# Multiresolution Topology Optimization of Large-Deformation Path-Generation Compliant Mechanisms with Stress Constraints

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## Abstract

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## 1. Introduction

**Problem****type I:**- The design of maximum-displacement mechanisms finds the topology for the largest displacement of predefined degrees of freedoms for a given load.
**Problem****type II:**- The design of path-generation mechanisms finds the topology in which certain degrees of freedom are designed to go through predefined points (also known as way points or precision points), which describe a trajectory or path.

## 2. Topology Optimization for the Synthesis of Compliant Mechanisms

#### 2.1. Nonlinear Finite-Element Analysis

#### 2.2. Formulation of Topology Optimization

#### 2.2.1. Design Variables and Their Stabilization

#### 2.2.2. Maximum Displacement Objective Function

#### 2.2.3. Path Generation Objective Function

#### 2.2.4. Volume Constraint

#### 2.2.5. Stress Constraint

#### 2.2.6. Multiresolution Topology Optimization

#### 2.3. Design Sensitivity Analysis via Adjoint Methodology

#### 2.3.1. Maximum Displacement Objective Sensitivity

#### 2.3.2. Path Generation Objective Sensitivity

#### 2.3.3. Volume Constraint Sensitivity

#### 2.3.4. Stress Constraint Sensitivity

#### 2.3.5. Sensitivity Analysis for Multiresolution Topology Optimization

## 3. Numerical Examples

#### 3.1. Material and Constitutive Law

#### 3.2. Numerical Example—Maximum Displacement Mechanism Design

#### 3.3. Numerical Example—Path-Generation Mechanism Design

#### 3.4. Engineering Example—Morphing Wing Design

## 4. Conclusions

- linear finite-element analysis with volume constraint;
- linear finite-element analysis with volume and stress constraints;
- nonlinear finite-element analysis with volume constraint;
- nonlinear finite-element analysis with volume and stress constraints.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Example of displacement element and density elements for multiresolution topology optimization with four Gauss integration points.

**Figure 2.**Example definition scheme for maximum output displacement gripper (gray: design space, black: non-design space of solid material, white: non-design space of voids).

**Figure 3.**Comparison of topology optimization results for the maximum output displacement gripper example (400 × 80 elements).

**Figure 4.**Stress field for gripper result based on Figure 3d with smoothed boundary (12,451 solid elements and 0 void elements).

**Figure 5.**Example definition scheme for an output path generating compliant mechanism (gray: design space, black: non-design space of solid material).

**Figure 6.**Comparison of the path generating example mechanism results for the three precision points in the load case $i=0$ with 150 × 150 elements, volume constrained on the left, stress and volume constrained on the right.

**Figure 7.**Path generating example mechanism topology with smoothed boundary based on Figure 6f: ${u}_{\mathrm{out},\mathrm{x}}=9.09$ mm, ${u}_{\mathrm{out},\mathrm{y}}=-1.65$ mm, $max\left(\tilde{\sigma}\right)=53.7$ MPa (5283 solid elements/0 void elements).

**Figure 12.**Morphing wing demonstrator based on topology optimization results showing undeformed and deformed states.

Parameter | Symbol | Value | Units |
---|---|---|---|

Young’s modulus | E | 4232 | MPa |

Poisson’s ratio | $\nu $ | 0.36 | − |

Limit stress | ${\sigma}_{\mathrm{limit}}$ | 100 | MPa |

Max allowable stress | ${\sigma}_{\mathrm{allow}}$ | 50 | MPa |

Parameter | Symbol | Value | Units |
---|---|---|---|

Minimum Young’s modulus | ${E}_{min}$ | 4.232 $\times {10}^{-6}$ | MPa |

Input force | ${P}_{\mathrm{in}}$ | 50 | N |

Length | ℓ | 200 | mm |

Height | h | 80 | mm |

Thickness | t | 5 | mm |

Input stiffness | ${k}_{\mathrm{in}}$ | 1.5 | N/mm |

Output stiffness | ${k}_{\mathrm{out}}$ | 4 | N/mm |

Filter radius | ${r}_{\mathrm{min}}$ | 4.7 | mm |

Volume fraction | ${\phi}_{\mathrm{allow}}$ | 0.3 | − |

Solution Topology | Validation Analysis Type | ${\mathit{u}}_{\mathbf{out}}$ [mm] | $max\tilde{\mathit{\sigma}}$ [MPa] |
---|---|---|---|

a (Figure 3a) | linear nonlinear | 9.76 7.23 | 528.28 624.65 |

b (Figure 3b) | linear nonlinear | 7.55 5.53 | 49.37 173.74 |

c (Figure 3c) | nonlinear | 7.84 | 571.28 |

d (Figure 3d) | nonlinear | 4.27 | 49.95 |

e (Figure 4) | nonlinear | 4.50 | 51.99 |

Precision Point j | ${\mathit{u}}_{\mathbf{in}}$ [mm] | ${\mathit{u}}_{\mathbf{out},\mathbf{x}}^{*}$ [mm] | ${\mathit{u}}_{\mathbf{out},\mathbf{y}}^{*}$ [mm] |
---|---|---|---|

1 | 1.5 | 3 | −0.18 |

2 | 3 | 6 | −0.73 |

3 | 4.5 | 9 | −1.67 |

Load Case i | $\mathit{\alpha}$ [-] | ${\mathit{P}}_{\mathbf{CL},\mathbf{x}}$ [N] | ${\mathit{P}}_{\mathbf{CL},\mathbf{y}}$ [N] |
---|---|---|---|

0 | 1 | 0 | 0 |

1 | 0.1 | 40 | 40 |

2 | 0.1 | -40 | 40 |

Precision Point j | ${\mathit{u}}_{\mathbf{in}}$ [mm] | ${\mathit{u}}_{\mathbf{out},\mathbf{x}}^{*}$ [mm] | ${\mathit{u}}_{\mathbf{out},\mathbf{y}}^{*}$ [mm] |
---|---|---|---|

1 | 4 | 5.53 | -14.18 |

Load Case i | $\mathit{\alpha}$ [-] | ${\mathit{P}}_{\mathbf{CL},\mathbf{x}}$ [N] | ${\mathit{P}}_{\mathbf{CL},\mathbf{y}}$ [N] |
---|---|---|---|

0 | 1 | 0 | 0 |

1 | 0.1 | 14.2 | 32.8 |

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**MDPI and ACS Style**

Reinisch, J.; Wehrle, E.; Achleitner, J.
Multiresolution Topology Optimization of Large-Deformation Path-Generation Compliant Mechanisms with Stress Constraints. *Appl. Sci.* **2021**, *11*, 2479.
https://doi.org/10.3390/app11062479

**AMA Style**

Reinisch J, Wehrle E, Achleitner J.
Multiresolution Topology Optimization of Large-Deformation Path-Generation Compliant Mechanisms with Stress Constraints. *Applied Sciences*. 2021; 11(6):2479.
https://doi.org/10.3390/app11062479

**Chicago/Turabian Style**

Reinisch, Joseph, Erich Wehrle, and Johannes Achleitner.
2021. "Multiresolution Topology Optimization of Large-Deformation Path-Generation Compliant Mechanisms with Stress Constraints" *Applied Sciences* 11, no. 6: 2479.
https://doi.org/10.3390/app11062479