Sakurai et al. (J Comput Phys, 2019) 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. 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.
翻译:Sakurai 等人(J Comput Phys, 2019年) 展示了一种以通量为基础的内流量惩罚法(VP), 用于在嵌入界面上强制实施不相容的 Neumann 边界条件。 以通量为基础的VP 等值法( VP 2019年) 改变了原始椭圆(Poisson) 方程式的传播系数, 并使用通量强制函数作为方程式的源词来强制实施Neumann 边界条件。 因此, 以通量为基础的VP 等值法可以很容易地融入现有的虚构造域代码。 Sakurai 等。 这限制了方法的实用性。 由于先前工作中采用的分析方法, 仅( SPatily) 常量性 Neucal 等值法( Poblicalalalal) 等离子( VP) 等值计算法( VP legencial) 等值的计算法(Orental- dentalal) 等离差法。 我们认为, 以两种平流法的进化法的进化法(Oral- divental) 的进化法(Oral) 的进化法(Oral) 等解法(Odal) 等式) 等解法(Odal) 等解法(Odal) ) 等解法(我们。