Collaborative Research: Development of Reduced Order Models for Poroelasticity and Related Problems

多孔弹性力学是研究多孔弹性介质和流体流动问题的连续介质力学模型框架。孔隙弹性问题在现实世界中有着广泛的应用，如石油工程中的碳氢化合物提取，人体中的血液流动等生理过程，环境工程中的地下水污染，以及地球物理学中的岩浆和地幔迁移建模。有一个真实的需要获得高分辨率的数值解的孔隙弹性，但这些需要大量的计算资源，即使理论上最优的算法。本计画将发展能以可行的计算成本，提供高精度解大型孔隙弹性问题的方法。学生将参与并接受跨学科应用的培训。 孔隙弹性问题是由一组具有不同物理参数的耦合偏微分方程描述的参数偏微分方程。为这些涉及复杂形式的物理参数的问题开发可靠和有效的数值方法对于在相对较短的时间内获得高分辨率的解决方案至关重要。该项目将使用缩减基方法，该方法可以为椭圆和抛物型参数偏微分方程提供快速算法，并具有坚实的数学框架，以推导具有严格的经认证的误差范围的降阶模型。然而，开发一个RBM耦合偏微分方程系统是不平凡的，因为它必须被设计为保持耦合系统的稳定性结构。本计画所发展之技术与分析，将增进对参数孔隙弹性力学及其相关问题之了解。本计画之研究结果亦可提供其他耦合参数偏微分方程之RBM发展之一般性指导。一旦开发，这些算法可以用于目前难以处理的复杂问题，与传统的数值算法。这些解决方案将导致石油工程、环境工程和计算生物力学等领域的进步，并对人类的生命和健康产生潜在的有益影响。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Poroelasticity is a framework of continuum mechanics models for problems involving a porous elastic medium and a fluid flow. Poroelasticity problems have real-world applications such as hydrocarbon extraction in petroleum engineering, physiological processes such as the blood flow in the human body, groundwater contamination in environmental engineering, and modeling magma and mantle migration in geophysics. There is a real need to obtain high-resolution numerical solutions for the poroelasticity but these require large computational resources, even with theoretically optimal algorithms. This project will develop methods that can provide high accuracy solutions for large poroelasticity problems with feasible computational costs. Students will be involved and trained in interdisciplinary applications. Poroelasticity problems are parametric partial differential equations (PDEs) which are described by a set of coupled PDEs with various physical parameters. Developing reliable and efficient numerical methods for these problems involving complex forms of physical parameters is crucial to obtain high-resolution solutions in a relatively short time. The project will use the reduced basis method (RBM) approach which can provide fast algorithms for parametric partial differential equations of elliptic and parabolic types and has a solid mathematical framework to derive reduced order models with rigorous, certified error bounds. However, developing an RBM for a system of coupled PDEs is nontrivial because it must be devised to preserve the stability structure of the coupled system. The techniques and analysis developed in this project will advance the understanding of parametric poroelasticity and its related problems. The results of this project can also provide a general guideline in developing RBMs for other coupled parametric PDEs. Once developed, these algorithms can be employed for complex problems that are currently intractable with traditional numerical algorithms. These solutions will lead to advances in fields such as petroleum engineering, environmental engineering, and computational biomechanics with potential beneficial impacts on human life and health.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.