Rodrigo Provasi

Paulo Akira F. Enabe, Rodrigo Provasi

Low-order virtual element methods (VEM) compute a consistent finite-strain contribution through polynomial projections and rely on stabilization to control the unresolved modes in the projector kernel. In current hyperelastic VEM practice, stabilization is often defined by integrating a nonlinear surrogate energy over an auxiliary sub-triangulation and scaled through modified Lamé parameters and incompressibility factors; this can introduce sensitivity to the arbitrary internal tessellation, complicate consistent Newton linearization, and, most critically, inject bulk-dependent proxies into shear-type kernel penalties, artificially stiffening isochoric missing modes in the nearly incompressible regime. This work develops a submesh-free, kernel-only stabilization that decouples deviatoric and volumetric channels and is explicitly designed to scale like the current Newton tangent energy on the kernel: the deviatoric term is scaled solely by a shear measure and enhanced by bounded geometry-driven directional weights, while the volumetric term is scaled by an independent bulk measure and can be capped or suppressed as $ν\to 1/2$. A spectral framework is established in which the canonical VEM stability requirement on the kernel is characterized by generalized Rayleigh quotients and eigenvalue bounds, and it is shown under standard polygon regularity assumptions that the deviatoric stabilization is uniformly equivalent to $μ_E|\cdot|_{1,E}^2$ on the kernel with constants independent of mesh size and Poisson ratio. Element-level diagnostics confirm that classical surrogate-based stabilizations assign bulk-driven energy to isochoric kernel modes as $ν\to 1/2$, whereas the proposed decoupled stabilization remains shear-scaled; kernel spectra and Cook's membrane simulations in the nearly incompressible regime further support improved robustness across polygonal mesh families.

Paulo Akira F. Enabe, Rodrigo Provasi

A hybrid framework integrating the Virtual Element Method (VEM) with deep learning is presented as an initial step toward developing efficient and flexible numerical models for one-dimensional Euler-Bernoulli beams. The primary aim is to explore a data-driven surrogate model capable of predicting displacement fields across varying material and geometric parameters while maintaining computational efficiency. Building upon VEM's ability to handle higher-order polynomials and non-conforming discretizations, the method offers a robust numerical foundation for structural mechanics. A neural network architecture is introduced to separately process nodal and material-specific data, effectively capturing complex interactions with minimal reliance on large datasets. To address challenges in training, the model incorporates Sobolev training and GradNorm techniques, ensuring balanced loss contributions and enhanced generalization. While this framework is in its early stages, it demonstrates the potential for further refinement and development into a scalable alternative to traditional methods. The proposed approach lays the groundwork for advancing numerical and data-driven techniques in beam modeling, offering a foundation for future research in structural mechanics.