# Combining Numerical Simulations, Artificial Intelligence and Intelligent Sampling Algorithms to Build Surrogate Models and Calculate the Probability of Failure of Urban Tunnels

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Evaluation of the Probability of Failure

- (a)
- Quantifying and assigning probabilistic distributions to the input uncertainties,
- (b)
- Sampling the distributions of these uncertain parameters in an iterative fashion using Monte Carlo methods,
- (c)
- Propagating the effects of uncertainties through the model, and
- (d)
- Predicting the outcomes in terms of probabilistic measures such as mean, variance and fractiles.

## 3. Proposed Methodology

#### 3.1. Deterministic Solutions Used to Build Surrogate Model

#### 3.2. Input Parameter Selection by Artificial Intelligence Techniques

^{®}package and its implemented method known as RFECV, which handles recursive elimination of variables with cross-validation. This method allows to find the informative input variables for a given AI algorithm by means of a recursive procedure, where features are iteratively removed and then the best subset of features is chosen based on the cross-validation score of the model. Particularly, sklearn’s algorithm begins the process with a model that includes all variables and then assigns an importance score to each one. Next, the random variables of lesser importance are removed, the model is reconstructed, and the importance scores are recalculated. In the present paper, for each iteration of the RFECV method, a single feature was removed after a 5-fold cross-validation strategy was considered (model parameters ‘step = 1, cv = 5’). Thus, the result of this analysis can lead to a reduction in the number of random variables involved in the problem.

- Bayesian ARD regression (‘ARDRegression’),
- AdaBoost regressor (‘AdaBoostRegressor’),
- Bayesian ridge regression (‘BayesianRidge’),
- Canonical correlation analysis (‘CCA’),
- Decision tree regressor (‘DecisionTreeRegressor’),
- Elastic net regressor (‘ElasticNet’), which is basically a linear regression with combined L1 and L2 priors as the regularizer,
- Elastic net CV regressor (‘ElasticNetCV’), which is an elastic net model with iterative fitting along a regularization path,
- Extremely randomized tree regressor (‘ExtraTreeRegressor’),
- Extra-trees regressor (‘ExtraTreesRegressor’),
- Gradient boosting for regression (‘GradientBoostingRegressor’),
- Linear regression model that is robust to outliers (‘HuberRegressor’),
- Least angle regression (‘Lars’),
- Cross-validated least angle regression (‘LarsCV’),
- Linear model trained with L1 prior as regularizer (‘Lasso’),
- Lasso linear model with iterative fitting along a regularization path (‘LassoCV’),
- Lasso model fit with least angle regression (‘LassoLars’),
- Cross-validated lasso, using the LARS algorithm model (‘LassoLarsCV’),
- Lasso model fit with Lars using Bayes information criterion or Akaike information criterion for model selection (‘LassoLarsIC’),
- Ordinary least squares linear regression (‘LinearRegression’),
- Linear support vector regression (‘LinearSVR’),
- Orthogonal matching pursuit model (‘OrthogonalMatchingPursuit’),
- Partial least squares transformer and regressor (‘PLSCanonical’),
- Partial least squares regressor (‘PLSRegression’),
- Passive aggressive regressor (‘PassiveAggressiveRegressor’),
- Generalized linear model with a Poisson distribution (‘PoissonRegressor’),
- A random forest regressor (‘RandomForestRegressor’),
- Linear least squares with L2 regularization (‘Ridge’),
- Ridge regression with built-in cross-validation (‘RidgeCV’),
- Linear model fitted by minimizing a regularized empirical loss with Stochastic gradient descent (‘SGDRegressor’),
- Theil-Sen estimator (‘TheilSenRegressor’), which is a robust multivariate regression model,
- Generalized linear model with a Tweedie distribution (‘TweedieRegressor’).

#### 3.3. Training and Testing Artificial Intelligence

#### 3.4. Monte Carlo Sampling Techniques to Calculate the Probability of Failure: Simple and Latin Hypercube Sampling

#### 3.5. Reliability of Sampling Techniques

## 4. Hypothetical Case of an Urban Tunnel

#### 4.1. Initial Model and Boundary Conditions

#### 4.1.1. Geotechnical Parameters

#### 4.1.2. Mesh and Domain

#### 4.1.3. Contact Properties

#### 4.2. Optimized Numerical Model

#### 4.2.1. The Building

#### 4.2.2. Tunnel’s Front and Back Boundary

- First lesson learned: the displacements and stresses were fixed as boundary conditions at the tunnel perimeter along the boundaries in the y-axis. Such stresses and displacements were exactly the ones observed at the mid-length cross-section. This attempt decreased the magnitude of the normalized displacements (Figure 14) but enlarged the affected zone, considering the 5% tolerance. Therefore, it was not useful.
- Second lesson learned: one-dimensional elements were included as reinforcement on the tunnel crown. This attempt was unsuccessful because the interference in the stress field was hardly controlled and affected even larger zones than initially foreseen.

#### 4.3. Numerical Model Features

#### 4.3.1. Strategy for Including Shotcrete

#### 4.3.2. Simplifications Adopted

- All stratigraphic layers were considered to be linear elastic with a Mohr–Coulomb plasticity criterion. Generally, in tunnel applications, using this constitutive model may lead to incorrect results, as it assumes the use of the same moduli to represent loadings and unloadings. Furthermore, it is also noteworthy that the linear elastic-Mohr–Coulomb model neglects other important aspects of soil behavior, such as the degradation of initial stiffness with strain, which is often important in tunnel applications [55].
- The building in the model does not include auxiliary structures such as the elevator shaft and/or stairwell;
- The building has no masonry enclosure, which may be necessary in some practical cases.

## 5. Results and Analyses

#### 5.1. Variable Sensitivity Analysis

#### 5.2. Training and Testing the Artificial Intelligence

#### 5.3. Reliability of Sampling Techniques

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Frequency histogram: (

**a**) Simple Monte Carlo sampling—left; and (

**b**) Latin hypercube sampling—right.

**Figure 2.**Methodological proposal with the combination of ABAQUS, a Python code and artificial intelligence techniques.

**Figure 13.**Pre-excavation conditions: (

**a**) geostatic condition; (

**b**) inclusion of the building on the surface and 1 kPa loading on the slabs.

**Figure 17.**Construction process using the sacrificial layer: (

**a**) intermediate excavation step; (

**b**) excavate 1.50 m and allow the soil to displace; and (

**c**) replace the deformed sacrificial layer with a new one, similar, attached to the shotcrete.

**Figure 20.**Training and testing artificial intelligence for predicting vertical displacements of Footing A.

**Figure 21.**Training and testing artificial intelligence for predicting vertical displacements of Footing D.

**Table 1.**Building damage as discussed in [34].

Damage Criteria | Angular Distortion ($\mathsf{\beta}=\mathsf{\delta}/\mathbf{L}$) |
---|---|

Machinery sensitive to settlements are to be feared | 1/750 |

Danger for frames with diagonals | 1/600 |

Safe limit for buildings where cracking is not permissible | 1/500 |

First cracking in panel walls is to be expected | 1/300 |

Difficulties with overhead cranes are to be expected | 1/300 |

Tilting of high, rigid buildings might become visible | 1/250 |

Considerable cracking in panel walls and brick walls | 1/150 |

Limit for flexible brick walls, h/L < 1/4 | 1/150 |

Structural damage of general buildings is to be feared | 1/150 |

Foundation | ||||

Structural Elements | Width | Length | Height | Diameter |

Base | - | - | 1.50 | 2.00 |

Shaft | - | - | 1.00 | 0.60 |

Block | 1.20 | 1.20 | 1.20 | - |

Beam | 0.40 | var. | 0.80 | - |

Slab | var. | var. | 0.10 | - |

First Floor | ||||

Structural Elements | Width | Length | Height | Diameter |

Pillar | 0.40 | 1.20 | 2.70 | - |

Beam | 0.40 | var. | 0.80 | - |

Slab | var. | var. | 0.10 | - |

2nd to 7th Floor | ||||

Structural Elements | Width | Length | Height | Diameter |

Pillar | 0.12 | 0.30 | 2.70 | - |

Beam | 0.12 | var. | 0.45 | - |

Slab | var. | var. | 0.10 | - |

Material/Statistical Truncated Distribution | Unit Weight (kN/m${}^{3}$) | Young’s Moduli (MPa) | Poisson Ratio | Friction Angle (°) | Dilatancy (°) | Cohesion (kPa) | k0 |
---|---|---|---|---|---|---|---|

Log-Normal | Log-Normal | Normal | Log-Normal | Log-Normal | Log-Normal | Normal | |

Layer H1 (0 to 15 m) | 16.7–17.6 | 14–40 | 0.30–0.40 | 35–40 | 1–2 | 10–15 | 0.4–0.7 |

Layer H2 (15 to 30 m) | 18.6–19.6 | 40–70 | 0.30–0.40 | 40–45 | 1–2 | 20–25 | 0.4–0.7 |

Layer H3 (30 to 45 m) | 20.6–21.6 | 70–110 | 0.30–0.40 | 45–50 | 5–10 | 30–35 | 0.4–0.7 |

Layer H4 (45 to 60 m) | 20.6–21.6 | 70–110 | 0.30–0.40 | 45–50 | 10–15 | 30–35 | 0.4–0.7 |

Reinforced concrete | 24.5–25.5 | 30–32 × ${10}^{3}$ | 0.20–0.25 | – | – | – | – |

Shotcrete | 23.5–24.5 | 30.5–32.5 × ${10}^{3}$ | 0.20 – 0.25 | – | – | – | – |

Footing | Influence Zone | Footing | Influence Zone | Footing | Influence Zone | Footing | Influence Zone |
---|---|---|---|---|---|---|---|

A | D-B | D | A-G-E | G | D-J-H | J | G-K |

B | A-E-C | E | B-D-H-F | H | E-G-K-I | K | H-J-L |

C | B-F | F | C-E-I | I | F-H-L | L | I-K |

Parameter | Number | Global Importance 1 | Local Importance 2 |
---|---|---|---|

of Choices | (%) | (%) | |

Layer 1—Young’s Modulus | 28 | 75.4 | 78.0 |

Layer 3—Young’s Modulus | 28 | 7.4 | 11.2 |

Layer 3—k0 | 26 | 2.7 | 5.7 |

Layer 1—Poisson | 23 | 1.0 | 3.0 |

Layer 2—Young’s Modulus | 21 | 0.6 | 2.1 |

∑ | 87.1 | 100.0 |

Parameter | AIC | Squared Error |
---|---|---|

Von Mises (vonmises) | −$1.50891\times {10}^{8}$ | $6.82\times {10}^{67}$ |

Gauss hypergeometric (gausshyper) | $-2.72042\times {10}^{7}$ | $3.29\times {10}^{13}$ |

Beta (beta) | $-1.96046\times {10}^{7}$ | $2.07\times {10}^{8}$ |

Johnson SB (johnsonsb) | $-1.96045\times {10}^{7}$ | $2.06\times {10}^{8}$ |

Pearson type III (pearson3) | $-1.96042\times {10}^{7}$ | $2.02\times {10}^{8}$ |

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**MDPI and ACS Style**

Domingues, V.R.; Ozelim, L.C.d.S.M.; Assis, A.P.d.; Cavalcante, A.L.B.
Combining Numerical Simulations, Artificial Intelligence and Intelligent Sampling Algorithms to Build Surrogate Models and Calculate the Probability of Failure of Urban Tunnels. *Sustainability* **2022**, *14*, 6385.
https://doi.org/10.3390/su14116385

**AMA Style**

Domingues VR, Ozelim LCdSM, Assis APd, Cavalcante ALB.
Combining Numerical Simulations, Artificial Intelligence and Intelligent Sampling Algorithms to Build Surrogate Models and Calculate the Probability of Failure of Urban Tunnels. *Sustainability*. 2022; 14(11):6385.
https://doi.org/10.3390/su14116385

**Chicago/Turabian Style**

Domingues, Vinícius Resende, Luan Carlos de Sena Monteiro Ozelim, André Pacheco de Assis, and André Luís Brasil Cavalcante.
2022. "Combining Numerical Simulations, Artificial Intelligence and Intelligent Sampling Algorithms to Build Surrogate Models and Calculate the Probability of Failure of Urban Tunnels" *Sustainability* 14, no. 11: 6385.
https://doi.org/10.3390/su14116385