# Modeling Cycle-to-Cycle Variations of a Spark-Ignited Gas Engine Using Artificial Flow Fields Generated by a Variational Autoencoder

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Engine Setup

^{3}. The engine is operated with a premixed gas–air mixture at lean conditions. Full load operating conditions at 1500 min

^{−1}are considered in the investigations described below.

#### 2.2. CFD Simulation Setup

^{−6}s, and a reduced NUI Galway reaction mechanism for natural gas is used [29].

#### 2.3. Multicycle Simulations

#### 2.4. Variational Autoencoder

#### 2.4.1. Principal Component Analysis and Latent Space Definition

#### 2.4.2. Architecture

#### 2.4.3. Data Preparation, Hyperparameters, and Training

#### 2.5. Artificial Flow Field Generation

## 3. Results

#### 3.1. Model Validation

#### 3.2. Artificial Cyclic Variations

_{50%}) values are 33.5 CAD for cycle 3 and 33.8 CAD for the artificial cycle.

^{2}/s

^{2}.

_{50%}ranges from 33.1 to 34.1 CAD. In terms of CoV, the original data set has a 1.69% peak pressure and 1.29% MFB

_{50%}, and the artificial cycles have 0.65% peak pressure and 0.71% MFB

_{50%}. From the CoV difference between the original and artificial cycles, it can be concluded that the VAE tends to generalize in the training process, and thus, extreme values are less conserved. This observed compression of the VAE’s generative degrees of freedom can be counteracted by measures such as early stopping, amplification of the discussed latent space disentanglement, or dedicated curation of the training data to contain a close to uniform distribution of more and less extreme samples. However, all of these strategies have to be carefully balanced to not interfere with the overall quality of reconstructed flow fields and thus require a large number of experiments. Nevertheless, the results prove the potential of the present approach.

## 4. Summary and Conclusions

_{50%}and have been evaluated, which yield 1.69% and 1.29% for the original multicycle data as well as 0.65% and 0.71% for the VAE-based cycles, respectively. These results indicate a trend for the VAE to generalize the training data, yielding the disappearance of extreme values. Further investigations are necessary to overcome this issue and obtain further agreement between the multicycle data used for training and the generated artificially data. However, the comparison between velocity and $TKE$ fields from the original simulation and the VAE-generated flow fields that apply histograms and flow field illustrations show how the VAE maintains the underlying in-cylinder physics. In comparison to state-of-the art methods that generate initial flow fields for multicycle simulation using random perturbation, the proposed method is able to maintain flow field properties to a great extent.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CoV | Coefficient of variation |

CCV | Cycle-to-cycle variations |

CFL | Courant–Friedrichs–Lewy |

SI | Spark-ignited |

RANS | Reynolds-averaged Navier–Stokes equations |

CFD | Computational fluid dynamics |

AMR | Adaptive mesh refinement |

CAD | Crank angle degree |

MFB | Mass fraction burned |

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**Figure 1.**Generation of the evaluation data. Here, the velocity and $TKE$ field is extracted from the fifth cycle and combined with all other quantities of the first cycle. Then, the simulation starts at the time that the fifth snapshot was taken.

**Figure 2.**VAE architecture. The format given for the data dimensions is $N\times C$ with N representing the number of variables and C representing the number of channels. k is the kernel size of the filters in the corresponding layer and s is the stride parameter, which defines the step size of the convolution operations within the respective layer.

**Figure 4.**Training of the VAE with velocity and $TKE$ fields in the specified domain of the ten cycles of the multicycle simulation.

**Figure 5.**Generation of artificial velocity and $TKE$ field using the trained VAE. The corresponding initial field is created by the new fields and the residual fields of cycle one.

**Figure 6.**Procedure to validate the ability of the VAE to generate meaningful artificial velocity and $TKE$ fields.

**Figure 7.**In-cylinder combustion process results started with the original and the VAE reconstructed velocity and $TKE$ fields.

**Figure 8.**Histogram of the velocity components $u,v,w$, and $TKE$ of the original and VAE reconstructed fields.

**Figure 9.**In-cylinder combustion process results of the common multicycle simulations and one artificial generated initialization set.

**Figure 10.**Histogram of the velocity components $u,v,w$, and $TKE$ of cycle 3 from the original multicycle dataset and the VAE generated fields.

**Figure 11.**Velocity magnitude and $TKE$ in a cut plane through the spark plug from the original multicycle dataset and the VAE generated fields.

**Figure 12.**In-cylinder combustion process results of the common multicycle simulation and 20 artificially generated initialization sets.

**Table 1.**Cumulative sum of explained variance ratios, calculated for the specified amount of principal components extracted via distinct PCAs of the original data’s quantities.

# of PCs | u | v | w | $\mathit{TKE}$ | Mean |
---|---|---|---|---|---|

1 | 0.503 | 0.350 | 0.357 | 0.404 | 0.404 |

2 | 0.689 | 0.625 | 0.573 | 0.568 | 0.614 |

3 | 0.806 | 0.778 | 0.702 | 0.702 | 0.747 |

4 | 0.870 | 0.851 | 0.803 | 0.821 | 0.836 |

5 | 0.918 | 0.907 | 0.869 | 0.889 | 0.895 |

6 | 0.951 | 0.941 | 0.920 | 0.934 | 0.937 |

7 | 0.971 | 0.969 | 0.963 | 0.972 | 0.969 |

8 | 0.987 | 0.989 | 0.986 | 0.987 | 0.987 |

9 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 |

10 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

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**MDPI and ACS Style**

Posch, S.; Gößnitzer, C.; Ofner, A.B.; Pirker, G.; Wimmer, A. Modeling Cycle-to-Cycle Variations of a Spark-Ignited Gas Engine Using Artificial Flow Fields Generated by a Variational Autoencoder. *Energies* **2022**, *15*, 2325.
https://doi.org/10.3390/en15072325

**AMA Style**

Posch S, Gößnitzer C, Ofner AB, Pirker G, Wimmer A. Modeling Cycle-to-Cycle Variations of a Spark-Ignited Gas Engine Using Artificial Flow Fields Generated by a Variational Autoencoder. *Energies*. 2022; 15(7):2325.
https://doi.org/10.3390/en15072325

**Chicago/Turabian Style**

Posch, Stefan, Clemens Gößnitzer, Andreas B. Ofner, Gerhard Pirker, and Andreas Wimmer. 2022. "Modeling Cycle-to-Cycle Variations of a Spark-Ignited Gas Engine Using Artificial Flow Fields Generated by a Variational Autoencoder" *Energies* 15, no. 7: 2325.
https://doi.org/10.3390/en15072325