The impact of the functional on the quality of the heat transfer coefficient mapping under fourth-kind boundary conditions using swarm algorithms

Abstract

The article aims to analyze three different functionals used as error norms in reconstructing the heat transfer coefficient κ under fourth-kind boundary conditions. The heat transfer coefficient κ is significant in modeling heat conduction at the interface of the casting and the mold. The study used two optimization algorithms (ABC – Artificial Bee Colony and ACO – Ant Colony Optimization) to assess how individual functionals respond to input data noise and how this affects the quality and stability of the parameter selection process. The studies were conducted for three levels of noise: 0%, 2%, and 5% of the input data and various population sizes: 5, 10, 15, and 20, with a constant number of 6 iterations. In each case, the average value of the functional and the compliance of the determined parameter κ with the reference value were analyzed. The algorithms’ efficiency was compared based on result stability and data error resistance. The (L₁) functional demonstrated high regularity and stability regardless of computational conditions. ABC and ACO produced consistent solutions, though ACO showed better repeatability. The (L₂) functional, more sensitive to noise, showed significant differences between the algorithms: ABC reacted strongly to noise, leading to a scatter of results, while ACO maintained high mapping quality. The most important differences were observed for the L_∞ functional, which was most susceptible to data noise. In this case, only ACO could maintain the stability and precision of the solutions, indicating its greater resilience in mapping the parameter κ. The analysis shows that the nature of the functional is crucial for the entire optimization process. It is not the algorithm, but the properties of the functional that determine the difficulty of the inverse problem and the scale of the final error. The selection of the functional should be treated as a strategic decision that affects the convergence, stability, and reliability of the solutions obtained. These conclusions are crucial in the context of engineering applications, where the precision and robustness of the computational method directly translate into the quality and safety of the technological process.