# Generating the Regular Axis from Irregular Column Grids through Genetic Algorithm

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## Abstract

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## Featured Application

**This study proposes an auto-generation method of the regular axis, applied in digital documentation of cultural heritage. The procedure is repeatable, the results show the displacement of columns with visual expression and numerical analysis.**

## Abstract

## 1. Introduction

#### 1.1. Review of Related Works

#### 1.2. Purpose and Significance

## 2. Materials and Methods

#### 2.1. Finite Element Modeling Using Genetic Algorithm

- Use the Circle—FitPoints command to manually select related points from the point cloud of a column head and then automatically draw a circle of the section profile. Repeat to go through all column heads (Figure 3).
- Launch Grasshopper, use the Curve component to collect all circles, use Project to obtain circles in the X-plane, use Area to obtain centroids of circles, use Deconstruct to obtain X, Y, and Z values, and use Sort List to sort centroids in a certain order (Figure 4).
- Use Number Slider to manually give a range for initial parameters including motions and rotations of axes, then connect them to the Genome of Galapagos as a candidate solution for the genetic algorithm (Figure 5).
- At this moment, a candidate solution and resulting axes are shown; the average distance is 144 mm (Figure 6).
- This solution could be optimized as long as a goal is given to the fitness of Galapagos, therefore Gauss sum is applied to calculate the applicability of axes, then the average distance between current centroids and rebuild centroids is applied to evaluate the result (Figure 7).
- Open the Galapagos, start the solver, and during the process, the result is displayed in real-time (Figure 8).
- When the result seems stable after 5 min of calculation, it could be output as the final solution, if the average distance is acceptable. Or wait until the minimum value is reached, though it may take longer (almost 1 h). The average distance is 68 mm (Figure 9).

**3 h 16 min**and stops automatically after reaching the

**minimum value**.

**initial angle**is 0.00002 degrees. The winding corridor has an average width of 3493 mm, and the four dimensions (3485 mm, 3488 mm, 3498 mm, 3500 mm) have a

**standard deviation**of 6 mm. Therefore, the winding corridor could be regarded as equidistantly offset from the second circle of columns. Similarly, the whole grid system could be regarded as symmetric.

#### 2.2. Hypothesis and Verification

**33 min**to reach the goal (Figure 11).

**54 min**to reach the

**minimum value**then stops automatically. However, when this grid system is asymmetric, the resulting axes have a smaller average distance (183 mm) than the initial pre-given axes (200 mm) and a smaller standard deviation (188 mm) than the initial pre-given axes (200 mm), but they fail to restore the original grids (Figure 14).

## 3. Results

- At least two rows.
- Each row has four symmetrically displaced columns or has more than four symmetrically displaced columns as long as the number is even (6, 8, 10, etc.).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Use Number Slider to give a candidate solution to the Genome (red wires) of Galapagos for genetic algorithm solution.

**Figure 11.**Verification of two columns (the disordered axes on the left, the result axes on the right).

**Figure 12.**Verification of four columns (the disordered axes on the left, the result axes on the right).

**Figure 13.**Verification of eight symmetric columns (the disordered axes on the left, the result axes on the right).

**Figure 14.**Verification of eight asymmetric columns (the disordered axes on the left, the result axes on the right).

**Figure 15.**Verification of six asymmetric columns (the disordered axes on the left, the result axes on the right).

**Figure 16.**Verification of six symmetric columns (the disordered axes on the left, the result axes on the right).

**Figure 17.**Verification of 10 asymmetric columns (the disordered axes on the left, the result axes on the right).

**Figure 18.**Verification of 10 symmetric columns (the disordered axes on the left, the result axes on the right).

Initial State of Pre-given Axes | Column Amount | 2 | 4 | 6 | 6 | 8 | 8 | 10 | 10 |

Single Axis or Rectangle Grids | Single Axis | Rectangle Grids | |||||||

Symmetry | Not Applicable | Yes | No | Yes | No | Yes | No | ||

Distribution Distance | Not Applicable | 200 mm | |||||||

Result of Generation | Restore the Movement | Yes | No | No | No | Yes | No | No | No |

Restore the Rotation | Yes | No | Yes | Yes | Yes | No | Yes | Yes | |

Average Distance | 0 mm | 182 mm | 177 mm | 177 mm | 200 mm | 183 mm | 192 mm | 192 mm | |

Standard Deviation | 0 mm | 372 mm | 188 mm | 188 mm | 200 mm | 188 mm | 196 mm | 196 mm | |

Duration | 33 min | 43 min | 54 min | 55 min | 54 min | 55 min | 55 min | 54 min | |

Outputs Evaluation | Right | Wrong | Applicable | Right | Applicable |

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**MDPI and ACS Style**

Wang, X.; Wu, C.; Bai, C.
Generating the Regular Axis from Irregular Column Grids through Genetic Algorithm. *Appl. Sci.* **2022**, *12*, 2109.
https://doi.org/10.3390/app12042109

**AMA Style**

Wang X, Wu C, Bai C.
Generating the Regular Axis from Irregular Column Grids through Genetic Algorithm. *Applied Sciences*. 2022; 12(4):2109.
https://doi.org/10.3390/app12042109

**Chicago/Turabian Style**

Wang, Xi, Cong Wu, and Chengjun Bai.
2022. "Generating the Regular Axis from Irregular Column Grids through Genetic Algorithm" *Applied Sciences* 12, no. 4: 2109.
https://doi.org/10.3390/app12042109