# Output Force Density Saturation in COMSOL Simulations of Biomimetic Artificial Muscles
## 4 Discussion  
The simulated numerical results in Figure 4 were validated by comparing them with the results of the previous simulations and first-principle back-of-the envelope calculations [25]. More specifically, the previous 1 \(\times\) 1 \(\times\) 1 simulation [25] predicted a 1.44 kPa force density, which is essentially in agreement with the 1 \(\times\) 10 \(\times\) 10 result here of \(\sim\)1.75 kPa. The \(\sim\)20% increase can be easily attributed to overall smaller edge effects. Simultaneously, back-of-the envelope calculations adjusted for effective muscle area predicted up to a 2.4 kPa force density [25], which is within a factor of two from both simulations' results. The observed saturation has several important consequences. It proves the hypothesis that as the size of the array increases, the relative influence of edge effects decreases, so the output force density approaches an asymptotic level at array sizes that are still small enough to be successfully simulated in COMSOL. This is an important conclusion both for these artificial muscle devices and as a general method of simulating macro-actuators made of arrays of individual micro-actuators. It significantly alleviates limitations imposed by computational capacity and allows important determinations to be achieved with relatively modest resources. For example, the presented N = 13 simulations took 4 days to complete on the Hamming cluster at NPS, while the presented N = 10 simulations took just 1 day. Knowing that N = 10 produces a sufficiently good estimate of the output force density already decreases computational time by at least 4\(\times\). Furthermore, for this type of artificial muscle [25; 26], the results show that the output force density at N = 10 is already approximately equal to the saturation value. Hence, simulations of laterally larger arrays should not be necessary to predict their behavior in terms of output force density. This array size can now be used as the optimal standard for simulations of such actuators, while the specifics of the individual micro capacitor can be varied. Next, the presented results show that the saturation level of the output force density appears to be \(\sim\)9 kPa for the 100 \(\upmu\)m scale of artificial muscles. Hence, a 10 cm \(\times\) 10 cm artificial bicep printed at the 100 \(\upmu\)m scale should output \(\sim\)90 Newtons of actuation force. Furthermore, as the COMSOL works with linear homogeneous materials, the calculations scale with physical size. Hence, the same architectures printed at the 10 \(\upmu\)m scale would have to produce 100\(\times\) the force produced at the 100 \(\upmu\)m scale, because the electrostatic force inside the capacitor scales as the inverse square of the plate distance. Consequently, the same artificial bicep from the above example should output \(\sim\)9 kN of actuation force. Such forces would be sufficient for all intended applications, including swimming robots [37]. The ability to quantitatively estimate the saturation force density using the shown method can be used to quantitatively estimate the performance of craft propelled by such artificial muscles. Hence, the presented results are an important step forward towards the practical implementation of this type of artificial muscle. More generally, the presented method of stepwise increases in matrix size N with nested parameter sweeps to maximize the force density for each N and searching for asymptotic or saturation behavior is a good general method to use for other types of devices, and other geometries as well. So long as the asymptotic behavior can be achieved before the simulations run out of available computational power, the established performance and asymptotic quantitative output can be used to predict the performance and behavior of macro-scale arrays of the same type. Hence, what has been demonstrated is a general method for a much larger range of devices and simulations, as well as a specific tool for artificial muscles, and quantitative predictions for these types of artificial muscles.