Virtual Element Method Applied to Two Dimensional Axisymmetric Elastic Problems

This work presents a Virtual Element Method (VEM) formulation tailored for two-dimensional axisymmetric problems in linear elasticity. By exploiting the rotational symmetry of the geometry and loading conditions, the problem is reduced to a meridional cross-section, where all fields depend only on the radial and axial coordinates. The method incorporates the radial weight $r$ in both the weak formulation and the interpolation estimates to remain consistent with the physical volume measure of cylindrical coordinates. A projection operator onto constant strain fields is constructed via boundary integrals, and a volumetric correction term is introduced to account for the divergence of the stress field arising from axisymmetry. The stabilization term is designed to act only on the kernel of the projection and is implemented using a boundary-based formulation that guarantees stability without affecting polynomial consistency. Furthermore, an a priori interpolation error estimate is established in a weighted Sobolev space, showing optimal convergence rates. The implementation is validated through patch tests that demonstrate the accuracy, consistency, and robustness of the proposed approach.