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

Extending CAS elements to remove shear and membrane locking from quadratic NURBS‐based discretizations of linear plane Timoshenko rods

“…In addition, Figure 6D,E reveals an anomalous behavior in the convergence of the membrane and shear forces using NURBS elements, namely, the relative error in $$$ {L}^2 $$$ norm of the membrane and shear forces increases as uniform $$$ h $$$‐refinement is performed multiple times (the relative error of the membrane and shear forces is greater than $$$ 100\% $$$ for most mesh resolutions and slenderness ratios). This type of anomalous behavior was reported for the membrane force in discretizations of Kirchhoff rods based on B‐splines 4 and NURBS 50 . However, to the authors knowledge, it has not been reported for Timoshenko rods before.…”

“…This type of anomalous behavior was reported for the membrane force in discretizations of Kirchhoff rods based on B-splines 4 and NURBS. 50 However, to the authors knowledge, it has not been reported for Timoshenko rods before. In contrast, the convergence curves of CAS elements for R∕t = 10 2 , R∕t = 10 3 , and R∕t = 10 4 overlap (with the exception of the fine meshes for R∕t = 10 4 which undergo slight differences).…”

“…Remark In Reference 50, the authors of this article proposed the element type CAS elements to remove membrane locking from NURBS‐based discretizations of Kirchhoff rods. In this section, the element type CAS elements was extended to remove shear and membrane locking from NURBS‐based discretizations of Timoshenko rods.…”

“…Solving systems of linear algebraic equations at the element level and performing matrix multiplications to obtain the stiffness matrix are also needed. In Reference 50, the membrane locking present in $$$ {C}^1 $$$‐continuous quadratic NURBS‐based discretizations of linear plane Kirchhoff rods is overcame constructing continuous assumed strains through the linear interpolation of the compatible strains at the knots. The resulting element type is named continuous‐assumed‐strain (CAS) elements.…”

“…[43][44][45] Due to the higher inter-element continuity of splines, a direct deployment of the numerical schemes mentioned above is not an effective strategy to overcome locking. Reduced/selective-reduced integration rules at the element level and assumed strain treatments with discontinuous strain across elements boundaries fail to properly vanquish locking [46][47][48][49][50] while the condensation of strain variables at the element level cannot be carried out in Bubnov-Galerkin discretizations of mixed formulations. 6,19 Thus, the higher inter-element continuity of splines demands to develop new numerical schemes to overcome locking.…”