A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity

TL;DR

The work targets locking in low-order conforming FEM for nearly incompressible elasticity by introducing a simple Lamé-parameter modification: λ_h = λ μ /(μ + λ h / L). A scaling-based choice α_h = α /(1 + α h / L) yields λ_h < λ and enables a provably locking-free scheme with uniform L^2 convergence of order O(h) and improved H^1 convergence of order O(λ_h^{1/2} h). Theoretical results are supported by regularity assumptions valid in 2D convex polygons and are extended to a dimension-agnostic presentation. Numerical experiments on Cook’s membrane and a rectangle-domain example confirm the predicted convergence behavior and show practical accuracy gains when λ is large relative to μ L / h, demonstrating the method’s robustness and simplicity compared to more complex locking-free approaches.

Abstract

Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lamé parameter $λ\to\infty$, or equivalently as the Poisson ratio $ν\to1/2$. This effect is known as {\itshape locking} or {\itshape non-robustness}. For the piecewise linear case, the error in the ${\bf L}^2$-norm of the standard Galerkin conforming FEM is bounded by~$Cλh^2$, resulting in poor accuracy for practical values of~$h$ if $λ$ is sufficiently large. In this short paper, we show that the locking phenomenon can be reduced by replacing $λ$ with~$λ_h=λμ/(μ+λh/L)<λ$ in the stiffness matrix, where $μ$ is the second Lamé parameter and $L$ is the diameter of the body $Ω$. We prove that with this modification, the error in the ${\bf L}^2$-norm is bounded by $Ch$ for a constant $C$ that does not depend on $λ$. Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if $λ$ is larger than about $μL/h$. Our analysis also shows that the error in the ${\bf H}^1$-norm is bounded by $Cλ_h^{1/2}\,h$, which improves the $Cλ^{1/2}\,h$ estimate for the case of conforming FEM.

Figures (3)

• Figure 1: The domain $\Omega$ for Cook's membrane problem.
• Figure 2: The deformed mesh obtained when each node is displaced by ${\bf u}_h$. Left: using the standard conforming method. Right: using the modified method \ref{['FEM new']}.
• Figure 3: Convergence of the computed value of the vertical displacement $u_2(A)$ for each choice of the bilinear form.

Theorems & Definitions (12)

• Theorem 2.1
• Theorem 2.3
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Remark 3.4