Solving the stationary Liouville equation via a boundary element method

TL;DR

This paper introduces a boundary element method for solving the stationary Liouville equation, bridging statistical energy analysis and ray tracing to efficiently predict wave energy distributions in complex structures.

Contribution

The authors develop an improved boundary element method based on the stationary Liouville equation, unifying statistical energy analysis and ray tracing techniques.

Findings

Efficiently handles large-scale complex problems.

Provides accurate energy distribution approximations.

Contains both statistical energy analysis and ray tracing as special cases.

Abstract

Intensity distributions of linear wave fields are, in the high frequency limit, often approximated in terms of flow or transport equations in phase space. Common techniques for solving the flow equations for both time dependent and stationary problems are ray tracing or level set methods. In the context of predicting the vibro-acoustic response of complex engineering structures, reduced ray tracing methods such as Statistical Energy Analysis or variants thereof have found widespread applications. Starting directly from the stationary Liouville equation, we develop a boundary element method for solving the transport equations for complex multi-component structures. The method, which is an improved version of the Dynamical Energy Analysis technique introduced recently by the authors, interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. We demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation.