Constructing well-behaved Laplacian and mass matrices is essential for tetrahedral mesh processing. Unfortunately, the \emph{de facto} standard linear finite elements exhibit bias on tetrahedralized regular grids, motivating the development of finite-volume methods. In this paper, we place existing methods into a common construction, showing how their differences amount to the choice of simplex centers. These choices lead to satisfaction or breakdown of important properties: continuity with respect to vertex positions, positive semi-definiteness of the implied Dirichlet energy, positivity of the mass matrix, and unbiased-ness on regular grids. Based on this analysis, we propose a new method for constructing dual-volumes which explicitly satisfy all of these properties via convex optimization.

翻译：构建性能良好的拉普拉斯矩阵和质量矩阵对于四面体网格处理至关重要。遗憾的是，\emph{事实上的}标准线性有限元方法在四面体化规则网格上表现出偏差，这推动了有限体积方法的发展。本文通过统一框架整合现有方法，揭示其差异本质上源于单纯形中心的选择策略。这些选择决定了若干关键属性的满足与否：顶点位置连续性、隐含狄利克雷能量的半正定性、质量矩阵的正定性，以及在规则网格上的无偏性。基于此分析，我们提出一种通过凸优化显式满足所有属性的新型双重体积构造方法。