Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity

Abstract

In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. We adopt the Hellinger--Reissner variational principle to construct a weak equilibrium condition and a stress based projection operator. On applying the divergence theorem to the weak strain-displacement relations, the stress projection operator is expressed in terms of the nodal displacements, which leads to a displacement-based formulation. This stress-hybrid approach assumes a globally continuous displacement field while the stress field is discontinuous across each element. The stress field is initially represented by divergence-free tensor polynomials based on Airy stress functions. However, for flexibility in choosing basis functions, we also present a formulation that uses a penalty term to enforce the element equilibrium conditions. This method is referred to as the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numerical results are presented for PSH-VEM and SH-VEM, and we compare their convergence to the composite triangle FEM and B-bar VEM on benchmark problems in linear elasticity. The SH-VEM converges optimally in the $L^2$ norm of the displacement, energy seminorm, and the $L^2$ norm of hydrostatic stress. Furthermore, the results reveal that PSH-VEM converges in most cases at a faster rate than the expected optimal rate, but it requires the selection of a suitably chosen penalty parameter.

Figures (50)

• Figure 1: Two-dimensional solid that occupies the region $\Omega$ with body force $\bm{b}$, and is subjected to displacement and traction boundary conditions.
• Figure 2: Examples of admissible six-noded elements (a) nonconvex element, (b) hexagonal element and (c) six-noded triangular element.
• Figure 3: (a)-(c) Sequence of six-noded triangular elements with vertices at $\{(-1,0), \, (1,0), \, (\alpha,\beta)\}$ and nodes placed at the midpoint of each edge.
• Figure 4: Contour plots of the fourth-lowest eigenvalue as a function of $(\gamma_1,\gamma_2)$ for (a) standard VEM, (b) B-bar VEM, (c) CT FEM, (d) $11\beta$ SH-VEM, (e) $13\beta$ SH-VEM, and (f) $15\beta$ SH-VEM.
• Figure 5: Contour plots of the fourth smallest eigenvalue as a function of $(\gamma_1,\gamma_2)$ using the penalty parameters (a) $\alpha = \frac{\ell_0^2}{10}$, (b) $\alpha=\ell_0^2$, (c) $\alpha =10\ell_0^2$, and (d) $\alpha=100\ell_0^2$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4