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 smooth solutions. Compared to solve a time-dependent form of eikonal equation, the Laplacian regularized eikonal equation 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 using domain decomposition with $1$-ring face neighborhood structure can be done straightforwardly by 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，用于在由多面体单元离散的三维计算域上计算距给定对象的距离。受粘性消失法的启发，求解了Laplacian正则化的eikonal方程，并在区域边界施加Soner边界条件以避免非粘性解。随着正则化参数（依赖于离散域特征长度）的减小，计算相应数值解。数值实验表明，当特征长度及对应的正则化参数逐渐减小时，该算法收敛至粘性解。对于光滑解，$L^1$范数误差的数值结果证实了二阶实验收敛阶。与求解时间依赖形式的eikonal方程相比，Laplacian正则化eikonal方程在单元数量较大或感兴趣区域远离给定对象时可显著降低计算成本。此外，采用含$1$环面邻域结构的区域分解并行计算可通过标准单元中心有限体积代码直接实现。