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

孔隙弹性是一个连续介质力学模型的框架，用于研究涉及多孔弹性介质和流体流动的问题。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。人们确实需要获得孔隙弹性的高分辨率数值解，但这需要大量的计算资源，即使是理论上最优的算法。该项目将开发能够以可行的计算成本为大型孔隙弹性问题提供高精度解决方案的方法。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系列具有不同物理参数的耦合偏微分方程来描述的参数偏微分方程。对于这些涉及复杂物理参数形式的问题，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至关重要。该项目将使用降基方法（RBM）方法，该方法可以为椭圆型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，可以推导具有严格的认证误差界限的降阶模型。然而，为耦合偏微分方程系统开发RBM并非易事，因为它必须设计成保持耦合系统结构的稳定性。本项目开发的技术和分析将促进对参数孔隙弹性及其相关问题的理解。本文的研究结果也可以为其它耦合参数偏微分方程的rbm开发提供一般指导。一旦开发出来，这些算法可以用于解决目前传统数值算法难以解决的复杂问题。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，对人类的生命和健康产生潜在的有益影响。该奖项反映了美国国家科学基金会的法定使命，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.