## 3 Materials and Methods
### Computational Simulation  
Material Analysis  
A simulation of the jellyfish body material behavior was carried out (Figure 5) to evaluate the displacement generated and the amount of water that can be evacuated with each contraction. The amount of water displaced was used as data for the CFD analysis. The SMA spring return to its extended form is due to the design of the tensioned silicone body. After the contraction is generated and the temperature in the SMA is reduced, its state becomes malleable, so that it is extended by the silicone body itself.  
Using Equation (1) [47,48] that relates the shear modulus of the SMA material (\(G\)) [N/m\({}^{2}\)], SMA diameter (\(d\)) [m], variation of displacement during contraction (\(x\)) [m], springs' turn diameter (\(D\)) [m] and the number of spring turns (\(N\)), we could calculate the maximum force that the central SMA will make during the application of contraction.  
\[F=\frac{\left(G*d^{4}*x\right)}{\left(8*D^{3}*N\right)} \tag{1}\]  
The equation only allows \"estimating\" the force, due to the complexity of modeling the entire actuation system. Although the stress strain produced is directly considered, the term in the equation \"\(G\)\" relates to both types of stress, since both are in function of the \(Y\) (Young's modulus) as established  
Figure 4: (**a**) Manufacturing of the SMAs. (**b**) Detailed section of encapsulated lateral SMA. Source: Author.  
by [PERSON] [42]. When working with Nitinol, there is shear stress and normal stress, but since both efforts are partially present at the same time, it is considered that \(G=Y\). The calculated previous force is represented by a cyclic force exerted by the central SMA and was applied to the CAD model (the force is represented by the brown arrows for the two cases in central Figure 5b and lateral Figure 5c), which is assigned the parameters of the material (Ecoflex Silicone 00-30): viscosity 300 cps, shore hardness 00-30, breaking strength of 900%, tear resistance of 38 _pli_ (6654 Nm/m), Shore hardness 00-, specific volume (26 _cu.in_/_lb_), maximum temperature 450 \({}^{\circ}\)F (232 \({}^{\circ}\)C) [49]. Simulation results show a maximum displacement of 70 mm in the central bell after contraction, which represents a volume of displaced water equal to 557 cm\({}^{3}\). The zones with a red color range represent the zones of greater displacement, produced in the bell by the central SMA.  
Figure 6 shows the behavior of jellyfish modeling; Figure 6a shows the contraction of the jellyfish bell for a reference of 10 cm, where the system response is extremely fast, achieving the reference in the first 2 s, and keeping it constant because of the fuzzy controller. For its part, Figure 6b shows the response of the force generated by the SMA to a 5 \(N\) reference signal; here, the sudden increase of the force is due to the sudden contraction of the SMA.  
Figure 7 shows a comparison of the diameter variation for two contractions. During the first 0.7 s, it quickly shrinks in size due to the force of contraction exerted by the SMA. After that, it returns to its relaxed state again while the SMA is cooled. The curve shown in red differs slightly from the red one, mainly due to the data capture and filters applied in the vision system.  
Figure 5: Simulation of the jellyfish body displacement when the central SMA contraction is applied: (**a**) before contraction; (**b**) after the central contraction; (**c**) after lateral contraction Source: Author.  
Figure 6: Responses to parametric modeling to a reference. (**a**) Contraction of the jellyfish bell. (**b**) Force produced by the SMA.