An isogeometric implicit G1 mixed finite element for Kirchhoff space rods

Abstract

configuration space is necessary for modelling the Kirchhoff rod (the centroid curve position vector and the torsional angle) [1,2]. In the literature this kind of manifold is also known as ribbon, see [3].Kirchhoff rod models have the advantage, with respect to shear deformable models, that shear locking is automatically avoided. Furthermore, it is used for modelling particular spatial structures, like cables with bending and torsional stiffness; for instance, in [4] the authors present a bending-stabilized cable model, while in [5,6] the authors consider the effect of torsional stiffness in aeroelastic analysis.In [1] a geometrically exact formulation of space rods based on a Lagrangian description was presented, that does not depend on the particular geometry of the centroid curve (differently from what has been done for instance in [7][8][9][10]). An orthogonal frame was introduced on the rod axis, that can be different from the natural frame (Bishop frame) used by Langer and Singer [11]. A pull-back of the strain along the directors was used to define the deformation of the cross section, analogously to what is done in non local and second gradient theory, i.e. [12][13][14][15][16][17].Since high continuity is required for the interpolation of the displacements in a Kirchhoff-Love rod model, the B-spline interpolation used in isogeometric analysis appears to be a natural choice for the development of numerical approximations of thin structural models. A first example of isogeometric interpolation for non polar rods can be found in [18] in which the authors have considered the polar formulation of rods developed in [19]. Many others numerical isogeometric formulations for rods have been proposed since (see, e.g., [20][21][22][23][24][25]).When multiple elements are used for discretizing the model, C 0 continuity for the end rotations of the element is needed, that for the Kirchhoff rod model means a G 1 continuity constraint on the deformation of the centroid curve, i.e., the unit tangent vectors have to coincide at the ends of adjacent elements; from an incremental point of view this means that the velocity of rotation at the ends of adjacent elements must be equal.In [26,27] G 1 continuity for space rods was obtained by means of a change of basis, generalizing Hermite's interpolation. The required geometric continuity was thus achieved without introducing Lagrangian or penalty terms in the formulation, as done for instance by the bending strip method [28]. An alternative strategy of a multi-patch approach for non polar shells can be found in [29].Although it has been claimed that sufficiently high degrees of interpolation avoid locking phenomena, isogeometric models, like all displacement based formulations, suffer from this pathology, that arises from the coupling terms appearing in the strain energy. Membrane and shear locking were observed in plane beam models [30][31][32][33] and membrane and flexural locking in space rod models [10,[34][35][36]. In [1,36] severe locking due to these interactions was fo...

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