense defined using the test functions. We present an analysis of our approach for an example nonlinear problem. Our unified framework plays an important role in designing various upscaled methods. Because local problems are directly related to the fine-grid problems, it simplifies the process of finding local solutions with appropriate constraints. Using machine learning (ML), we identify the complex map from macroscopic variablesto fine-grid solution. We present numerical results for several porous media applications, including two-phase flow and transport.

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