An Investigation of Stabilization Scaling in Finite-Strain Virtual Element Methods for Hyperelasticity

Abstract

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.

Figures (15)

• Figure 1: Example of an auxiliary triangulation $\mathcal{T}_E$ used to evaluate the stabilization contribution without introducing additional degrees of freedom.
• Figure 2: Schematic definition of an ellipse-based aspect ratio for a polygonal element.
• Figure 3: Single-element isochoric kernel-mode diagnostic: raw stabilization energy $U_{h,E}^s(\alpha\,\mathbf{u}^{\ker})$ as a function of Poisson ratio $\nu$ on the unit square. The same kernel mode $\mathbf{u}^{\ker}\in\ker(\Pi^\nabla_{E})$ is used for all $\nu$, with fixed amplitude $\alpha=10^{-2}$ under a fixed Young modulus $E_Y$ (hence $\mu=E_Y/(2(1+\nu))$ varies mildly with $\nu$). The classical stabilization exhibits a monotone increase in energy as $\nu\to 1/2$, despite the mode being isochoric, whereas the decoupled stabilization follows only the shear-scale variation and remains bounded.
• Figure 4: Single-element isochoric kernel-mode diagnostic for the classical stabilization: normalized stabilization energy versus $\nu$. The normalization by the physical shear scale, $U_{h,E}^s/(\mu\,\alpha^2)$, increases strongly as $\nu\to 1/2$, indicating bulk-driven stiffening of an isochoric kernel mode. The same energy normalized by the effective parameter used internally by the classical stabilization, $U_{h,E}^s/(\hat{\mu}\,\alpha^2)$, remains approximately constant, confirming that the classical stabilization scales this isochoric mode with $\hat{\mu}$ rather than with $\mu$.
• Figure 5: Comparison of classical and decoupled stabilization on an isochoric kernel mode under the physically relevant normalization $U_{h,E}^s/(\mu\,\alpha^2)$. The decoupled stabilization remains essentially constant over the full range of $\nu$, consistent with a shear-scaled penalty on isochoric missing modes. The classical stabilization grows substantially as $\nu\to 1/2$, providing element-level evidence of volumetric-proxy leakage into the shear-type stabilization channel.

Theorems & Definitions (18)

• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.1
• Proposition 4.1