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

孔弹性是一种连续介质力学模型的框架，用于解决涉及多孔弹性介质和流体流动的问题。孔隙弹性问题在石油工程中具有实际应用，如石油工程中的油气提取、人体内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中的岩浆和地幔迁移模拟等。确实需要获得孔弹性的高分辨率数值解，但这需要大量的计算资源，即使使用理论上最优的算法也是如此。该项目将开发能够以可行的计算成本为大的孔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。孔弹性问题是由一组物理参数不同的耦合偏微分方程组描述的参数偏微分方程组。为了在较短的时间内获得高分辨率的解，发展可靠和高效的数值方法对于这些涉及复杂形式的物理参数的问题是至关重要的。该项目将使用减基方法(RBM)方法，该方法可以为椭圆型和抛物型参数偏微分方程组提供快速算法，并具有坚实的数学框架来推导具有严格的、经证明的误差界的降阶模型。然而，为耦合偏微分方程组系统开发RBM并非易事，因为它必须被设计成保持耦合系统的稳定结构。本项目所开发的技术和分析将促进对参数孔弹性及其相关问题的理解。本项目的研究结果也可以为其他耦合参数偏微分方程组的RBM的开发提供一般的指导。一旦开发出来，这些算法就可以用来解决目前用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智力优势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.