A differential quadrature hierarchical finite element method using Fekete points for triangles and tetrahedrons and its applications to structural vibration
A differential quadrature hierarchical finite element method using Fekete points for triangles and tetrahedrons and its applications to structural vibration
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三角形和四面体Fekete点的微分求积分层有限元方法及其在结构振动中的应用
DOI：
10.1016/j.cma.2018.10.051
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发表时间：
2019-06
影响因子：
7.2
通讯作者：
Ferreira A J M
中科院分区：
文献类型：
--
作者：
Liu Bo;Liu Cuiyun;Lu Shuai;Wu Yang;Xing Yufeng;Ferreira A J M
针对三角形和四面体，提出了一种基于Fekete点的微分求积分层有限元方法，并将其应用于结构振动分析。首先，推导了可作为分层有限元方法基础的三角形和四面体上的正交多项式，并给出了将一维非均匀结点转化为单纯形的简单公式。然后以非均匀结点为初始猜想，利用牛顿-拉夫森方法结合正交多项式求解单形上的Fekete点。利用高次有限元基和Fekete点，建立了单形上新的微分求积规则。新的DQ元素的不同边和面上以及主体内部的节点或基础的数量并不像HFE那样相互关联，后者可以在不同的边和面上和主体内部自由地分配不同数量的基础。因此，新的DQ方法被命名为DQH方法，它使用插值函数或正交多项式作为单元内的基。其弱形式被命名为DQHFE。除了DQH方法及其弱形式外，还提出了一种从单个NURBS面片生成高质量线性和高阶三角、四面体网格的简单方法。通过结构振动分析的数值试验表明，即使在曲线域上使用DQH基，也可以在物理场和几何场上得到高精度的结果。结果表明，DQH方法和DQHFE在科学和工程上的广泛应用是可能的，并值得开发基于它们的商业代码。
A differential quadrature hierarchical finite element method (DQHFEM) using Fekete points was formulated for triangles and tetrahedrons and applied to structural vibration analyses. First, orthogonal polynomials on triangles and tetrahedrons that can be used as bases of the hierarchical finite element method (HFEM) were derived and simple formulas of transforming one dimensional non-uniform nodes to simplexes were presented. Then the non-uniform nodes were used as initial guesses to solve the Fekete points on simplexes through Newton–Raphson’s method together with the orthogonal polynomials. New differential quadrature (DQ) rules on simplexes were formulated using the HFEM bases and the Fekete points. The numbers of nodes or bases on different edges and faces and inside the body of the new DQ elements do not relate with each other like the HFEM that can freely assign different numbers of bases on different edges and faces and inside the body. So the new DQ method was named as a differential quadrature hierarchical (DQH) method that uses either interpolation functions or orthogonal polynomials as bases inside the element. Its weak form was named as the DQHFEM. Besides the DQH method and its weak form, a simple method of generating high quality linear and high order triangular and tetrahedral meshes from a single NURBS patch was presented. Numerical tests of the DQHFEM through structural vibration analyses showed that high accuracy results can be obtained using only a few nodes even on curvilinear domains using the DQH bases on both physical and geometric fields. It was concluded that wide applications of the DQH method and the DQHFEM to science and engineering are possible and commercial codes based on them are deserved to be developed.
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DOI：
10.1016/j.amc.2012.12.037
发表时间：
2013-02
期刊：
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影响因子：
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