In this paper, we propose a robust low-order stabilization-free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress-hybrid principle. We refer to this approach as the Stress-Hybrid Virtual Element Method (SH-VEM). In this method, the Hellinger$-$Reissner variational principle is adopted, wherein both the equilibrium equations and the strain-displacement relations are variationally enforced. We consider small-strain deformations of linear elastic solids in the compressible and near-incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are $\textit{virtual}$. The stress field, similar to the stress-hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5-parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain-displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH-VEM is free of volumetric and shear locking, and it converges optimally in the $L^2$ norm and energy seminorm of the displacement field, and in the $L^2$ norm of the hydrostatic stress.
翻译:本文提出了一种基于应力混合原理的鲁棒低阶稳定的四边形网格上线性弹性的虚拟元素方法。该方法被称为应力混合虚拟元素法(SH-VEM)。该方法采用Hellinger-Reissner变分原理,同时对平衡方程和应变-位移关系进行变分强制。我们考虑四边形(凸和非凸)网格上可压和近不可压的线性弹性固体的小应变变形。在一个单元中,位移场被近似为一个由“虚拟”基函数线性组合而成的形状函数。应力场,类似于Pian和Sumihara的应力混合有限元方法,采用线性组合的对称张量多项式进行表示。在每个单元中采用5参数扩展的应力场,对于扭曲的四边形采用应力变换方程。在应变-位移关系的变分陈述中,使用散度定理,将应力系数表示为节点位移的函数。这导致一个仅含有节点位移作为未知数的公式。我们展示了在几个线性弹性基准问题中的数值结果。我们证明了SH-VEM不会出现体积锁定和剪切锁定,并在位移场的$L^2$范数和能量半范数以及水平应力的$L^2$范数中达到最优收敛性。