# Output Force Density Saturation in COMSOL Simulations of Biomimetic Artificial Muscles
## 2 Materials and Methods  
Figure 1 shows the basic geometry and parameter definitions used in the simulations. The overall bulk material (outer rectangular prism) and alternating pairs of electrode chambers (inner rectangular prisms) were situated in parallel within the material [26]. The structures were joined together using the union function. Tendon thicknesses e (at device edges) and h (between fibers) were locked to be equal and swept together between 100 and 200 \(\upmu\)m, in steps of 10 \(\upmu\)m. Top and bottom edge thickness c was set to 100 \(\upmu\)m. The non-flexed lateral widths (plateX, plateY) of micro capacitor plates were set equal to 1000 \(\upmu\)m. The non-flexed plate thickness (plateZ) and plate distance D were set equal to 100 \(\upmu\)m. Silicone (polydimethylsiloxane (PDMS)) was assigned to the bulk dielectric. Liquid water was assigned to the electrode chambers [25]. To set up the simulations, three physics nodes were used: electrostatics, solid mechanics, and moving mesh set to \"normal\". The solid mechanics node was applied to the bulk dielectric. The moving mesh node was applied to the electrode chambers for efficient computation in finite element analysis applications [27]. The electrostatics node was applied to all domains. Within the electrostatic node, a potential of 3000 V was applied to all surfaces of one set of electrodes and a ground potential was applied to all surfaces of the other set of electrodes. This choice was made for consistency with the previous work [25]. This voltage also ensures that even if the electrostatically induced deformation would shrink the distance between the plates of the micro capacitor, the resulting field would still be well-below the typical dielectric breach value of several hundred volts per micron. Within the solid mechanics node, a fixed-boundary node was applied to one side of the bulk dielectric that was parallel to the electrode plates (the bottom outer surface in Figures 1 and 2), while the opposite side (top outer surface in Figures 1 and 2) was allowed to deform. This method [25] significantly simplified the calculations for COMSOL while still producing valid results. Further, a boundary load from the electrostatic force was applied to all surfaces of the model with the outputs for each component force being equated to their respective Maxwell Surface Tensor Equations [28]. The boundary node allowed the electromagnetic forces to be included in the physical model. A quadrilateral linear mesh of uniform seed size was applied to the model. A table of parameters was created to avoid manually building each individual array and instead make proportional relationships between the parameters, so that increasing the array size would expand the overall structure to scale. The parameter N was utilized to define the lateral size of the model during our study in the number of micro capacitors deployed in each of the two lateral dimensions. Therefore, an N \(\times\) N \(\times\) 10 device contained N \(\times\) N parallel columns, wherein each column contained 10 micro capacitors. The model was iterated from N = 1 to N = 13 at a step of 1, while the size of the model in the z direction was kept constant (10 micro capacitors tall). Figure 1: Basic architecture and parameter definitions. Tendon thicknesses e (at device edges) and h (between fibers) were locked equal. Top and bottom edge thickness c was set to 100 \(\upmu\)m. Lateral widths (plateX, plateY) of micro capacitor plates were set to 1000 \(\upmu\)m. Plate thickness (plateZ) and plate distance D were set to 100 \(\upmu\)m.  
For each N value, the tendon width between adjacent muscle fibers (h = u) was set equal to the tendon width e at the edges of the overall array (Figure 1), and the resulting parameter e was swept from 100 to 200 \(\upmu\)m, in steps of 10 \(\upmu\)m. Heat maps (as in Figure 2) of the resulting deformations were generated to visually and qualitatively confirm the correct behavior of the simulation.