Comparison of Lagrangian and Isogeometric Boundary Element Formulations for Orthotropic Heat Conduction Problems

Orthotropic materials are increasingly employed in advanced thermal systems due to their direction-dependent heat transfer characteristics. Accurate numerical modeling of heat conduction in such media remains challenging, particularly for 3D geometries with nonlinear boundary conditions and internal heat generation. In this study, conventional boundary element method (BEM) and isogeometric boundary element method (IGABEM) formulations are developed and compared for steady-state orthotropic heat conduction problems. A coordinate transformation is adopted to map the anisotropic governing equation onto an equivalent isotropic form, enabling the use of classical Laplace fundamental solutions. Volumetric heat generation is incorporated via the radial integration method (RIM), preserving the boundary-only discretization, while nonlinear Robin boundary conditions are treated using variable condensation and a Newton–Raphson iterative scheme. The performance of both methods is evaluated using a hollow ellipsoidal benchmark problem with available analytical solutions. The results demonstrate that IGABEM provides higher accuracy and smoother convergence than conventional BEM, particularly for higher-order discretizations, which is owing to its exact geometric representation and higher continuity. Although IGABEM involves additional computational overhead due to NURBS evaluations, both methods exhibit similar quadratic scaling with respect to the degrees of freedom.