# Multi-Directional Shape Change Analysis of Biotensegrity Model Mimicking Human Spine Curvature
## 4 Numerical Examples
### Convergence History  
Figures 9-11 show the plots of the normalized objective function (NOF) versus the iteration for 54 cases of shape change analysis with different target displacements in the uni-directional mode for SB1, SB2 and SB3 models. The NOF plots for all of the series show either linear or nonlinear patterns until reaching a plateau towards the end of the optimization process. Obvious differences in trends were noticed between the NOF plots  
Figure 8: Computational steps in series under tri-directional mode for SB2 model (**left**) series XY200 and (**right**) series XY400. Figure 7: Computational steps in series under bi-directional mode. for series Zr, Zn and the remaining series, particularly for cases with target displacements at 200 mm and 400 mm. For cases with target displacements of 600 mm, the pattern of plots for series Zr remains different from the others, while series Zn exhibits a similar NOF plot pattern to series Xn, Yp, Yn and Yp. Series Zr shows the highest iterations which are followed by series Zn and other series in most of the cases. Figure 10: Convergence history for 18 shape change analysis cases in the uni-directional mode with target displacement of 400 mm. Figure 9: Convergence history for 18 shape change analysis cases in the uni-directional mode with target displacement of 200 mm. The slope of curves starts to change at an NOF lower than 0.05 in almost all of the cases. It is also clearly seen at an NOF lower than 0.05 that NOF plots of few analysis cases reach a plateau extending greater than a quarter of the total iteration (i.e., cases Xp400 and Yp400). Many applications such as inspections and maintenance works do not require high accuracy of the shape change results. Therefore, the efficiency of the proposed algorithm is rechecked by allowing a 2% error (where the value of 2% is based on the recorded successful rate of the overall analysis). It is found that, with this allowance, reduction of up to 40% total iteration in cases of target displacements in the uni-directional mode can be achieved. Figure 12 shows the plots of the NOF versus iteration for 36 analysis cases with different target displacements in the bi-directional mode. The convergence of NOF in a nonlinear manner was observed in all of the cases with target displacements of 200 mm (series XY200), 400 mm (series XY400) and 600 mm (series XY600). A slower rate of convergence was generally observed at the end of the optimization process. Particularly, an unusual long plateau was observed in the analysis case SB1 XnYn600 (up to 485 iterations), where the NOF of this case remained at the value of 0.01 for almost 60% of the total computational step. An allowance for a 2% error in the proposed algorithm can reduce up to 66% of the total computational step in analysis case SB1 XnYn600 and up to 40% for the remaining cases of target displacements with the bi-directional mode. Figures 13 and 14 show the plots of the NOF versus iteration for 56 analysis cases with different target displacements in the tri-directional mode of SB2. Only the results of the analysis of model SB2 are shown, as similar trends of results are observed in the SB1 and SB3 models. The convergence of NOF in a nonlinear manner is observed in all of the cases in the tri-directional mode. There are also significant changes in slope observed in these NOF plots. For instance, series XY200Z and XY400Z show a significant change in slope at NOF of ranges of 0.1-0.3 and 0.3-0.5, respectively. The NOF in series Zp begins with a short and significant drop, which is then followed by a long curve with a linear decreasing pattern. Series XY200 Zp and XY400 Zp show a significant change in slope at NOF of ranges of 0.7-0.95 and 0.6-0.9, respectively. Generally, a more gradual slope in NOF is observed in series Zp compared to both series Z and Zn for all of the target displacement magnitudes of 200 mm, 400 mm and 600 mm. Series Zn only shows a change in slope at an NOF of less than 0.1 for all target displacements of 200, 400 and 600 mm.