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ABSTRACT
Firefly algorithm combined with Newton method (FA–NM) is proposed for identifying time-dependent boundary conditions of two-dimensional transient heat conduction problems with a heat source. The dual reciprocity boundary element method (DRBEM) is applied to solve the direct problem. The improved firefly algorithm has not only a good global search ability, but also a good local search ability. FA–NM can acquire accurate results with much less iterations. Furthermore, different measurement points and noises are also considered. A small number of measurement points cannot obtain precise results. With the increase in the measurement points to a certain value, the results are more accurate. When the measurement noises are not very large, FA–NM can get desirable results. With the decrease in measurement noises, the results are in better agreement with the exact solutions. Numerical results also indicate that the more front the time substep is, the more accurate the estimated boundary conditions are.
Nomenclature
| c | = | specific heat |
| d | = | number of undetermined variables |
| I | = | brightness of firefly |
| k | = | thermal conductivity |
| q | = | heat flux |
| = | normal gradient of | |
| Q | = | heat source |
| t | = | time |
| T | = | temperature |
| T0 | = | initial temperature |
| = | see in Eq. (8) | |
| T | = | vector of node temperature |
| xi | = | Cartesian coordinate of nodes |
| x | = | vector of unknown boundary conditions |
| α | = | see in Eq. (5) |
| β | = | attractiveness in firefly algorithm |
| ε0 | = | the stop criterion |
| Γ | = | boundary of the domain Ω |
| ξ | = | see in Eq. (6) |
| γ | = | light absorption coefficient in firefly algorithm |
| ρ | = | density |
| ϕ(R) | = | radial basis function |
| Φ | = | vector of ϕ(R) |
| Ω | = | domain of problem |
| ζ | = | see in Eq (18) |
| Superscript | = | |
| c | = | calculated result |
| exa | = | exact result |
| inv | = | inverse result |
| m | = | measurement result |
| Abbreviation | = | |
| DRBEM | = | dual reciprocity boundary element method |
| FA–NM | = | firefly algorithm combined with Newton method |
Nomenclature
| c | = | specific heat |
| d | = | number of undetermined variables |
| I | = | brightness of firefly |
| k | = | thermal conductivity |
| q | = | heat flux |
| = | normal gradient of | |
| Q | = | heat source |
| t | = | time |
| T | = | temperature |
| T0 | = | initial temperature |
| = | see in Eq. (8) | |
| T | = | vector of node temperature |
| xi | = | Cartesian coordinate of nodes |
| x | = | vector of unknown boundary conditions |
| α | = | see in Eq. (5) |
| β | = | attractiveness in firefly algorithm |
| ε0 | = | the stop criterion |
| Γ | = | boundary of the domain Ω |
| ξ | = | see in Eq. (6) |
| γ | = | light absorption coefficient in firefly algorithm |
| ρ | = | density |
| ϕ(R) | = | radial basis function |
| Φ | = | vector of ϕ(R) |
| Ω | = | domain of problem |
| ζ | = | see in Eq (18) |
| Superscript | = | |
| c | = | calculated result |
| exa | = | exact result |
| inv | = | inverse result |
| m | = | measurement result |
| Abbreviation | = | |
| DRBEM | = | dual reciprocity boundary element method |
| FA–NM | = | firefly algorithm combined with Newton method |
Acknowledgments
The research is supported by the National Natural Science Foundation of China (Nos. 11672098, 11502063, and 11602076) and the Natural Science Foundation of Anhui Province (No. 1608085QA07).