Geometrically exact beam finite element with generalized B-spline interpolation on the special Euclidean group SE(3)

This work aims to address the challenge of achieving continuity in beam element interpolation by introducing a geometrically exact beam finite element based on generalized B-spline interpolation on the special Euclidean group $SE\left(3\right)$. The beam’s configuration is represented within the $SE\left(3\right)$ framework and interpolated using a generalized B-spline approach, enabling high-order interpolation with enhanced $C^1$ continuity. The interpolation is implemented through an extension of the classical De Boor algorithm to Lie groups via the pyramid algorithm. A systematic method is proposed for computing derivatives and linearizations essential for finite element formulations. Both static and dynamic equilibrium equations are derived on $SE\left(3\right)$, and the finite element formulations are established accordingly. Numerical examples validate the proposed element, confirming its correctness and adherence to critical properties, including objectivity, path-independence, and the absence of locking. The combination of $C^1$ continuity and the high degree of the interpolation substantially enhances the convergence performance. In particular, the degree-2 beam element achieves improved accuracy with sixth-order convergence rates when the degrees of freedom are relatively low. These characteristics make the proposed element highly suitable for high-accuracy simulations of beam structures undergoing large deformations and rotations.