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

研究启动奖为传统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。期望该奖项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。摩根州立大学获得的奖项在许多领域具有潜在的更广泛和社会影响。该项目侧重于从许多重要应用中产生的数值方法的发展。这项研究跨越了数学、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。本项目的重点是发展变系数孔隙弹性模型的数值方法。本研究将提出、分析并实现空间离散化条件下的几种快速、最优、可扩展的孔隙弹性模型数值算法，包括有限元法、有限体积法和浸入界面法。具体来说，新的数值方法，如预处理方法、多重网格方法、区域分解方法和传统迭代方法的加速技术，将被开发和研究。数值算法将大大提高孔隙弹性求解器的效率。该研究具有改进传统的有限差分法和有限元法在各种应用中的线性和非线性求解方法的潜力。例如，所开发的数值算法可用于模拟地下蓄能，这需要大规模的数值计算；此外，所开发的数值方法可用于模拟脑肿胀模型，从而量化脑水肿评估。结合图像数据和脑血流状况等患者特异性数据，数值方法可用于模拟缺血性或创伤性脑损伤后的脑肿胀。这些算法将以开源软件包的形式实现。

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.