## 5 Cross-Section Model Based on Polygonal Discretization
### Cross-Section Stiffness Matrix
#### 6.1.4 Split of the load-step increment for Convergence of the FEM Algorithm  
The most usual numerical problems that occur in the practical application of similar FEM formulations and lead the algorithm to no convergence for equilibrium, are the following:  
* The iterations of the algorithm do not lead to the solution monotonically. * The algorithm is getting \"trapped\" between two values, one on each side of the solution (\"flip-flop\" phenomenon). * Unacceptable convergence tolerance. In the proposed model, a simple methodology for handling such problems has been introduced to the main FEM algorithm. It is based on the fact that small loading steps \(\Delta\mathbf{P}\) usually lead to easier convergence. As for the computational cost due to the possible use of many small steps, this is discussed in Section 7. Thus, for convergence purposes, a maximum number of Newton-Raphson iterations, \(max\_iter\), is defined in which the method is expected to converge without problems. If, in a particular load step \(k\), this value is reached without convergence of the algorithm, then the initial load increment \(\Delta\mathbf{P}^{k}\) is divided by 3 and the computational process of load step \(k\) starts from the beginning (setting the iteration counter \(i\) to zero), with a \"new\" \(\Delta\mathbf{P}\) value equal to \(\Delta\mathbf{P}_{1/3}^{k}=\Delta\mathbf{P}^{k}/3\). If the algorithm fails again to converge, having reached the maximum number of iterations \(max\_iter\), the previous process is repeated by dividing the already reduced value of \(\Delta\mathbf{P}^{k}/3\) by 3. The new value \(\Delta\mathbf{P}_{1/3}^{k}=\Delta\mathbf{P}^{k}/3^{2}\) is now used for the recomputation of step \(k\). This process of dividing the initial load-step increment \(\Delta\mathbf{P}^{k}\) by three is continued until a final appropriate value \(\Delta\mathbf{P}_{1/3}^{k}=\frac{\Delta\mathbf{P}^{k}}{3^{spil3}}<\Delta\mathbf{P}^{k}\) is found for which convergence occurs for load step \(k\). Superscript _split3_ is the counter of the number of divisions by 3. Therefore, for _split3_ number of successive divisions, the current value of \(\Delta\mathbf{P}_{1/3}^{k}\) is given by the following equation:  
\[\Delta\mathbf{P}_{1/3}^{k}=\frac{\Delta\mathbf{P}^{k}}{3^{spil3}} \tag{65}\]  
The decision for the splitting of load increment \(\Delta\mathbf{P}^{k}\) is made in step 22 of the algorithm. Split is activated if \(i=max\_iter\) and the whole process follows the backward step order 22-25-26-6 of flowchart FC1. Thus, step \(k\) is divided into two parts, the first \(\Delta\mathbf{P}_{1/3}^{k}\) and the second \(\Delta\mathbf{P}_{res}^{k}\) that is equal to:  
\[\Delta\mathbf{P}_{res}^{k}=\Delta\mathbf{P}^{k}-\Delta\mathbf{P}_{1/3}^{k}= \Delta\mathbf{P}^{k}-\frac{\Delta\mathbf{P}^{k}}{3^{spil3}} \tag{66}\]  
If the application of the first part leads to convergence, then the application of the second part follows without changing the load-step counter \(k\), i.e., this second part is applied as a supplementary load portion in the current step \(k\). The process of application of the second part \(\Delta\mathbf{P}_{res}^{k}\) is implemented in steps 17-21-6 of flowchart FC1. The decision for this application is made in step 17 that gives a \"reminder\" that a previous split has occurred in step 22. As mentioned previously, the two convergence criteria of step 15 are used, when the split-load increments \(\Delta\mathbf{P}_{1/3}^{k}\) and \(\Delta\mathbf{P}_{res}^{k}\) are applied. In flowchart FC1, the magenta paths refer to the bifurcation of steps 22 and 17 of the algorithm, when the process of applying a reduced load increment \(\Delta\mathbf{P}_{1/3}^{k}\) and its supplementary value \(\Delta\mathbf{P}_{res}^{k}\) is activated. This methodology of local reductions of the loading increments \(\Delta\mathbf{P}\), when needed, proved to be very effective.