有限元几何上的非负矩坐标
Nonnegative moment coordinates on finite element geometries
AI总结：
本文提出矩坐标，通过附加线性矩方程构造满秩线性系统，在非凸四边形和平面凸六面体上获得非负形函数，并建立其与均值坐标和 Wachspress 坐标的联系。
摘要：
本文在非凸四边形以及具有平面面的凸六面体上引入了新的广义重心坐标（称为“矩坐标”）。这项工作借鉴了在非光滑动力系统中构造插值函数以描述 Filippov 滑模向量场运动的最新进展，其中研究了基于（部分）距离的带符号矩阵的非负解。对于在 $\mathbb{R}^2$ 中具有 $n$ 个顶点（节点）的有限元，在常数和线性再生条件之外补充额外的线性矩方程，以建立一个满秩为 $n$ 的线性方程组，其解产生非负形函数。在简单（凸或非凸）四边形上，使用带符号距离的矩坐标与均值坐标相同。对于基于到与顶点关联的边的距离及其边长乘积的带符号权重，我们在凸四边形上恢复了 Wachspress 坐标。矩坐标也被构造在具有平面面的凸六面体上。我们给出了支持该构造的证明以及形函数的图像，以证实其性质。
英文摘要：
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.
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