复边界上具有一般边界条件的偏微分方程的扩展光滑边界法
Extended Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries
摘要：
在这篇文章中，我们描述了一种方法来解决偏微分方程的一般边界条件施加在任意形状的边界。一个连续的功能，域参数，被用来修改原始微分方程，使方程求解的区域中的域参数采取一个指定的值，而边界条件施加在该区域中的域参数的值在很短的距离内平滑地变化。数学推导是简单的，一般适用于各种各样的偏微分方程。为了证明该方法的普遍适用性，我们在此提供了四个例子：（1）具有Neumann和Dirichlet边界条件的扩散方程;（2）具有表面扩散和反应的扩散方程;（3）力学平衡方程;以及（4）存在附加边界的相变方程。这些情况下的解决方案进行了验证，对相应的解析和半解析解决方案。该方法的潜力被证明与五个应用程序：表面反应扩散动力学与复杂的几何形状，Kirkendall效应引起的变形，在一个复杂的几何形状的热应力，相变影响基板表面，和自推进液滴。
英文摘要：
In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original differential equations such that the equations are solved in the region where a domain parameter takes a specified value while boundary conditions are imposed on the region where the value of the domain parameter varies smoothly across a short distance. The mathematical derivations are straightforward and generically applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples herein: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions; (2) the diffusion equation with both surface diffusion and reaction; (3) the mechanical equilibrium equation; and (4) the equation for phase transformation with the presence of additional boundaries. The solutions for several of these cases are validated against corresponding analytical and semi-analytical solutions. The potential of the approach is demonstrated with five applications: surface-reaction-diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces, and a self-propelled droplet.
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