## THERMAL SCIENCE

International Scientific Journal

### RESEARCH ON GEOMETRIC INVERSE PROBLEM BASED ON THE FIREFLY CONJUGATE GRADIENT METHOD

**ABSTRACT**

In this paper the 2-D steady-state heat transfer geometric inverse problem is solved using the finite element method, conjugate gradient method, and firefly algorithm. Based on the finite element method for solving the forward heat transfer model, and based on continuous iterative optimisation of the conjugate gradient method, the accuracy of the error function of the measured and estimated values is kept within a certain range, so that the geometry of the object under test can be calculated in an inverse way. In the study of the forward problem, the temperature field distribution is solved using circular pipes as the research objects, and the feasibility of applying the finite element method to heat transfer problems is verified. The inverse problem takes the circular tube as the object and considers two different corrosion defects on the inner wall of the circular tube. Simultaneously, the firefly algorithm is introduced based on the conjugate gradient method to stochastically optimize the temperature data and suppress the fluctuation of the inversion result. Numerical experiment results indicate that the method in this paper can perform more accurate geometric shape recognition when there is a certain temperature measurement error or the temperature measurement information is relatively complete.

**KEYWORDS**

PAPER SUBMITTED: 2022-12-30

PAPER REVISED: 2023-01-30

PAPER ACCEPTED: 2023-02-08

PUBLISHED ONLINE: 2023-03-11

**THERMAL SCIENCE** YEAR

**2023**, VOLUME

**27**, ISSUE

**Issue 5**, PAGES [4145 - 4159]

- Wang, S. B., et al., Solving of 2-D Unsteady-State Heat transfer Inverse Problem Using Finite Difference Method and Model Prediction Control Method, Complexity, 2019 (2019), 7, pp. 1-12
- Gonzalez, M., et al., Inverse Geometry Heat Transfer Problem Based on a Radial Basis Functions Geometry Representation, International Journal for Numerical Methods in Engineering, 65 (2006), 8, pp. 1243-1268
- Lu, T., et al., A 2-D Inverse Heat Conduction Problem in Estimating the Fluid Temperature in a Pipe-Line, Applied Thermal Engineering, 30 (2010), 13, pp. 1574-1579
- Szenasi, S., et al., Configuring Genetic Algorithm to Solve the Inverse Heat Conduction Problem, Acta Polytechnica Hungarica, 14 (2017), 6, pp. 133-152
- Zhang, T., et al., Seeking Heat Source in Inverse Heat Conduction Problem by Using Particle Swarm Optimization, University of Shanghai for Science and Technology, 35 (2013), 4, pp. 377-381
- Peker, H. A., et al., Application of Kashuri Fundo Transform and Homotopy Perturbation Methods to Fractional Heat Transfer and Porous Media Equations, Thermal Science, 26 (2022), 4A, pp. 2877-2884
- Cuha, F. A., et al., Solution of Abel's Integral Equation by Kashuri Fundo Transform, Thermal Science, 26 (2022), 4A, pp. 3003-3010
- Peker, H. A., et al., Solving Steady Heat Transfer Problems Via Kashuri Fundo Transform, Thermal Science, 26 (2022), 4A, pp. 3011-3017
- Jiang, S., et al., Geometry Estimation of Furnace Inner Wall Based on BEM and Decentralized Fuzzy Inference Method, World Sci-Tech R and D, 65 (2006), 8, pp. 122-126
- Luo, Z., et al., Fuzzy Estimation for Irregular Configuration of Furnace, Journal of Engineering Thermophysics, 10 (2013), 8, pp. 1906-1909
- Wang, D., Thermal Conducting Geometric Inversion Method Based on the Third Boundary Condition (in Chinese), Ph. D. thesis, Harbin Institute of Technology, Harbin, China, 2018
- Li, Y., Non-Iterative Algorithm for Identifying Boundary Conditions and Geometry Shapes in Heat Conduction Problems (in Chinese), M. Sc. thesis, Hefei University of Technology, Hefei, China, 2018
- Wang, S., et al., The 2-D Steady-State Boundary Shape Inversion of CGM-SPSO Algorithm on Temperature Information, Advances in Materials Science and Engineering, 2017 (2017), ID2461498
- Mei, L., et al., A New Method to Evaluate the Subsurface Defect by Thermal Non-destructive Testing, Infrared Millim Waves, 19 (2000), 6, pp. 457-459
- Sun, S., et al., Back Calculation for Cold-state Geometric Shape of Turbine Blade Based on Modification of Nodal Co-Ordinates and Mesh Smoothing, Journal Of Mechanical Engineering, 50 (2014), 10, pp. 143-148
- Fazeli, H., et al., Shape Identification Problems on Detecting of Defects in a Solid Body Using Inverse Heat Conduction Approach, Journal of Mechanical Science and Technology, 26 (2012), pp. 3681-3690
- Fazeli, H., et al., Estimation of Location and Size of Defects in a Solid Body via Inverse Heat Conduction Problem, Proceedings, 14th International Heat Transfer Conference, Washington DC, USA, 2010, pp. 387-396
- Li, Y., et al., A Decentralized Fuzzy Inference Method for the Inverse Geometry Heat Conduction Problem, Applied Thermal Engineering, 106 (2016), 8, pp. 109-116
- Cheng, C. H., Cheng, M. H., Shape Identification by Inverse Heat Transfer Method, Journal of Heat Transfer, 125 (2013), 2, pp. 224-231
- Mosavati, M., et al., A Novel, Non-iterative Inverse Boundary Design Regularized Solution Technique Using the Backward Monte Carlo Method, Journal of Heat Transfer, 135 (2013), 4, pp. 130-136
- Mosavati, B., et al., Solution of Radiative Inverse Boundary Design Problem in a Combined Radiating-Free Convecting Furnace, International Communications in Heat and Mass Transfer, 45 (2013), July, pp. 130-136
- Mosavati, B., et al., Inverse Boundary Design Solution in a Combined Radiating-Free Convecting Furnace Filled with Participating Medium Containing Specularly Reflecting Walls, International Communications in Heat and Mass Transfer, 76 (2016), Aug., pp. 69-76
- Chen, W., Yang, Y., Inverse Estimation for Unknown Fouling-Layer Profiles with Arbitrary Geometries on the Inner Wall of a Forced-Convection Duct, International Journal of Thermal Sciences, 49 (2010), 1, pp. 86-98
- Fan, C., et al., Research on the Recognition Algorithm of Irregular Boundary of Inner Wall of Circular Tube Based on Infrared Temperature Measurement, Journal of Thermal Science and Technology, 02 (2006), pp. 112-117
- Partridge, P. W., Wrobel, L. C., An Inverse Geometry Problem for the Localisation of Skin Tumours by Thermal Analysis, Engineering Analysis with Boundary Elements, 31 (2007), 10, pp. 803-811
- Zhu, L., et al., Estimating Steady Multi-variables Inverse Heat Conduction Problem by Using Conjugate Gradient Method, Proceedings of the CSEE, 31 (2011), 8, pp. 58-61
- Zhang, Y. W., et al., Inverse Heat Conduction Problem of Deducing Inner Wall Temperature by Using Conjugate Gradient Method, Journal Of Engineering Thermophysics, 30 (2009), 7, pp. 1188-1190
- Yang, X. S., Firefly Algorithm, Stochastic Test Functions and Design Optimization, International Journal of BioInspired Computation, 2 (2010), 2, pp. 78-84
- Ma, X., et al., Energy-Saving Operation Optimization of High-Speed Trains Using Firefly Algorithm, Journal of Huaqiao University (Natural Science), 4 (2019), pp. 452-456
- Zuo, Z., et al., An Improved Swarm Optimization Alogorithm, Microelectronics and Computer, 35 (2018), 2, pp. 61-66
- Gao, W., Study on the Firefly Algorithm and Application (in Chinese), Lanzhou University, Lanzhou, China, 2013
- Lu, S., et al., Identification of Irregular Pipe-Line Geometry Boundary Using Infrared Transient Inspection Based on Finite Element Discretization, CIESC Journal, 63 (2012), 12, pp. 3805-381