2024/07/09 by Kassem Mustapha, Mustapha, K., William McLean +5 · 1 citation
Engineering · #FOS: Mathematics #Mechanical Engineering and Vibrations Research #Metal Forming Simulation Techniques #Metallurgy and Material Forming #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2407.06831
openalex publication_date 2024/07/09 · openalex created_date 2024/07/11 · openalex updated_date 2026/07/28
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→∞, or equivalently as the Poisson ratio ν→1/2. This effect is known as \itshape locking or \itshape non-robustness. For the piecewise linear case, the error in the \bf L2-norm of the standard Galerkin conforming FEM is bounded by~Cλh2, 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~λh=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 L2-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 H1-norm is bounded by Cλh1/2 h, which improves the Cλ1/2 h estimate for the case of conforming FEM.