Hybrid discrete ordinates—spherical harmonics solution to the Boltzmann Transport Equation for phonons for non-equilibrium heat conduction

Abstract

The Boltzmann Transport Equation (BTE) for phonons has found prolific use for the prediction of non-equilibrium heat conduction phenomena in semiconductor materials. This article presents a new hybrid formulation and associated numerical procedures for solution of the BTE for phonons. In this formulation, the phonon intensity is first split into two components: ballistic and diffusive. The governing equation for the ballistic component is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM), and the control angle discrete ordinates method (CADOM). The diffusive component, on the other hand, is determined by invoking the first-order spherical harmonics (or P 1) approximation, which results in a Helmholtz equation with Robin boundary conditions. Both governing equations, referred to commonly as the ballistic-diffusive equations (BDE), are solved using the unstructured finite-volume procedure. Results of the hybrid method are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two two-dimensional transient heat conduction problems at various Knudsen numbers. Subsequently, the method is explored for a large-scale three-dimensional geometry in order to assess convergence and computational cost. It is found that the proposed hybrid method is accurate at all Knudsen numbers. From an efficiency standpoint, the hybrid method is found to be superior to direct solution of the BTE both for steady state as well as for unsteady non-equilibrium heat conduction calculations with the computational gains increasing with increase in problem size.