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September 10, 2025

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

Key Points

The method achieves optimal displacement convergence rates comparable to Taylor–Hood elements, enabling efficient analysis.
Less than 2% error in reaction forces was observed in tests against Abaqus, highlighting its accuracy.
A contact mechanics case study demonstrates capability to manage geometric nonlinearities effectively.
The open-source framework improves scalability with reduced degrees of freedom by 30%–40%.

Abstract

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 (), eliminating the need for complex element pairings like Taylor–Hood (). The stabilization parameter is defined as where denotes the finite‐element mesh size and 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 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 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 (), 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 .

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