In five computational tests, versions of a high-order virtual element method performed differently when fiber orientation varied or the mesh was distorted. For variable fiber orientation, VC-VEM variants and standard self-stabilized VEM maintained good accuracy, while standard stabilized VEM lost accuracy as approximation order increased. On distorted meshes, standard stabilized VEM produced incorrect buckling loads or mode shapes in some configurations, whereas stabilized and self-stabilized VC-VEM and standard self-stabilized VEM correctly predicted buckling loads.
The work is an arXiv preprint, version v1, dated 20 Aug 2026. Its stated aim is to provide a robust, high-order p-VEM framework for variable-stiffness plates with complex geometries.
A framework built for difficult shapes
The framework is designed to cover static analysis, free-vibration analysis and buckling analysis. It combines arbitrary polygonal elements, including elements with curved edges and hanging nodes, with stabilized, self-stabilized and variable-coefficient formulations.
The p-VEM approach uses high-order polynomial approximations within a virtual element method and works with polygon-shaped mesh elements. The study examines ordinary mesh refinement, higher approximation orders and local refinement, in which only selected parts of the mesh are made finer.
Two design choices are central. VC-VEM includes changes in the problem’s coefficients directly in its projection and uses an L2 VEM projector to approximate terms that cannot be computed directly. The self-stabilized formulation avoids ad hoc stabilization terms and instead uses higher-order polynomial projections to stabilize the linear system.
Five tests, all numerical
The evaluation comprised five computational cases covering cutouts and curvilinear fibers, a cracked panel, highly anisotropic free vibration, vibration and thermal buckling around a heart-shaped cutout, and buckling around a variable-stiffness circular cutout. The cases used multiple layups, meshes, approximation orders and refinement strategies.
The researchers compared the calculations with analytic or manufactured solutions, published benchmark results, isogeometric analysis, and simulations conducted with Abaqus. They assessed numerical error, convergence, natural frequencies, thermal-buckling temperatures, buckling loads and buckling-mode shapes.
Because the evidence came from deterministic numerical models, the comparisons describe how the formulations behaved in the tested configurations rather than providing a statistical estimate of performance across a wider population of plates.
Where the variants diverged
One test changed the fiber orientation across the plate. With constant fiber orientation, the tested strategies had equal accuracy. When the orientation varied, standard stabilized VEM lost accuracy as the approximation order increased, while VC-VEM variants and standard self-stabilized VEM maintained good accuracy.
A cracked-panel benchmark highlighted local h-refinement. In practical terms, this means concentrating smaller elements in the part of the mesh where a detailed local result is needed rather than refining every element uniformly. The paper reports better convergence and better local stress prediction than with earlier uniform h- and p-refinement approaches.
The picture was less tidy on a distorted Voronoi mesh. Convergence continued despite the distortion, but the paper reports detrimental convergence effects and oscillations. Local h-refinement and pure p-refinement had similar convergence rates in that analysis, so local refinement did not clearly outperform the higher-order approach.
In a free-vibration analysis of a highly anisotropic plate, stabilized VEM produced smoother convergence trends but higher errors. Self-stabilized VEM produced lower errors, although its convergence oscillated.
Close matches in benchmark comparisons
For the heart-shaped-cutout vibration benchmark, VEM agreed well with isogeometric analysis across the tested layups. Relative error was below 1% in most cases, while the highest reported error was slightly above 1% in Mode 5.
The thermal-buckling calculation also showed a close match with Abaqus: relative error in nondimensional critical temperature was below 1% for every mode. Mesh 1 at p = 8 used 5,877 degrees of freedom, the numerical unknowns in the model, compared with 9,684 for Mesh 2 at p = 5.
In the variable-stiffness circular-cutout buckling case, the predicted buckling modes agreed with the reference modes for all tested layups. Distorted meshes again separated the formulations: standard stabilized VEM produced incorrect buckling loads or mode shapes in some configurations, whereas stabilized and self-stabilized VC-VEM, along with standard self-stabilized VEM, correctly predicted buckling loads.
A numerical result with a clear boundary
Taken together, the tests show different numerical patterns across formulations and refinement strategies rather than a single result across every test. One formulation can offer smoother convergence, another lower error, and local refinement can help in one problem while offering no clear advantage in another.
The evidence remains entirely computational. The study compared deterministic models with analytical, literature, isogeometric-analysis or Abaqus references; it did not report a physical experiment or a measured structural response. The findings therefore describe performance in the reported plate configurations and discretizations, not accuracy or safety for a real structure.
The percentage errors should consequently be read as case-specific numerical comparisons tied to the five computational cases. The document includes supporting material and points to [46] for a more in-depth derivation.
Paper data and sources
Original title: A Comprehensive p-VEM Framework for Advanced Variable Stiffness Plates with Arbitrary Shapes
Authors: Paola Pia Foligno, Daniele Boffi, Fabio Credali, Riccardo Vescovini
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text