In this work, we develop a general high-order virtual element method for three-dimensional linear and nonlinear elastic problems. Applications of the virtual element method (VEM) in three-dimensional mechanics include linear elasticity problems, finite elastic strain problems, finite deformation plasticity problems, etc. But besides linear elastic problems, see e.g. Visinoni, 2024, the numerical schemes were all based on a first-order approximation of the displacement. We derive three-dimensional elastic problems, including linear elastic problems and for the first time hyperelastic problems. Similar to previous work, we discuss the calculation method of three-dimensional high-order projection operators of vector fields and calculate the tangent stiffness matrix of elastic problems according to the variational scheme. Since traditional VEM requires the use of stabilization terms to ensure the correctness of the rank of the stiffness matrix, we give suggestions for the selection of stabilization terms for high-order virtual element methods in both linear and nonlinear elasticity. Finally, we illustrate the accuracy, convergence, and stability of the high-order VEM for elastic problems by means of some classic elastic and hyperelastic examples. In addition, we also apply the developed methodology to some complex and difficult problems which illustrate the adaptability of the method to engineering problems.

中文翻译：

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用于线性和非线性弹性的高阶 3D 虚拟单元法


在这项工作中，我们开发了一种用于三维线性和非线性弹性问题的通用高阶虚拟元素方法。虚拟元法（VEM）在三维力学中的应用包括线弹性问题、有限弹性应变问题、有限变形塑性问题等。但除了线弹性问题外，参见Visinoni，2024，数值方案均基于位移的一阶近似。我们推导了三维弹性问题，包括线性弹性问题和首次超弹性问题。与之前的工作类似，我们讨论了向量场三维高阶投影算子的计算方法，并根据变分格式计算弹性问题的切线刚度矩阵。由于传统VEM需要使用稳定项来保证刚度矩阵秩的正确性，因此我们对线性和非线性弹性中的高阶虚拟单元方法的稳定项选择提出了建议。最后，我们通过一些经典的弹性和超弹性例子说明了高阶VEM对于弹性问题的准确性、收敛性和稳定性。此外，我们还将所开发的方法应用于一些复杂、困难的问题，说明了该方法对工程问题的适应性。