We explore a new way to handle flux boundary conditions imposed on level sets. The proposed approach is a diffuse interface version of the shifted boundary method (SBM) for continuous Galerkin discretizations of conservation laws in embedded domains. We impose the interface conditions weakly and approximate surface integrals by volume integrals. The discretized weak form of the governing equation has the structure of an immersed boundary finite element method. A ghost penalty term is included to extend the weak solution into the external subdomain. The calculation of interface forcing terms requires (i) construction of an approximate delta function and (ii) extrapolation of embedded boundary data into quadrature points. We accomplish these tasks using a level set function, which is given analytically or evolved numerically. A globally defined averaged gradient of this approximate signed distance function is used to construct a simple map to the closest point on the interface. The normal and tangential derivatives of the numerical solution at that point are calculated using the interface conditions and/or interpolation on uniform stencils. Similarly to SBM, extrapolation back to the quadrature points is performed using Taylor expansions. The same strategy is used to construct ghost penalty functions and extension velocities. Computations that require extrapolation are restricted to a narrow band around the interface. Numerical results are presented for elliptic, parabolic, and hyperbolic test problems, which are specifically designed to assess the error caused by the numerical treatment of interface conditions on fixed and moving boundaries in 2D.
翻译:我们探索了一种新的方法来处理对水平设置强加的通量边界条件。 提议的方法是一个分散的介质版本, 用于对嵌入域的保存法进行连续的 Galerkin 分解的转换边界法( SBM) 。 我们通过体积内分解将界面条件以弱和近似表面整体体设置。 管理方程式的分散化弱化形式是隐蔽的边界限定要素法的结构。 包含一个幽灵惩罚术语, 将薄弱的解决方案扩展到外部子域。 界面强制条件的计算要求 (一) 近似三角形函数的处理和 (二) 将嵌入边界数据外推到等离子点。 我们完成这些任务时使用一个水平设定的函数, 以数字集函数为分析或进化。 使用全球定义的平均渐近距离函数来构造一个简单地图, 至界面的最接近点。 数字解决方案的正常和相近衍生衍生物在使用界面上的界面和/ 。 与 SBBM 类似, 外推导到二次曲线扩展时, 将精确的边框内测测测算结果,, 将使用固定的测算结果 。