Handling Neumann and Robin boundary conditions in a fictitious domain volume penalization framework
Handling Neumann and Robin boundary conditions in a fictitious domain volume penalization framework
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DOI：
10.1016/j.jcp.2021.110726
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发表时间：
2021-01
期刊：
影响因子：
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通讯作者：
R. Thirumalaisamy;N. Patankar;A. Bhalla
中科院分区：
文献类型：
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作者：
R. Thirumalaisamy;N. Patankar;A. Bhalla
Sakurai等人（2019）[14]提出了一种基于通量的体积惩罚（VP）方法，用于在嵌入界面上施加非均匀Neumann边界条件。基于通量的VP方法修改了原始椭圆（泊松）方程的扩散系数，并使用通量强迫函数作为方程中的源项来施加Neumann边界条件。因此，基于通量的VP方法可以很容易地并入现有的虚拟域代码。Sakurai等人依赖于通量强迫函数的分析构造，这限制了该方法的实用性。由于在以前的工作中采取的分析方法，只有（空间）恒定通量值的简单接口被认为是。在本文中，我们提出了一种数值技术，用于构造任意复杂边界的通量强迫函数。在我们的方法中，还允许施加的通量值在空间上变化。此外，基于通量的VP方法扩展到包括（空间变化）罗宾边界条件，这使得基于通量的VP方法更一般。通量强迫函数的数值构造仅依赖于描述网格点到界面的距离的有符号距离函数，并且可以针对任何不规则边界构造。我们考虑了几个二维和三维的测试例子来访问的数值解的空间精度。该方法也被用来模拟通量驱动的热对流在同心环形区域。我们正式推导了满足强形式Neumann/Robin边界条件的基于通量的体积惩罚泊松方程;这样的推导在Sakurai等人中没有提出，方程第一次出现在诺依曼问题中。推导表明，基于通量的VP方法依赖于表面δ函数施加非齐次Neumann/Robin边界条件。然而，基于通量的VP方法不需要显式构造δ函数，这使得它不同于文献中提出的其他扩散域方程。
Sakurai et al. (2019) [14] presented a flux-based volume penalization (VP) approach for imposing inhomogeneous Neumann boundary conditions on embedded interfaces. The flux-based VP method modifies the diffusion coefficient of the original elliptic (Poisson) equation and uses a flux-forcing function as a source term in the equation to impose the Neumann boundary conditions. As such, the flux-based VP method can be easily incorporated into existing fictitious domain codes. Sakurai et al. relied on an analytical construction of flux-forcing functions, which limits the practicality of the approach. Because of the analytical approach taken in the prior work, only (spatially) constant flux values on simple interfaces were considered. In this paper, we present a numerical technique for constructing flux-forcing functions for arbitrarily complex boundaries. The imposed flux values are also allowed to vary spatially in our approach. Furthermore, the flux-based VP method is extended to include (spatially varying) Robin boundary conditions, which makes the flux-based VP method even more general. The numerical construction of the flux-forcing functions relies only on a signed distance function that describes the distance of a grid point from the interface and can be constructed for any irregular boundary. We consider several two- and three-dimensional test examples to access the spatial accuracy of the numerical solutions. The method is also used to simulate flux-driven thermal convection in a concentric annular domain. We formally derive the flux-based volume penalized Poisson equation satisfying Neumann/Robin boundary condition in strong form; such a derivation was not presented in Sakurai et al., where the equation first appeared for the Neumann problem. The derivation reveals that the flux-based VP approach relies on a surface delta function to impose inhomogeneous Neumann/Robin boundary conditions. However, explicit construction of the delta function is not necessary for the flux-based VP method, which makes it different from other diffuse domain equations presented in the literature.