A Stabilized Finite Element Framework for Incompressible Hyperelastic Materials at Finite Strains: Analysis, Implementation, and Applications

Abstract

This paper presents a stabilized finite element method (FEM) for incompressible hyperelastic materials at finite strains, addressing the computational and implementational challenges of traditional inf-sup-stable mixed FEMs. By augmenting the weak formulation with Galerkin/Least-Squares (GLS) stabilization terms, the method enables equal-order Lagrange elements (${\mathbb{P}}_k/{\mathbb{P}}_k$), eliminating the need for complex element pairings like Taylor–Hood (${\mathbb{P}}_2/{\mathbb{P}}_1$). The stabilization parameter is defined as $\tau =0.1\frac{h^2}{\mu }$ where $h$ denotes the finite-element mesh size and $\mu$ is the shear modulus; this choice, derived via eigenvalue analysis-ensures robustness against volumetric locking while preserving consistency. The nonlinear system of algebraic equations is solved via a Newton–Raphson scheme with a block-preconditioned GMRES solver, achieving optimal $𝒪\left(h^2\right)$ displacement convergence rates (comparable to Taylor–Hood elements). Validated on benchmarks including Cook's membrane and a pressurized neo-Hookean cylinder, the method demonstrates optimal $𝒪\left(h^2\right)$ displacement convergence and less than 2% error in reaction forces compared to Abaqus. It reduces degrees of freedom by 30%–40% and solve times by 35% relative to Taylor–Hood elements, as shown in weak scaling tests. An industrial case study on seal compression under 4 mm displacement highlights its capability to handle contact mechanics and geometric nonlinearities, with pressure oscillations suppressed below 5%. The framework's open-source implementation, combined with its accuracy, efficiency, and robustness under mesh distortion (aspect ratios up to 5:1) and near-incompressibility ($\nu =0.499999$), makes it a versatile tool for engineering applications. Key advancements include a theoretically grounded stabilization strategy, computational efficiency gains, and seamless integration into industrial workflows. The open-source implementation is available at https://github.com/ujwalwarbhe/A-Stabilized-Finite-Element-Framework-for-Incompressible-Hyperelastic-Materials.