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。该项目的动机是主要研究者对心脏电波传播建模的研究，其中必须求解变系数和各向异性椭圆抛物偏微分方程（PDE）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）的移动边界。用标准的有限元或有限体积法求解动边界问题时，随着区域边界的演变，需要频繁地重新生成贴体非结构体网格，这通常使得模拟非常昂贵。本计画旨在发展一个有效且二阶精度的演算法，以求解具移动边界的复杂区域上的一般变系数及各向异性椭圆型偏微分方程。最近，主要研究者开发了一种无核边界积分（KFBI）方法来求解二维空间（2D）中的椭圆型偏微分方程。KFBI方法是一种基于结构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。该方法适用于求解复杂区域上具有移动边界的一般椭圆型偏微分方程。本计画将进一步发展KFBI方法，以求解二维及三维空间中复杂区域上之变系数及各向异性椭圆型偏微分方程。为了进一步提高效率，将作为项目的一部分开发KFBI方法的适应性版本。在他的博士和博士后研究期间，主要研究者还开发了一种自适应网格细化（AMR）算法，该算法具有2D和3D复杂但稳定的椭圆/抛物偏微分方程的贴体网格。然而，基于贴体网格的AMR算法不适合于移动边界问题。KFBI方法和AMR技术的结合和进一步发展将克服这个问题，并显着提高算法的效率和鲁棒性。无核边界积分方法是一种二阶精度的尖界面方法。建议的研究是一个开拓性的努力，在应用二阶精确的尖锐界面方法结合自适应网格加密技术来解决一般椭圆型偏微分方程，其系数是空间可变的和各向异性的，复杂的区域与移动边界。此外，所提出的研究对工程应用具有明显的广泛影响。该项目的成果将使人们有可能对临床重要现象进行更有效，更准确的建模和模拟，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.