A Stabilisation-Free Hybrid Virtual Triangular Element with Drilling Rotations for Two-Dimensional Elasticity

A stabilisation-free hybrid virtual triangular element for the analysis of two-dimensional elasticity problems is presented. The formulation is developed within a hybrid framework in which the kinematic field is defined along the element boundary, while the stress field is independently approximated within the element domain through an energy-consistent projection procedure. The stress approximation employs a divergence-free, iso-stable polynomial basis, i.e., one in which the number of stress parameters matches the number of deformational kinematic modes. The basis isspecifically constructed to avoid spurious energy modes, thereby eliminating the need for additional stabilisation terms.The proposed element possesses nine degrees of freedom, namely two translational components and one drilling rotation at each vertex. The introduction of drilling rotations as independent kinematic variables within a hybrid virtual element framework constitutes a novel aspect of the formulation.The resulting element, denoted as HVT3-6, is assessed through standard benchmark problems in two-dimensional elasticity. The numerical results demonstrate optimal convergence rates of order h2 for displacement, stress, and complementary energy measures. Moreover, the proposed formulation yields improved accuracy compared to a displacement-based triangular finite element with the same number of degrees of freedom on both structured and unstructured meshes, including coarse discretisations.Finally, while the present formulation is developed for triangular elements, the proposed drilling-based hybrid framework constitutes a basis for future extensions of hybrid virtual elements with independent drilling rotations to general polygonal meshes.