An improved isogeometric collocation formulation for spatial multi-patch shear-deformable beams with arbitrary initial curvature

The present paper presents a robust multi-patch formulation based on the isogeometric collocation (IGA-C) method for the solution of linear, spatial Timoshenko beam structures with complex geometry. The proposed approach is based on the combination of the (local) Bishop frame with the exponential map for SO(3) (Rodrigues formula) to compute the beam curvature and its derivative. This choice permits bypassing known issues related to the Serret–Frenet (SF) frame (in cases where the local beam curvature vanishes) and does not require the Darboux vector and its derivative, which are both affected by limitations of the SF frame. Moreover, in contrast to existing formulations mostly derived in the local SF frame, here the formulation is consistently derived by linearizing the governing equations in the material setting, where the multi-patch coupling can be enforced in a straightforward way. Numerical tests are performed on complex spatial curved beams, including a demanding biomechanical problem of a braided stent, proving the accuracy and the robustness of the formulation.