Analytical solutions of 2D orthotropic transient heat conduction problems under Robin boundary conditions within the symplectic framework

Abstract

The Robin boundary conditions-based orthotropic transient heat conduction problems are frequently encountered in engineering applications, which are characterized by the coupled effect of temperature and heat flux. Although numerous attentions have been paid to this topic, the current focus is mainly on isotropic materials, and analytical solutions are still scarce due to mathematical challenges. This study presents novel analytical solutions of 2D orthotropic transient heat conduction problems under Robin boundary conditions by the symplectic superposition method. The Laplace transform is first employed to convert the problems to the frequency domain, and they are then transferred into the symplectic framework. The process effectively divides an original problem into two subproblems, and they are solved through the variable separation method and symplectic eigen expansion, and ultimately the solutions of the original problem are obtained through superposition. The results obtained in this study agree well those by the finite element analysis. The effects of heat convection coefficient and thermal conductivity are discussed. The present solutions are derived rigorously within the symplectic framework, without assuming any fundamental solution forms, such that more analytical solutions are attainable within the same framework.