[论文解读] Numerical method for Darcy flow derived using Discrete Exterior Calculus

Anil N. Hirani, K. B. Nakshatrala+1 more|arXiv (Cornell University)|Oct 19, 2008

Black Holes and Theoretical Physics参考文献 36被引用 12

一句话总结

本文提出了一种基于离散外微分（DEC）的坐标不变数值方法，用于求解达西流（Darcy flow），通过单纯形网格对泊松方程的混合形式进行离散化，采用通量和压力变量。该方法确保了局部质量守恒，对通量和压力均实现了最优收敛率，并通过一个对角、正定的霍奇星（Hodge star）离散化处理了空间变化的渗透率，在二维、三维及曲面问题上均表现出良好性能。

ABSTRACT

We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solution in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. We also show numerical evidence of convergence of the flux and the pressure. A convergence experiment is also included for Darcy flow on a surface. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this paper. We also include a discussion of the boundary condition in terms of exterior calculus.

研究动机与目标

• 开发一种保持局部质量守恒与几何不变性的达西流数值方法。
• 将离散外微分（DEC）扩展至处理多孔介质流中非均匀、空间变化的渗透率。
• 通过基于网格的内在离散化，为二维、三维及曲面上的达西流提供统一的公式化表达。
• 在标准测试问题中，证明通量和压力变量均达到最优收敛率。
• 建立与有限元外微分计算兼容的框架，并为未来扩展至各向异性渗透率做好准备。

提出的方法

• 利用外微分形式重写达西流的控制方程，将通量表示为微分形式，压力表示为0-形式。
• 应用离散外微分（DEC）对微分形式和算子进行离散化，保持拓扑与几何性质（如斯托克斯定理）。
• 为非均匀空间变化的渗透率提出一种新型霍奇星算子离散化，得到一个对角矩阵且所有元素为正。
• 采用交错网格配对方式，使通量与边（或面）关联，压力与顶点（或单元）关联，从而保证局部质量守恒。
• 所得线性系统为鞍点型，左上角块为对角矩阵，支持高效的求解策略。
• 通过外微分形式的语言实现边界条件，保持与底层几何结构的一致性。

实验结果

研究问题

• RQ1 基于DEC的方法是否能在一般单纯形网格上对达西流的通量和压力均实现最优收敛？
• RQ2 如何对非均匀、空间变化的渗透率离散化霍奇星算子，同时保持正定性与对角结构？
• RQ3 DEC框架能否在不依赖嵌入计算的前提下，扩展至曲面上的达西流求解？
• RQ4 当存在不连续或各向异性渗透率时，该方法是否仍能保持局部质量守恒？
• RQ5 在标准基准问题上，该DEC方法的性能与标准有限元法或有限体积法相比如何？

主要发现

• 该方法在二维和三维中通过了所有标准patch测试，验证了其一致性和稳定性。
• 数值实验中观察到通量和压力均具有收敛性，且在曲面上的收敛性实验确认了最优收敛率。
• 针对非均匀渗透率的离散霍奇星算子为对角矩阵且所有元素为正，确保了线性系统的良好条件性。
• 由于DEC中精确的离散斯托克斯定理，该方法在构造上即保证了局部质量守恒。
• 该公式的内在性与坐标无关性，使其可直接应用于曲面（如半球面）而无需依赖嵌入的修改。
• 该方法在具有交替渗透率的层状介质中以及沿流动方向存在渗透率梯度的流动问题中均表现出鲁棒性能。