Direct serendipity finite elements on cuboidal hexahedra

IF 6.9 1区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY
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引用次数: 0

Abstract

We construct direct serendipity finite elements on general cuboidal hexahedra, which are H1-conforming and optimally approximate to any order. The new finite elements are direct in that the shape functions are directly defined on the physical element. Moreover, they are serendipity by possessing a minimal number of degrees of freedom satisfying the conformity requirement. Their shape function spaces consist of polynomials plus (generally nonpolynomial) supplemental functions, where the polynomials are included for the approximation property and supplements are added to achieve H1-conformity. The finite elements are fully constructive. The shape function spaces of higher order r3 are developed first, and then the lower order spaces are constructed as subspaces of the third order space. Under a shape regularity assumption, and a mild restriction on the choice of supplemental functions, we develop the convergence properties of the new direct serendipity finite elements. Numerical results with different choices of supplements are compared on two mesh sequences, one regularly distorted and the other one randomly distorted. They all possess a convergence rate that aligns with the theory, while a slight difference lies in their performance.
立方体六面体上的直接偶然性有限元
我们在一般立方体六面体上构建了直接的偶然性有限元,这些有限元具有 H1 顺应性,并能以最佳方式逼近任何阶次。新的有限元是直接的,因为形状函数是直接定义在物理元素上的。此外,它们还具有偶然性,因为它们拥有满足一致性要求的最少自由度。它们的形状函数空间由多项式加上(一般为非多项式)补充函数组成,其中多项式是为了逼近特性而加入的,而补充函数则是为了实现 H1 一致性而加入的。有限元是完全构造的。首先建立高阶 r≥3 的形状函数空间,然后作为三阶空间的子空间构建低阶空间。在形状正则性假设和对补充函数选择的温和限制下,我们发展了新的直接偶然性有限元的收敛特性。在两个网格序列(一个是规则变形网格,另一个是随机变形网格)上比较了不同补充函数选择的数值结果。它们都具有与理论一致的收敛速度,但在性能上略有不同。
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来源期刊
CiteScore
12.70
自引率
15.30%
发文量
719
审稿时长
44 days
期刊介绍: Computer Methods in Applied Mechanics and Engineering stands as a cornerstone in the realm of computational science and engineering. With a history spanning over five decades, the journal has been a key platform for disseminating papers on advanced mathematical modeling and numerical solutions. Interdisciplinary in nature, these contributions encompass mechanics, mathematics, computer science, and various scientific disciplines. The journal welcomes a broad range of computational methods addressing the simulation, analysis, and design of complex physical problems, making it a vital resource for researchers in the field.
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