# Model-Based Design Optimization of Soft Polymeric Domes Used as Nonlinear Biasing Systems for Dielectric Elastomer Actuators

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. DEA Operating Principle

#### 2.1. DEA Operating Principle

_{e}is generated, quantified as follows:

- ϵ
_{0}is the vacuum permittivity; - ϵ
_{r}is the DE relative permittivity; - V is the applied voltage;
- t is the thickness of the dielectric film.

#### 2.2. Bias Elements for DEAs

_{1}, p

_{2}for the PBS and d

_{1}, d

_{2}for the NBS, respectively. By comparing those two cases, the performance gain achieved with the NBS can readily be observed.

#### 2.3. Bistable and Monostable NBS Elements

## 3. Nonlinear Biasing Dome

#### 3.1. Dome FE Model

^{®}environment, is shown on the right-hand side of Figure 7.

_{b}permits avoiding taking into account tolerance errors in the dome clamping phase, resulting in more repeatable experimental results.

#### 3.2. Post-Buckling Analysis and Numerical Implementation Aspects

- Load control mode: the load force must necessarily be prescribed via nonlinear dynamic solver to solve the singularity of the problem, thus performing a time-dependent study. If a time-based analysis is conducted, there is a balance between the applied external load and elastic forces (note that all dynamic forces are neglected in this study). After that, the axial displacement represents the quantity calculated as the output. The final result is equivalent to the dashed red curve depicted in Figure 11, in which the dynamic jump from state 1 to state 2 is clearly visible.
- Displacement control mode: as the deformation represents the quantity increasing monotonically, it can be used as an input control parameter. In this way, the description of the load softening effect (occurring after the critical point) is derived unambiguously based on the simulation output.

- Principal stretches λ
_{1}, λ_{2}, λ_{3}; - Material constitutive parameters c
_{i}_{0}, i = 1, 2, 3; - Bulk modulus κ, which allows practical accounting of the material incompressibility in a numerically efficient way;
- Volume ratio J, equal to the determinant of the deformation gradient.

_{1}, A

_{2}, A

_{3}in Figure 8. The Comsol integrated direct solver MUMPS (MUltifrontal Massively Parallel Sparse) is selected, with a nonlinear Automatic (Newton) method, to solve the system of equations. The solver parameters are set as follows: maximum number of iterations of 25, initial damping factor of 10-4, and a tolerance factor of 10-2. Representative values of c

_{10}= 0.05 MPa, c

_{20}= 1.52 kPa, c

_{3}

_{0}= 5.11 kPa, κ = 22 × 10

^{6}, are chosen for the study.

## 4. Dome Calibration Based on the Experimental Characterization Process

#### 4.1. Dome Experimental Characterization Process

^{®}612 EH silicone [50], through a process of casting into 3D-printed molds. The experiments are conducted by deforming the flat upper part of the dome with a 0.1 Hz sinusoidal displacement, via an indenter connected to a linear actuator (Aerotech, Inc., Model: ANT-25LA). During the deformation, the force is acquired via a load cell (ME-Meßsysteme GmbH, KD40s). In this way the force–displacement characterization curve of the dome is obtained. Due to the tolerances of the 3D printer for mold manufacture and the manual mixture of the silicone used for the domes, several batches representative of the same geometry but made with different molds are created. In this way, it is possible to quantify the reliability and repeatability of the experimental curves, which will be included in the dataset shown in the next section. For further details regarding the experimental characterization of the dome, please refer to [38].

#### 4.2. Dome Identification and Validation

_{i}

_{0}, the bulk modulus κ, as well as the rounding parameters A

_{1}, A

_{2}, A

_{3}introduced for numerical robustness purpose (see Section 3.2 for details). Clearly, the numerical values of those coefficients are chosen identically for each dome geometry. Once the constitutive parameters are known, we can use the model to reproduce the experimental trends obtained for different geometric configurations of the dome structure, in order to make reliable and accurate predictions about its behavior.

## 5. Dome Design Optimization and Experimental Validation

_{x}, A

_{y}) and B = (B

_{x}, B

_{y}). These variables can be uniquely related to the following performance measures, which allow equivalently describe the target biasing curve in Figure 14 in a more intuitive way:

- Stroke, computed as the distance between states A and B along the x-axis, i.e., B
_{x}− A_{x}; - Slope, defined as the angular coefficient of the line connecting equilibrium points A and B, i.e., (B
_{y}− A_{y})/(B_{x}− A_{x}); - Maximum force, defined as B
_{y}; - Horizontal shift, defined as a constant offset applied to both A
_{x}and B_{x}.

#### 5.1. Optimal Parameter Selection

- The calibrated FE model (described in the Section 4) is used to realize a dataset of simulated force-strain curves for different dome geometries. For the considered case study, the ranges of H and r are chosen in a physically meaningful way as follows: H ∈ [3, 5], r ∈ [2, 4];
- The entire design algorithm is implemented in MATLAB
^{®}, based on the obtained simulation dataset. For each simulated force-strain curve, the minimum and maximum force points defining the unstable branch of the dome characteristic are calculated and collected, in order to determine corresponding slope, stroke, and maximum force. Those minimum and maximum points are therefore considered as representatives of A and B, where the intersection with the DE characteristic curves occurs; - Surface fitting functions are generated to express H, r, and maximum force as a function of the stroke and slope, based on the computations performed in the previous step. Resulting functions H = g
_{1}(stoke, slope), r = g_{2}(stoke, slope), and maximum force = g_{3}(stoke, slope) are shown in Figure 15. As it can be seen, such surfaces allow unique determination of H, r, and the maximum force, once the target stroke and slope are known.

#### 5.2. Design Optimization Algorithm

- Select a target DE membrane, and characterize it experimentally under quasi-static conditions in order to obtain the characteristic curves for minimum and maximum applied voltage;
- Based on the obtained DE curves, estimate an ideal biasing behavior, and compute the coordinates of the corresponding intersection points A and B (cf. Figure 14);
- Based on the coordinates of A and B, determine the desired stroke, the slope, and the maximum force values, i.e., the features that must be satisfied by the dome force–displacement curve in order to ensure the desired performance;
- If the maximum force is not satisfactory, one can eventually start again rom point 2 and try different combinations of stroke and slope values, until an overall desirable behavior is obtained.

#### 5.3. Design Procedure Validation

_{DE}= [80–100] V/µm) [46]. The results are shown in Figure 18. For the maximum voltage, a stroke of about 3.1 mm is observed. This quantity is smaller than the theoretical value of 3.8 mm predicted with the graphical method, possibly due to unavoidable tolerances, misalignments, and inaccuracies occurring during the manufacturing process.

## 6. Discussion and Future Developments

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Examples of NBS elements: (

**a**) pre-compressed buckled beam; (

**b**) attracting permanent magnet; (

**c**) buckling polymeric dome.

**Figure 2.**DE working principle: undeformed configuration on the left-hand side, deformed one due to electrical actuation on the right-hand side. Due to the combination of Maxwell stress σ

_{e}and incompressibility of the elastomer, the reduction in thickness results in an expansion of the area.

**Figure 3.**Three-dimensional (3D) rendering of a DEA: (

**a**) undeformed flat configuration; (

**b**) deformed out-of-the plane configuration.

**Figure 4.**(

**a**–

**c**) Sketch of the DEA actuator composed by a DE membrane coupled with a PBS element, for three different configurations. (

**d**) Actuator force equilibrium analysis, intersection points p1 and p2 represents the equilibrium states obtained by switching the voltage.

**Figure 5.**(

**a**–

**c**) Sketch of the DEA actuator composed by a DE membrane coupled with a NBS element, for three different configurations. (

**d**) Actuator force equilibrium analysis, intersection points d

_{1}and d

_{2}represents the equilibrium states obtained by switching the voltage.

**Figure 6.**A sketch of (

**a**) the bi-stable NBS and (

**b**) the monostable structure is presented on the left-hand side, while the corresponding mechanical characteristics are shown on the right-hand side, respectively.

**Figure 7.**Fully polymeric dome: (

**a**) real-life prototype; (

**b**) 3D rendering implemented in COMSOL Multiphysics

^{®}.

**Figure 8.**Two-dimensional (2D) FE model of the dome. Boundary conditions are explicitly reported as black solid lines. The non-colored image indicates the undeformed configuration, while the colored one shows the dome deformation through the colormap on the right-hand side. Note: due to radial symmetry, only the part of the dome which is located at positive y-coordinates is actually modeled.

**Figure 9.**Three-dimensional (3D) FE of the dome, obtained from a rotation of the 2D model in Figure 8 along the z axis: (

**a**) Undeformed configuration; (

**b**) Deformed configuration.

**Figure 11.**Experimental force–displacement curve of polymeric dome. The red and blue lines are derived by applying the load and displacement control methods, respectively. Points 1 and 2 indicate the dynamic jump occurring under load control mode, from state 1 to 2.

**Figure 12.**Mesh convergence study: (

**a**) quantifies the average percentage error between the curve obtained by setting the mesh extremely fine (considered as reference) and the other default types listed on the x-axis; (

**b**) 2D FEM, with mesh defined as normal and customized along the blue edges. (

**c**) quantifies the average percentage error between the characteristics curves resulting from varying the maximum size of the mesh elements along the blue edges.

**Figure 13.**Matrix of plots showing the comparison among experimental (depicted in black) and simulated (depicted in red) curves, for different dome geometries.

**Figure 14.**Intersection between the DE mechanical characteristics without (solid blue line) and with (solid red line) applied voltage, and ideal biasing behavior ensuring large stroke (solid black line). Points A e B represent the system equilibrium points for low and high voltage, respectively. Their distance along the horizontal axis defines the corresponding stroke.

**Figure 15.**Surface fitting functions created based on the simulations dataset, with the final goal of deriving the dome geometric values for H and r based on the desired performance: (

**a**) H as a function of stroke and slope; (

**b**) r as a function of stroke and slope; (

**c**) maximum force as a function of stroke and slope.

**Figure 16.**Design algorithm experimental validation. The input is represented by the desired intersection points (black circles), while the dashed black curve and solid gray curve represent the simulated and experimental results based on the optimized dome geometry.

**Figure 18.**The upper part shows a step voltage signal applied as input, applied for a time of 10 s. The different colors represent signals of different amplitudes set as the input, which vary in the range 3–3.5 kV with increments of 0.1 kV. In the lower part, the corresponding stroke reached by the DEA is depicted.

**Figure 19.**The upper part shows a sinusoidal voltage signal applied as input, at the frequency of 0.1 Hz. The different colors represent signals of different amplitudes set as the input, which vary in the range 3–3.5 kV with increments of 0.1 kV. In the lower part, the corresponding stroke reached by the DEA is depicted.

**Figure 20.**Stroke reached by the DEA as a function of the applied voltage, considering sinusoidal voltage inputs having frequency of 0.1 Hz and different maximum amplitudes.

r = 2 mm | r = 3 mm | r = 4 mm | |
---|---|---|---|

H = 3 mm | Validated | Validated | Validated |

H = 3.5 mm | Validated | Validated | Validated |

H = 4 mm | Identified | Identified | Identified |

c_{10} | c_{20} | c_{30} | κ | A_{1} | A_{2} | A_{3} |
---|---|---|---|---|---|---|

0.11 MPa | 3.29 kPa | 5.73 kPa | 0.25 × 10^{6} | 0.44 mm | 0.78 mm | 2.2 mm |

**Table 3.**Normalized stroke reached with different bias elements: positive-rate biasing spring (PBS), attracting permanent magnets (PM), negative-rate biasing spring (NBS), and novel silicone dome.

PBS | PM | NBS + PBS | Silicone Dome |
---|---|---|---|

0.1 [-] | 0.31 [-] | 0.4 [-] | 0.62 [-] |

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**MDPI and ACS Style**

Croce, S.; Neu, J.; Hubertus, J.; Seelecke, S.; Schultes, G.; Rizzello, G. Model-Based Design Optimization of Soft Polymeric Domes Used as Nonlinear Biasing Systems for Dielectric Elastomer Actuators. *Actuators* **2021**, *10*, 209.
https://doi.org/10.3390/act10090209

**AMA Style**

Croce S, Neu J, Hubertus J, Seelecke S, Schultes G, Rizzello G. Model-Based Design Optimization of Soft Polymeric Domes Used as Nonlinear Biasing Systems for Dielectric Elastomer Actuators. *Actuators*. 2021; 10(9):209.
https://doi.org/10.3390/act10090209

**Chicago/Turabian Style**

Croce, Sipontina, Julian Neu, Jonas Hubertus, Stefan Seelecke, Guenter Schultes, and Gianluca Rizzello. 2021. "Model-Based Design Optimization of Soft Polymeric Domes Used as Nonlinear Biasing Systems for Dielectric Elastomer Actuators" *Actuators* 10, no. 9: 209.
https://doi.org/10.3390/act10090209