Inverse Analysis of Structural Damage Based on the Modal Kinetic and Strain Energies with the Novel Oppositional Unified Particle Swarm Gradient-Based Optimizer
Abstract
:1. Introduction and Literature Review
2. Theory of the Inverse Analysis of Structural Damage
2.1. The Model-Based Inverse Method for Structural Damage Identification
2.2. Proof of the Principle of Damage Identification (Sensitivity Analysis)
2.3. Formulation of the Objective Function
3. The Theory of the Proposed OL-UPSGBO Algorithm
3.1. Oppositional-Based Learning (OL)
3.2. The UPSO
3.3. The GBO Algorithm
3.3.1. The Initialization Stage
3.3.2. The GSR Stage
3.3.3. The Local Escaping Stage
3.4. The Proposed OL-UPSGBO Algorithm
3.5. Benchmarking of the OL-UPSGBO Algorithm
4. Inverse Analysis of Structural Damage Using the Developed Approach
- A.
- Initialization stage
- 1.
- Develop the FE model of the structure using a commercial software or a self-coded model.
- 2.
- Set a population of search agents, which are the damage indicators related to the overall structural or substructural elements, where each search agent represents one configuration of the FE model of the structure.
- 3.
- Extract the modal features related to the intact and damaged structure, and calculate the MSEn-, MKEn-, and mode shape-based sub-objectives using Equations (37)–(39). Thereafter, calculate the objective function using Equation (40) for each corresponding FE model configuration related to each candidate search agent.
- 4.
- Evaluate all the search agents using the developed objective function (as in Equation (40)).
- 5.
- Initialize all the stochastic parameters of the OL-UPSOGBO algorithm, as in Table 1.
- 6.
- Define the initial best global and local solutions, worst global solution, and the best performance of each search agent.
- 7.
- Apply the OL paradigm as in Equation (42).
- B.
- Iterative stage
- 1.
- Start the UPSO framework by calculating the global and local velocities as in Equation (45).
- 2.
- Update the population using Equation (46).
- 3.
- Update the best global and local solutions, worst global solution, and the best performance of each search agent.
- 4.
- Start the GBO stage by employing Equation (61).
- 5.
- Apply the local escaping operator, as in Equation (63).
- 6.
- Apply the OL paradigm, as in Equation (42).
- 7.
- Update the best global and local solutions, worst global solution, and the best performance of each search agent.
- 8.
- Break if termination criteria are satisfied.
- C.
- Damage identification stage
- 1.
- After termination of the iterative process, the best performed search agent is selected and registered.
- 2.
- The best solution contains the damage parameters corresponding to each element .
- 3.
- Elicit the damage locations and calculate the damage severities using Equation (41), and analyze the results.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Function Type | No. | Function | Optimum |
---|---|---|---|
Unimodal | Shifted and Rotated Bent Cigar Function | 100 | |
NA | NA | ||
Shifted and Rotated Zakharov Function | 300 | ||
Multimodal | Shifted and Rotated Rosenbrock’s Function | 400 | |
Shifted and Rotated Rastrigin’s Function | 500 | ||
Shifted and Rotated Expanded Scaffer’s Function | 600 | ||
Shifted and Rotated Lunacek Bi_Rastrigin Function | 700 | ||
Shifted and Rotated Non-Continuous Rastrigin’s Function | 800 | ||
Shifted and Rotated Levy Function | 900 | ||
Shifted and Rotated Schwefel’s Function | 1000 | ||
Hybrid | Hybrid Function 1 (N = 3) | 1100 | |
Hybrid Function 2 (N = 3) | 1200 | ||
Hybrid Function 3 (N = 3) | 1300 | ||
Hybrid Function 4 (N = 4) | 1400 | ||
Hybrid Function 5 (N = 4) | 1500 | ||
Hybrid Function 6 (N = 4) | 1600 | ||
Hybrid Function 6 (N = 5) | 1700 | ||
Hybrid Function 6 (N = 5) | 1800 | ||
Hybrid Function 6 (N = 5) | 1900 | ||
Hybrid Function 6 (N = 6) | 2000 | ||
Composite | Composition Function 1 (N = 3) | 2100 | |
Composition Function 2 (N = 3) | 2200 | ||
Composition Function 3 (N = 4) | 2300 | ||
Composition Function 4 (N = 4) | 2400 | ||
Composition Function 5 (N = 5) | 2500 | ||
Composition Function 6 (N = 5) | 2600 | ||
Composition Function 7 (N = 6) | 2700 | ||
Composition Function 8 (N = 6) | 2800 | ||
Composition Function 9 (N = 3) | 2900 | ||
Composition Function 10 (N = 3) | 3000 |
Algorithm | Parameters |
---|---|
GWO | Default stochastic parameter descending from 2 to 0. |
WOA | Default stochastic parameter one descending from 2 to 0. Default stochastic parameter two descending from 2 to 0. |
SCA | Default stochastic parameter descending from 2 to 0. |
MVO | Traveling distance rate (default) Wormhole existence probability (default) |
AOA | Default stochastic parameters: MOP_Max = 1; MOP_Min = 0.2; Alpha = 5; Mu = 0.499; |
EO | Default stochastic parameters: a1 = 5; a2 = 1; GP = 0.5; |
AO | Default stochastic parameters: alpha = 0.1; delta = 0.1; |
OL-UPSGBO | (default) (default) (default) |
Function | Property | GWO | WOA | MVO | SCA | GBO | AOA | EO | AO | OLPSGBO |
---|---|---|---|---|---|---|---|---|---|---|
Ave | 1.1 × 109 | 2.6 × 106 | 3.7 × 108 | 1.6 × 1010 | 1.9 × 103 | 2.0 × 108 | 5.6 × 103 | 3.0 × 102 | 1.5 × 102 | |
Min | 5.5 × 108 | 7.2 × 105 | 2.8 × 108 | 1.0 × 1010 | 3.2 × 102 | 9.4 × 103 | 1.2 × 102 | 1.1 × 102 | 1.0 × 102 | |
Max | 2.7 × 109 | 5.3 × 106 | 4.2 × 108 | 2.1 × 1010 | 5.1 × 103 | 8.2 × 108 | 1.7 × 104 | 9.2 × 102 | 2.9 × 102 | |
STD | 6.5 × 108 | 1.2 × 106 | 4.5 × 107 | 2.8 × 109 | 1.9 × 103 | 3.0 × 108 | 5.3 × 103 | 2.5 × 102 | 7.4 × 101 | |
Rank | 8 | 5 | 7 | 9 | 3 | 6 | 4 | 2 | 1 | |
NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | |
Ave | 2.9 × 104 | 2.0 × 105 | 1.8 × 103 | 4.5 × 104 | 3.0 × 102 | 7.8 × 104 | 7.3 × 102 | 3.0 × 104 | 3.0 × 102 | |
Min | 8.3 × 103 | 9.9 × 104 | 1.6 × 103 | 4.0 × 104 | 3.0 × 102 | 6.0 × 104 | 4.0 × 102 | 2.1 × 104 | 3.0 × 102 | |
Max | 3.9 × 104 | 3.5 × 105 | 2.0 × 103 | 5.6 × 104 | 3.0 × 102 | 8.8 × 104 | 1.4 × 103 | 3.5 × 104 | 3.0 × 102 | |
STD | 9.0 × 103 | 7.6 × 104 | 1.4 × 102 | 5.3 × 104 | 2.3 × 10−2 | 9.5 × 103 | 3.3 × 102 | 4.9 × 103 | 1.3 × 10−3 | |
Rank | 8 | 5 | 7 | 9 | 3 | 6 | 4 | 2 | 1 | |
Ave | 5.8 × 102 | 5.6 × 102 | 5.2 × 102 | 1.6 × 103 | 4.5 × 102 | 1.0 × 104 | 5.0 × 102 | 5.5 × 102 | 4.0 × 102 | |
Min | 5.4 × 102 | 4.8 × 102 | 5.0 × 102 | 1.2 × 103 | 4.0 × 102 | 5.6 × 103 | 4.7 × 102 | 5.2 × 102 | 4.0 × 102 | |
Max | 6.3 × 102 | 6.5 × 102 | 5.3 × 102 | 2.3 × 103 | 4.8 × 102 | 1.3 × 104 | 5.2 × 102 | 6.1 × 102 | 4.0 × 102 | |
STD | 3.4 × 101 | 4.6 × 101 | 1.2 × 101 | 3.5 × 102 | 3.1 × 101 | 2.6 × 103 | 1.6 × 101 | 2.9 × 101 | 2.1 × 102 | |
Rank | 8 | 5 | 7 | 9 | 3 | 6 | 4 | 2 | 1 | |
Ave | 5.7 × 102 | 7.9 × 102 | 7.0 × 102 | 8.0 × 102 | 6.7 × 102 | 8.2 × 102 | 5.6 × 102 | 6.7 × 102 | 6.3 × 102 | |
Min | 5.6 × 102 | 7.1 × 102 | 6.7 × 102 | 7.7 × 102 | 6.2 × 102 | 7.5 × 102 | 5.3 × 102 | 6.4 × 102 | 5.8 × 102 | |
Max | 6.0 × 102 | 9.5 × 102 | 7.3 × 102 | 8.2 × 102 | 7.1 × 102 | 8.6 × 102 | 6.0 × 102 | 7.0 × 102 | 6.6 × 102 | |
STD | 1.4 × 101 | 7.1 × 101 | 2.1 × 101 | 1.7 × 101 | 2.4 × 101 | 3.4 × 101 | 2.0 × 101 | 2.0 × 101 | 2.9 × 101 | |
Rank | 2 | 7 | 6 | 8 | 5 | 9 | 1 | 4 | 3 | |
Ave | 6.1 × 102 | 6.7 × 102 | 6.2 × 102 | 6.6 × 102 | 6.2 × 102 | 6.7 × 102 | 6.0 × 102 | 6.5 × 102 | 6.1 × 102 | |
Min | 6.1 × 102 | 6.6 × 102 | 6.1 × 102 | 6.5 × 102 | 6.1 × 102 | 6.5 × 102 | 6.0 × 102 | 6.4 × 102 | 6.0 × 102 | |
Max | 6.2 × 102 | 6.8 × 102 | 6.3 × 102 | 6.7 × 102 | 6.2 × 102 | 6.7 × 102 | 6.0 × 102 | 6.5 × 102 | 6.2 × 102 | |
STD | 2.0 × 101 | 7.1 × 101 | 7.3 × 101 | 6.0 × 101 | 4.9 × 101 | 6.4 × 101 | 8.0 × 10−2 | 5.4 × 101 | 6.0 × 101 | |
Rank | 3 | 9 | 4 | 7 | 5 | 8 | 1 | 6 | 2 | |
Ave | 9.3 × 102 | 1.3 × 103 | 9.8 × 102 | 1.1 × 103 | 9.3 × 102 | 1.3 × 103 | 8.0 × 102 | 1.0 × 103 | 8.8 × 102 | |
Min | 9.1 × 102 | 1.1 × 103 | 9.4 × 102 | 1.1 × 103 | 8.9 × 102 | 1.2 × 103 | 7.6 × 102 | 9.4 × 102 | 8.2 × 102 | |
Max | 9.6 × 102 | 1.5 × 103 | 1.0 × 102 | 1.2 × 103 | 9.8 × 102 | 1.4 × 103 | 8.3 × 102 | 1.1 × 103 | 9.2 × 102 | |
STD | 1.6 × 101 | 1.1 × 102 | 2.4 × 101 | 2.4 × 101 | 3.3 × 101 | 4.8 × 101 | 2.0 × 101 | 5.1 × 101 | 3.0 × 101 | |
Rank | 4 | 8 | 5 | 7 | 3 | 9 | 1 | 6 | 2 | |
Ave | 9.4 × 102 | 1.0 × 103 | 1.0 × 103 | 1.1 × 103 | 9.3 × 102 | 1.1 × 103 | 8.7 × 102 | 9.3 × 102 | 9.1 × 102 | |
Min | 9.1 × 102 | 9.6 × 102 | 9.7 × 102 | 1.0 × 103 | 8.9 × 102 | 1.0 × 103 | 8.4 × 102 | 8.9 × 102 | 8.8 × 102 | |
Max | 9.6 × 102 | 1.1 × 103 | 1.1 × 103 | 1.1 × 103 | 9.8 × 102 | 1.1 × 103 | 9.0 × 102 | 9.6 × 102 | 9.3 × 102 | |
STD | 1.4 × 101 | 6.3 × 101 | 4.0 × 101 | 1.6 × 101 | 3.3 × 101 | 3.4 × 101 | 1.9 × 101 | 2.3 × 101 | 1.8 × 101 | |
Rank | 5 | 6 | 7 | 9 | 4 | 8 | 1 | 3 | 2 | |
Ave | 1.7 × 103 | 7.5 × 103 | 1.3 × 103 | 5.6 × 103 | 2.0 × 103 | 5.8 × 103 | 9.0 × 102 | 5.2 × 103 | 1.6 × 103 | |
Min | 1.2 × 103 | 5.1 × 103 | 1.0 × 103 | 4.4 × 103 | 1.4 × 103 | 4.5 × 103 | 9.0 × 102 | 3.5 × 103 | 1.2 × 103 | |
Max | 2.4 × 103 | 1.0 × 104 | 2.5 × 103 | 6.5 × 103 | 3.3 × 103 | 7.2 × 103 | 9.1 × 102 | 7.1 × 103 | 2.1 × 103 | |
STD | 3.5 × 102 | 1.8 × 103 | 4.7 × 102 | 6.8 × 102 | 6.2 × 102 | 8.0 × 102 | 4.5 × 101 | 1.1 × 103 | 3.1 × 102 | |
Rank | 5 | 6 | 7 | 9 | 4 | 8 | 1 | 3 | 2 | |
Ave | 5.9 × 103 | 6.4 × 103 | 7.2 × 103 | 8.2 × 103 | 5.1 × 103 | 6.8 × 103 | 4.6 × 103 | 4.9 × 103 | 4.5 × 103 | |
Min | 4.7 × 103 | 5.3 × 103 | 6.6 × 103 | 7.4 × 103 | 4.5 × 103 | 6.0 × 103 | 3.3 × 103 | 4.0 × 103 | 3.8 × 103 | |
Max | 6.5 × 103 | 8.4 × 103 | 8.1 × 103 | 8.7 × 103 | 7.0 × 103 | 7.7 × 103 | 5.8 × 103 | 5.7 × 103 | 5.2 × 103 | |
STD | 5.4 × 102 | 1.0 × 103 | 4.8 × 102 | 4.0 × 102 | 7.5 × 102 | 5.7 × 102 | 8.6 × 102 | 5.4 × 102 | 4.1 × 102 | |
Rank | 5 | 6 | 7 | 9 | 4 | 8 | 1 | 3 | 2 | |
Ave | 1.5 × 103 | 1.5 × 103 | 1.4 × 103 | 2.4 × 103 | 1.2 × 103 | 4.4 × 103 | 1.4 × 103 | 1.4 × 103 | 1.2 × 103 | |
Min | 1.3 × 103 | 1.3 × 103 | 1.3 × 103 | 2.0 × 103 | 1.2 × 103 | 1.8 × 103 | 1.3 × 103 | 1.3 × 103 | 1.2 × 103 | |
Max | 2.1 × 103 | 1.7 × 103 | 1.5 × 103 | 2.9 × 103 | 1.3 × 103 | 8.6 × 103 | 1.4 × 103 | 1.5 × 103 | 1.2 × 103 | |
STD | 2.7 × 102 | 1.5 × 102 | 4.9 × 101 | 3.3 × 102 | 2.7 × 101 | 2.3 × 103 | 6.5 × 101 | 6.3 × 101 | 1.1 × 101 | |
Rank | 7 | 6 | 5 | 8 | 2 | 9 | 3 | 4 | 1 | |
Ave | 9.6 × 107 | 4.1 × 107 | 4.7 × 107 | 1.5 × 109 | 2.9 × 104 | 9.2 × 109 | 7.5 × 105 | 1.5 × 107 | 2.9 × 104 | |
Min | 5.5 × 107 | 9.3 × 106 | 3.4 × 107 | 7.6 × 108 | 1.3 × 104 | 4.0 × 109 | 3.0 × 105 | 3.0 × 106 | 1.3 × 104 | |
Max | 1.8 × 108 | 1.4 × 108 | 7.0 × 107 | 2.2 × 109 | 4.2 × 104 | 1.2 × 1010 | 2.0 × 106 | 2.6 × 107 | 4.3 × 104 | |
STD | 4.1 × 107 | 3.8 × 107 | 1.2 × 107 | 5.0 × 108 | 1.0 × 104 | 2.6 × 109 | 5.1 × 105 | 6.9 × 106 | 1.1 × 104 | |
Rank | 7 | 6 | 5 | 8 | 2 | 9 | 3 | 4 | 1 | |
Ave | 2.5 × 107 | 1.1 × 105 | 1.7 × 107 | 6.4 × 108 | 1.0 × 104 | 9.3 × 107 | 4.3 × 104 | 3.2 × 105 | 7.6 × 103 | |
Min | 1.2 × 107 | 4.3 × 104 | 8.0 × 106 | 3.8 × 108 | 3.0 × 103 | 2.4 × 104 | 9.4 × 103 | 7.6 × 104 | 2.5 × 103 | |
Max | 5.2 × 107 | 2.5 × 105 | 2.1 × 107 | 9.5 × 108 | 1.8 × 104 | 6.1 × 108 | 1.0 × 105 | 5.4 × 105 | 1.5 × 104 | |
STD | 1.2 × 107 | 6.4 × 104 | 5.0 × 106 | 2.0 × 108 | 6.3 × 103 | 1.9 × 108 | 2.8 × 104 | 1.3 × 105 | 4.5 × 103 | |
Rank | 7 | 4 | 6 | 9 | 2 | 8 | 3 | 5 | 1 | |
Ave | 8.0 × 104 | 4.7 × 105 | 1.9 × 104 | 2.1 × 105 | 1.8 × 103 | 7.1 × 104 | 6.2 × 104 | 2.9 × 105 | 1.7 × 103 | |
Min | 1.6 × 104 | 1.6 × 104 | 4.2 × 103 | 4.6 × 104 | 1.6 × 103 | 1.7 × 104 | 2.4 × 104 | 3.7 × 104 | 1.6 × 103 | |
Max | 2.5 × 105 | 2.0 × 106 | 3.2 × 104 | 5.6 × 105 | 3.2 × 103 | 1.4 × 105 | 1.0 × 105 | 8.3 × 105 | 1.7 × 103 | |
STD | 7.0 × 104 | 6.1 × 105 | 7.9 × 103 | 1.9 × 105 | 5.0 × 102 | 3.9 × 104 | 2.6 × 104 | 2.3 × 105 | 4.9 × 101 | |
Rank | 6 | 9 | 3 | 7 | 2 | 5 | 4 | 8 | 1 | |
Ave | 4.9 × 105 | 9.5 × 104 | 1.8 × 106 | 2.0 × 107 | 1.2 × 104 | 2.1 × 104 | 2.3 × 104 | 7.6 × 104 | 2.5 × 103 | |
Min | 1.4 × 105 | 1.7 × 104 | 9.3 × 105 | 1.9 × 106 | 1.9 × 103 | 1.6 × 104 | 7.5 × 103 | 3.1 × 104 | 1.6 × 103 | |
Max | 1.1 × 106 | 1.9 × 105 | 2.6 × 106 | 4.4 × 107 | 4.0 × 104 | 4.4 × 104 | 5.9 × 104 | 1.5 × 105 | 5.3 × 103 | |
STD | 2.7 × 105 | 5.0 × 104 | 6.3 × 105 | 1.2 × 107 | 1.3 × 104 | 8.9 × 103 | 1.7 × 104 | 3.8 × 104 | 1.3 × 103 | |
Rank | 7 | 6 | 8 | 9 | 2 | 3 | 4 | 5 | 1 | |
Ave | 2.6 × 103 | 3.3 × 103 | 2.9 × 103 | 3.8 × 103 | 2.7 × 103 | 3.9 × 103 | 2.6 × 103 | 3.1 × 103 | 2.4 × 103 | |
Min | 2.3 × 103 | 2.5 × 103 | 2.4 × 103 | 3.5 × 103 | 2.4 × 103 | 3.0 × 103 | 2.0 × 103 | 2.6 × 103 | 2.1 × 103 | |
Max | 2.8 × 103 | 3.8 × 103 | 3.3 × 103 | 4.0 × 103 | 3.1 × 103 | 4.8 × 103 | 3.2 × 103 | 4.0 × 103 | 2.6 × 103 | |
STD | 1.9 × 102 | 4.3 × 102 | 2.6 × 102 | 1.6 × 102 | 2.5 × 102 | 5.2 × 102 | 3.7 × 102 | 4.1 × 102 | 1.9 × 102 | |
Rank | 3 | 7 | 5 | 8 | 4 | 9 | 2 | 6 | 1 | |
Ave | 2.0 × 103 | 2.5 × 103 | 2.1 × 103 | 2.5 × 103 | 2.3 × 103 | 2.7 × 103 | 2.2 × 103 | 2.3 × 103 | 2.1 × 103 | |
Min | 1.9 × 103 | 2.1 × 103 | 2.0 × 103 | 2.3 × 103 | 2.0 × 103 | 2.5 × 103 | 2.0 × 103 | 1.8 × 103 | 1.9 × 103 | |
Max | 2.1 × 103 | 2.9 × 103 | 2.3 × 103 | 2.7 × 103 | 2.7 × 103 | 3.2 × 103 | 2.3 × 103 | 2.6 × 103 | 2.3 × 103 | |
STD | 7.7 × 101 | 2.4 × 102 | 1.0 × 102 | 1.7 × 102 | 2.5 × 102 | 2.6 × 102 | 9.6 × 101 | 2.4 × 102 | 1.6 × 102 | |
Rank | 1 | 8 | 3 | 6 | 5 | 9 | 4 | 7 | 2 | |
Ave | 1.3 × 106 | 2.8 × 106 | 3.3 × 105 | 5.1 × 106 | 3.3 × 104 | 1.1 × 106 | 4.0 × 105 | 1.6 × 106 | 1.7 × 104 | |
Min | 9.8 × 104 | 2.7 × 105 | 1.8 × 105 | 2.8 × 106 | 1.1 × 104 | 1.6 × 105 | 1.5 × 105 | 2.2 × 105 | 2.7 × 103 | |
Max | 7.2 × 106 | 9.7 × 106 | 4.6 × 105 | 8.3 × 106 | 9.4 × 104 | 2.6 × 106 | 7.7 × 105 | 5.6 × 106 | 3.3 × 104 | |
STD | 2.1 × 106 | 3.1 × 106 | 1.1 × 105 | 1.7 × 106 | 2.9 × 104 | 6.7 × 105 | 1.9 × 105 | 2.0 × 106 | 1.1 × 104 | |
Rank | 5 | 8 | 3 | 9 | 2 | 6 | 4 | 7 | 1 | |
Ave | 1.5 × 106 | 2.8 × 106 | 2.4 × 106 | 3.8 × 107 | 6.4 × 103 | 2.7 × 103 | 3.2 × 104 | 8.0 × 105 | 2.3 × 103 | |
Min | 3.0 × 105 | 2.9 × 105 | 1.2 × 106 | 1.1 × 107 | 2.0 × 103 | 2.5 × 103 | 3.2 × 103 | 8.7 × 104 | 2.0 × 103 | |
Max | 4.5 × 106 | 7.4 × 106 | 6.2 × 106 | 8.6 × 107 | 2.3 × 104 | 2.9 × 103 | 6.6 × 104 | 1.8 × 106 | 3.0 × 103 | |
STD | 1.2 × 106 | 2.2 × 106 | 1.4 × 106 | 2.4 × 107 | 6.1 × 103 | 1.3 × 102 | 2.5 × 104 | 6.1 × 105 | 2.6 × 102 | |
Rank | 6 | 8 | 7 | 9 | 3 | 2 | 4 | 5 | 1 | |
Ave | 2.4 × 103 | 2.8 × 103 | 2.5 × 103 | 2.7 × 103 | 2.4 × 103 | 2.7 × 103 | 2.4 × 103 | 2.5 × 103 | 2.3 × 103 | |
Min | 2.2 × 103 | 2.4 × 103 | 2.2 × 103 | 2.5 × 103 | 2.2 × 103 | 2.5 × 103 | 2.3 × 103 | 2.2 × 103 | 2.2 × 103 | |
Max | 2.7 × 103 | 3.0 × 103 | 2.9 × 103 | 2.9 × 103 | 2.5 × 103 | 2.9 × 103 | 2.8 × 103 | 2.8 × 103 | 2.5 × 103 | |
STD | 1.3 × 102 | 1.9 × 102 | 1.9 × 102 | 1.3 × 102 | 1.2 × 102 | 1.1 × 102 | 2.0 × 102 | 1.8 × 102 | 1.2 × 102 | |
Rank | 3 | 9 | 6 | 8 | 2 | 7 | 4 | 5 | 1 | |
Ave | 2.4 × 103 | 2.5 × 103 | 2.5 × 103 | 2.6 × 103 | 2.4 × 103 | 2.6 × 103 | 2.4 × 103 | 2.5 × 103 | 2.4 × 103 | |
Min | 2.4 × 103 | 2.5 × 103 | 2.4 × 103 | 2.5 × 103 | 2.4 × 103 | 2.6 × 103 | 2.4 × 103 | 2.4 × 103 | 2.4 × 103 | |
Max | 2.5 × 103 | 2.7 × 103 | 2.5 × 103 | 2.6 × 103 | 2.5 × 103 | 2.6 × 103 | 2.5 × 103 | 2.5 × 103 | 2.5 × 103 | |
STD | 2.7 × 101 | 6.3 × 101 | 2.7 × 101 | 1.6 × 101 | 3.0 × 101 | 2.7 × 101 | 4.0 × 101 | 3.9 × 101 | 4.1 × 101 | |
Rank | 4 | 7 | 6 | 8 | 3 | 9 | 1 | 5 | 2 | |
Ave | 4.3 × 103 | 6.0 × 103 | 6.8 × 103 | 9.1 × 103 | 3.2 × 103 | 8.5 × 103 | 6.1 × 103 | 2.8 × 103 | 2.3 × 103 | |
Min | 2.5 × 103 | 2.3 × 103 | 2.4 × 103 | 4.0 × 103 | 2.3 × 103 | 6.7 × 103 | 2.3 × 103 | 2.3 × 103 | 2.3 × 103 | |
Max | 7.5 × 103 | 8.5 × 103 | 9.7 × 103 | 1.0 × 104 | 8.0 × 103 | 9.3 × 103 | 7.9 × 103 | 6.6 × 103 | 2.3 × 103 | |
STD | 2.2 × 103 | 2.5 × 103 | 3.1 × 103 | 1.8 × 103 | 2.0 × 103 | 7.3 × 102 | 2.1 × 103 | 1.4 × 103 | 3.0 × 101 | |
Rank | 4 | 5 | 7 | 9 | 3 | 8 | 6 | 2 | 1 | |
Ave | 2.8 × 103 | 3.1 × 103 | 2.9 × 103 | 3.0 × 103 | 2.8 × 103 | 3.4 × 103 | 2.9 × 103 | 2.8 × 103 | 2.8 × 103 | |
Min | 2.8 × 103 | 2.9 × 103 | 2.8 × 103 | 2.9 × 103 | 2.7 × 103 | 3.3 × 103 | 2.7 × 103 | 2.8 × 103 | 2.8 × 103 | |
Max | 2.9 × 103 | 3.3 × 103 | 3.0 × 103 | 3.0 × 103 | 2.8 × 103 | 3.6 × 103 | 4.0 × 103 | 2.9 × 103 | 2.8 × 103 | |
STD | 1.7 × 101 | 1.4 × 103 | 4.9 × 101 | 3.4 × 101 | 2.8 × 101 | 8.4 × 101 | 3.9 × 103 | 4.0 × 101 | 1.3 × 101 | |
Rank | 3 | 8 | 5 | 7 | 2 | 9 | 6 | 4 | 1 | |
Ave | 3.0 × 103 | 3.2 × 103 | 3.0 × 103 | 3.2 × 103 | 3.0 × 103 | 3.7 × 103 | 3.0 × 103 | 3.1 × 103 | 2.9 × 103 | |
Min | 3.0 × 103 | 3.1 × 103 | 3.0 × 103 | 3.1 × 103 | 2.9 × 103 | 3.6 × 103 | 2.9 × 103 | 2.9 × 103 | 2.9 × 103 | |
Max | 3.0 × 103 | 3.5 × 103 | 3.0 × 103 | 3.2 × 103 | 3.0 × 103 | 4.0 × 103 | 3.9 × 103 | 3.3 × 103 | 3.0 × 103 | |
STD | 8.4 × 101 | 1.5 × 102 | 2.4 × 101 | 3.4 × 101 | 2.7 × 101 | 1.4 × 102 | 3.2 × 102 | 9.9 × 101 | 3.2 × 101 | |
Rank | 4 | 7 | 3 | 8 | 2 | 9 | 5 | 6 | 1 | |
Ave | 3.0 × 103 | 2.9 × 103 | 2.9 × 103 | 3.2 × 103 | 2.9 × 103 | 4.5 × 103 | 2.9 × 103 | 2.9 × 103 | 2.9 × 103 | |
Min | 2.9 × 103 | 2.9 × 103 | 2.9 × 103 | 3.1 × 103 | 2.9 × 103 | 4.1 × 103 | 2.9 × 103 | 2.9 × 103 | 2.9 × 103 | |
Max | 3.0 × 103 | 2.9 × 103 | 2.9 × 103 | 3.3 × 103 | 2.9 × 103 | 5.4 × 103 | 2.9 × 103 | 2.9 × 103 | 2.9 × 103 | |
STD | 3.0 × 101 | 1.5 × 101 | 1.0 × 101 | 6.0 × 101 | 9.8 × 101 | 3.8 × 102 | 1.5 × 101 | 1.8 × 101 | 9.0 × 101 | |
Rank | 7 | 6 | 5 | 8 | 3 | 9 | 2 | 4 | 1 | |
Ave | 5.4 × 103 | 7.8 × 103 | 5.2 × 103 | 6.8 × 103 | 4.8 × 103 | 9.5 × 103 | 4.9 × 103 | 3.7 × 103 | 4.6 × 103 | |
Min | 5.0 × 103 | 6.2 × 103 | 3.0 × 103 | 5.0 × 103 | 2.8 × 103 | 8.0 × 103 | 4.0 × 103 | 2.9 × 103 | 2.8 × 103 | |
Max | 5.7 × 103 | 8.8 × 103 | 6.0 × 103 | 7.7 × 103 | 5.7 × 103 | 1.1 × 1004 | 6.2 × 103 | 5.3 × 103 | 5.8 × 103 | |
STD | 2.0 × 102 | 9.2 × 102 | 1.1 × 103 | 7.4 × 102 | 1.0 × 103 | 7.7 × 102 | 7.3 × 102 | 7.5 × 102 | 1.3 × 103 | |
Rank | 6 | 8 | 5 | 7 | 3 | 9 | 4 | 1 | 2 | |
Ave | 3.3 × 103 | 3.3 × 103 | 3.2 × 103 | 3.4 × 103 | 3.2 × 103 | 4.2 × 103 | 3.2 × 103 | 3.3 × 103 | 3.2 × 103 | |
Min | 3.2 × 103 | 3.3 × 103 | 3.2 × 103 | 3.3 × 103 | 3.2 × 103 | 3.9 × 103 | 3.2 × 103 | 3.2 × 103 | 3.2 × 103 | |
Max | 3.3 × 103 | 3.5 × 103 | 3.2 × 103 | 3.4 × 103 | 3.3 × 103 | 4.6 × 103 | 3.3 × 103 | 3.3 × 103 | 3.3 × 103 | |
STD | 1.7 × 101 | 7.6 × 101 | 1.0 × 101 | 2.6 × 101 | 2.2 × 101 | 2.2 × 102 | 2.2 × 101 | 2.6 × 101 | 1.6 × 101 | |
Rank | 5 | 7 | 1 | 8 | 4 | 9 | 3 | 6 | 2 | |
Ave | 3.4 × 103 | 3.3 × 103 | 3.3 × 103 | 4.0 × 103 | 3.2 × 103 | 6.3 × 103 | 5.4 × 103 | 3.3 × 103 | 3.1 × 103 | |
Min | 3.3 × 103 | 3.3 × 103 | 3.2 × 103 | 3.8 × 103 | 3.1 × 103 | 5.5 × 103 | 3.2 × 103 | 3.3 × 103 | 3.1 × 103 | |
Max | 3.4 × 103 | 3.3 × 103 | 3.3 × 103 | 4.3 × 103 | 3.2 × 103 | 7.5 × 103 | 6.4 × 103 | 3.4 × 103 | 3.1 × 103 | |
STD | 3.3 × 101 | 1.8 × 101 | 2.1 × 101 | 1.6 × 102 | 6.1 × 101 | 6.7 × 102 | 1.5 × 103 | 3.8 × 101 | 2.6 × 10−9 | |
Rank | 6 | 3 | 5 | 7 | 2 | 9 | 8 | 4 | 1 | |
Ave | 3.9 × 103 | 5.0 × 103 | 3.9 × 103 | 4.7 × 103 | 3.9 × 103 | 6.3 × 103 | 4.0 × 103 | 4.3 × 103 | 3.8 × 103 | |
Min | 3.6 × 103 | 4.2 × 103 | 3.7 × 103 | 4.3 × 103 | 3.6 × 103 | 5.5 × 103 | 3.8 × 103 | 3.8 × 103 | 3.5 × 103 | |
Max | 4.2 × 103 | 5.9 × 103 | 4.2 × 103 | 5.1 × 103 | 4.3 × 103 | 7.5 × 103 | 4.2 × 103 | 4.8 × 103 | 3.9 × 103 | |
STD | 1.6 × 102 | 5.2 × 102 | 1.6 × 102 | 2.1 × 102 | 2.2 × 102 | 6.7 × 102 | 1.4 × 102 | 3.7 × 102 | 1.4 × 102 | |
Rank | 4 | 8 | 3 | 7 | 2 | 9 | 5 | 6 | 1 | |
Ave | 8.0 × 106 | 1.0 × 107 | 4.1 × 106 | 1.2 × 108 | 7.7 × 103 | 3.1 × 107 | 1.4 × 105 | 7.2 × 106 | 6.1 × 103 | |
Min | 3.9 × 106 | 1.6 × 106 | 2.2 × 106 | 6.1 × 106 | 5.4 × 103 | 5.3 × 106 | 5.6 × 103 | 3.4 × 106 | 5.5 × 103 | |
Max | 1.3 × 107 | 3.3 × 107 | 9.6 × 106 | 2.1 × 108 | 1.4 × 104 | 9.0 × 107 | 7.3 × 105 | 1.2 × 107 | 6.5 × 103 | |
STD | 3.1 × 106 | 8.9 × 106 | 2.1 × 106 | 5.2 × 107 | 3.4 × 103 | 2.4 × 107 | 2.2 × 105 | 2.9 × 106 | 3.5 × 102 | |
Rank | 6 | 7 | 4 | 9 | 2 | 8 | 3 | 5 | 1 | |
Friedman average rank | 4.9655 | 6.8966 | 4.931 | 7.8621 | 2.9655 | 7.7931 | 3.2414 | 4.8276 | 1.3793 | |
Friedman total rank | 144 | 200 | 143 | 228 | 86 | 226 | 94 | 140 | 40 |
Function | GWO | WOA | MVO | SCA | GBO | AOA | EO | AO |
---|---|---|---|---|---|---|---|---|
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.193359375 | |
NA | NA | NA | NA | NA | NA | NA | NA | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.0078125 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.03125 | 0.001953125 | 0.001953125 | 0.013671875 | |
0.275390625 | 0.001953125 | 0.064453125 | 0.001953125 | 0.01953125 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.0078125 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.01953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.193359375 | 0.001953125 | 0.00390625 | 0.232421875 | |
0.4921875 | 0.001953125 | 0.130859375 | 0.001953125 | 0.193359375 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.048828125 | 0.001953125 | 0.921875 | 0.193359375 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.625 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.845703125 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.556640625 | 0.001953125 | 0.00390625 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.625 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.064453125 | 0.001953125 | 0.00390625 | 0.001953125 | 0.01953125 | 0.001953125 | 0.130859375 | 0.001953125 | |
0.064453125 | 0.013671875 | 0.845703125 | 0.009765625 | 0.275390625 | 0.001953125 | 0.625 | 0.01953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.375 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.009765625 | 0.013671875 | 0.001953125 | 0.001953125 | |
1 | 0.00390625 | 0.01953125 | 0.001953125 | 0.921875 | 0.001953125 | 0.6953125 | 0.064453125 | |
0.064453125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.6953125 | 0.001953125 | 0.42578125 | 0.037109375 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.6875 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.4921875 | 0.001953125 | 0.6953125 | 0.009765625 | |
0.001953125 | 0.001953125 | 0.048828125 | 0.001953125 | 0.048828125 | 0.001953125 | 0.625 | 0.01953125 | |
0.001953125 | 0.00390625 | 0.009765625 | 0.001953125 | 0.275390625 | 0.001953125 | 0.921875 | 0.048828125 | |
0.02734375 | 0.001953125 | 0.083984375 | 0.00390625 | 0.845703125 | 0.001953125 | 0.921875 | 0.064453125 | |
0.048828125 | 0.001953125 | 0.375 | 0.001953125 | 0.4921875 | 0.001953125 | 0.556640625 | 0.00390625 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.232421875 | 0.001953125 | 0.001953125 | 0.001953125 | |
0.037109375 | 0.001953125 | 0.16015625 | 0.001953125 | 0.375 | 0.001953125 | 0.048828125 | 0.005859375 | |
0.001953125 | 0.001953125 | 0.001953125 | 0.001953125 | 0.322265625 | 0.001953125 | 0.00390625 | 0.001953125 |
Damage Scenario | Damage Locations | Damage Severity |
---|---|---|
Scenario 1 | Brace elements 83, and 114 | 25% |
Scenario 2 | Brace elements 26, and 55 | 25% |
Scenario 3 | Brace elements 52, and 109 | 25% |
Scenario 4 | Brace elements 24, 82, and 112 | 25% |
Scenario 5 | Brace elements 23, 51, and 111 | 25% |
Scenario 6 | Brace elements 22, 51, 80, and 109 | 25% |
Damage Scenario | Algorithm | Mean | Standard Deviation | Min | Max | Wilcoxon Sign Rank (p-Value) |
---|---|---|---|---|---|---|
Scenario 1 | UPSO | 5.7 × 10−4 | 5.7 × 10−4 | 6.5 × 10−6 | 1.2 × 10−3 | 3.9 × 10−3 |
GBO | 6.3 × 10−3 | 2.3 × 10−4 | 6.1 × 10−3 | 6.5 × 10−3 | 9.0 × 10−3 | |
OL-UPSGBO | 1.1 × 10−6 | 2.2 × 10−6 | 6.2 × 10−7 | 6.5 × 10−6 | ||
Scenario 2 | UPSO | 1.3 × 10−2 | 1.4 × 10−5 | 1.3 × 10−2 | 1.3 × 10−2 | 9.0 × 10−3 |
GBO | 1.6 × 10−2 | 4.8 × 10−3 | 1.1 × 10−2 | 2.1 × 10−2 | 9.0 × 10−3 | |
OL-UPSGBO | 9.3 × 10−8 | 1.1 × 10−7 | 4.9 × 10−7 | 2.8 × 10−7 | ||
Scenario 3 | UPSO | 5.3 × 10−3 | 1.1 × 10−4 | 5.2 × 10−3 | 5.4 × 10−3 | 9.0 × 10−3 |
GBO | 6.8 × 10−2 | 5.4 × 10−4 | 6.8 × 10−2 | 6.9 × 10−2 | 3.9 × 10−3 | |
OL-UPSGBO | 5.5 × 10−7 | 8.2 × 10−7 | 3.2 × 10−8 | 2.0 × 10−6 | ||
Scenario 4 | UPSO | 1.3 × 10−2 | 3.0 × 10−5 | 1.3 × 10−2 | 1.3 × 10−2 | 3.9 × 10−3 |
GBO | 5.3 × 10−2 | 1.7 × 10−2 | 4.1 × 10−2 | 7.6 × 10−2 | 3.9 × 10−3 | |
OL-UPSGBO | 3.9 × 10−6 | 4.6 × 10−6 | 8.8 × 10−8 | 1.2 × 10−5 | ||
Scenario 5 | UPSO | 3.5 × 10−2 | 3.7 × 10−4 | 3.5 × 10−2 | 3.5 × 10−2 | 3.9 × 10−3 |
GBO | 2.8 × 10−2 | 1.9 × 10−3 | 2.7 × 10−2 | 3.3 × 10−2 | 9.1 × 10−3 | |
OL-UPSGBO | 2.2 × 10−5 | 3.1 × 10−6 | 1.9 × 10−5 | 2.6 × 10−5 | ||
Scenario 6 | UPSO | 1.3 × 10−2 | 1.9 × 10−5 | 1.3 × 10−2 | 1.3 × 10−2 | 3.9 × 10−3 |
GBO | 2.3 × 10−2 | 7.3 × 10−4 | 2.3 × 10−2 | 2.4 × 10−2 | 3.9 × 10−3 | |
OL-UPSGBO | 1.6 × 10−6 | 1.9 × 10−6 | 1.9 × 10−8 | 5.2 × 10−6 | ||
Scenario 6 (3% noise) | UPSO | 1.3 × 10−2 | 2.5 × 10−5 | 1.3 × 10−2 | 1.3 × 10−2 | 3.9 × 10−3 |
GBO | 2.5 × 10−2 | 1.1 × 10−5 | 2.5 × 10−2 | 2.5 × 10−2 | 3.9 × 10−3 | |
OL-UPSGBO | 4.0 × 10−4 | 7.5 × 10−7 | 4.0 × 10−4 | 4.0 × 10−4 | ||
Scenario 6 (5% noise) | UPSO | 1.3 × 10−2 | 2.7 × 10−5 | 1.3 × 10−2 | 1.3 × 10−2 | 3.9 × 10−3 |
GBO | 2.5 × 10−2 | 5.3 × 10−4 | 2.5 × 102 | 2.6 × 10−4 | 3.9 × 10−3 | |
OL-UPSGBO | 5.8 × 10−4 | 4.0 × 10−6 | 5.8 × 10−4 | 5.9 × 10−4 |
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Alkayem, N.F.; Shen, L.; Al-hababi, T.; Qian, X.; Cao, M. Inverse Analysis of Structural Damage Based on the Modal Kinetic and Strain Energies with the Novel Oppositional Unified Particle Swarm Gradient-Based Optimizer. Appl. Sci. 2022, 12, 11689. https://doi.org/10.3390/app122211689
Alkayem NF, Shen L, Al-hababi T, Qian X, Cao M. Inverse Analysis of Structural Damage Based on the Modal Kinetic and Strain Energies with the Novel Oppositional Unified Particle Swarm Gradient-Based Optimizer. Applied Sciences. 2022; 12(22):11689. https://doi.org/10.3390/app122211689
Chicago/Turabian StyleAlkayem, Nizar Faisal, Lei Shen, Tareq Al-hababi, Xiangdong Qian, and Maosen Cao. 2022. "Inverse Analysis of Structural Damage Based on the Modal Kinetic and Strain Energies with the Novel Oppositional Unified Particle Swarm Gradient-Based Optimizer" Applied Sciences 12, no. 22: 11689. https://doi.org/10.3390/app122211689
APA StyleAlkayem, N. F., Shen, L., Al-hababi, T., Qian, X., & Cao, M. (2022). Inverse Analysis of Structural Damage Based on the Modal Kinetic and Strain Energies with the Novel Oppositional Unified Particle Swarm Gradient-Based Optimizer. Applied Sciences, 12(22), 11689. https://doi.org/10.3390/app122211689