Adaptive Kernel-free Boundary Integral Method for Elliptic PDEs

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本项目是由主要研究人员对心脏中电波传播建模的研究激发的，该研究需要求解一个变系数和各向异性椭圆-抛物型偏微分方程系统。研究中的一个主要挑战是在模拟过程中考虑几何复杂域（跳动的心脏）的运动边界。用标准的有限元或有限体积法求解运动边界问题时，由于域边界的演变需要频繁地重新生成贴体非结构化体网格，这使得仿真成本非常高。本课题旨在开发一种求解具有移动边界的复杂区域上的一般变系数和各向异性椭圆偏微分方程的高效二阶精确算法。最近，主要研究者开发了一种求解二维椭圆偏微分方程的无核边界积分（KFBI）方法。KFBI方法是一种基于结构化网格的边界积分方法。所涉及的结构化网格不需要与域边界对齐。KFBI方法不需要积分算子核的解析表达式。该方法适用于求解具有移动边界的复杂域上的一般椭圆偏微分方程。本项目将进一步发展求解二维和三维复杂域上变系数和各向异性椭圆偏微分方程的KFBI方法。为了进一步提高效率，将开发KFBI方法的自适应版本作为该项目的一部分。在他的博士和博士后研究期间，首席研究员还开发了一种自适应网格细化（AMR）算法，该算法使用贴体网格用于椭圆/抛物型偏微分方程在复杂但稳定的二维和三维区域上。然而，基于体拟合网格的AMR算法不适用于移动边界问题。KFBI方法与AMR技术的结合和进一步发展将克服这一问题，显著提高算法的效率和鲁棒性。无核边界积分法是一种二阶精确锐界面法。将二阶精确锐界面法与自适应网格细化技术相结合，在具有移动边界的复杂域上求解系数为空间可变和各向异性的一般椭圆型偏微分方程，是一项开创性的研究。此外，所提出的研究对工程应用具有明显的广泛影响。该项目的成果将使对临床重要现象进行更有效和准确的建模和模拟成为可能，例如心脏病的心脏电动力学、肿瘤生成、肿瘤诱导的血管生成和肿瘤内药物输注，仅举几例。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is motivated by the principal investigator's research on modeling the electrical wave propagation in the heart, for which a system of variable coefficient and anisotropic elliptic-parabolic partial differential equations (PDEs) must be solved. A major challenge in the research is to take into account the moving boundary of the geometrically complicated domain (beating heart) during the simulation. With the standard finite element or finite volume method to solve the moving boundary problems, the need to frequently regenerate body-fitted unstructured volume grids as the domain boundary evolves usually makes the simulation very expensive. This project aims to develop an efficient and second-order accurate algorithm for solving the general variable coefficient and anisotropic elliptic PDEs on complex domains with moving boundaries. Recently, the principal investigator developed a kernel-free boundary integral (KFBI) method for solving elliptic PDEs in two space dimensions (2D). The KFBI method is a structured grid based boundary integral method. The structured grids involved are not required to be aligned with the domain boundary. The KFBI method does not need the analytical expression for the kernel of the integral operator. It is applicable for solving general elliptic PDEs on complex domains with moving boundaries. This project will further develop the KFBI method for solving variable coefficient and anisotropic elliptic PDEs on complex domains in both 2D and three space dimensions (3D). To further improve the efficiency, an adaptive version of the KFBI method will be developed as part of the project. During his doctoral and post-doctoral studies, the principal investigator has also developed an adaptive mesh refinement (AMR) algorithm with body-fitted grids for elliptic/parabolic PDEs on complex but stationary domains in both 2D and 3D. The body-fitted grid based AMR algorithm, however, is not suitable for moving boundary problems. The combination and further development of the KFBI method and the AMR technique will overcome this issue and significantly improve both the efficiency and the robustness of the algorithm. The kernel-free boundary integral method is a second-order accurate sharp interface method. The proposed research is a pioneering effort in applying the second-order accurate sharp interface method in combination with an adaptive mesh refinement technique to solve general elliptic partial differential equations, whose coefficients are spatially variable and anisotropic, on complex domains with moving boundaries. In addition, the proposed research has clear broad impacts to engineering applications. The outcome of the project will make it possible to perform more efficient and accurate modeling and simulation of clinically important phenomena, such as the cardiac electrical dynamics for heart diseases, tumor generation, tumor-induced angiogenesis and the intra-tumoral infusion of drugs, just to name a few.