In this paper, we propose a numerical algorithm based on a cell-centered finite volume method to compute a distance from given objects on a three-dimensional computational domain discretized by polyhedral cells. Inspired by the vanishing viscosity method, a Laplacian regularized eikonal equation is solved and the Soner boundary condition is applied to the boundary of the domain to avoid a non-viscosity solution. As the regularization parameter depending on a characteristic length of the discretized domain is reduced, a corresponding numerical solution is calculated. A convergence to the viscosity solution is verified numerically as the characteristic length becomes smaller and the regularization parameter accordingly becomes smaller. From the numerical experiments, the second experimental order of convergence in the $L^1$ norm error is confirmed for a smooth solution. Compared to an algorithm to solve a time-dependent form of eikonal equation, the proposed algorithm has the advantage of reducing computational cost dramatically when a more significant number of cells is used or a region of interest is far away from the given objects. Moreover, the implementation of parallel computing on decomposed domains with $1$-ring face neighborhood structure can be done straightforwardly in a standard cell-centered finite volume code.

翻译：本文提出一种基于单元中心有限体积法的数值算法，用于计算由多面体单元离散的三维计算域中给定目标物的距离。受消失粘度法启发，通过求解Laplacian正则化eikonal方程并在域边界施加Soner边界条件以避免非粘性解。当正则化参数随离散域特征长度减小而调整时，可计算对应数值解。数值验证表明，随着特征长度减小及正则化参数相应减小，解收敛至粘性解。数值实验证实，对于光滑解，$L^1$范数误差具有二阶实验收敛阶。相较于求解eikonal方程时间依赖形式的算法，当单元数量显著增加或关注区域远离给定目标物时，本算法可大幅降低计算成本。此外，基于$1$环面邻域结构的分解域并行计算可便捷地嵌入标准单元中心有限体积代码。