SOLVING PDES IN COMPLEX GEOMETRIES: A DIFFUSE DOMAIN APPROACH.
SOLVING PDES IN COMPLEX GEOMETRIES: A DIFFUSE DOMAIN APPROACH.
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DOI：
10.4310/cms.2009.v7.n1.a4
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发表时间：
2009-03-01
影响因子：
1
通讯作者：
Voigt A
中科院分区：
文献类型：
--
作者：
Li X;Lowengrub J;Rätz A;Voigt A
我们推广了前人的工作，提出了一种求解具有Dirichlet、Neumann和Robin边界条件的复杂、静止或运动几何偏微分方程解的一般方法。使用通过辅助相场函数的几何的隐式表示，该函数用漫射层(例如漫射域)代替域的尖锐边界，在更大的正则域上重新表示方程。由此得到的偏微分方程与原始方程具有相同的阶数，并附加了低阶项来逼近边界条件。改写后的方程可以用标准的数值方法求解。我们使用匹配渐近展开的方法来证明重新表述的方程的解收敛于原始方程的解。我们提供的数值模拟证实了这一分析。我们还介绍了该方法在生长结构域和复杂三维结构方面的应用，并讨论了它在细胞生物学和异质外延方面的应用。
We extend previous work and present a general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions. Using an implicit representation of the geometry through an auxilliary phase field function, which replaces the sharp boundary of the domain with a diffuse layer (e.g. diffuse domain), the equation is reformulated on a larger regular domain. The resulting partial differential equation is of the same order as the original equation, with additional lower order terms to approximate the boundary conditions. The reformulated equation can be solved by standard numerical techniques. We use the method of matched asymptotic expansions to show that solutions of the re-formulated equations converge to those of the original equations. We provide numerical simulations which confirm this analysis. We also present applications of the method to growing domains and complex three-dimensional structures and we discuss applications to cell biology and heteroepitaxy.