Simulations

Basic Device

Theoretical feasibility was shown through back-of-the-envelope calculations of force output under reasonable assumptions for device dimensions. Hence, it made sense to spend the time to set up simulations in COMSOL for models of basic devices.

The simplest device was a single microcapacitor defined in a bulk dielectric material as two conductive plates. Applying voltage across the plates generated an attractive electric force between the plates, which COMSOL calculated and applied to the mechanical arrangement, then calculated the resulting deformation and stress. COMSOL then plotted the results in a heat map, where highest stress was shown in red and lowest stress in blue. The deformation from the original shape (shown in black contour) was also readily observable, showing correct contractive behavior.

Basic Device Optimization

The above simulation was repeated for multiple values of the parameter e, which signifies the width of the sides of the device. Such series of simulations are called a parameter sweep. The point of the sweep is to compare results as a function of the parameter and determine the optimal value that typically maximizes some important output parameter. Here, that output is the generated force per unit cross-sectional area, or force density.

The basic idea is that the capacitor generates force, but the force needs to be transferred to the outside world. In biological muscles, muscle fibers contract and pull on their sheathes made of connective tissue. These sheathes become ligaments and tendons. With these artificial muscles, the plates generate the attractive force, which pulls them closer, so they pull on the surrounding material trying to contract it. Hence, it is the surrounding material that transfers the force to the outside world.

Ideally, it is desirable to have as much force generation and as efficient force transfer as possible. However, both generation and transfer compete for the same real estate on the device, in this case cross-sectional area. If a larger percentage of the area is devoted to generation, less becomes available for transfer, and vice versa. This tradeoff suggests an optimal arrangement should exist that maximizes the output force density.

The conducted parameter sweeps produced the expected behavior. As the plate dimensions are kept the same while the side thickness e is increased, the generated force remains the same but its transfer to the outside world becomes more and more efficient. However, this process has diminishing returns and beyond a certain point, the output force essentially saturates. On the other hand, the overall area keeps increasing, so the force density, i.e. force per unit cross-sectional area, will first increase, then peak, then eventually start decreasing.

The simulations confirmed these predictions and showed an optimal value for e for the chosen geometries.

Simulations of Small Arrays

The success of the simulations of individual devices led to simulations of arrays of devices. The arrays were the next logical step, because in the end, the microcapacitors are on the micro scale, whereas artificial muscles are on the macro scale, so the use of arrays was inevitable. Hence, it was necessary to confirm that the same desirable basic behavior would be reproduced with multiple devices along the longitudinal direction as well as the two lateral directions. 

This led to 2x2x1 devices, which means two capacitors wide in each of the lateral directions but only one capacitor high along the longitudinal direction. When polarities were applied, it became evident that the traditional wiring of all plates in the same plane sharing the same polarity would produce contractive muscles. 

On the other hand, if the polarities were to be applied a checkered formation within the same plane, then horizontal attraction would be larger than the vertical one, leading to a bulge up instead of contraction. This led to the invention of counter-muscles, i.e. devices that expand when actuated, instead of contracting. 

Both types of muscles can be very useful, depending on the application. For example, if a particular actuator must do a reciprocal motion that must be powered in both directions, the ready solution would be the employment of both muscles and counter-muscles along the same axis. 

In contrast, biological muscles are strictly contractive. To reach out, a biological arm must contract a tricep on the opposite side of a joint, compared to using a bicep to pull the arm in. Biologically, both triceps and biceps are contractive. It is the joint that inverts the motion. 

Alternatively, it can be said that employing artificial muscles and counter-muscles may avoid the use of joints in some applications, and thus make the devices more compact and potentially more efficient.

Larger Array Simulations

The success of the small arrays led to the idea to build larger and larger arrays and simulate them, to confirm the general behavior remains the same and is valid as the array size increased. The resulting simulations confirmed these expectations. 

Larger arrays required increasing computational resources, so eventually the simulations had to be migrated from a single powerful desktop PC to the Hamming cluster at NPS. The PC took 4 days to run COMSOL on the 10x10x10 array, and it ran out of memory when it tried to load a 11x11x10 simulation. Even at the cluster, the simulations started taking too long with 13x13x10 arrays. So, going beyond 14x14x10 became apparent to be impractical even when using the computer cluster.