## 5 Cross-Section Model Based on Polygonal Discretization
### Algorithm of the Proposed Model on the Element Level  
In classic FEM formulation, in analysis step-iteration \(k\)-\(i\), the following displacement vector is applied to the structural end nodes of an element:  
\[\mathbf{q}^{k,i}=\mathbf{q}^{k-1,i}+\Delta\mathbf{q}^{k,i} \tag{67}\]  
As proposed by [PERSON] et al. [5], internal to iteration \(i\), another iterative procedure (with iteration index \(j\)) is taking place with the aim of minimizing the unbalanced forces in the cross-section level. The procedure starts with the computation of the nodal-force increments \(\Delta\mathbf{Q}^{i}\) of each element in iteration \(j\) as  
\[\Delta\mathbf{Q}^{i}=\left[\mathbf{F}_{E}^{j-1}\right]^{-1}\cdot\Delta \mathbf{q}^{j}=\mathbf{K}_{E}^{j-1}\cdot\Delta\mathbf{q}^{j}\text{ if }j=1 \tag{68}\]  
Herein, the element's flexibility matrix, \(\mathbf{F}_{E}\), results from integrating the individual flexibility matrices \(\mathbf{f}_{s}(x)\) of the control cross-sections over the element length using the Gauss-Lobato integration scheme as follows:  
\[\mathbf{F}_{E}^{j-1}=\int\limits_{0}^{L}\mathbf{b}^{\mathrm{T}}(x)\cdot \mathbf{f}_{s}^{j-1}(x)\cdot\mathbf{b}(x)dx\Rightarrow\mathbf{K}_{E}^{j-1}= \left[\mathbf{F}_{E}^{j-1}\right]^{-1} \tag{69}\]  
Thus, the element-stiffness matrix \(\mathbf{K}_{E}\) can be computed by inverting its flexibility matrix \(\mathbf{F}_{E}\). Next, from the element nodal-force increments \(\Delta\mathbf{Q}^{j}\) the control cross-section force increment vector \(\Delta\mathbf{D}^{j}(x)\) can be derived using Equation (55), i.e.,  
\[\Delta\mathbf{D}^{j}(x)=\mathbf{b}(x)\cdot\Delta\mathbf{Q}^{j} \tag{70}\]  
For each control cross-section \(x\), the corresponding deformation vector \(\Delta\mathbf{d}^{j}(x)\) is computed from the following equation. \[\Delta\mathbf{d}^{j}(x)=\mathbf{r}^{j-1}(x)+\mathbf{f}_{\mathrm{s}}^{j-1}(x) \cdot\Delta\mathbf{D}^{j}(x)\,x\in[0,L] \tag{71}\]  
In Equation (71), \(\dot{\mathbf{f}}_{\mathrm{s}}^{j-1}(x)\) is the cross-section-flexibility matrix at the previous iteration \(j-1\), and \(\mathbf{r}^{j-1}(x)\) is the corresponding vector of the residual deformations that must be minimized during the \(j\) iterations. This vector was computed in the previous iteration \(j-1\), as  
\[\mathbf{r}^{j-1}(x)=\mathbf{f}_{\mathrm{s}}^{j-1}(x)\cdot\mathbf{D}_{U}^{j-1} (x)\,\mathbf{D}_{U}^{j-1}(x)=\mathbf{D}^{j-1}(x)-\mathbf{D}_{int}^{j-1}(x) \tag{72}\]  
where \(\mathbf{D}_{U}^{j-1}(x)\) is the cross-section's unbalanced forces and \(\mathbf{D}^{j-1}(x)\), \(\mathbf{D}_{int}^{j-1(x)}\) are its external and internal forces, respectively. Then, for the current iteration \(j\), the internal cross-section forces \(\mathbf{D}_{int}^{j(x)}\) are computed. The computation is based on the procedure of Section 5 and is implemented in the embedded flowchart FC3 (see Section 6.3). Also, new values of quantities \(\mathbf{D}_{U}^{j}(x)\) and \(\mathbf{r}^{j}(x)\) are computed from Equation (72). The cross-section-flexibility matrix \(\dot{f}_{\mathrm{s}}(x)\) is computed by inverting the corresponding stiffness matrix \(\mathbf{k}_{\mathrm{s}}^{j}(x)\) of Equation (62). The execution of the above computations for all control cross-sections of an element is followed by the update of its flexibility and stiffness matrices, \(\mathbf{F}_{E}^{j}\) and \(\mathbf{K}_{E}^{j}\), respectively, using Equation (69), for use in the next iteration \(j+1\). The final step of the above procedure is the check for minimization of the unbalanced forces \(\mathbf{D}_{U}^{j}(x)\) for every control cross-section \(x\), using the convergence criteria of Section 6.1. If convergence is not achieved, a new iteration \(j+1\) starts with the element's residual displacements \(\mathbf{s}_{E}^{j}\) in place of the original \(\Delta\mathbf{q}^{j}\) of Equation (68).