An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods

Mixed formulations of the B̄ type, when applied to B-spline representations, bear significant computational cost and lead to a full stiffness matrix. In this work we present an efficient mixed formulation for avoiding membrane locking of curved non-polar plane rods in the context of isogeometric analysis. An efficient spline reconstruction of the assumed axial strains obtained by means of a local projection at the element level is performed using the method proposed by Thomas et al. (2015) for the reconstruction of the geometry. In this way a much smaller bandwidth of the stiffness matrix is obtained with respect to the non local B̄-formulation, with significant reduction of the computational cost. The blended mixed formulation splits in two steps: in the first step the local B̄ operators are defined at the element level via projection of the strain measure (both L2-projection and discrete collocation approaches are considered). Successively, by means of the spline reconstruction algorithm, the global projection of the strain measures at the patch level is defined. Numerical experiments show that the proposed method, in addition to completely remove membrane locking, yields the same accuracy and rate of convergence as the non local B̄-method.