A hybrid singularity superposition/boundary element-based inverse problem method for the reconstruction of multi-dimensional heat flux distributions is developed. Cauchy conditions are imposed at exposed surfaces that are readily reached for measurements while convective boundary conditions are unknown at surfaces that are not amenable to measurements such as the edges of cooling slots. The purpose of the inverse analysis is to determine the heat flux distribution along cooling slot surfaces. This is accomplished in an iterative process by distributing a set of singularities at the vicinity of the cooling slot surface inside a fictitious extension of the physical domain with a given initial strength distribution. A forward steady-state heat conduction problem is solved using the boundary element method (BEM), and an objective function is defined to measure the difference between the heat flux measured at the exposed surfaces and the heat flux predicted by the BEM under the current strength distribution of the singularities. A genetic algorithm iteratively alters the strength distribution of the singularities until the measuring surfaces heat fluxes are matched, thus, satisfying Cauchy conditions. Subsequent to the solution of the inverse problem, the heat flux at the inaccessible surface is computed using the BEM. The hybrid singularity superposition/BEM approach thus eliminates the need to mesh the surface of the film cooling slot and the need to parametrize the heat flux over that surface. Rather, the heat flux is determined in a post-processing stage after the inverse problem is solved. This constitutes a tremendous advantage in solving the inverse problem, particularly in three-dimensional applications.

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