Accurate-geometry-embodied finite element method for phonon Boltzmann transport equation

Abstract

Modeling nano- and micro-scale heat conduction based on the phonon Boltzmann transport equation has gained increasing research interest due to the demand for better thermal performance of semiconductors. Nevertheless, the high dimensionality of the Boltzmann equation results in the so-called curse of dimensionality, presenting a bottleneck for efficient numerical solutions based on direct discretization. In practice, high-order numerical schemes such as discontinuous Galerkin finite element methods are preferable to reduce the degrees of freedom, thereby reducing the computational cost. However, when complex geometries emerge, cumbersome refinement is required to approximate the boundary of the computational domain if spatial meshes with straight-sided elements are employed, concealing the advantage of a high-order scheme. In this work, we extend the idea of the non-uniform rational B-splines enhanced finite element method. By embodying accurate geometric information, including parametric descriptions for curved boundaries and sampling information of rough surfaces reconstructed from scanning electron microscope images or by a random growth approach, into the faces of the elements adjacent to the physical boundary, the geometric inaccuracies and heavy refinement can be eliminated in a very coarse mesh. Strategies to define the polynomial basis and compute the integrals over the geometry-embodied elements are investigated. Numerical results, including heat conduction in a silicon ring, nano-porous media with circular pores, and a square domain with a rough boundary, show that to obtain solutions with the same order of accuracy, the discontinuous Galerkin method performed on accurate-geometry-embodied meshes can be 10-100 times faster than that implemented on straight-sided meshes. The efficiency of higher-order discretization methods is fully promoted, where fewer spatial elements combined with higher-order approximating polynomials are preferable to obtain solutions with high accuracy and reduced computational cost.