Space-Time Nonlinear Upscaling Framework Using Non-local Multi-continuum Approach
Space-Time Nonlinear Upscaling Framework Using Non-local Multi-continuum Approach
复制标题
使用非局部多连续体方法的时空非线性升级框架
DOI：
10.1615/intjmultcompeng.2019031829
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
M. Wheeler
中科院分区：
文献类型：
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作者：
W. Leung;Eric T. Chung;Y. Efendiev;M. Vasilyeva;M. Wheeler
在本文中，我们开发了一个时空尺度提升框架，可用于许多具有挑战性的多孔介质应用，没有尺度分离和高对比度。我们的主要重点是多尺度系数的非线性微分方程。该框架是建立在非线性非局部多连续尺度上的概念，并显着扩展了在前面的文件中的结果。
我们的方法从一个粗糙的时空分区，并确定测试功能，每个分区，这起着多连续的作用。测试函数是通过优化定义的，在非线性放大中起着至关重要的作用。在第二阶段，我们解决过采样区域的非线性局部问题，通过测试函数定义的一些约束。这些局部解定义了一个从宏观变量到细网格场的非线性映射。该图可以被认为是从宏观变量到细网格解的缩小图。在最后一个阶段，我们寻求宏观变量在整个域中，这样的降尺度场解决了一个弱意义上的全局问题定义使用测试功能。我们提出了一个例子非线性问题的分析我们的方法。
我们的统一框架在设计各种升级方法中起着重要的作用。由于局部问题与细网格问题直接相关，因此它简化了在适当约束下寻找局部解的过程。使用机器学习（ML），我们识别从宏观变量到细网格解决方案的复杂映射。我们提出了几个多孔介质的应用，包括两相流和运输的数值结果。
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 nonlinear nonlocal multi-continuum upscaling concept and significantly extends the results in the proceeding paper.
Our approach starts with a coarse space-time partition and identifies test functions for each partition, which plays a role of multi-continua. 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 variables to fine-grid solution. We present numerical results for several porous media applications, including two-phase flow and transport.