An updated Lagrangian Bézier finite element formulation for the analysis of slender beams

Abstract

A G1-conforming finite element formulation based on the Kirchhoff beam model suitable for the analysis of structures composed by coupling of slender beams is presented. A new set of kinematic parameters is introduced in order to account for the continuity required by the rod model. This new set of kinematic parameters defines the G1-map that guarantees continuity of the rotations at the ends of the beam. The tangent stiffness matrix for the proposed Kirchhoff beam model is derived in a consistent way. It is shown that an additional geometric term, specific for the G1-conforming formulation, appears in the tangent stiffness matrix. In order to avoid the singularities arising with the introduction of the G1-map, an updated Lagrangian formulation is adopted. In this way, a G1-conforming Bézier finite element based on the Kirchhoff beam model able to model large deformations of space rod systems is obtained. Several numerical examples show the high accuracy and the robustness of the proposed conforming formulation.

An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods

“…First, in order to perform the variations of the strain energy functional, it is necessary to introduce the connection for the rotation manifold. The proper form of the Levi-Civita connection was found by Greco et al [9, 10], who showed that the consistent use of the covariant derivatives yields the proper symmetric forms of the tangent operators. Second, an objective interpolation of the finite rotations cannot be obtained with the standard polynomial interpolations [11].…”

“…The model employs the implicit G 1 continuity introduced by Greco and Cuomo [6, 7] and Greco [8]. Furthermore, the use of the Levi-Civita connection for the configuration manifold allows to naturally obtain symmetric tangent operators as shown by Greco et al [9, 10].…”

“…Their expressions were derived by Greco et al [9, 10] and are reported for completeness in section “Strain measures” and are symmetric bi-linear forms of the configuration quantities.…”

“…In particular is proposed an interpolation for the rotation angle ϕ able to preserve objectivity of the strain measures. In addition, a re-parametrization of the configuration space is adopted, which uses a new set of conforming parameters q that account for the G 1 continuity required by the variational formulation of the KL rod model as done by Greco [8] and Greco et al [9, 10].…”

“…The G 1 implicit conforming formulation proposed by Greco and Cuomo [6, 7], Greco [8], and Greco et al [9, 10] is used in this work. A generalization of this idea to the IGA for the static non-linear case can be found in Vo et al’s study [55, 56] while for the dynamic linear case in another Vo et al’s study [55].…”