# Black Hole Algorithm for Sustainable Design of Counterfort Retaining Walls

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## Abstract

**:**

## 1. Introduction

- An adaptation of the BH optimization technique to work in discrete environments. Naturally, BH works in continuous spaces. Here, an adaptation is proposed using the concept of optimal approach velocity and a min–max normalization, which allows the velocity to be transformed into a transition probability.
- The application of the discrete version of BH to the counterfort retaining walls optimization problem. This optimization considers the objective function, the costs, and the CO${}_{2}$ separately.
- The impact of the relevant design variables is studied, both in costs and in CO${}_{2}$ emission.

## 2. Problem Definition

#### 2.1. Optimization Problem

#### 2.2. Problem Design Variables

#### 2.3. Problem Design Parameters

#### 2.4. Problem Constraints

## 3. The Discrete Black Hole Algorithm

Algorithm 1 Black Hole Algorithm | |

1: | Initialize solutions |

2: | Select black hole ${x}_{BH}$ |

3: | while$Iteration<MaxIteration$do |

4: | for all $X\in particle$ do |

5: | apply objective movement operator |

6: | apply objective function ${f}_{X}$ |

7: | if ${f}_{X}<{f}_{BH}$ then |

8: | Replace the black hole with the new best solution |

9: | end if |

10: | if $R>d(X,BH)$ then |

11: | Generate random new solution |

12: | end if |

13: | end for |

14: | end while |

#### The Discrete Algorithm

Algorithm 2 Discrete Algorithm | |

1: | movement = 0 |

2: | nx <–normalize solution |

3: | if${r}_{1}<n{x}^{d}$then |

4: | if ${r}_{2}>0.5$ then |

5: | movement = 1 |

6: | else |

7: | movement = −1 |

8: | end if |

9: | ${x}^{d}$ = max(1,min(${x}_{bh}^{d}$,${x}^{d}$+movement)) |

10: | end if |

## 4. Results and Discussion

#### 4.1. Wall Height Analysis

**Emission from optimum%**= $100\ast \frac{1270-1259}{1259}$.

#### 4.2. Design Variable Analysis

#### 4.3. Algorithm Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Set of reinforcement variables. Source: [10].

**Figure 2.**Buttressed wall. Cross-section of the floor. Source: [10].

**Figure 3.**Problem design parameters. Source: [10].

**Figure 4.**Variation and dispersion of best costs and CO${}_{2}$ emissions by the height of the wall.

**Figure 5.**Variation and dispersion of average costs and CO${}_{2}$ emissions by the height of the wall.

**Table 1.**Unit breakdown by unit cost, emissions, and cost. Source: [10].

Unit | Cost (€) | Emissions (CO${}_{2}$-eq) |
---|---|---|

kg steel B400 | 0.56 | 3.02 |

kg steel B500 | 0.58 | 2.82 |

m${}^{3}$ of concrete in stem | ||

C25/30 | 56.66 | 224.34 |

C30/37 | 60.80 | 224.94 |

C35/45 | 65.32 | 265.28 |

C40/50 | 70.41 | 265.28 |

C45/55 | 75.22 | 265.91 |

C50/60 | 80.03 | 265.95 |

m${}^{2}$ stem formwork | 21.61 | 1.92 |

m${}^{3}$ of backfill | 5.56 | 28.79 |

m${}^{3}$ of concrete in foundation | ||

C25/30 | 50.65 | 224.34 |

C30/37 | 54.79 | 224.94 |

C35/45 | 59.31 | 265.28 |

C40/50 | 64.40 | 265.28 |

C45/55 | 69.21 | 265.91 |

C50/60 | 74.02 | 265.95 |

**Table 2.**Design variables. Source: [10].

Variables | Lower Bound | Upper Bound | Increment | N of Values |
---|---|---|---|---|

c | H/20 | H/5 | 5 cm | f(H) ${}^{1}$ |

d | H/5 cm | 2H/3 | 5 cm | f(H) ${}^{1}$ |

b | 25 cm | 122.5 | 2.5 cm | 40 |

p | 20 cm | 610 | 10 cm | 60 |

t | 20 cm | 905 | 15 cm | 60 |

${e}_{c}$ | 25 cm | 122.5 | 2.5 cm | 40 |

${f}_{ck}$ | 25, 20, 25, 40, 45, 50 | 7 | ||

${f}_{yk}$ | 400, 500 | 2 | ||

${A}_{1}$ to ${A}_{10}$ | 6, 8, 10, 12, 16, 20, 25, 32 | 8 | ||

1 steel rebar | 12 rebars | 2 rebars | 6 | |

${A}_{11}$ to ${A}_{12}$ | 6, 8, 10, 12, 16, 20, 25, 32 | 8 | ||

1 steel rebar | 4 rebars | 10 rebars | 7 |

Parameter Considered | Value | |
---|---|---|

Bearing capacity | 0.3 MPa | |

Fill slope | 0 | |

Foundation depth, H2 | 2 m | |

Uniform load on top of the fill, $\gamma $ | 10 kN/m${}^{2}$ | |

Base-friction coefficient, $\mu $ | tg ${30}^{\circ}$ | |

Wall-fill friction angle, $\delta $ | ${0}^{\circ}$ | |

Safety coefficient: | ||

against sliding, ${\gamma}_{fs}$ | 1.5 | |

against overturning, ${\gamma}_{fo}$ | 1.8 | |

for loading (EHE) | Normal | |

of steel (ULS) | 1.15 | |

of concrete (ULS) | 1.5 | |

Ambient exposure | IIa |

Height (m) | Opt. Cost | Emissions | Cost | Opt. Emissions | Cost from Optimum | Emissions from Optimum |
---|---|---|---|---|---|---|

6 | 593 | 1270 | 607 | 1259 | 2.36% | 0.87% |

7 | 679 | 1455 | 694 | 1441 | 2.21% | 0.97% |

8 | 774 | 1706 | 795 | 1659 | 2.71% | 2.83% |

9 | 912 | 2046 | 930 | 1998 | 1.97% | 2.40% |

10 | 1096 | 2589 | 1135 | 2478 | 3.56% | 4.48% |

11 | 1309 | 3249 | 1407 | 3060 | 7.49% | 6.18% |

12 | 1533 | 3841 | 1600 | 3716 | 4.37% | 3.36% |

13 | 1783 | 4602 | 1876 | 4471 | 5.22% | 2.93% |

14 | 2052 | 5523 | 2163 | 5291 | 5.41% | 4.38% |

15 | 2363 | 7116 | 2491 | 6788 | 5.42% | 4.83% |

Cost | Emissions | |||||||
---|---|---|---|---|---|---|---|---|

Optimization | Optimization | |||||||

Height | Steel in | Steel in | Concrete in | Concrete in | Steel in | Steel in | Concrete in | Concrete in |

Stem (Kg) | Base (Kg) | Stem (m${}^{\mathbf{3}}$) | Base (m${}^{\mathbf{3}}$) | Stem (Kg) | Base (Kg) | Stem (m${}^{\mathbf{3}}$) | Base (m${}^{\mathbf{3}}$) | |

6 | 53.27 | 30.51 | 1.88 | 0.86 | 48.48 | 23.96 | 1.97 | 0.86 |

7 | 79.91 | 45.90 | 2.13 | 0.88 | 68.73 | 32.80 | 2.24 | 0.88 |

8 | 121.48 | 40.39 | 2.38 | 1.01 | 92.81 | 37.42 | 2.55 | 1.01 |

9 | 134.06 | 66.03 | 2.75 | 1.30 | 110.55 | 59.20 | 2.91 | 1.30 |

10 | 187.74 | 98.09 | 3.16 | 1.78 | 139.04 | 77.53 | 3.51 | 1.77 |

11 | 234.57 | 156.61 | 3.70 | 2.37 | 167.98 | 103.98 | 4.21 | 2.41 |

12 | 272.33 | 197.82 | 4.17 | 3.02 | 220.12 | 160.13 | 4.69 | 3.01 |

13 | 372.58 | 293.64 | 4.80 | 3.78 | 270.65 | 230.85 | 5.16 | 3.80 |

14 | 411.92 | 365.27 | 5.13 | 4.63 | 312.40 | 316.09 | 5.77 | 4.69 |

15 | 481.51 | 503.15 | 5.87 | 5.58 | 356.85 | 424.82 | 6.02 | 6.09 |

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## Share and Cite

**MDPI and ACS Style**

Yepes, V.; Martí, J.V.; García, J.
Black Hole Algorithm for Sustainable Design of Counterfort Retaining Walls. *Sustainability* **2020**, *12*, 2767.
https://doi.org/10.3390/su12072767

**AMA Style**

Yepes V, Martí JV, García J.
Black Hole Algorithm for Sustainable Design of Counterfort Retaining Walls. *Sustainability*. 2020; 12(7):2767.
https://doi.org/10.3390/su12072767

**Chicago/Turabian Style**

Yepes, Víctor, José V. Martí, and José García.
2020. "Black Hole Algorithm for Sustainable Design of Counterfort Retaining Walls" *Sustainability* 12, no. 7: 2767.
https://doi.org/10.3390/su12072767