论文标题
$ c^1 $四边形有限元素的家庭
A family of $C^1$ quadrilateral finite elements
论文作者
论文摘要
我们提出了一个新颖的家族，包括$ c^1 $四边形有限元素，该元素定义了全球$ c^1 $ spaces，并在普通的四边形网格上，并带有任意价值的顶点。这些元素通过（Brenner and Sung，J。Sci。Comput。，2005）扩展了构造，该元素基于张量产品的多项式元素$ p \ geq 6 $，均为$ p \ geq 3 $。因此，我们称$ c^1 $有限元素的家族Brenner-Sung四边形。提出的$ c^1 $四边形可以看作是Argyris iSOEGEOMETRIC元素的特殊情况（Kapl，Sangalli和Takacs，Cagd，2019年）。四边形元素具有与经典的Argyris三角形相似的自由度。就像Argyris三角形一样，我们还在顶点强加了$ C^2 $连续性。在本文中，我们专注于较低的案例，这对于较低的计算成本和更好的基础条件可能是可取的：我们确实考虑了（Bi-）度〜$ 5 $的多项式四边形，而多项式学位$ p = 3 $和$ p = 4 $，通过使用a用$ 3 \ $ 3 \ $ 3 \ $ 2 $ 2 $ 2 \ $ 2 \ $ 2 \ polty tirs 2 $ 2 \ polty 2 $ $ 2 $ $ p = 4 $。 拟议的元素繁殖总数$ p $。我们表明该空间提供了最佳的近似顺序。由于插值属性，误差边界在每个元素上都是局部的。此外，我们描述了一个简单的本地基础的构建，并在\ {3,4,5 \}中提供$ p \ in \ {3,4,5 \} $ explicit公式，用于bézier或B-Spline系数的基础功能。通过求解Biharmonic方程的数值实验证明了提出的$ C^1 $四边形有限元元素对第四阶问题的数值分析的潜力，还表明（对于$ p = 5 $），所提出的元素（对于自由度的数量）相对于Argyris Triangle，该元素的表现可比或一般而言。
We present a novel family of $C^1$ quadrilateral finite elements, which define global $C^1$ spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by (Brenner and Sung, J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product degree $p\geq 6$, to all degrees $p \geq 3$. Thus, we call the family of $C^1$ finite elements Brenner-Sung quadrilaterals. The proposed $C^1$ quadrilateral can be seen as a special case of the Argyris isogeometric element of (Kapl, Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles. Just as for the Argyris triangle, we additionally impose $C^2$ continuity at the vertices. In this paper we focus on the lower degree cases, that may be desirable for their lower computational cost and better conditioning of the basis: We consider indeed the polynomial quadrilateral of (bi-)degree~$5$, and the polynomial degrees $p=3$ and $p=4$ by employing a splitting into $3\times3$ or $2\times2$ polynomial pieces, respectively. The proposed elements reproduce polynomials of total degree $p$. We show that the space provides optimal approximation order. Due to the interpolation properties, the error bounds are local on each element. In addition, we describe the construction of a simple, local basis and give for $p\in\{3,4,5\}$ explicit formulas for the Bézier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed $C^1$ quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for $p=5$) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.
