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# Meshless Local Petrov-Galerkin Mixed Collocation Method for Solving Cauchy Inverse Problems of Steady-State Heat Transfer

Tao Zhang1,2, Yiqian He3, Leiting Dong4, Shu Li1, Abdullah Alotaibi5, Satya N. Atluri2,5

School of Aeronautic Science and Engineering, Beihang University, China.
Center for Aerospace Research & Education, University of California, Irvine, USA.
State Key Lab of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, China.
Corresponding Author. Department of Engineering Mechanics, Hohai University, China. Email: dong.leiting@gmail.com
King Abdulaziz University, Jeddah, Saudi Arabia.

Computer Modeling in Engineering & Sciences 2014, 97(6), 509-533. https://doi.org/10.3970/cmes.2014.097.509

## Abstract

In this article, the Meshless Local Petrov-Galerkin (MLPG) Mixed Collocation Method is developed to solve the Cauchy inverse problems of Steady- State Heat Transfer In the MLPG mixed collocation method, the mixed scheme is applied to independently interpolate temperature as well as heat flux using the same meshless basis functions The balance and compatibility equations are satisfied at each node in a strong sense using the collocation method. The boundary conditions are also enforced using the collocation method, allowing temperature and heat flux to be over-specified at the same portion of the boundary. For the inverse problems where noise is present in the measurement, Tikhonov regularization method is used, to mitigate the inherent ill-posed nature of inverse problem, with its regularization parameter determined by the L-Curve method. Several numerical examples are given, wherein both temperature as well as heat flux are prescribed at part of the boundary, and the data at the other part of the boundary and in the domain have to be solved for. Through these numerical examples, we investigate the accuracy, convergence, and stability of the proposed MLPG mixed collocation method for solving inverse problems of Heat Transfer.

## Keywords

Zhang, T., He, Y., Dong, L., Li, S., Alotaibi, A. et al. (2014). Meshless Local Petrov-Galerkin Mixed Collocation Method for Solving Cauchy Inverse Problems of Steady-State Heat Transfer. CMES-Computer Modeling in Engineering & Sciences, 97(6), 509–533.

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