Removing membrane locking in quadratic NURBS-based discretizations of linear plane Kirchhoff rods: CAS elements

Abstract

NURBS-based discretizations of the Galerkin method suffer from membrane locking when applied to primal formulations of curved thin-walled structures. We consider linear plane curved Kirchhoff rods as a model problem to study how to remove membrane locking from NURBS-based discretizations. In this work, we propose continuous-assumed-strain (CAS) elements, an assumed strain treatment that removes membrane locking from quadratic NURBS for an ample range of slenderness ratios. CAS elements take advantage of the C1 inter-element continuity of the displacement vector given by quadratic NURBS to interpolate the membrane strain using linear Lagrange polynomials while preserving the C0 inter-element continuity of the membrane strain. To the authors’ knowledge, CAS elements are the first NURBS-based element type able to remove membrane locking for a broad range of slenderness ratios that combines the following distinctive characteristics: (1) No additional degrees of freedom are added, (2) No additional systems of algebraic equations need to be solved, and (3) The nonzero pattern of the stiffness matrix is preserved. Since the only additional computations required by the proposed element type are to evaluate the derivatives of the basis functions and the unit tangent vector at the knots, the proposed scheme barely increases the computational cost with respect to the locking-prone NURBS-based discretization of the primal formulation. The benchmark problems show that the convergence of CAS elements is independent of the slenderness ratio up to 104 while the convergence of quadratic NURBS elements with full and reduced integration, local B̄ elements, and local ANS elements depends heavily on the slenderness ratio and the error can even increase as the mesh is refined. The numerical examples also show how CAS elements remove the spurious oscillations in stress resultants caused by membrane locking while quadratic NURBS elements with full and reduced integration, local B̄ elements, and local ANS elements suffer from large-amplitude spurious oscillations in stress resultants. In short, CAS elements are an accurate, robust, and computationally efficient numerical scheme to overcome membrane locking in quadratic NURBS-based discretizations.