Research Initiation Award: Fast Solvers for Variable-Coefficient Poroelastic Models

研究启动奖为历史上黑人学院和大学的教师提供支持，他们正在建立一个研究项目。预计该奖项将有助于进一步提高教师的研究能力和效率，改善其所在机构的研究和教学，并使本科生参与研究经验。授予摩根州立大学的奖项在许多领域具有潜在的更广泛的社会影响。该项目的重点是从许多重要的应用所产生的数值方法的发展。该研究跨越了数学、计算物理和材料科学领域。本科生和研究生将获得研究经验。本项目侧重于发展变系数多孔弹性模型的数值方法。本研究将提出、分析并实现几种快速、优化和可扩展的空间离散下的多孔弹性模型数值算法，包括有限元法、有限体积法和浸入界面法。具体而言，新的数值方法，如预处理方法，多重网格方法，区域分解方法，和加速技术的传统迭代方法，将开发和研究。该数值算法将大大提高求解效率。该研究具有改进传统的有限差分和有限元方法的线性和非线性求解器的多孔弹性模型在各种应用的潜力。作为一个例子，所开发的数值算法可以用于模拟能量存储在地下，这需要大规模的数值计算;此外，所开发的数值方法可以应用于模拟脑肿胀模型，从而量化脑水肿评估。结合图像数据和患者特定的数据，如脑血流状况，数值方法可用于模拟缺血条件下或创伤性脑损伤后的脑肿胀。这些算法将作为开放源码软件包实施。

Research Initiation Awards provide support for faculty at Historically Black Colleges and Universities who are building a research program. It is expected that the award helps to further the faculty member's research capability and effectiveness, improves research and teaching at his home institution, and involves undergraduate students in research experiences. The award to Morgan State University has potential broader and societal impact in a number of areas. The project focuses on the development of numerical methods arising from many important applications. The research crosses the fields of mathematics, computational physics and material sciences. Undergraduate and graduate students will gain research experiences.This project focuses on the development of numerical methods for variable-coefficient poroelastic models. The research will propose, analyze and implement several fast, optimal and scalable numerical algorithms for poroelastic models under the spatial discretizations including the finite element method, finite volume method and immersed interface method. Specifically, novel numerical approaches, such as preconditioning methods, Multigrid methods, domain decomposition methods, and the acceleration techniques for the conventional iterative methods, will be developed and investigated. The numerical algorithms will greatly improve the efficiency of poroelastic solvers. The research has the potential of improving the linear and nonlinear solvers for the conventional Finite Difference and Finite Element methods for poroelastic models in various applications. As an example, the developed numerical algorithms can be used in simulating energy storage in subsurface, which requires large-scale numerical computations; furthermore, the developed numerical methods can be applied to simulate a brain swelling model and therefore quantify brain edema assessment. Combined with image data and patient-specific data such as cerebral blood flow conditions, the numerical methods can be used for simulating brain swelling under ischemic conditions or after traumatic brain injury. The algorithms will be implemented as open source software packages.