Co-rotational 3D shell element using quaternion algebra to account for large rotations: Static and dynamic applications

IF 3.2 Q2 MATERIALS SCIENCE, MULTIDISCIPLINARY

Forces in mechanics Pub Date : 2025-06-25 DOI:10.1016/j.finmec.2025.100315

Stéphane Grange, David Bertrand

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Abstract

This paper presents a new co-rotational shell element based on quaternion algebra as a means of parameterizing large rotations. The co-rotational framework is suitable for beam or shell elements and has been extensively studied in the literature. It is based on a decomposition between rigid body motion and local displacements that generate deformation.

The advantage of this framework lies in the fact that the internal element defined in the co-rotational frame can be derived from a library of elements (possibly in small or large deformations and even with material nonlinearities).

The present formulation constitutes an extension of a previous work devoted to a beam finite element using quaternion algebra and applied to shell finite elements. Quaternion algebra is used throughout the kinematic chain, and such parameterization offers an alternative to classical co-rotational formulations. The model is developed within the framework of incremental rotation formulations. Once the decomposition between rigid body motion and local displacements has been performed, the principle of virtual work is introduced to calculate the element response projected onto large displacements and rotations.

The adopted methodology is then exposed for a three-node triangular shell element. For the sake of simplicity and to demonstrate the capabilities of the co-rotational frame with quaternions, DKT (bending) and OPT (membrane) triangular shell elements with small strains are chosen as the internal element.

One of the main advantages of using quaternions for parameterization lies in their efficiency for dynamic applications, as they allow for a relative straightforward computation of gyroscopic terms. The numerical simulations show a stable mechanical energy of the systems and a good numerical stability.

Ten distinct static and dynamic numerical applications are also presented and compared to the literature.

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共旋转三维壳单元使用四元数代数，以说明大旋转：静态和动态应用

本文提出了一种新的基于四元数代数的共旋转壳元，作为参数化大旋转的一种手段。共转框架适用于梁或壳单元，在文献中得到了广泛的研究。它是基于刚体运动和产生变形的局部位移之间的分解。这种框架的优点在于，在共旋转框架中定义的内部单元可以从单元库中推导出来（可能是小变形或大变形，甚至是材料非线性）。目前的公式构成了以前的工作的延伸，专门用于梁有限元使用四元数代数，并适用于壳有限元。四元数代数在整个运动链中使用，这种参数化提供了经典共旋转公式的替代方案。该模型是在增量旋转公式的框架内开发的。一旦进行了刚体运动和局部位移的分解，就引入虚功原理来计算大位移和大旋转上的单元响应。采用的方法，然后暴露为一个三节点三角形壳单元。为了简单起见，也为了展示四元数共旋转框架的能力，我们选择了具有小应变的DKT（弯曲）和OPT（膜）三角形壳单元作为内单元。使用四元数进行参数化的主要优点之一在于它们对动态应用的效率，因为它们允许相对直接的陀螺仪项计算。数值模拟结果表明，系统具有稳定的机械能和良好的数值稳定性。十种不同的静态和动态数值应用也提出和比较文献。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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