Frame-invariance in finite element formulations of geometrically exact rods

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• 作者：Peinan Zhong ; Guojun Huang ; Guowei Yang
• 关键词：Key wordsgeometrically exact rod ; finite element method ; interpolation ; equivariance ; frame invariance
• 刊名：Applied Mathematics and Mechanics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：37
• 期：12
• 页码：1669-1688
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Applications of Mathematics; Classical Mechanics; Mathematical Modeling and Industrial Mathematics;
• 出版者：Shanghai University
• ISSN：1573-2754
• 卷排序：37

文摘

This article is concerned with finite element implementations of the threedimensional geometrically exact rod. The special attention is paid to identifying the condition that ensures the frame invariance of the resulting discrete approximations. From the perspective of symmetry, this requirement is equivalent to the commutativity of the employed interpolation operator I with the action of the special Euclidean group SE(3), or I is SE(3)-equivariant. This geometric criterion helps to clarify several subtle issues about the interpolation of finite rotation. It leads us to reexamine the finite element formulation first proposed by Simo in his work on energy-momentum conserving algorithms. That formulation is often mistakenly regarded as non-objective. However, we show that the obtained approximation is invariant under the superposed rigid body motions, and as a corollary, the objectivity of the continuum model is preserved. The key of this proof comes from the observation that since the numerical quadrature is used to compute the integrals, by storing the rotation field and its derivative at the Gauss points, the equivariant conditions can be relaxed only at these points. Several numerical examples are presented to confirm the theoretical results and demonstrate the performance of this algorithm.