In this paper, we investigate the capability of the recently proposed extended virtual element method (X-VEM) to efficiently and accurately solve the problem of a cracked prismatic beam under pure torsion, mathematically described by the Poisson equation in terms of a scalar stress function. This problem is representative of a wide class of elliptic problems for which classic finite element approximations tend to converge poorly, due to the presence of singularities. The X-VEM is a stabilized Galerkin formulation on arbitrary polygonal meshes derived from the virtual element method (VEM) by augmenting the standard virtual element space with an additional contribution that consists of the product of virtual nodal basis functions with a suitable enrichment function. In addition, an extended projector that maps functions lying in the extended virtual element space onto linear polynomials and the enrichment function is employed. Convergence of the method on both quadrilateral and polygonal meshes for the cracked beam torsion problem is studied by means of numerical experiments. The computed results affirm the sound accuracy of the method and demonstrate a significantly improved convergence rate, both in terms of energy and stress intensity factor, when compared to standard finite element method and VEM.

中文翻译：

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裂纹棱柱梁扭转问题的扩展虚元法

在本文中，我们研究了最近提出的扩展虚拟元法 (X-VEM) 有效且准确地解决纯扭转下有裂纹棱柱梁问题的能力，该问题由泊松方程根据标量应力函数进行数学描述. 这个问题代表了一大类椭圆问题，由于奇点的存在，经典的有限元近似往往收敛性很差。X-VEM 是从虚拟元素方法 (VEM) 派生的任意多边形网格上的稳定 Galerkin 公式，通过增加标准虚拟元素空间与附加贡献，该贡献由虚拟节点基函数与合适的富集函数的乘积组成。此外，扩展投影仪将位于扩展虚拟元素空间中的函数映射到线性多项式，并使用富集函数。通过数值实验研究了该方法在四边形和多边形网格上求解裂纹梁扭转问题的收敛性。与标准有限元方法和 VEM 相比，计算结果证实了该方法的准确性，并证明了在能量和应力强度因子方面的收敛速度显着提高。