# Machine-Learning-Based Design Optimization of Chassis Bushings

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Characterization and Finite Element Modeling of the Reference Bushing

#### 2.1. Reference Bushing

#### 2.2. Experimental Characterization

#### 2.3. Virtual Modeling

## 3. Material Parameter Identification

#### 3.1. Fundamentals of Particle Swarm Optimization

#### 3.2. Setup and Results of the Material Parameter Identification

## 4. Data Generation—Design of Experiment

#### 4.1. Data Sampling Method

#### 4.2. Simplification of the Geometrical Model

#### 4.3. Geometrical Modeling and Parameterization

#### 4.4. Constraints of the Geometrical Model

#### 4.5. Data Generation

## 5. Machine-Learning-Based Surrogate Model

#### 5.1. Fundamentals of Neural Networks

#### 5.2. Training of the Neural Network

#### 5.3. Setup of the Neural Network Used in This Study

#### 5.4. Hyper-Parameter Optimization

## 6. Design Optimization

#### 6.1. Methodology and Problem Description

#### 6.2. Numerical Examples

#### 6.3. Results of the Design Optimization

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Example simulation results of the reference bushing, inhomogeneous von Mises stress distribution on the deformed geometry of the elastomer under (

**a**) radial, (

**b**) axial, and (

**c**,

**d**) torsional loads scaled by factor 2 for radial load and factor 4 for axial and torsional loads.

**Figure 4.**Convergence behavior of the PSO algorithm for material parameter identification. The crosses represent particles. The encircled crosses show the particle with the smallest mean absolute percentage error (MAPE) in each iteration.

**Figure 5.**Model simplification and parameterization of the free elastomer contour: (

**a**) represents the reference bushing and (

**b**) shows the simplified bushing model.

**Figure 6.**Geometrical modeling of the reference bushing: (

**a**) represents the parameters varied within the design study, and (

**b**) shows fixed and resulting parameters from boundary conditions.

**Figure 7.**Schematic illustration of a neural network consisting of an input layer with three input parameters, a hidden layer with six neurons, and an output layer with one output parameter.

**Figure 10.**Performance of the ranking term based on the number of valid particles of all iterations for both usecases.

Stiffness | Boundary Condition | Target Stiffness Value |
---|---|---|

radial | force | 3.5 kN |

axial | force | 0.6 kN |

torsional | displacement | 5° |

Yeoh Material Parameter | ${\mathit{C}}_{10}$ (MPa) | ${\mathit{C}}_{20}$ (MPa) | ${\mathit{C}}_{30}$ (MPa) |
---|---|---|---|

lower bounds | 0.457 | −0.0457 | 0.00457 |

upper bounds | 1.12 | −0.112 | 0.0112 |

Yeoh Material Parameter | ${\mathit{C}}_{10}$ (MPa) | ${\mathit{C}}_{20}$ (MPa) | ${\mathit{C}}_{30}$ (MPa) |
---|---|---|---|

[lower, upper] bounds | [0.457, 1.12] | [−0.0457, −0.112] | [0.00457, 0.0112] |

0.4954 | −0.0555 | 0.0086 |

**Table 4.**Comparison of the simulation results from the reference bushing model with the test rig measurement.

Load Direction | Radial | Axial | Torsional |
---|---|---|---|

Stiffness measurement | 6.265 (kN/mm) | 0.68 (kN/mm) | 3.93 (Nm/°) |

Stiffness simulation | 5.668 (kN/mm) | 0.68 (kN/mm) | 4.60 (Nm/°) |

Relative deviation | 9.5% | 0% | −17% |

DP | $\mathit{\alpha}$ (°) | $\mathit{\delta}$ (°) | $\mathit{c}$ (mm) | $\mathit{d}$ (mm) | $\mathit{e}$ (mm) | $\mathit{f}$ (mm) |
---|---|---|---|---|---|---|

97.96 | 91.55 | 0.95 | 0.73 | 6.7 | 14.8 |

${\mathit{T}}_{\mathit{E},\mathit{m}\mathit{i}\mathit{n}}$ (mm) | ${\mathit{T}}_{\mathit{E},\mathit{m}\mathit{a}\mathit{x}}$ (mm) | ${\mathit{T}}_{\mathit{M}}$ (mm) | ${\mathit{T}}_{\mathit{E},\mathit{A}}$ (mm) |
---|---|---|---|

3.5 | 8 | 5 | 1 |

${\mathit{R}}_{1}$ (mm) | ${\mathit{R}}_{3}$ (mm) | $\frac{1}{2}{\mathit{L}}_{\mathit{E}}$ (mm) | ${\mathit{y}}_{\mathit{m}1}$ (mm) | ${\mathit{y}}_{\mathit{m}3}$ (mm) | ${\mathit{R}}_{\mathit{E}\mathit{A}}$ (mm) | |
---|---|---|---|---|---|---|

min | 30 | 30 | 12 | −20 | 45 | 20.5 |

max | 40 | 40 | 18 | −15 | 50 | 24.45 |

Hyper-Parameter | Description | Options |
---|---|---|

${\mathit{N}}_{\mathit{H}\mathit{L}}$ | number of hidden layers | ${N}_{HL}=\left[1:1:10\right]$ |

${\mathit{N}}_{\mathit{N}}$ | number of neurons per layer | ${N}_{N}=\left[8:4:128\right]$ |

${\mathit{d}}_{\mathit{r}}$ | dropout rate | ${d}_{r}=\left[0:0.005:0.02\right]$ |

${\mathit{f}}_{\mathit{a}}$ | activation function | ${f}_{a}=\left\{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r},\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{m}\mathrm{o}\mathrm{i}\mathrm{d},\mathrm{R}\mathrm{e}\mathrm{L}\mathrm{U},\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\right\}$ |

$\stackrel{\u02c7}{\mathit{\eta}}$ | learning rate | $\stackrel{\u02c7}{\eta}=\left\{0.01,\text{}0.001,\text{}0.0001\right\}$ |

Hyper-Parameter | ${\mathit{c}}_{\mathit{r}\mathit{a}\mathit{d}}$ | ${\mathit{c}}_{\mathit{a}\mathit{x}}$ | ${\mathit{c}}_{\mathit{t}\mathit{o}\mathit{r}\mathit{s}}$ |

${\mathit{N}}_{\mathit{H}\mathit{L}}$ | 8 | 8 | 10 |

${\mathit{N}}_{\mathit{N}}$ | 8 in each hidden layer | {104, 16, 20, 116, 100, 32, 56, 8} | {128, 96, 8, 8, 12, 80, 60, 8, 8, 8} |

${\mathit{d}}_{\mathit{r}}$ | 0 in each hidden layer | 0.005 in each hidden layer | 0 in each hidden layer |

${\mathit{f}}_{\mathit{a}}$ | ReLU in each hidden layer | {tanh, tanh, ReLU, tanh, σ, σ, tanh, ReLU} | {ReLU, ReLU, tanh, ReLU, ReLU, σ, ReLU, ReLU, ReLU, ReLU} |

$\stackrel{\u02c7}{\mathit{\eta}}$ | 0.01 | 0.001 | 0.001 |

${\mathit{c}}_{\mathit{r}\mathit{a}\mathit{d}}$ | ${\mathit{c}}_{\mathit{a}\mathit{x}}$ | ${\mathit{c}}_{\mathit{t}\mathit{o}\mathit{r}\mathit{s}}$ | |
---|---|---|---|

MAPE (%) | 2.46 | 0.55 | 0.3 |

${\mathit{c}}_{\mathit{r}\mathit{a}\mathit{d}}^{\mathbf{r}\mathbf{e}\mathbf{q}}$ (kN/mm) | ${\mathit{c}}_{\mathit{a}\mathit{x}}^{\mathbf{r}\mathbf{e}\mathbf{q}}$ (kN/mm) | ${\mathit{c}}_{\mathit{t}\mathit{o}\mathit{r}\mathit{s}}^{\mathbf{r}\mathbf{e}\mathbf{q}}$ (Nm/°) | $\frac{1}{2}{\mathit{L}}_{\mathit{E}}$ (mm) | ${\mathit{R}}_{\mathit{E}\mathit{A}}$ (mm) | |
---|---|---|---|---|---|

Usecase 1 | 8 | 0.6 | 3.5 | - | - |

Usecase 2 | 10 | 0.7 | 4 | ≤13 | ≤21 |

${\mathit{c}}_{\mathit{r}\mathit{a}\mathit{d}}$ (kN/mm) | ${\mathit{c}}_{\mathit{a}\mathit{x}}$ (kN/mm) | ${\mathit{c}}_{\mathit{t}\mathit{o}\mathit{r}\mathit{s}}$ (Nm/°) | |
---|---|---|---|

Requirement | 8 | 0.6 | 3.5 |

Results DO | 7.9301 | 0.6447 | 3.5004 |

Results FEM | 7.7025 | 0.6477 | 3.5039 |

DP | ${\mathit{R}}_{1}$ (mm) | ${\mathit{R}}_{3}$ (mm) | $\frac{1}{2}{\mathit{L}}_{\mathit{E}}$ (mm) | ${\mathit{y}}_{\mathit{m}1}$ (mm) | ${\mathit{y}}_{\mathit{m}3}$ (mm) | ${\mathit{R}}_{\mathit{E}\mathit{A}}$ (mm) |
---|---|---|---|---|---|---|

[lower, upper] bounds | [30, 40] | [30, 40] | [12, 18] | [−20, −15] | [45, 50] | [20.5, 24.45] |

34.4504 | 32.7475 | 12.5964 | −17.7151 | 47.9825 | 21.7282 |

${\mathit{c}}_{\mathit{r}\mathit{a}\mathit{d}}$ (kN/mm) | ${\mathit{c}}_{\mathit{a}\mathit{x}}$ (kN/mm) | ${\mathit{c}}_{\mathit{t}\mathit{o}\mathit{r}\mathit{s}}$ (Nm/°) | $\frac{1}{2}{\mathit{L}}_{\mathit{E}}$ (mm) | ${\mathit{R}}_{\mathit{E}\mathit{A}}$ (mm) | |
---|---|---|---|---|---|

Requirement | 10 | 0.7 | 4 | ≤13 | ≤21 |

Results DO | 10.006 | 0.7416 | 3.9968 | 12.552 | 20.9137 |

Results FEM | 10.2002 | 0.7491 | 4.007 | - | - |

DP | ${\mathit{R}}_{1}$ (mm) | ${\mathit{R}}_{3}$ (mm) | $\frac{1}{2}{\mathit{L}}_{\mathit{E}}$ (mm) | ${\mathit{y}}_{\mathit{m}1}$ (mm) | ${\mathit{y}}_{\mathit{m}3}$ (mm) | ${\mathit{R}}_{\mathit{E}\mathit{A}}$ (mm) |
---|---|---|---|---|---|---|

[lower, upper] bounds | [30, 40] | [30, 40] | [12, 13] | [−20, −15] | [45, 50] | [20.5, 21] |

35.421 | 33.6226 | 12.552 | −18.2982 | 47.3173 | 20.9137 |

MAPE (%) (Requirement-DO) | MAPE (%) (DO-FEM) | MAPE (%) (Requirement-FEM) | |
---|---|---|---|

Usecase 1 | 2.78 | 1.15 | 3.93 |

Usecase 2 | 2.03 | 1.07 | 3.06 |

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## Share and Cite

**MDPI and ACS Style**

Töpel, E.; Fuchs, A.; Büttner, K.; Kaliske, M.; Prokop, G.
Machine-Learning-Based Design Optimization of Chassis Bushings. *Vehicles* **2024**, *6*, 1-21.
https://doi.org/10.3390/vehicles6010001

**AMA Style**

Töpel E, Fuchs A, Büttner K, Kaliske M, Prokop G.
Machine-Learning-Based Design Optimization of Chassis Bushings. *Vehicles*. 2024; 6(1):1-21.
https://doi.org/10.3390/vehicles6010001

**Chicago/Turabian Style**

Töpel, Eric, Alexander Fuchs, Kay Büttner, Michael Kaliske, and Günther Prokop.
2024. "Machine-Learning-Based Design Optimization of Chassis Bushings" *Vehicles* 6, no. 1: 1-21.
https://doi.org/10.3390/vehicles6010001