Topology Optimisation Using MPBILs and Multi-Grid Ground Element
Abstract
:1. Introduction
2. Topological Designs with Single-and Multi-Ground Design Approaches
2.1. Topological Designs with Ground Element Filtering
2.2. Single-and Multi-Grid Ground Elements
Algorithm 1. Encoding and decoding scheme for a MG approach. |
Initialization: Generate four sets of ground elements and define the threshold value of ε for each set. Inputx sized (N41 + 1) × 1. Output: Thicknesses of ground elements. Encoding x1 ∈ [1, 4] is used for selecting a set of ground elements. x2 to are used for element thicknesses. Decoding 1: Find n = round(x1) where round(.) is a round-off operator. 2: If n = 1: x2 to are set as N11 element thicknesses and ε = ε1. 3: If n = 2: x2 to are set as N21 element thicknesses and ε = ε2. 4: If n = 3: x2 to are set as N31 element thicknesses and ε = ε3. 5: If n = 4: x2 to are set as N41 element thicknesses and ε = ε4. |
3. Performance Enhancements of Multi-Objective, Population-Based Incremental Learning
3.1. Multi-Objective, Population-Based Incremental Learning
3.2. Opposite-Based MPBIL
3.3. Multi-Learning Rate
Algorithm 2. MPBIL with multi-learning rate. |
Initialization Probability matrix P = [0.5]l×nb, Probability matrix Pi = [0.5]l/M×nb where i = 1, …, M = 3, external Pareto archive Pareto = {}. 1: Generate a binary population B from P. 2: Decode the binary population to be xn×Np and find the objective values fm×Np. 3: Update Pareto by replacing it with non-dominated solutions of union set Pareto ∪ x. 4: If the number of members in Pareto exceeds the predefined archive size NA, remove some of them by using an archiving technique. 5: If the termination criterion is fulfilled, stop the procedure. Otherwise, go to step 6: 6: Update P and create a binary population 6.1: Set a binary population B = {}. 6.2: For i = 1 to l/M. 6.2.1: Select n0 binary solutions from Pareto randomly. 6.2.2: Use LRk = 0.25, 0.5, 0.75, for each k = 1, …, M. (For this research M = 3) 6.2.3: Update the ith row of P by using (1). 6.2.4: Generate the ith row of probability matrix Pi using (2) and each LRk. 6.2.5: Generate rand ∈ [0, 1] a uniform random number. 6.2.6: If rand < the predefined mutation probability, update the ith row of P1, P2 and P3 using similar equation in [3]. 6.2.7: Generate binary subpopulations SB1, SB2 and SB3 from the ith row of P1, P2 and P3, respectively. 6.2.8: Set B = B∪SB1∪SB2∪SB3 6.3: Next i. 7: Go to step 2. |
3.4. Adaptive Learning Rate
Algorithm 3. MPBIL with adaptive learning rate. |
Initialization probability matrix P = [0.5]l×nb, external Pareto archive Pareto = {}. 1: Generate a binary population B from P. 2: Decode the binary population to be xn×Np and find the objective values fm×Np. 3: Update Pareto by replacing it with non-dominated solutions of union set Pareto∪x. 4: If the number of members in Pareto exceeds the predefined archive size NA, remove some of them by using an archiving technique. 5: If the termination criterion is fulfilled, stop the procedure. Otherwise, go to step 6: 6: Update P. 6.1: For i = 1 to l. 6.1.1: Select n0 binary solutions from Pareto randomly. 6.1.2: Generate LR using (3). 6.1.3: Update the ith row of P by using (1). 6.1.4: Generate rand ∈ [0, 1] a uniform random number. 6.1.5: If rand <the predefined mutation probability, update the ith row of P using similar equation in [3]. 6.2: Next i. 7: Go to step 1. |
3.5. The Performance Index and Non-Parametric Statistical Test
4. Numerical Experiment
0.2 ≤ r ≤ 0.8
ρi ∈ {0.0001, 1}
c2 ≤ 5c2,min
0.2 ≤ r ≤ 0.8
ρi ∈ {0.0001, 1}
5. Design Results
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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MOP1 | MOP2 | MOP3 | MOP4 | |||||
---|---|---|---|---|---|---|---|---|
WMG | WOMG | WMG | WOMG | WMG | WOMG | WMG | WOMG | |
MPBIL | 0.8553 | 0.8255 | 0.7255 | 0.6420 | 0.7951 | 0.7229 | 0.7195 | 0.6219 |
OMPBIL | 0.8556 | 0.8426 | 0.7259 | 0.6430 | 0.7968 | 0.7438 | 0.6723 | 0.6403 |
MPBILMLR | 0.8115 | 0.7739 | 0.7212 | 0.6285 | 0.7016 | 0.5976 | 0.6167 | 0.5651 |
MPBILADLR | 0.8543 | 0.8371 | 0.7240 | 0.6385 | 0.7954 | 0.7407 | 0.6404 | 0.6292 |
Average Ranking of Each AlgorithmFriedman | p-Value | |||
---|---|---|---|---|
MPBIL | OMPBIL | MPBILMLR | MPBILADLR | |
2.6250 | 3.8750 | 1 | 2.5000 | 0.0002 |
(2) | (1) | (4) | (3) |
Design Problems | Average Hypervolume | |
---|---|---|
WMG | WOMG | |
MOP1 | 0.8442 | 0.8198 |
MOP2 | 0.7242 | 0.6380 |
MOP3 | 0.7722 | 0.7013 |
MOP4 | 0.6622 | 0.6141 |
Average Ranking (p-value = 0.0455) | 2 (1) | 1 (2) |
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Sleesongsom, S.; Bureerat, S. Topology Optimisation Using MPBILs and Multi-Grid Ground Element. Appl. Sci. 2018, 8, 271. https://doi.org/10.3390/app8020271
Sleesongsom S, Bureerat S. Topology Optimisation Using MPBILs and Multi-Grid Ground Element. Applied Sciences. 2018; 8(2):271. https://doi.org/10.3390/app8020271
Chicago/Turabian StyleSleesongsom, Suwin, and Sujin Bureerat. 2018. "Topology Optimisation Using MPBILs and Multi-Grid Ground Element" Applied Sciences 8, no. 2: 271. https://doi.org/10.3390/app8020271
APA StyleSleesongsom, S., & Bureerat, S. (2018). Topology Optimisation Using MPBILs and Multi-Grid Ground Element. Applied Sciences, 8(2), 271. https://doi.org/10.3390/app8020271