# Prediction of Maximum Pressure at the Roofs of Rectangular Water Tanks Subjected to Harmonic Base Excitation Using the Multi-Gene Genetic Programming Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Modelling

_{c}is the mass of the convective part of the liquid (c = convective), M is the total liquid mass, L is half of the tank length, h is the total liquid height, k

_{c}is the stiffness of the assumed spring that connects the convective mass to the tank’s walls in the direction of movement, g is ground acceleration equal to 9.81 m/s

^{2}, and ω

_{c}and T

_{c}are the resonance frequency and resonance period of the first (fundamental) mode of the oscillating liquid, respectively. In lieu of Housner’s method to determine the natural frequency of the tank, Lamb’s formula can be used for simplicity [38]. In Table 2, the resonance frequencies that were applied to each tank based on the size and liquid height are presented. Each tank size–liquid height combination was simulated at four different orientations of 0°, 30°, 60°, and 90°. Since the direction of an earthquake is not predictable, the maximum pressure among all orientations was used as the input for the GP section. In other words, the maximum of maximums was found and applied to the GP. The excitation orientations of 0°, 30°, and 60° are presented in Figure 2.

^{3}) and total pressure (Pa) respectively; u, v, and w are the particle speeds in the x, y, and z directions (m/s); t is time (s); and $g=9.81$ m/s

^{2}is the gravity acceleration and

_{1}and ρ

_{2}are the densities of air and water, respectively, and α indicates the volume of each particle that is filled with each of the fluids. The value of α varies between 0.0 and 1.0, with 1.0 meaning the cell is filled with water and 0.0 indicating air. A value of 0.5 is allocated to the free surface. Any value between 0.0 and 0.5 indicates air, and a value between 0.5 and 1.0 indicates water.

#### 2.1.1. Computational Setup

#### 2.1.2. CFD Details

^{−4}m

^{2}/s was found to provide the best results compared to the experimental data.

#### 2.2. Genetic Programming

_{(ij)}is the calculated value for jth data based on ith function, T

_{j}is the actual value for the jth data, and $\overline{T}$ is the average of the T

_{j}values.

_{1}, x

_{2}, etc. (Pandey et al. [41]).

_{1}and x

_{2}(i.e., $y=f\left({x}_{1},{x}_{2}\right)$, y is dependent on two variables of x

_{1}and x

_{2}), A

_{1}and B

_{1}are randomly created parent genes as follows:

_{1}is switched with a sub-tree of the parent gene B

_{1}, resulting in second generation genes, A

_{2}and B

_{2}:

_{2}and B

_{2}is replaced by a new randomly chosen sub-tree, creating the third-generation genes, A

_{3}and B

_{3}:

- A set of variables is initiated.
- The chromosomes’ architecture is defined.
- The chromosomes are randomly formulated.

## 3. Results and Discussion

#### 3.1. Numerical Modelling

_{d}is the dimensionless pressure, P

_{max}is the maximum pressure on the roof, a is the maximum acceleration of the harmonic excitation, ρ is the density of water, h is the liquid height in the tank, H is the height of the tank, L is half of the length of the tank (i.e., the tank’s length is 2L), and Fb is the available freeboard. The parameters a and Fb can be calculated by Equations (26) and (27):

#### 3.2. Genetic Programming

#### 3.2.1. Single-Gene Solution

_{d,S}is the dimensionless maximum pressure obtained by the Single-Gene solution.

_{d}(i.e., dimensionless liquid height) has a value between 0.3179 and 3.2211, and hence the results are valid for tanks with dimensionless liquid height in that range. Since this relationship is obtained based on the maximum pressure in all tank orientations, it is not affected by the angle of tank orientation. Figure 11 presents the complexity of the model plotted against its accuracy level (1 − R

^{2}) for the population on the training set of data. In this figure, green dots represent Pareto models, and blue dots represent non-Pareto models. The green dot with a red circle shows the best model in terms of R

^{2}on the training data.

#### 3.2.2. Multi-Gene Solution (MGGP)

_{d,M}is the dimensionless maximum roof pressure obtained by the Multi-Gene program. The number of generations was set to 500 with a population of 300. Equation (29) was obtained in generation 473. This equation is composed of the following genes:

^{2}) for the population on the training set of data.

#### 3.2.3. Error Estimations

#### a. R-Squared (R^{2})

^{2}, is calculated as

^{2}values for the Single-Gene and Multi-Gene solutions.

#### b. Root Mean Squared Error (RMSE)

#### c. Mean Absolute Deviation (MAD)

#### d. Mean Absolute Error (MAE)

#### e. Mean Absolute Percentage Error (MAPE)

#### f. Akaike Information Criterion (AIC):

#### g. Performance Index (PI):

^{2}and higher MAPE, which can be indicators of higher errors and overfitting of the model. However, the RMSE and MAE values provide comparable results for the test and trained data sets with fewer errors. In other words, two of the four error indicators show better results in test data sets, while the other two may indicate overfitting. Given the circumstances, the results for both Single-Gene and Multi-Gene models are reasonably acceptable.

#### 3.3. Uncertainty Analysis and Confidence Bands

_{d}were generated in the range of 0.3179 to 3.2211. Then, the equation was run for each random number. To generate random data with normal-shaped distribution in a specific range, a truncated Gaussian function was used. The histogram of the generated data using the truncated Gaussian function is shown in Figure 14.

_{d}, namely P

_{mc}, were calculated. The mean absolute deviation (MAD) was calculated around the average using Equation (27)

_{avg}is the average of the pressures calculated by the Monte Carlo simulation [20], thus leading to

_{SG}and U

_{MG}are the uncertainty percentages for the Single-Gene and Multi-Gene equations, respectively. Due to the high slope of the graph of the equation in the beginning, these amounts of uncertainty are reasonable.

#### 3.4. Sensitivity Analysis

_{dp}is the 10% perturbed mean dimensionless liquid height, h

_{dm}is the actual mean dimensionless liquid height, ∆P

_{d}is the perturbation that appears in the dimensionless pressure due to the 10% perturbation in the dimensionless liquid height, P

_{dp}is the change in the value of the dimensionless pressure when the dimensionless liquid height changes, P

_{dm}is the value of the dimensionless pressure at mean dimensionless liquid height (h

_{dm}) calculated based on Equations (28) and (29) for Single-Gene and Multi-Gene modes, and S

_{n}is the normal sensitivity of those equations.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Computational fluid dynamics (CFD) outputs for tank size 2, with 800 mm water depth at 0° orientation and time $t=9.50$ s; (

**a**) liquid surface and (

**b**) pressure.

**Figure 10.**Dimensionless maximum pressure versus dimensionless liquid height for the observed (CFD) data.

**Figure 13.**Observed dimensionless maximum pressure plotted against the dimensionless maximum pressure obtained by (

**a**) Single-Gene procedure and (

**b**) Multi-Gene procedure, for the overall data sets.

**Figure 15.**Graph of the proposed equation for dimensionless pressure plotted against dimensionless liquid height with 95% confidence bounds for (

**a**) Single-Gene and (

**b**) Multi-Gene modes.

Length (mm) | Width (mm) | Height (mm) | |
---|---|---|---|

Size 1 | 755 | 300 | 300 |

Size 2 | 1978 | 779 | 1200 |

Size 3 | 1283 | 327 | 1200 |

Size 4 | 683 | 342 | 1200 |

Length | Width | Tank Height | Liquid Height | Dimensionless Liquid Height | ω_{i} | T_{i} | |
---|---|---|---|---|---|---|---|

(mm) | (mm) | (mm) | (mm) | (h_{l}/L) | (rad/s) | (s) | |

Size 1 | 755 | 300 | 300 | 100 | 0.265 | 4.023 | 1.562 |

120 | 0.318 | 4.354 | 1.443 | ||||

145 | 0.384 | 4.705 | 1.335 | ||||

200 | 0.53 | 5.282 | 1.190 | ||||

230 | 0.609 | 5.510 | 1.140 | ||||

250 | 0.662 | 5.636 | 1.115 | ||||

280 | 0.742 | 5.792 | 1.085 | ||||

Size 2 | 1978 | 779 | 1200 | 1100 | 1.112 | 3.819 | 1.645 |

1000 | 1.011 | 3.777 | 1.663 | ||||

900 | 0.910 | 3.721 | 1.689 | ||||

800 | 0.809 | 3.644 | 1.724 | ||||

700 | 0.708 | 3.540 | 1.775 | ||||

600 | 0.607 | 3.400 | 1.848 | ||||

Size 3 | 1283 | 327 | 1200 | 1100 | 1.714 | 4.858 | 1.293 |

1000 | 1.559 | 4.845 | 1.297 | ||||

900 | 1.403 | 4.824 | 1.303 | ||||

800 | 1.247 | 4.789 | 1.312 | ||||

700 | 1.091 | 4.732 | 1.328 | ||||

600 | 0.935 | 4.640 | 1.354 | ||||

Size 4 | 683 | 327 | 1200 | 1100 | 3.221 | 6.686 | 0.940 |

1000 | 2.928 | 6.686 | 0.940 | ||||

900 | 2.635 | 6.685 | 0.940 | ||||

800 | 2.343 | 6.682 | 0.940 | ||||

700 | 2.05 | 6.677 | 0.941 | ||||

600 | 1.757 | 6.662 | 0.943 |

Data Set | MAD | ||
---|---|---|---|

Observed Data | Single-Gene Results | Multi-Gene Results | |

Trained | 30.63 | 30.27 | 30.44 |

Test | 6.32 | 17.30 | 5.90 |

Overall | 26.08 | 25.90 | 25.56 |

Data Set | R-Squared | RMSE | MAE | MAPE (%) | |||||
---|---|---|---|---|---|---|---|---|---|

Value | % of Maximum Dimensionless Pressure | % of Mean Dimensionless Pressure | AIC | PI | |||||

Single-Gene | Trained | 0.989 | 4.54 | 2.69 | 17.09 | 3.64 | 68% | 21.15 | 0.086 |

Test | 0.844 | 3.23 | 14.17 | 46.30 | 3.03 | 260% | 10.55 | 0.241 | |

Overall | 0.989 | 4.31 | 2.55 | 19.03 | 3.52 | 107% | 23.87 | 0.114 | |

Multi-Gene | Trained | 0.992 | 3.89 | 2.30 | 14.63 | 3.28 | 76% | 21.8 | 0.073 |

Test | 0.889 | 2.73 | 11.99 | 39.18 | 2.19 | 302% | 12.18 | 0.202 | |

Overall | 0.992 | 3.69 | 2.18 | 16.26 | 3.06 | 121% | 24.17 | 0.082 |

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**MDPI and ACS Style**

Bahreini Toussi, I.; Mohammadian, A.; Kianoush, R. Prediction of Maximum Pressure at the Roofs of Rectangular Water Tanks Subjected to Harmonic Base Excitation Using the Multi-Gene Genetic Programming Method. *Math. Comput. Appl.* **2021**, *26*, 6.
https://doi.org/10.3390/mca26010006

**AMA Style**

Bahreini Toussi I, Mohammadian A, Kianoush R. Prediction of Maximum Pressure at the Roofs of Rectangular Water Tanks Subjected to Harmonic Base Excitation Using the Multi-Gene Genetic Programming Method. *Mathematical and Computational Applications*. 2021; 26(1):6.
https://doi.org/10.3390/mca26010006

**Chicago/Turabian Style**

Bahreini Toussi, Iman, Abdolmajid Mohammadian, and Reza Kianoush. 2021. "Prediction of Maximum Pressure at the Roofs of Rectangular Water Tanks Subjected to Harmonic Base Excitation Using the Multi-Gene Genetic Programming Method" *Mathematical and Computational Applications* 26, no. 1: 6.
https://doi.org/10.3390/mca26010006