Transport of phase space densities through tetrahedral meshes using discrete flow mapping

TL;DR

This paper extends discrete flow mapping, a ray-based method for wave energy distribution, from polygonal surfaces to three-dimensional tetrahedral meshes, enabling efficient analysis in complex 3D domains.

Contribution

It introduces a 3D extension of discrete flow mapping using tetrahedral meshes and develops a hybrid basis approximation approach for phase space.

Findings

Efficient computation of ray transfer operators in 3D.

Successful application to complex 3D wave energy problems.

Maintains computational feasibility with low-order and high-order basis functions.

Abstract

Discrete flow mapping was recently introduced as an efficient ray based method determining wave energy distributions in complex built up structures. Wave energy densities are transported along ray trajectories through polygonal mesh elements using a finite dimensional approximation of a ray transfer operator. In this way the method can be viewed as a smoothed ray tracing method defined over meshed surfaces. Many applications require the resolution of wave energy distributions in three-dimensional domains, such as in room acoustics, underwater acoustics and for electromagnetic cavity problems. In this work we extend discrete flow mapping to three-dimensional domains by propagating wave energy densities through tetrahedral meshes. The geometric simplicity of the tetrahedral mesh elements is utilised to efficiently compute the ray transfer operator using a mixture of analytic and spectrally accurate numerical integration. The important issue of how to choose a suitable basis approximation in phase space whilst maintaining a reasonable computational cost is addressed via low order local approximations on tetrahedral faces in the position coordinate and high order orthogonal polynomial expansions in momentum space.