Adaptive Multiscale Simulation Framework for Reduced-Order Modeling in Perforated Domains

穿孔域中的过程存在于许多重要的应用中。这包括土壤、膜和过滤器中的复杂过程。利用目前的成像技术，可以构建这些穿孔材料的详细的微尺度几何形状。然而，由于规模的丰富层次结构，在这些穿孔领域中解决复杂过程的成本高得令人望而却步。为此，需要某些类型的降阶计算技术。这个项目的目标是开发和分析新的计算技术，以解决穿孔领域中具有挑战性的多尺度问题。新的方法将把来自细节几何的信息带到大规模模拟中，并将改善模拟中的预测。这将进一步允许设计新材料和优化工艺。目前解决穿孔域问题的许多多尺度方法都局限于均化，这种方法适用于介质具有尺度分离的情况。然而，在许多现实的射孔介质中，没有鳞片分离，即孔隙大小可以有各种各样的鳞片。这个项目的多尺度方法开发了一个总体框架，允许严格和系统地减少。PI的目标是：(1)开发用于计算多尺度基函数的系统的局部模型降阶工具；(2)开发和分析使用这些基函数的新的有限元技术；(3)研究基函数的局部化和全局耦合机制之间的相互作用；(4)将所开发的方法应用于各种具有非线性和多物理的三维流动；(5)测试和展示它们解决工程和地球科学问题的能力。

Processes in perforated domains occur in many important applications. These include complex processes in soil, membranes, and filters. With current imaging techniques, detailed microscale geometries of these perforated materials can be constructed. However, it is prohibitively expensive to solve complex processes in these perforated domains due to a rich hierarchy of scales. For this reason, some types of reduced-order computational techniques are needed. The goal of this project is to develop and analyze novel computational techniques for solving challenging multiscale problems in perforated domains. The new approaches will bring the information from the detailed geometries to large-scale simulations and will improve the predictions in the simulations. This will further allow deigning new materials and optimize processes. Many current approaches for multiscale methods for problems in perforated domains have been restricted to homogenization, which is applicable when the media has scale separation. However, in many realistic perforated media, there is no scale separation, i.e., pore sizes can have a wide variety of scales. The multiscale methods of this project develop a general framework that allows rigorous and systematic reduction. The PI's goals are: (1) to develop systematic local model reduction tools for computing multiscale basis functions; (2) to develop and analyze new finite element techniques using these basis functions; (3) to study the interplay between localization of the basis functions and the global coupling mechanism; (4) to apply the developed methods to a wide variety of flows with nonlinearities and multiphysics in 3D; (5) to test and demonstrate their capabilities for solving problems in engineering and geosciences.