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arXiv · 2609.35878

PoliVEM: a Python-driven virtual element framework for computational solid mechanics

Paulo Akira F. Enabe·Rodrigo Provasi
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Abstract

This work presents PoliVEM, a software framework for the Virtual Element Method (VEM) in computational solid and structural mechanics. A C++17 computational core and a Python interface place one-dimensional beams, two- and three-dimensional elasticity, axisymmetric elasticity, transient diffusion, and finite-strain hyperelasticity in a common implementation. The framework stores vertex, edge, face, and cell degrees of freedom in one hierarchy, constructs the energy, strain, and $L^2$ projections from common polynomial data, and retains the consistency--stabilization split at the element level. The core separates the mesh, material, element, assembler, and solver responsibilities and combines them by composition. A new formulation supplies its projection, discrete form, and stabilization while reusing the mesh representation, degree-of-freedom numbering, boundary-condition treatment, sparse assembly, algebraic solvers, and Python binding pattern. The numerical infrastructure includes polygonal and polyhedral mesh input, higher-order entity numbering, cached projection operators, static condensation, coloured sparse assembly, linear solver selection based on the matrix structure, an incremental Newton method with line search and regularization, and explicit and implicit time integration. Three numerical experiments examine the shared implementation through polynomial reproduction, recovery after static condensation, higher-order degree-of-freedom handling, polygonal and polyhedral meshes, patch tests, stiffness spectra, and comparisons with independent finite element calculations. The results verify the common core for the beam and two- and three-dimensional elasticity formulations. They also identify the scaling of high-order polynomial bases on anisotropic cells and the memory required by sparse direct factorizations as the main limitations of the current implementation.

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Paulo Akira F. Enabe, Rodrigo Provasi. 2026-09-26. PoliVEM: a Python-driven virtual element framework for computational solid mechanics. https://arxiv.org/abs/2609.35878

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