# Fluid-Structure Interaction Modelling of a Soft Pneumatic Actuator

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fluid-Structure Interaction

- Navier–Stokes equations for the laminar, incompressible fluid flow:$$\rho \frac{\partial u}{\partial t}+\rho \left(u\xb7\nabla \right)u=\nabla \xb7\left[-pI+\mu \left(\nabla u+{\left(\nabla u\right)}^{\mathrm{T}}\right)\right]+F+\rho \mathrm{g}$$$$\rho \nabla \xb7u=0$$
- Equations of solid mechanics given by Newton’s second law:$$\rho \frac{{\partial}^{2}u}{\partial {t}^{2}}=\nabla \xb7{\left(FS\right)}^{\mathrm{T}}+{F}_{v}$$$$F=I+\nabla u$$
_{v}is the body force.

#### 2.2. Hyperelastic Material Model

_{1}, C

_{2}, and C

_{3}are model parameters and I

_{1}, I

_{2}, and I

_{3}are strain invariants of Cauchy–Green deformation tensors. For modelling the material behaviour of silicone rubbers, one can use the third-order Yeoh model parameters. In this paper, elastosil was used as material of the soft actuator, and the Yeoh model parameters were C

_{1}= 0.11, C

_{2}= 0.02, and C

_{3}= 0 [54]. The FSI simulation model can also be implemented for other silicone materials using suitable hyperelastic model parameters.

#### 2.3. PneuNets Bending Actuator

#### 2.4. Coupled 2D FSI Modelling and Simulation

#### 2.5. Coupled 3D FSI Modelling and Simulation

## 3. Results and Discussion

#### 3.1. Deformation and Comparison with FEM Model

#### 3.2. Initial Time Step Condition

#### 3.3. Middle and Last Time Step Condition

#### 3.4. Average Velocity of Fluid Domain

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- Required modules in COMSOL: COMSOL Multiphysics Module, Structural Mechanics/MEMS Module, Nonlinear Structural Materials Module, CFD Module, and CAD Import Module
- Space dimension: three-dimensional model
- Physics: fluid–solid interaction
- Study type: stationary (gravity) and time-dependent (air pressure)
- Fluid mechanics: Air is chosen as material. Incompressible and laminar flow are selected. Gravity is included. Walls are treated with no slip condition. Material properties like dynamic viscosity and density of air are considered using a built-in function in COMSOL.
- Solid mechanics:
- ○
- Elastomer: Elastosil is assigned to the top and bottom parts, which is created as user-defined material with a density of 1130 kg/m
^{3}. The Yeoh hyperelastic material model is assigned using the model parameters given in Section 2.2. - ○
- Paper: Paper is assigned to the interface boundary, and linear elastic material behaviour is applied. Paper is created as user defined material with required material properties of density = 750 kg/m
^{3}, Poisson’s ratio = 0.2, and Young’s modulus = 6.5 GPa [54]. - ○
- Contacts: The sidewalls of the chambers (except the left sidewall of the first chamber and the right sidewall of the last chamber) are assigned as contact pairs for the deformed configuration. All the left sidewalls are considered as source and the right sidewalls as destination for contact modelling. For the thin paper layer, another contact pair is created for the initial configuration. Contacts are modelled using the penalty condition.

- Moving mesh: The fluid domain is selected as deforming domain under a moving mesh. The Navier–Stokes equations given in (1) and (2) are solved in the deforming domain using Yeoh smoothing with C
_{2}= 10. The prescribed mesh displacement of zero in all directions is applied to the inlet and the wall boundaries of the air supply tube above the first chamber. - Boundary conditions: An inlet air pressure of 0.5 bar is applied at the inlet boundary. The pressure is applied using the ramp function, gradually increasing the pressure from 0 bar to 0.5 bar from t = 0 s to t = 1 s. The side and top walls of the first chamber are fixed, hence there is no displacement in all directions.
- Gravity is taken into account for the solid domain, and it acts in negative y direction.
- Fluid–structure interaction: The fluid domain is selected as deforming domain under a moving mesh and the walls of the fluid chambers are automatically assigned as the interface for fluid and solid. The fully coupled solver is selected for simulating fluid and solid domains.
- Mesh: The physics-controlled mesh is selected. The fine mesh is created predominantly with tetrahedral elements. The mesh is generated specifically in each domain and FSI interface. The total number of mesh elements is 1,022,609, in which 598,947 elements represent the fluid domain and are calibrated for fluid mechanics and 311,810 elements represent the solid domains. The minimum element quality based on skewness is 0.06.
- Study: Two study steps are implemented. Study Step 1 denotes a stationary study, which solves the model for gravity. Regarding this, the parameter t is defined with value zero in global definition. Study Step 2 denotes the time-dependent study, in which the inlet pressure is applied in time steps. The time steps are given as 0, 0.01, and 1 s. The inlet pressure increases for each time step by 500 Pa.
- Solver: The segregated solver is selected for the time-dependent study. Compared to the fully coupled solver, the segregated solver approach requires less memory and computation time.
- Cluster specifications: The model is computed on 24 cores running at 2.5 GHz. It takes 18 h and 43 min to compute.

## Appendix B

**Figure A1.**Pressure difference in the actuator at lower rates of pressurization: P is the applied air pressure at the inlet; ΔP

_{ave}is the difference between P and the average fluid pressure distributed on the boundaries of the fluid domain; ΔP

_{first}is the difference between P and the average pressure distributed on the walls of the first chamber; ΔP

_{last}is the difference between P and the average pressure distributed on the walls of the last chamber.

_{last}= 50 Pa) relatively larger than in the later time steps. Nevertheless, the magnitude of the pressure difference is not significant (ΔP

_{ave}predominantly lies between 10 Pa and 20 Pa, and the highest value of ΔP

_{last}is 50 Pa at the initial time steps). Therefore, there is no significant deviation in the bending behaviour of the actuator between the FSI and FEM models in the initial time steps as well as throughout the simulation time. The chambers of the actuator at this rate of pressurization inflate uniformly. Therefore, in this case, a comparison of the FSI model with the FEM model is acceptable and is treated as verification, as there are no significant pressure changes from the applied pressure to the walls inside the actuator.

_{ave}= 1500 Pa), as well as throughout the simulation time (ΔP

_{ave}predominantly lies in the range above 350 Pa), is significant for higher rates of pressurization. It also shows that the pressure distributed on the walls of the first chamber is higher than the applied air pressure at the inlet and hence the pressure difference (ΔP

_{first}) shows negative values in Figure A2.

**Figure A2.**Pressure difference in the actuator at higher rates of pressurization: P is the applied air pressure at the inlet; ΔP

_{ave}is the difference between P and the average fluid pressure distributed on the boundaries of the fluid domain; ΔP

_{first}is the difference between P and the average pressure distributed on the walls of the first chamber; ΔP

_{last}is the difference between P and the average pressure distributed on the walls of the last chamber.

**Figure A3.**Pressure distribution on the boundaries of the chambers in the PneuNets bending actuator at higher rates of pressurization: (

**a**) At 500 Pa inlet pressure; (

**b**) At 5000 Pa inlet pressure; (

**c**) At 12,500 Pa inlet pressure; (

**d**) At 25,000 Pa inlet pressure; (

**e**) At 37,500 Pa inlet pressure; (

**f**) At 50,000 Pa inlet pressure.

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**Figure 2.**Simulation results of the coupled 2D FSI problem: (

**a**) Fluid velocity streamline at the initial time step; (

**b**) Fluid velocity streamline and solid stress at the final time step; (

**c**) Fluid pressure and solid deformation at the final time step.

**Figure 4.**Deformation comparison of the PneuNets bending actuator from the FSI and FEM simulation models: (

**a**) Initial deformation from the FSI model due to gravity at 0 kPa pressure; (

**b**) Final deformation from the FSI model at 50 kPa pressure; (

**c**) Initial deformation from the FEM model due to gravity at 0 kPa pressure; (

**d**) Final deformation from the FEM model at 50 kPa pressure.

**Figure 5.**Comparison of the displacement path measured at the actuator tip from the FSI and FEM simulation models: (

**a**) Pressure vs. displacement in y direction; (

**b**) Pressure vs. displacement in x direction.

**Figure 6.**FSI simulation results of the PneuNets bending actuator at the initial time step of 0.01 s: (

**a**) Velocity magnitude at the mid surface; (

**b**) Flow streamlines with velocity magnitude (cross sectional view); (

**c**) Pressure distribution in the fluid domain; (

**d**) Stress distribution in the actuator (cross sectional view).

**Figure 7.**FSI simulation results of the PneuNets bending actuator at the middle time step of 0.5 s: (

**a**) Velocity magnitude at the mid surface; (

**b**) Flow streamlines with velocity magnitude (cross sectional view); (

**c**) Pressure distribution in the fluid domain; (

**d**) Stress distribution in the actuator (cross sectional view).

**Figure 8.**FSI simulation results of the PneuNets bending actuator at the final time step of 1 s: (

**a**) Velocity magnitude at the mid surface; (

**b**) Flow streamlines with velocity magnitude (cross sectional view); (

**c**) Pressure distribution in the fluid domain; (

**d**) Stress distribution in the actuator (cross sectional view).

**Figure 9.**Average velocity plot for the complete simulation: (

**a**) Velocity plot at the inlet boundary; (

**b**) Velocity plot for the whole fluid domain.

Sylgard | Elastosil | Ecoflex | |
---|---|---|---|

Manufacturer | Dow Corning | Wacker Chemie | Smooth-On |

Type | 184 | M4601 | 00–30 |

Colour | transparent | reddish brown | translucent |

Shore hardness | 50 A | 28 A | 00–30 |

Tensile strength | 6.76 MPa | 6.50 MPa | 1.38 MPa |

Density | 1.04 g/cm^{3} | 1.13 g/cm^{3} | 1.07 g/cm^{3} |

Elongation at break | 150% | 700% | 900% |

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**MDPI and ACS Style**

Maruthavanan, D.; Seibel, A.; Schlattmann, J.
Fluid-Structure Interaction Modelling of a Soft Pneumatic Actuator. *Actuators* **2021**, *10*, 163.
https://doi.org/10.3390/act10070163

**AMA Style**

Maruthavanan D, Seibel A, Schlattmann J.
Fluid-Structure Interaction Modelling of a Soft Pneumatic Actuator. *Actuators*. 2021; 10(7):163.
https://doi.org/10.3390/act10070163

**Chicago/Turabian Style**

Maruthavanan, Duraikannan, Arthur Seibel, and Josef Schlattmann.
2021. "Fluid-Structure Interaction Modelling of a Soft Pneumatic Actuator" *Actuators* 10, no. 7: 163.
https://doi.org/10.3390/act10070163