Quadratic Serendipity Element Shape Functions on General Planar Polygons
Quadratic Serendipity Element Shape Functions on General Planar Polygons
复制标题
一般平面多边形上的二次偶然元形函数
DOI：
10.1016/j.cma.2013.04.009
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发表时间：
2022
影响因子：
7.2
通讯作者：
Yongjie Jessica Zhang
中科院分区：
文献类型：
--
作者：
Juan Cao;Yi Xiao;Yanyang Xiao;Zhonggui Chen;Fei Xue;Xiaodong Wei;Yongjie Jessica Zhang
本文给出了平面凸多边形和非凸多边形上二次偶然性形函数的发展。借鉴Bompadre等人的工作。（2012）[1]和Hormann和Sukumar（2008）[2]，我们采用符号（正或负）形状函数的相对熵度量，节点先验权重函数在多边形的边界上具有适当的零集。我们最大化的目标功能的约束下提出的二次完备性兰德等人。（2013）[3]。沿着多边形的沿着边，近似与单变量伯恩斯坦多项式相同：节点先验权重函数的选择确保形状函数在每条边上满足弱克罗内克-δ性质。对于任意平面多边形，形状函数是定义良好的。在使用一个修改的数值积分方案，我们表明，二次补丁测试是通过多边形网格与凸和非凸元素。在自相似梯形网格和准均匀多边形网格上对Poisson方程进行了数值试验，结果表明该方法具有良好的精度，并在L2范数和H1范数下获得了最优收敛速度.
In this paper, we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons. Drawing on the work of Bompadre et al.(2012)[1] and Hormann and Sukumar (2008)[2], we adopt a relative entropy measure for signed (positive or negative) shape functions, with nodal prior weight functions that have the appropriate zero-set on the boundary of the polygon. We maximize the objective functional subject to the constraints for quadratic completeness proposed by Rand et al.(2013)[3]. Along an edge of a polygon, the approximation is identical to univariate Bernstein polynomials: the choice of the nodal prior weight function ensures that the shape functions satisfy a weak Kronecker-delta property on each edge. The shape functions are well-defined for arbitrary planar polygons without self-intersections. On using a modified numerical integration scheme, we show that the quadratic patch test is passed on polygonal meshes with convex and nonconvex elements. Numerical tests for the Poisson equation on self-similar trapezoidal meshes and quasiuniform polygonal meshes are presented, which reveal the sound accuracy of the method, and optimal rates of convergence in the L 2 norm and the H 1 seminorm are established.