Direct and Inverse Steady-State Heat Conduction in Materials with Discontinuous Thermal Conductivity: Hybrid Difference/Meshless Monte Carlo Approaches

This study investigates steady-state heat conduction in materials with stepwise discontinuities in thermal conductivity, a phenomenon frequently encountered in layered composites, thermal barrier coatings, and electronic packaging. The problem is formulated for a 2D two-domain region, where each subdomain has a distinct constant conductivity. Both the direct problem-determining the temperature field from known conductivities-and the inverse problem-identifying conductivities and the internal heat source from limited temperature measurements-are addressed. To this end, three deterministic finite-difference-type models are developed: two for the standard formulation and one for a meshless formulation based on Moving Least Squares (MLS), all derived within a local framework that efficiently enforces interface conditions. In addition, two Monte Carlo models are proposed-one for the standard and one for the meshless setting-providing pointwise estimates of the solution without requiring computation over the entire domain. Finally, an algorithm for solving inverse problems is introduced, enabling the reconstruction of material parameters and internal sources. The performance of the proposed approaches is assessed through 2D benchmark problems of varying geometric complexity, including both structured grids and irregular node clouds. The numerical experiments cover convergence studies, sensitivity of inverse reconstructions to measurement noise and input parameters, and evaluations of robustness across different conductivity contrasts. The results confirm that the hybrid difference-meshless Monte Carlo framework delivers accurate temperature predictions and reliable inverse identification, highlighting its potential for engineering applications in thermal design optimization, material characterization, and failure analysis.