A Locking-free modified conforming FEM for planar elasticity

Due to the divergence-instability, low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations suffer from the so-called locking phenomenon as the Lam\'e parameter λ→∞ and consequently the material becomes more and more incompressible. For the piecewise linear case, the error in the L²-norm of the standard Galerkin conforming FEM is bounded by C_λ h². However, C_λ → ∞ as λ → ∞, resulting in poor accuracy for practical values of h if λ is sufficiently large. In this short paper, we show that for 2D problems the locking phenomenon can be controlled by replacing λ with λ^α in the stiffness matrix, for a certain choice of α=α_*(h,λ) in the range 0<α≤ 1. We prove that for this optimal choice of α, the error in the L²-norm is bounded by $Ch$ where C does not depend on λ. Numerical experiments confirm the expected convergence behaviour and show that, for practical meshes, our locking-free method is more accurate than the standard method if the material is nearly incompressible. Our analysis also shows that the error in the H¹-norm is bounded by Ch1/2, but our numerical experiments suggest that this bound is not sharp.