# Machine Learning Based Surrogate Models for the Thermal Behavior of Multi-Plate Clutches

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

^{−5}were achieved. Anandan Kumar et al. [19] developed a surrogate model for the thermal FE simulation of a powder bed fusion process using Gaussian processes.

#### 1.2. Research Objectives

## 2. Methodology

#### 2.1. FE-Model and Use Cases

#### 2.2. Dataset Generation

#### 2.3. Model Development

- Polynomial Regression (PR): Polynomial regression is a subclass of linear regression, in which a basis function expansion is performed with polynomial functions [8,21]. The use of higher order polynomial functions enables non-linear relationships to be modeled. The PR model is given by:$$f\left(x\right)={w}^{T}\varphi \left(x\right)$$$$\varphi \left(x\right)=\left[1,x,{x}^{2},\dots ,{x}^{d}\right]$$
- Decision Tree (DT): Decision trees are methods that divide the input space into several areas. These subdivisions take place along the individual axes of the input space by means of the CART algorithm and can be represented by means of a tree. The mean value ${w}_{i}$ is calculated for each of the regions ${R}_{i}$ created after the subdivision. The output value of the model is given by the following expression:$$f\left(x\right)={\displaystyle \sum}_{i=1}^{n}{w}_{i}\mathbb{I}\left(x\in {R}_{i}\right)$$
- Support Vector Regression (SVR): Support vector regression is a parametric model that uses kernels and considers only a portion of the training dataset to generate predictions. SVR models for a predefined kernel $\kappa $ can generically be defined by the following equation:$$f\left(x\right)={w}_{0}+{\displaystyle \sum}_{i=1}^{n}{w}_{i}\kappa \left({x}_{i},x\right)$$

- 4.
- Gaussian Process (GP): Gaussian processes are non-parametric methods having the inference of distributions over functions as a basic principle [21]. If the data is noise-free, then GPs have the capability to interpolate the data points exactly. This is advantageous when creating surrogate models of deterministic FE models [23]. For the measured values $f$ at the sample points $X$ and the values ${f}_{*}$ being predicted at the points ${X}_{*}$, the joint probability distribution is given by:$$\left(\begin{array}{c}f\\ {f}_{*}\end{array}\right)~\mathcal{N}\left(\left(\begin{array}{c}\mu \\ {\mu}_{*}\end{array}\right),\left(\begin{array}{cc}K& {K}_{*}\\ {K}_{*}^{T}& {K}_{**}\end{array}\right)\right)$$$$K=\kappa \left(X,X\right),\text{}{K}_{*}=\kappa \left(X,{X}_{*}\right),\text{}{K}_{**}=\kappa \left({X}_{*},{X}_{*}\right)$$$$p\left({f}_{*}|{X}_{*},X,f\right)=\mathcal{N}({f}_{*}|{\mu}_{*},{\mathsf{\Sigma}}_{*})$$$${\mu}_{*}=\mu \left({X}_{*}\right)+{K}_{*}^{T}{K}^{-1}\left(f-\mu \left(X\right)\right)$$$${\mathsf{\Sigma}}_{*}={K}_{**}-{K}_{*}^{T}{K}^{-1}{K}_{*}$$

- 5.
- Backpropagation Neural Network (BPNN): Neural networks are a set of models based on the structure and function of biological neurons and consist of a number of layered and interconnected units (neurons) [8]. The output $h$ of a neuron consists of a linear combination of the inputs, which is then subjected to a nonlinear activation function:$${h}_{W,b}=\varphi \left(XW+b\right)$$

#### 2.4. Model Evaluation

## 3. Results

#### 3.1. Use Case 1—Axial Force + Rotational Speed

#### 3.2. Use Case 2—Axial Force + Rotational Speed + Lining Thickness

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Flow chart for the simulation process [20].

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**Figure 4.**Exemplary curve of the axial force and differential rotational speed [20].

Use Case | Description | Varied Parameters |
---|---|---|

1 | Load | Axial force Rotational speed |

2 | Load + lining wear | Axial force Rotational speed Lining thickness |

Parameter | Lower Bound | Upper Bound |
---|---|---|

Axial force in kN | 9.292 | 37.168 |

Rotational speed in rpm | 80 | 140 |

Lining thickness in mm | 0.2 | 0.8 |

Simulation | Use Case 1 | Use Case 2 |
---|---|---|

Input variables | Axial force Rotational speed Time | Axial force Rotational force Lining thickness Time |

Output variable | T_{max} | T_{max} |

Model | Hyperparameter | Values |
---|---|---|

Polynomial Regression | Degree | [1, 10] |

Decision Tree | Depth | {inf, 5, 10, 15, 20} |

Criterion | {‘squared_error’, ‘friedman’, ‘poisson’} | |

Support Vector Regression | Kernel | Linear, poly, RBF |

C | 1, 10, 100, 1000 | |

Epsilon | 1 × 10 ^{−4}, 1 × 10 ^{−3}, 1 × 10 ^{−2} | |

Gaussian Process | Kernel | DotProduct, RationalQuadratic, RBF |

Alpha | [1 × 10 ^{−5}, 1 × 10 ^{−2}] | |

Backpropagation Neural Networks | Number of hidden layers Number of neurons per layers Learning rate | [1, 10] [1, 50] [1 × 10 ^{−5}, 1 × 10 ^{−1}] |

PR | DT | SVR | GP | BPNN | FE Model | |
---|---|---|---|---|---|---|

RMSE | 9.535 | 17.564 | 28.22 | 6.772 | 4.290 | - |

MAPE | 1.36% | 2.15% | 3.69% | 0.74% | 0.51% | - |

Training time in s | <1 | <1 | $~15$ | $~110$ | $~350$ | - |

Inference time in s | 0.018 | 0.007 | 0.233 | 0.277 | 0.948 | $~1100$ |

PR | DT | SVR | GP | BPNN | FE Model | |
---|---|---|---|---|---|---|

RMSE | 24.614 | 19.706 | 24.756 | 14.434 | 6.312 | - |

MAPE | 4.02% | 2.18% | 3.65% | 2.20% | 0.99% | - |

Training time in s | <1 | <1 | $~15$ | $~120$ | $~400$ | - |

Inference time in s | 0.026 | 0.001 | 0.260 | 0.306 | 0.311 | $~\text{}1100$ |

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**MDPI and ACS Style**

Schneider, T.; Bedrikow, A.B.; Dietsch, M.; Voelkel, K.; Pflaum, H.; Stahl, K.
Machine Learning Based Surrogate Models for the Thermal Behavior of Multi-Plate Clutches. *Appl. Syst. Innov.* **2022**, *5*, 97.
https://doi.org/10.3390/asi5050097

**AMA Style**

Schneider T, Bedrikow AB, Dietsch M, Voelkel K, Pflaum H, Stahl K.
Machine Learning Based Surrogate Models for the Thermal Behavior of Multi-Plate Clutches. *Applied System Innovation*. 2022; 5(5):97.
https://doi.org/10.3390/asi5050097

**Chicago/Turabian Style**

Schneider, Thomas, Alexandre Beiderwellen Bedrikow, Maximilian Dietsch, Katharina Voelkel, Hermann Pflaum, and Karsten Stahl.
2022. "Machine Learning Based Surrogate Models for the Thermal Behavior of Multi-Plate Clutches" *Applied System Innovation* 5, no. 5: 97.
https://doi.org/10.3390/asi5050097