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：本文在具有平面边界的非凸四边形和凸六面体上引入了新型广义重心坐标（称为"矩坐标"）。该工作借鉴了近期在非光滑动力系统中构建插值函数以描述Filippov滑移向量场运动的研究进展，该类研究基于（部分）距离的符号矩阵非负解。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基础上补充附加线性矩方程，建立满秩为$n$的线性方程组，其解即为非负形函数。在简单（凸或非凸）四边形上，基于符号距离的矩坐标与平均坐标等价。对于基于顶点关联边距离乘积及其边长的符号权重，我们在凸四边形上恢复了Wachspress坐标。此外，在具有平面边界的凸六面体上也构造了矩坐标。本文给出了支持该构造的数学证明，并通过形函数图解验证了其性质。