# Multi-Objective Lightweight Optimization of Parameterized Suspension Components Based on NSGA-II Algorithm Coupling with Surrogate Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Parametric Modeling of Control Arm and Torsion Beam

#### 2.1. Mesh Morphing Technology

**D**is the control node displacement variable matrix. $C{N}_{i}^{current}$ and $C{N}_{i}^{new}$ are the coordinate matrices of control nodes before and after deformation, respectively. $D{N}_{i}^{current}$ and $D{N}_{i}^{new}$ are the coordinate matrices of deformable nodes before and after deformation, respectively. $f\left(D,\varphi ,\phi \right)$ is the deformation shape function.

#### 2.2. Parametric Modeling

## 3. Establishment of Vehicle Rigid–Flexible Coupling Model

#### 3.1. Flexible Body Models of Control Arm and Torsion Beam

#### 3.2. Establishment and Verification of Rigid–Flexible Coupling Vehicle Model

## 4. Lightweight Design of Control Arm and Torsion Beam

#### 4.1. Optimization Formulation

**x**is the design vector; ${m}_{1}\left(x\right)$ and ${m}_{2}\left(x\right)$ are the mass of control arm and torsion beam, respectively; ${a}_{v}\left(x\right)$ is the total weighted root mean square of seat rail acceleration at the speed of 60 km/h under grade B road excitation; ${\phi}_{\mathrm{max}}\left(x\right)$ is the maximum vehicle roll angle in double lane change simulation with vehicle speed of 80 km/h; ${N}_{1}\left(x\right)$ and ${N}_{2}\left(x\right)$ are the minimum fatigue life of control arm and torsion beam, respectively; ${K}_{11}\left(x\right)$ and ${K}_{12}\left(x\right)$ represent the longitudinal stiffness and lateral stiffness of control arm, and K

_{L}is set as 3.76 kN/m while K

_{T}is set as 40.6 kN/m; ${K}_{2}\left(x\right)$ denote the torsional stiffness of torsion beam and K

_{N}is set as 40.8 N·m/°; ${f}_{1}\left(x\right)$ and ${f}_{2}\left(x\right)$ are first order modal frequencies of control arm and torsion beam, F

_{1}= 212.0 Hz, F

_{2}= 40.6 Hz;

**DV**

_{L}and

**DV**

_{U}are the lower and upper limits of design vector

**x**, respectively.

#### 4.2. Surrogate Model

#### 4.3. Multi-Objective Optimization Based on NSGA-II Algorithm

_{t}(t = 0) from P

_{t}using GA operators of selection, crossover, and mutation. Then calculate the fitness value of each individual.

_{t}and Q

_{t}to create a new population with size of 2 N. Perform non-dominated sorting of population R

_{t}and classify them into several fronts (F

_{1}, F

_{2}, F

_{3}, …). Then calculate the crowding distance for a set of population individuals, where the definition of crowding distance for the ith individual is shown in Figure 9. It can be calculated as

_{i}is the crowding distance of the ith individual; k is the number of objective functions; ${f}_{j}^{i+1}$ and ${f}_{j}^{i-1}$ denote the jth objective function of the (i − 1)th and (i + 1)th individual, respectively; ${f}_{j}^{\mathrm{max}}$ and ${f}_{j}^{\mathrm{min}}$ represent the maximum and minimum of the jth objective function, respectively.

_{t+}

_{1}. Select the individuals of fronts with low order of dominance first, and then select the individuals with a large crowding distance in the same front.

_{t+}

_{1}to generate a new offspring Q

_{t+}

_{1}with size of N.

## 5. Results and Discussion

#### 5.1. Optimization Results

#### 5.2. Structural Performance of Control Arm and Torsion Beam

^{6}, which is converted into the kilometrage of the combined durability pavement of 236,800 km. Moreover, the minimum life position of the torsion beam appears at the crossbeam. The kilometrage of the combined durability pavement is 212,800 km, calculated by the minimum cycle life of 1.33 × 10

^{6}. It indicates that both of them satisfy the requirement of fatigue life.

#### 5.3. Vehicle Dynamic Performance

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 7.**Accuracy verification results of Kriging surrogate models: (

**a**) weight of control arm; (

**b**) weight of torsion beam; (

**c**) total weighted acceleration RMS; (

**d**) maximum vehicle roll angle.

Variables | Description | Deformation Mode | Initial Value | Upper Limit | Lower Limit |
---|---|---|---|---|---|

DV1/mm | Side height | Free-form | 0 | −5.0 | 5.0 |

DV2 | Scaling of front width | Control block | 1.0 | 0.90 | 1.1 |

DV3 | Scaling of rear width | Control block | 1.0 | 0.90 | 1.1 |

DV4 | Scaling of small hole diameter | Free-form | 1.0 | 0.90 | 1.1 |

DV5 | Scaling of big hole diameter | Free-form | 1.0 | 0.90 | 1.1 |

DV6 | Scaling of groove length | Control block | 1.0 | 0.95 | 1.05 |

DV7/mm | Groove depth | Free-form | 0 | −5.0 | 5.0 |

DV8/mm | Thickness | 4.0 | 2.0 | 6.0 |

Variables | Description | Deformation Mode | Initial Values | Upper Limit | Lower Limit |
---|---|---|---|---|---|

DV9 | Scaling of beam width | Control block | 1.0 | 0.90 | 1.1 |

DV10 | Scaling of beam height | Control block | 1.0 | 0.90 | 1.1 |

DV11 | Scaling of bottom circle arc | Free-form | 1.0 | 0.95 | 1.05 |

DV12 | Scaling of outer circle arc | Free-form | 1.0 | 0.95 | 1.05 |

DV13 | Scaling of inner circle arc | Free-form | 1.0 | 0.95 | 1.05 |

DV14 | V-beam length | Control block | 0 | −10.0 | 10.0 |

DV15/mm | Transition zone length | Control block | 0 | −10.0 | 10.0 |

DV16/mm | Thickness | 3.0 | 2.0 | 4.0 |

Order | Control Arm/Hz | Relative Error | Torsion Beam/Hz | Relative Error | ||
---|---|---|---|---|---|---|

Simulation | Test | Simulation | Test | |||

1 | 212.0 | 204.7 | 3.4% | 40.6 | 42.1 | 3.7% |

2 | 246.1 | 235.7 | 4.2% | 65.7 | 68.5 | 4.3% |

3 | 396.9 | 378.8 | 4.6% | 99.0 | 99.9 | 0.9% |

4 | 715.6 | 674.1 | 5.8% | 102.3 | 106.1 | 3.7% |

5 | 928.4 | 909.5 | 2.0% | 137.2 | 139.8 | 1.9% |

6 | 994.9 | 970.2 | 2.5% | 169.1 | 167.2 | 1.1% |

Design Variable | Optimal Results | Modified Value |
---|---|---|

DV1/mm | 3.3577 | 3.4 |

DV2 | 0.9109 | 0.91 |

DV3 | 1.0579 | 1.06 |

DV4 | 1.0894 | 1.09 |

DV5 | 1.0387 | 1.04 |

DV6 | 0.9914 | 0.99 |

DV7/mm | −2.3306 | −2.3 |

DV8/mm | 3.5374 | 3.5 |

DV9 | 1.0227 | 1.02 |

DV10 | 0.9657 | 0.97 |

DV11 | 0.9878 | 0.99 |

DV12 | 1.0291 | 1.03 |

DV13 | 1.0328 | 1.03 |

DV14/mm | −9.1089 | −9.1 |

DV15/mm | 3.0972 | 3.1 |

DV16/mm | 2.5329 | 2.5 |

Stiffness | Original | Optimum | Variation | |
---|---|---|---|---|

Control arm | Longitudinal stiffness(kN/mm) | 3.76 | 3.05 | −0.71 |

Lateral stiffness(kN/mm) | 40.60 | 34.84 | −5.76 | |

Torsion beam | Torsional stiffness(N·m/°) | 40.8 | 42.1 | +1.3 |

Mode | Control Arm/Hz | Torsion Beam/Hz | ||||
---|---|---|---|---|---|---|

Original | Optimum | Variation | Original | Optimum | Variation | |

1 | 212.0 | 224.7 | +12.7 | 40.6 | 44.1 | +3.5 |

2 | 246.1 | 328.4 | +82.3 | 65.7 | 86.8 | +21.1 |

3 | 396.9 | 502.6 | +105.7 | 99.0 | 130.6 | +31.6 |

4 | 715.6 | 878.1 | +162.5 | 102.3 | 202.6 | +100.3 |

5 | 928.4 | 1024.1 | +95.7 | 137.2 | 260.1 | +122.9 |

6 | 994.9 | 1227.4 | +232.5 | 169.1 | 335.4 | +166.3 |

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**MDPI and ACS Style**

Jiang, R.; Jin, Z.; Liu, D.; Wang, D.
Multi-Objective Lightweight Optimization of Parameterized Suspension Components Based on NSGA-II Algorithm Coupling with Surrogate Model. *Machines* **2021**, *9*, 107.
https://doi.org/10.3390/machines9060107

**AMA Style**

Jiang R, Jin Z, Liu D, Wang D.
Multi-Objective Lightweight Optimization of Parameterized Suspension Components Based on NSGA-II Algorithm Coupling with Surrogate Model. *Machines*. 2021; 9(6):107.
https://doi.org/10.3390/machines9060107

**Chicago/Turabian Style**

Jiang, Rongchao, Zhenchao Jin, Dawei Liu, and Dengfeng Wang.
2021. "Multi-Objective Lightweight Optimization of Parameterized Suspension Components Based on NSGA-II Algorithm Coupling with Surrogate Model" *Machines* 9, no. 6: 107.
https://doi.org/10.3390/machines9060107