# 3D Evolutionary Reconstruction of Scalar Fields in the Gas-Phase

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Evolutionary Reconstruction Scheme

Algorithm 1 Evolutionary reconstruction technique. | |

1: Start | |

2: $[\mathit{GApars},\mathit{IMpars}]\leftarrow \mathit{read}\_\mathit{settings}\left(\right)$ | ▹ Read GA and image settings |

3: $\mathit{IMG}\leftarrow \mathit{read}\_\mathit{images}\left(\right)$ | ▹ Read reference images |

4: $\mathit{MSK}\leftarrow \mathit{create}\_\mathit{mask}\left(\mathit{IMG}\right)$ | ▹ Create stochastic mask |

5: $\mathit{POP}\leftarrow \mathit{init}\_\mathit{population}(\mathit{GApars}.\mathit{PopSize})$ | ▹ Initialize population |

6: $\mathit{POP}.\mathit{Fit}\leftarrow \mathit{eval}\_\mathit{fitness}\left(\mathit{POP}\right)$ | ▹ Evaluate Fitness |

7: $\mathit{POP}.\mathit{Rank}\leftarrow \mathit{rank}\_\mathit{chromosomes}\left(\mathit{POP}\right)$ | ▹ Calculate ranks |

8: $\mathit{BST}\leftarrow \mathit{POP}(\mathit{POP}.\mathit{Rank}==\mathit{1})$ | ▹ Get best chromosome |

9: while $i<\mathit{GApars}.\mathit{MaxGenerations}$ and $\mathit{BST}.\mathit{Fit}<\mathit{GApars}.\mathit{FitThreshold}$ do | |

10: $\mathit{POP}\leftarrow \mathit{evolution}\_\mathit{step}(\mathit{POP},\mathit{MSK})$ | ▹ Perform evolution step |

11: $\mathit{POP}.\mathit{Fit}\leftarrow \mathit{eval}\_\mathit{fitness}\left(\mathit{POP}\right)$ | ▹ Evaluate fitness |

12: $\mathit{POP}.\mathit{Rank}\leftarrow \mathit{rank}\_\mathit{chromosomes}\left(\mathit{POP}\right)$ | ▹ Calculate ranks |

13: $\mathit{BST}\leftarrow \mathit{POP}(\mathit{POP}.\mathit{Rank}==\mathit{1})$ | ▹ Get best chromosome |

14: if $\mathit{modulo}(\mathit{i},\mathit{GApars}.\mathit{Steps})==0$ and $i>\mathit{GApars}.\mathit{MAstart}$ then | |

15: $\mathit{MSK}\leftarrow \mathit{recreate}\_\mathit{mask}\left(\mathit{BST}\right)$ | ▹ Re-create stochastic mask |

16: end if | |

17: $i=i+1$ | |

18: end while | |

19: End |

#### 2.1. Ray-Tracing

#### 2.2. The Genetic Algorithm

Algorithm 2 Evolution Step | |

1: function Evolution_Step(POP, MSK) | ▹ Input is the population and mask |

2: for $i=1:\mathit{GApars}.\mathit{PopSize}$ do | |

3: $[\mathit{MOM},\mathit{DAD}]\leftarrow \mathit{select}\left(\mathit{POP}\right)$ | ▹ Select two chromosomes |

4: $\mathit{OFF}\left(\mathit{i}\right)\leftarrow \mathit{average}(\mathit{MOM},\mathit{DAD})$ | ▹ Merge to offspring |

5: ${r}_{1}=\mathit{rand}\left(\right),{r}_{2}=\mathit{rand}\left(\right),{r}_{3}=\mathit{rand}\left(\right)$ | |

6: if ${r}_{1}<\mathit{GApars}.\mathit{mrate}$ then | |

7: $\mathit{OFF}\left(\mathit{i}\right)\leftarrow \mathit{mutate}\left(\mathit{OFF}\right(\mathit{i}),\mathit{MSK})$ | ▹ Apply mutation operator |

8: end if | |

9: if ${r}_{2}<\mathit{GApars}.\mathit{arate}$ then | |

10: $\mathit{OFF}\left(\mathit{i}\right)\leftarrow \mathit{annihilate}\left(\mathit{OFF}\right(\mathit{i}),\mathit{MSK})$ | ▹ Apply annihilation operator |

11: end if | |

12: if ${r}_{3}<\mathit{GApars}.\mathit{frate}$ then | |

13: $\mathit{OFF}\left(\mathit{i}\right)\leftarrow \mathit{filter}\left(\mathit{OFF}\right(\mathit{i}),\mathit{MSK})$ | ▹ Apply filter operator |

14: end if | |

15: end for | |

16: $\mathit{POP}\leftarrow \mathit{copy}\left(\mathit{OFF}\right)$ | ▹ Copy to new population |

17: return POP | |

18: end function |

#### 2.3. The Stochastic Mask

Algorithm 3 First stage Metropolis sampling step. | |

1: function MetropolisStep1(${c}_{0}=({x}^{0},{y}^{0},{z}^{0})$) | ▹ input is a start location ${c}_{0}$ |

2: ${c}^{p}=({x}^{p},{y}^{p},{z}^{p})\leftarrow \mathit{rand}\left(\mathit{3}\right)$ | ▹${c}^{p}\in U[1,NX]\times U[1,NY]\times U[1,NZ]$ |

3: $r\leftarrow \mathit{rand}\left(\right)$ | ▹$r\in U[0,1]$ |

4: for $i=1:\mathit{NoViews}$ do | |

5: $[{X}^{0},{Y}^{0}]\leftarrow \mathit{getPixel}(\mathit{camera}\left(i\right),{c}^{0})$ | |

6: $[{X}^{p},{Y}^{p}]\leftarrow \mathit{getPixel}(\mathit{camera}\left(i\right),{c}^{p})$ | |

7: if $r\le \mathit{image}({X}^{p},{Y}^{p},i)/\mathit{image}({X}^{0},{Y}^{0},i)$ then | |

8: if $i==\mathit{NoViews}$ then | |

9: return ${c}^{n}={c}^{p}$ | |

10: end if | |

11: else | |

12: return ${c}^{n}={c}^{0}$ | |

13: end if | |

14: end for | |

15: end function |

Algorithm 4 Second stage Metropolis sampling step. | |

1: function MetropolisStep2(${c}_{0}=({x}^{0},{y}^{0},{z}^{0})$) | ▹ input is a start location ${c}_{0}$ |

2: ${c}^{p}=({x}^{p},{y}^{p},{z}^{p})\leftarrow \mathit{rand}\left(\mathit{3}\right)$ | ▹${c}^{p}\in U[1,NX]\times U[1,NY]\times U[1,NZ]$ |

3: $r\leftarrow \mathit{rand}\left(\right)$ | ▹$r\in U[0,1]$ |

4: if $r\le \mathit{box}({x}^{p},{y}^{p},{z}^{p})/\mathit{box}({x}^{0},{y}^{0},{z}^{0}))$ then | |

5: return ${c}^{n}={c}^{p}$ | |

6: else | |

7: return ${c}^{n}={c}^{0}$ | |

8: end if | |

9: end function |

## 3. Phantom Study on Numerical Data

#### 3.1. Parameter Study on Canonical Phantom Data

#### 3.2. Phantom Study on Three Generic Flame Types

#### 3.3. The Bunsen Flame Phantom

#### 3.4. The Swirl Flame Phantom

#### 3.5. The Cambridge–Sandia Stratified Flame Phantom

## 4. Applications to Experimental Data

#### 4.1. The Bunsen Flame

#### 4.2. The Swirl Flame

#### 4.3. The Cambridge-Sandia Stratified Flame

## 5. Quantitative Comparisons—Phantoms and Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**First stage stochastic mask generated with 15 rendered images, image size is 52 by 55 pixels, three million samples were drawn.

**Figure 8.**Slices normal to each spatial axis and height above burner exit (in mm), the correlation/normalized Manhattan distance to the phantom of each pair of slices is displayed between the images.

**Figure 9.**Slices normal to each spatial axis and height above burner exit (in mm), the correlation/normalized Manhattan distance to the phantom of each pair of slices is displayed between the images.

**Figure 11.**Slices normal to each spatial axis and height above burner exit (in mm), the correlation/normalized Manhattan distance to the phantom of each pair of slices is displayed between the images.

**Figure 13.**Vertical slices at the flame center (x-normal and z-normal), and horizontal slices at different heights above burner exit (in mm).

**Figure 14.**Vertical slices at the flame center (x-normal and z-normal), and horizontal slices at different heights above burner exit (in mm).

**Figure 15.**Vertical slices at the flame center (x-normal and z-normal), and horizontal slices at different heights above burner exit (in mm).

**Figure 16.**Calculated correlations based on horizontal slices. (

**a**) Bunsen flame phantom; (

**b**) Bunsen flame experiment; (

**c**) Swirl flame phantom; (

**d**) Swirl flame experiment; (

**e**) Cambridge-Sandia stratified flame phantom; (

**f**) Cambridge–Sandia stratified flame experiment.

**Figure 17.**Scatter plots of the phantom (

**left**) column and experimental (

**right**) column reconstructions. (

**a**) Bunsen flame phantom; (

**b**) Bunsen flame; (

**c**) Swirl flame phantom; (

**d**) Swirl flame; (

**e**) Cambridge–Sandia stratified flame phantom; (

**f**) Cambridge–Sandia stratified flame.

**Table 1.**Flow rates (in SLPM) and equivalence rations of the Cambridge–Sandia stratified burner for the SwB1 non swirling operating condition.

Parameter | Air | ${\mathbf{CH}}_{4}$ | ${\mathsf{\Phi}}_{1}$/${\mathsf{\Phi}}_{2}$ |
---|---|---|---|

outer-flow | 441.7 | 34.8 | 0.75 |

inner-flow | 144.0 | 11.4 | 0.75 |

co-flow | 765.6 |

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**MDPI and ACS Style**

Unterberger, A.; Kempf, A.; Mohri, K.
3D Evolutionary Reconstruction of Scalar Fields in the Gas-Phase. *Energies* **2019**, *12*, 2075.
https://doi.org/10.3390/en12112075

**AMA Style**

Unterberger A, Kempf A, Mohri K.
3D Evolutionary Reconstruction of Scalar Fields in the Gas-Phase. *Energies*. 2019; 12(11):2075.
https://doi.org/10.3390/en12112075

**Chicago/Turabian Style**

Unterberger, Andreas, Andreas Kempf, and Khadijeh Mohri.
2019. "3D Evolutionary Reconstruction of Scalar Fields in the Gas-Phase" *Energies* 12, no. 11: 2075.
https://doi.org/10.3390/en12112075