POLI-VEM

POLI-VEM is my main research codebase — a C++17 library implementing the Virtual Element Method (VEM) and Finite Element Method (FEM) for problems in solid mechanics, with a pybind11 Python interface and a FastAPI REST middleware.

Solvers implemented

• 1D Euler-Bernoulli beam VEM — arbitrary-order solver with static condensation and distributed-load assembly.
• 2D linear elastic VEM (k ≥ 1) — energy projection $\Pi^\nabla$ for $k=1$ and L²-strain projection $\Pi^\varepsilon$ for $k \geq 1$ (Artioli et al. 2018), with verified convergence rates $O(h^k)$ on Voronoi and distorted polygonal meshes ($k = 1, 2, 3$).
• 2D axisymmetric VEM — rotational-symmetry formulation incorporating the radial weight in the weak form; supports standard, divergence, and boundary stabilization strategies.
• Parabolic VEM — time-dependent heat conduction with multiple time-integration schemes.
• 2D nonlinear VEM — finite-deformation hyperelasticity with a Neo-Hookean energy and two stabilization families:
  • Coupled (Van Huyssteen) — triangulation-based, autodiff tangent, modified Lamé parameters to prevent volumetric locking.
  • Decoupled (kernel-based) — submesh-free, constant precomputed tangent, deviatoric/volumetric decomposition in the principal frame.
• Hyperelastic FEM reference — Q1, Q8 (2D) and H8 (3D) elements, Newton solver, benchmarked against Cook’s membrane.

Stack

C++17 · Eigen3 · nlohmann/json · OpenCV · autodiff · pybind11 · FastAPI. JSON mesh format, manufactured-solution convergence harnesses, Cook’s-membrane benchmarks for k = 1, 2, 3.

This codebase backs several of the papers on the Publications page (axisymmetric VEM, parabolic VEM with SSP-RK time stepping, finite-strain stabilization scaling, and the hybrid VEM + deep-learning Euler-Bernoulli work).