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Automatic finite element formulation and assembly of hyperelastic higher order structural models

机译：超弹性高阶结构模型的自动有限元公式化和组装

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摘要

The formulation of higher order structural models and their discretization using the finite element method is difficult owing to their complexity, especially in the presence of nonlin-earities. In this work a new algorithm for automating the formulation and assembly of hyperelastic higher-order structural finite elements is developed. A hierarchic series of kinematic models is proposed for modeling structures with special geometries and the algorithm is formulated to automate the study of this class of higher order structural models. The algorithm developed in this work sidesteps the need for an explicit derivation of the governing equations for the individual kinematic modes. Using a novel procedure involving a nodal degree-of-freedom based automatic assembly algorithm, automatic differentiation and higher dimensional quadrature, the relevant finite element matrices are directly computed from the variational statement of elasticity and the higher order kinematic model. Another significant feature of the proposed algorithm is that natural boundary conditions are implicitly handled for arbitrary higher order kinematic models. The validity algorithm is illustrated with examples involving linear elasticity and hyperelasticity.
机译：由于它们的复杂性，特别是在存在非线性空间的情况下，很难使用有限元方法来制定高阶结构模型及其离散化。在这项工作中，开发了一种用于自动完成超弹性高阶结构有限元的配方和组装的新算法。提出了一系列运动学模型，用于对具有特殊几何形状的结构进行建模，并制定了算法以自动研究此类高阶结构模型。在这项工作中开发的算法避开了对各个运动模式的控制方程式进行明确推导的需要。使用涉及基于节点自由度的自动装配算法，自动微分和高维正交的新颖过程，可以直接从弹性的变分表述和高阶运动学模型中计算出相关的有限元矩阵。所提出算法的另一个显着特征是，对于任意高阶运动学模型，隐式处理了自然边界条件。通过涉及线性弹性和超弹性的示例说明了有效性算法。

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