# Comparative Analysis between Genetic Algorithm and Simulated Annealing-Based Frameworks for Optimal Sensor Placement and Structural Health Monitoring Purposes

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Genetic Algorithm

#### 2.2. Simulated Annealing

#### 2.3. Ensemble Kalman Filter

_{i}(n represents the modal state vector size) [25,34]:

#### 2.4. GA-EnKF Methodology for OSP

#### 2.5. SA-EnKF Methodology for OSP

## 3. Numerical Example

**M**is the mass matrix;**F**is the nonlinear restoring force. The ith element of**F**is expressed as:$${F}_{i}\left(u,\dot{u}\right)={a}_{i}\left({u}_{i}-{u}_{i-1}\right)+{a}_{i+1}\left({u}_{i}-{u}_{i+1}\right)+{b}_{i}{\left({u}_{i}-{u}_{i-1}\right)}^{3}+{b}_{i+1}{\left({u}_{i}-{u}_{i+1}\right)}^{3}\phantom{\rule{0ex}{0ex}}+{c}_{i}\left({\dot{u}}_{i}-{\dot{u}}_{i-1}\right)+{c}_{i+1}\left({\dot{u}}_{i}-{\dot{u}}_{i+1}\right)+{d}_{i}\left({u}_{i}-{u}_{i-1}\right)\left({\dot{u}}_{i}-{\dot{u}}_{i-1}\right)\phantom{\rule{0ex}{0ex}}+{d}_{i+1}\left({u}_{i}-{u}_{i+1}\right)\left({\dot{u}}_{i}-{\dot{u}}_{i+1}\right)$$- i represents the floor number, varying from 1 to 10 for this specific numerical problem, and {a
_{i}}, {b_{i}}, {c_{i}}, {d_{i}} represent the damping and stiffness data coefficients corresponding to floor i. Table 1 summarizes the relations between these coefficients and the stiffness and damping corresponding to floor i.

- a = 1 × 10
^{8} - b = 2 × 10
^{5} - c = 9.4 × 10
^{5} - d = 4.5 × 10
^{5}

- a = 1 × 10
^{7} - b = 1 × 10
^{4} - c = 4.7 × 10
^{4} - d = 1.5 × 10
^{4}

- If the number of generations exceeds 100 generations (MaxIterations), the algorithm stops immediately;
- If the mean difference in fitness function is lower than the tolerance function (10
^{−10}) for 20 successive generations (MaxStallIterations), the algorithm also stops immediately.

- i represents the number of floors;
- u
^{i}represents the displacement of floor i; - v
^{i}corresponds to the velocity of floor i; - p corresponds to the predicted values of the displacements and velocities;
- m represents the measured values of the displacements and velocities.

## 4. Results and Discussions

#### 4.1. SA-EnKF Framework

#### 4.1.1. Case 1: Two Available Sensors

#### 4.1.2. Case 2: Three Available Sensors

#### 4.2. GA-EnKF Framework

#### 4.2.1. Case 1: Two Available Sensors

#### 4.2.2. Case 2: Three Available Sensors

^{3}s, the time taken for the GA-EnKF framework to determine the optimal sensor placements was around 2.5 × 10

^{4}s on the same machine. Consequently, the proposed SA-EnKF framework outperforms the GA-EnKF methodology in terms of computational expediency.

#### 4.3. Brute-Force Optimal Results

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**Two-sensors case: (

**a**) Dots representing the best fitness value at every iteration and (

**b**) stopping criteria.

**Figure 3.**Two-sensors case: Estimates of the 8th floor displacement and velocity at initial sensor locations: (

**a**) using SA-EnKF and (

**b**) using GA-EnKF (results taken from [25]).

**Figure 4.**Two-sensors case: Estimates of the 8th floor displacement and velocity at final iteration: (

**a**) at SA-EnKF Optimal sensor locations (Floors 3 and 10) and (

**b**) at GA-EnKF Optimal Sensor Locations (Floors 1 and 10) (Results taken from [25]).

**Figure 5.**Three-sensors case: (

**a**) Dots representing the best fitness value at every iteration and (

**b**) stopping criteria.

**Figure 6.**Three-sensors case: Estimates of the 8th floor displacement and velocity at initial sensor locations: (

**a**) using SA-EnKF and (

**b**) using GA-EnKF (results taken from [25]).

**Figure 7.**Three-sensors case: Estimates of the 8th floor displacement and velocity at final iteration: (

**a**) at SA-EnKF optimal sensor locations (Floors 3, 6 and 10) and (

**b**) at GA-EnKF optimal sensor locations (Floors 1, 7 and 10) (results taken from [25]).

**Figure 8.**GA-EnKF and SA-EnKF penalty values corresponding to the optimal sensor locations versus computational time.

Components | Multiplying | Related to |
---|---|---|

{a_{i}} | The inter-story drift of floor i | the stiffness of floor i (directly) |

{b_{i}} | The cube of the inter-story drift of floor i | the stiffness of floor i (indirectly) |

{c_{i}} | The inter-story velocity of floor i | the damping of floor i (directly) |

{d_{i}} | The product of the inter-story drift and the velocity of floor i | the damping and the stiffness of the floor i (indirectly) |

Number of Sensors | Total Number of Combinations | Brute-Force Approach Optimal Results {Floors} | SA-EnKF Results {Floors} | GA-EnKF Results {Floors} |
---|---|---|---|---|

2 | 45 | {1 10} | {3 10} | {1 10} |

3 | 120 | {1 7 10} | {3 6 10} | {1 7 10} |

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**MDPI and ACS Style**

Nasr, D.; Dahr, R.E.; Assaad, J.; Khatib, J.
Comparative Analysis between Genetic Algorithm and Simulated Annealing-Based Frameworks for Optimal Sensor Placement and Structural Health Monitoring Purposes. *Buildings* **2022**, *12*, 1383.
https://doi.org/10.3390/buildings12091383

**AMA Style**

Nasr D, Dahr RE, Assaad J, Khatib J.
Comparative Analysis between Genetic Algorithm and Simulated Annealing-Based Frameworks for Optimal Sensor Placement and Structural Health Monitoring Purposes. *Buildings*. 2022; 12(9):1383.
https://doi.org/10.3390/buildings12091383

**Chicago/Turabian Style**

Nasr, Dana, Reina El Dahr, Joseph Assaad, and Jamal Khatib.
2022. "Comparative Analysis between Genetic Algorithm and Simulated Annealing-Based Frameworks for Optimal Sensor Placement and Structural Health Monitoring Purposes" *Buildings* 12, no. 9: 1383.
https://doi.org/10.3390/buildings12091383