Locking-free isogeometric discretizations of linear plane Timoshenko rods: LAS elements

Abstract

We make use of linear plane curved Timoshenko rods as a model problem to study how to overcome shear and membrane locking in NURBS-based discretizations. In this work, we propose lumped-assumed-strain (LAS) elements, a projection-based assumed-strain treatment that removes shear and membrane locking for a very broad range of slenderness ratios. For a NURBS patch with basis functions of degree p and Cp−1 continuity across element boundaries, the space of the assumed strains is defined on a B-spline patch with the same elements, but using basis functions of degree p−1 and Cp−2 continuity across element boundaries. The assumed strains are obtained by performing a L2 projection of the compatible strains at the patch level in which the consistent mass matrix is substituted with the lumped mass matrix. Our numerical investigations suggest that LAS elements need either the same or slightly finer mesh resolutions than B̄ elements in order for the relative errors in L2 norm of the unknowns and the stress resultants to be all below 1%. However, LAS elements are significantly more computationally efficient than B̄ elements for a given mesh. The use of C1-continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element exhibit superconvergence of the stress resultants for most slenderness ratios, making it a particularly cost-effective choice of obtaining accurate results. When compared with linear Lagrange B̄ elements with 2 Gauss–Legendre quadrature points per element, C1-continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element require far fewer elements in order to obtain relative errors in L2 norm of the unknowns and the stress resultants smaller than 1%.