Effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear-deformable beams with varying curvature and torsion

We present a displacement-based and a mixed isogeometric collocation (IGA-C) formulation for free-form, three-dimensional, shear-deformable beams with high and rapidly-varying curvature and torsion. When such complex shapes are concerned, the approach used to build the IGA geometric model becomes relevant. Although IGA-C has been so far successfully applied to a wide range of problems, the effects that different parameterization and knot placement techniques may have on the accuracy of collocation-based formulations is still an unexplored field. To fill this gap, primal and mixed formulations are used combining two parameterization methods (chord-length and equally spaced) with two knot placement techniques (uniformly spaced and De Boor). With respect to the space-varying Frenet local frame, we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations. This approach is very efficient and easy to implement within a collocation-based scheme. Several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques, revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of, at least, p=6, although some caution must be adopted when very high Jacobians and small curvatures occur. The same holds for the mixed formulation, with the difference that p=4 is enough to yield accurate results.