Conforming Discretizations of Mixed-Dimensional Partial Differential Equations
Conforming Discretizations of Mixed-Dimensional Partial Differential Equations
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混合维偏微分方程的一致离散化
DOI：
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发表时间：
2018
期刊：
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通讯作者：
W. Boon
中科院分区：
文献类型：
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作者：
W. Boon
混合维偏微分方程是定义在不同维数连通流形上的耦合方程。本文主要考虑了两个混合维偏微分方程的例子，即裂隙多孔介质中的渗流和复合材料力学。我们专注于这些例子的离散化，使用层次有限元定义的余维一耦合流形上，先后。通过揭示它们的底层结构，我们使用相应的工具来定义，分析和离散混合维偏微分方程。我们的第一个例子有关的混合维偏微分方程出现在断裂流的背景下。在这里，平面裂缝，相交线，以及交叉点表示为低维流形。反过来，整个嵌入式裂缝网络形成混合维几何形状。我们继续通过定义的混合维几何的守恒和本构律，导致分层耦合系统的偏微分方程。接下来，我们将这些概念从流动中延伸出来，以类似的方式推导出关于薄夹杂物材料力学的控制方程。嵌入的特征及其周围环境一起形成混合维几何形状，并且可以通过规定显著不同的材料参数来捕获系统的行为。这些系统的分析引入了一些新的概念，包括混合维函数空间和半离散微分算子。以离散化为目的，我们利用有限元外演算在混合维几何上构造混合有限元格式。我们专注于两个家庭的混合维有限元，层次有序的维度。我们指的是这些家庭的第一和第二类，并表明，这两个感兴趣的背景下，裂缝流，不同的行为方面的收敛性和计算成本。另一方面，力学方程的混合形式需要第二类元素族。对于裂缝流，稳定性和最佳收敛的离散化方法显示与使用加权，混合维Sobolev空间。一种新的方法，将裂缝孔径导致一个计划能够处理任意小的和空间变化的孔径。在裂缝尖灭的情况下，方程的退化消除了导致裂缝自然终止的流动的可能性。在一个基准研究有关通过裂缝多孔介质的流动，所提出的计划相比，其他各种数值方法。四个不同复杂度的二维测试用例被考虑，专门设计来突出典型的困难
Mixed-dimensional partial differential equations (PDEs) are coupled equations defined on connectedmanifolds of different dimensionalities. Twomain examples ofmixed-dimensional PDEs are considered in this dissertation, namely flow in fractured porous media and mechanics of composite materials. We focus on the discretization of these examples using hierarchical finite elements defined on coupled manifolds of codimension one, successively. By uncovering their underlying structure, we use the corresponding tools to define, analyze and discretize mixed-dimensional partial differential equations. Our first example concerning mixed-dimensional PDEs arises in the context of fracture flow. Here, the planar fractures, intersection lines, as well as intersection points are represented as lower-dimensional manifolds. In turn, the entire embedded fracture network forms a mixed-dimensional geometry. We continue by defining the conservation and constitutive laws on the mixed-dimensional geometry, leading to a hierarchically coupled system of partial differential equations. Next, we extend these concepts from flow to derive the governing equations concerning mechanics of materials with thin inclusions in an analogous manner. Together, the embedded features and their surroundings form the mixed-dimensional geometry and the behavior of the system can be captured by prescribing significantly different material parameters. The analysis of these systems introduces several new concepts including mixed-dimensional function spaces and semi-discrete differential operators. With the aim of discretization, we use finite element exterior calculus to construct mixed finite element schemes on the mixed-dimensional geometry. We focus on two families of mixed-dimensional finite elements, hierarchically ordered by dimensionality. We refer to these families as the first and second kind and show that both are of interest in the context of fracture flow, with different behavior in terms of convergence and computational cost. On the other hand, the mixed formulation of the mechanics equations requires the family of elements of the second kind. For fracture flow, stability and optimal convergence of the discretization method are shown with the use of weighted, mixed-dimensional Sobolev spaces. A novel way of incorporating the fracture aperture leads to a scheme capable of handling arbitrarily small and spatially varying apertures. In case of fractures pinching out, the degeneration of the equations eliminates the possibility for flow resulting in a natural termination of fractures. In a benchmark study concerning flow through fractured porous media, the proposed scheme is compared to various other numerical methods. Four two-dimensional test cases of varying complexity are considered, specifically designed to highlight the typical difficulties