http://orcid.org/0000-0003-3678-7688
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ABSTRACT
Cuckoo Search (CS) algorithm has shortcomings of weak local search ability, slow convergence speed, and low accuracy. In order to overcome these disadvantages, an improved CS algorithm based on Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm (CS-BFGS) is proposed for solving inverse geometry heat conduction problems, and the physical field is the steady-state heat conduction. Firstly, the unknown initial boundary is evolved by Lévy flights and elimination mechanism. Then the BFGS algorithm is applied to minimize the objective function. Finally, the influences of random errors, measurement point number, and measurement point position on the inverse results are investigated. The results show that the CS-BFGS algorithm has higher accuracy and faster convergence speed than BFGS and CS algorithm. With the decrease of measurement errors, the increase of measurement point number, and measurement point position closer to the inverse boundary, the results become more accurate.
Funding
The Natural Science Foundation of Anhui Province. 1608085QA07; the National Natural Science Foundation of China. 11672098, 11502063.
Acknowledgments
The research is supported by the National Natural Science Foundation of China (Nos. 11672098, 11502063) and the Natural Science Foundation of Anhui Province (No. 1608085QA07).
Nomenclature
| A | = | coefficient matrix of the equation |
| ARE | = | average relative error |
| B | = | right-hand side of the equation |
| BEM | = | boundary element method |
| BFGS | = | Broyden–Fletcher–Goldfarb–Shanno |
| CGM | = | conjugate gradient method |
| CPU | = | central processing unit |
| CS | = | cuckoo search |
| D | = | number of individuals in the test function |
| FEM | = | finite element method |
| f(x) | = | exact upper boundary shape |
| = | estimated upper boundary shape | |
| G | = | estimated matrix of Hessian matrix |
| H | = | Hessian matrix |
| I | = | unit matrix |
| IHCP | = | inverse heat conduction problem |
| J | = | objective function |
| L | = | length of the geometry model |
| LMM | = | Levenberg–Marquardt method |
| m | = | number of measurement points |
| M | = | maximum generation |
| MFS | = | method of fundamental solution |
| n | = | outward normal of the boundary |
| n | = | number of observation points |
| pa | = | possibility of finding |
| q | = | differential coefficient of T with respect to n |
| = | heat flux (W/m2) | |
| q* | = | differential coefficient of T* with respect to n |
| r | = | distance between source and field points (m) |
| R | = | real number field |
| S | = | iteration number of BFGS |
| STD | = | standard deviation |
| T | = | temperature vector (°C) |
| T* | = | exact temperature vector (°C) |
| = | temperature on the upper boundary (°C) | |
| T* | = | fundamental solution of Laplace equations |
| Ti | = | measured temperature (°C) |
| = | computed temperature (°C) | |
| TRM | = | Tikhonov regularization method |
| T(x,y) | = | temperature at point (x,y) (°C) |
| u, v | = | normal stochastic variable |
| w | = | random variable vector |
| x | = | x-coordinate |
| x | = | vector of unknown boundary values |
| y | = | y-coordinate |
| y | = | vector of y-coordinate of estimated boundary |
Greek symbols
| α | = | Lévy exponent |
| Γ | = | boundary of the domain |
| Γ(z) | = | gamma function |
| γ | = | regularization parameter |
| ϵ | = | error criterion |
| λ | = | thermal conductivity (W/(m · °C)) |
| μ | = | step size |
| ξ | = | a number with the range from 0 to 1 |
| σ | = | standard deviation |
| ϕ(x, y) | = | interior angle at the point (x, y) |
| Ω | = | domain |
| ∂0 | = | step factor |
| ∇2 | = | Laplace operator |
Subscripts
| best | = | best nest |
| est | = | estimated boundary |
| exa | = | exact boundary |
| i | = | i-th node |
| j | = | j-th node |
Superscripts
| k | = | iteration number |
| ^ | = | computed values |
Additional information
Notes on contributors
Hao-Long Chen
Hao-long Chen is a Ph.D. student at the School of Civil Engineering, Hefei University of Technology, China. He received his bachelor's degree from Northwest A&F University in 2013. His research interests include inverse heat transfer problem based on the boundary element method.
Bo Yu
Bo Yu is an associate professor at the School of Civil Engineering, Hefei University of Technology, China. He received his Ph.D. degree in computational mechanics from Dalian University of Technology in 2014. His research focuses on the boundary element method, thermoelastic analysis, and inverse problems. He has published more than 15 international journal papers.
Huan-Lin Zhou
Huanlin Zhou is a professor at the School of Civil Engineering, Hefei University of Technology, China. He received his Ph.D. degree in solid mechanics from University of Science and Technology of China in 2003. His main research interests include heat transfer, elasticity, thermo-elasticity, and fracture mechanics based on the boundary element method. He has published more than 80 journal papers.
Zeng Meng
Zeng Meng is a lecturer at the School of Civil Engineering, Hefei University of Technology, China. He received his Ph.D. degree in engineering mechanics from Dalian University of Technology of China in 2015. His main research interests include inverse problem, optimization, and reliability analysis.