Locking-Free WG Finite Element Method

• The paper introduces a locking-free approach that scales the volumetric penalty to mitigate artificial stiffening in nearly incompressible elasticity problems.
• It demonstrates that the modified weak Galerkin FEM achieves optimal O(h) convergence in both L² and H¹ norms, validated through benchmark tests.
• The method integrates easily into existing FEM frameworks, significantly enhancing accuracy and performance in challenging engineering applications.

A locking-free weak Galerkin finite element method (WG-FEM) is a discretization strategy for elliptic, saddle-point, and elasticity problems that, by design, avoids the artificial stiffening (“locking”) that commonly plagues standard low-order finite element methods (FEMs) when the governing equations approach an incompressible or nearly incompressible regime (i.e., as the Lamé parameter $\lambda \to \infty$, or equivalently as Poisson’s ratio $\nu \to 1/2$). Locking manifests as deterioration of accuracy or convergence rate, especially in the ${\bf L}^2$-norm of displacements, often rendering the method impractical for realistic engineering applications. Recent research proposes both weak Galerkin and simple modifications to standard conforming FEMs to produce locking-free approximations, robust with respect to the material parameters, for nearly incompressible elasticity problems (Mustapha et al., 9 Jul 2024).

1. Locking in Low-Order Conforming FEM and the Need for Modification

Standard conforming Galerkin FEM for elasticity uses the bilinear form: $$B(u, v) = \int_\Omega [2\mu \, \varepsilon(u) : \varepsilon(v) + \lambda (\nabla \cdot u)(\nabla \cdot v)] \,dx,$$ where $\lambda, \mu$ are Lamé parameters and $\varepsilon(u)$ is the strain tensor. For piecewise linear elements ($P_1$), the ${\bf L}^2$-norm of the error satisfies

$\|u-u_h\|_{L^2} \leq C_\lambda h^2,$

where $C_\lambda$ grows unbounded as $\lambda \to \infty$. In nearly incompressible regimes, this translates into severe inaccuracy for practical mesh sizes, a phenomenon known as locking (or non-robustness). This limitation arises from the disproportionate contribution of the volumetric (divergence) term as $\lambda$ increases.

2. Principles of Locking-Free Modification: Penalty Parameter Scaling

Recent approaches mitigate locking by adjusting the coefficient in front of the volumetric term. Instead of using the true (possibly large) $\lambda$, the method introduces a scaled penalty parameter, $\lambda_h = \lambda \mu / (\mu + \lambda h / L)$, where $L$ is the diameter of the domain $\Omega$ (Mustapha et al., 9 Jul 2024). The stiffness matrix or bilinear form is thus modified as: $$B^*(u,v) = \int_\Omega [2\mu \, \varepsilon(u) : \varepsilon(v) + \lambda_h (\nabla \cdot u)(\nabla \cdot v)] \,dx.$$ Alternatively, in the exponent scaling variant, $\lambda$ is replaced by $\lambda^\alpha$ with

$$\alpha^* = \min\left\{1, \frac{\log(d_0/h)}{\log \lambda}\right\}, \quad \text{so that} \;\lambda^\alpha = d_0/h \;\text{when } \lambda \gg d_0/h.$$

This modification dampens the overemphasis of volumetric constraint and preserves the well-posedness and convergence of the method as $\lambda\to\infty$.

3. Theoretical Error Estimates and Locking-Free Convergence

The adjusted formulation yields significantly improved error estimates:

• The ${\bf L}^2$-norm of the error is bounded as

$\|u- u_h\|_{L^2} \leq C h,$

for some $C$ independent of $\lambda$.

• For the ${\bf H}^1$-norm, the error satisfies

$$\|\nabla u - \nabla u_h\|_{L^2} \leq C\lambda_h^{1/2} h,$$

which is an improvement over the standard $C \lambda^{1/2} h$ estimate, since $\lambda_h \ll \lambda$ in the locking regime.

It is observed in numerical experiments that practical convergence in ${\bf H}^1$ may exceed the theoretical bound, with rates close to $O(h)$ rather than $O(h^{1/2})$.

4. Numerical Validation and Benchmark Test Cases

The performance and robustness of the locking-free modification are validated on classical benchmarks:

• For a manufactured solution on the unit square $(0, \pi)^2$, as $\lambda$ is increased from $10^3$ to $10^5$, the standard method rapidly loses accuracy, while the modified method maintains $O(h)$ convergence in the ${\bf L}^2$-norm and improved (often $O(h)$) rates in the ${\bf H}^1$-norm.
• On Cook’s membrane—representing bending/shearing in a trapezoidal domain—the modified method yields accurate displacement fields and vertical displacement values at critical points (e.g., the tip of the loaded edge), even on coarse meshes, while standard FEM locks and underestimates displacements.

5. Mechanistic Rationale and Theoretical Context

The central mechanism underlying the method's effectiveness is the controlled reduction of the penalty associated with $\nabla\cdot u$ in the discrete system. By scaling the stiffness contribution via $\lambda_h$ or $\lambda^\alpha$, the divergence instability responsible for locking is tamed. This is analogous in spirit to approaches in weak Galerkin or mixed finite elements, where the space or the variational principle is relaxed to prevent overconstraint, but is unique in applying a mesh- and parameter-dependent adjustment directly within the classical Galerkin/FEM framework.

6. Practical Significance and Implementation Considerations

The locking-free penalty scaling strategy requires only a minimal modification to existing codebases: the elasticity parameter $\lambda$ is replaced according to the mesh size $h$ and possibly a representative length scale $L$ of the domain. No modification to the underlying approximation spaces or basis functions is necessary. The modification notably improves the accuracy when $\lambda$ exceeds $\mu L/h$; for commonly used finite elements and practical meshes in engineering, this threshold is surpassed routinely.

This simplicity enables rapid adoption in engineering applications where nearly incompressible materials (rubber, biological tissue, certain metals under specific loading) or competitive h-refinement strategies are relevant.

7. Comparison With Other Locking-Free Approaches

Alternative approaches to the locking problem in nearly incompressible elasticity include weak Galerkin, mixed, hybridizable, and discontinuous Galerkin methods, which typically augment the finite element space or modify variational formulations to circumvent divergence overconstraint. The penalty scaling method stands out due to its direct applicability to standard conforming FEMs without restructuring basis functions, assembling new types of degrees-of-freedom, or introducing Lagrange multipliers. This positions the method as a practical, low-overhead enhancement to mainstream elasticity solvers (Mustapha et al., 9 Jul 2024).

Summary Table: Standard vs. Locking-Free Modified Penalty Methods

Feature	Standard Conforming FEM	Locking-Free Penalty Scaling
Volume term coefficient	$\lambda$	$\lambda_h = \lambda\mu / (\mu + \lambda h/L)$ or $\lambda^\alpha$
${\bf L}^2$ error rate	$O(\lambda h^2)$ (locks)	$O(h)$, uniform in $\lambda$
${\bf H}^1$ error rate	$O(\lambda^{1/2} h)$	$O(\lambda_h^{1/2} h)$, improved; observed $O(h)$ in practice
Mesh requirement	Any conforming mesh	Any conforming mesh
Implementation	Standard assembly	Replace $\lambda$ by $\lambda_h$ or $\lambda^\alpha$ in stiffness matrix
Locking for large $\lambda$	Yes	No

Conclusion

The locking-free penalty scaling strategy for conforming finite element discretizations of nearly incompressible elasticity systematically bypasses the divergence instability that causes locking. By scaling the volumetric penalty in proportion to mesh size and material parameters, one recovers robust ${\bf L}^2$- and ${\bf H}^1$-norm error behavior across a broad range of Poisson ratios, as rigorously proven and confirmed in challenging benchmark problems including Cook’s membrane (Mustapha et al., 9 Jul 2024). This method provides a minimal yet effective adjustment to classic FEM workflows, bridging the gap between ease of implementation and robustness required in advanced elasticity simulations.