In this paper, we introduce new generalized barycentric coordinates (coined as {\em moment coordinates}) on nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with $n$ vertices (nodes) in $\mathbb{R}^2$, the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank $n$, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties.

翻译：本文在非凸四边形和具有平面面的凸六面体上引入了一种新型广义重心坐标（称为“矩坐标”）。该工作借鉴了近期在非光滑动力系统中构造插值函数以描述Filippov滑动向量场运动的研究进展，其中研究了基于（部分）距离的符号矩阵的非负解。对于$\mathbb{R}^2$中具有$n$个顶点（节点）的有限元，在常数和线性再生条件的基础上补充线性矩方程，建立了一个满秩为$n$的线性方程组，其解生成非负形函数。在简单（凸或非凸）四边形上，使用有符号距离的矩坐标与平均坐标一致。对于基于顶点相邻边距离乘积及边长的有符号权重，我们在凸四边形上恢复了Wachspress坐标。此外，我们还在具有平面面的凸六面体上构造了矩坐标。文中给出了支持该构造的证明，并通过形函数图像验证了其性质。