Direct Solution of Inverse Steady-State Heat Transfer Problems by Improved Coupled Radial Basis Function Collocation Method

This paper presents an improved coupled radial basis function (ICRBF) approach for solving inverse steady-state heat conduction problems. The proposed method combines infinitely smooth Gaussian radial basis functions with a real-valued mth-order conical spline, where m serves as a coupling index. Unlike the original coupled RBF approach, which relied on multiquadric RBFs paired with a fixed fifth-order spline or later integer-order extensions, our real-order spline generalization enhances accuracy and simplifies the tuning of m. We present a particle swarm optimization approach to optimize the coupling index m. This work represents the first application of the CRBF framework to inverse steady-state heat conduction problems. The ICRBF methodology addresses three key limitations of traditional RBF frameworks: (1) it resolves the persistent issue of shape parameter selection in global RBF methods; (2) it inherently produces well-posed linear systems that can be solved directly, avoiding the need for the regularization typically required in inverse problems; and (3) it delivers superior accuracy compared to existing approaches. Extensive numerical experiments on benchmark problems demonstrate that the proposed method achieves high accuracy and robust numerical stability in solving steady-state heat conduction Cauchy inverse problems, even under significant noise contamination.