2024/01/11 by Alvin Chen, Joseph E. Bishop, Chen, Alvin +3
Computer Science · Engineering · #Advanced Numerical Methods in Computational Mathematics #Composite Structure Analysis and Optimization #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2401.06280
openalex publication_date 2024/01/11 · openalex created_date 2024/01/16 · openalex updated_date 2026/07/28
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 L2 norm of the displacement, energy seminorm, and the L2 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.