## 3 Finite Element Formulation
### Discretization of the Linearized Equations  
It is worth noting that separate interpolations of the displacement and rotation fields are conducted to make the geometric representations uniform in the finite element analysis and CAD. Since the displacement field takes values in linear space \(\mathbb{R}^{3}\) while the rotation field in nonlinear manifold \(SO(3)\), the interpolations of the displacement and rotation field are conducted in different methods. To satisfy the objectivity of strain measures, the Crisfield and Jelenic interpolation algorithm [12] is adopted, in which the relative rotation is interpolated. The interpolation formulas are given by:  
\[\mathbf{x}=N_{I}\mathbf{x}_{I},\qquad\mathbf{x}^{{}^{\prime}}=N^{\prime}{}_{I }\mathbf{x}_{I} \tag{39}\]  
\[\boldsymbol{\Theta}_{I}^{r}=N_{I}\log(\mathbf{R}_{r}^{-1}\mathbf{R}_{I}) \tag{40}\]  
where \(N_{I}\) is the NURBS basis function corresponding to control point \(I\), and a summation is extended to all the control points of the element. The notation \(\boldsymbol{\Theta}_{I}^{r}\) is the relative rotation, and \(\mathbf{R}_{r}\) is the rotation matrix of the reference point, which is typically chosen as the first control point on an element. The strain measures can thus be interpolated as:  
\[\varepsilon=\left[\begin{array}{c}\gamma\\ \kappa\end{array}\right]=\left[\begin{array}{c}\mathbf{R}^{\mathrm{T}}N^{ \prime}\mathbf{i}\mathbf{x}_{I}-\mathbf{A}_{1}\\ \mathbf{T}(N_{I}\mathbf{\Theta}_{I}^{\mathrm{r}})N^{\prime}\mathbf{\Theta}_{I} ^{\mathrm{r}}\end{array}\right] \tag{41}\]  
With  
\[\mathbf{R}=\mathbf{R}_{\mathrm{r}}\exp(\widetilde{\mathbf{\Theta}}^{\mathrm{r }}) \tag{42}\]  
Hence, the generalized displacement variation and its derivative can be discretized into:  
\[\delta\mathbf{h}^{\star}=N_{I}\delta\mathbf{h}_{I}^{\star},\qquad\delta \mathbf{h}^{\star\prime}=N^{\prime}\mathbf{\delta h}_{I}^{\star} \tag{43}\]  
where \(\delta\mathbf{h}_{I}^{\star}\) is the generalized displacement variation of the control point. The strain-displacements matrix \(\mathbf{B}_{I}^{\star}\) is the discrete counterpart of the strains operator \(\mathbf{B}^{\star}\), as shown in Equation (22), giving:  
\[\mathbf{B}_{I}^{\star}=\left[\begin{array}{cc}\mathbf{R}^{\mathrm{T}}N^{ \prime}\mathbf{}_{I}&(\widetilde{\mathbf{R}^{\mathrm{T}}\mathbf{x}^{\prime}}) \mathrm{T}N_{I}\\ 0&\mathrm{T}N^{\prime}\mathbf{}_{I}+(\mathbf{T}^{\prime}+\widetilde{\mathbf{ \kappa}}\mathbf{T})N_{I}\end{array}\right] \tag{44}\]  
Consequently, the strain measures can be written as:  
\[\begin{array}{c}\text{-}\\ \delta\mathbf{E}=\mathbf{B}_{I}^{\star}\delta\mathbf{h}_{I}^{\star}\end{array} \tag{45}\]  
The generalized internal force and inertial force can be discretized from Equations (23) and (30) as:  
\[\mathbf{F}_{I}^{\mathrm{int}}=\int_{I}\mathbf{B}_{I}^{\mathrm{r}}\widetilde{ \mathbf{S}}\mathrm{d}\xi_{1} \tag{46}\]  
\[\mathbf{F}_{I}^{\mathrm{ine}}=\int_{I}A_{\rho}N_{I}^{\mathrm{T}}\left[ \begin{array}{c}\ddot{\mathbf{x}}\\ \mathbf{T}^{\mathrm{T}}\mathbf{J}\mathbf{\alpha}+\mathbf{T}^{\mathrm{T}} \widetilde{\mathbf{\Omega}}(\mathbf{J}\mathbf{\Omega})\end{array}\right] \mathrm{d}\xi_{1} \tag{47}\]  
In this work, as a separate interpolation is employed, shear locking exists in 1-order elements and is circumvented by the selective reduced integration. The one-point quadrature strategy is employed to calculate the shear strain, while the full quadrature, i.e., two-point quadrature for 1-order element is used to calculate the membrane and bending strains. Besides, high-order elements will not suffer from the shear locking problem and the full quadrature is employed to obtain all the strains in the Equations (46) and (47).