Solving the stationary Liouville equation via a boundary element method

Energy 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 in both time dependent and stationary problems are ray tracing and level set methods. In the context of predicting the vibro-acoustic response of complex engineering structures, related methods such as Statistical Energy Analysis or variants thereof have found widespread applications. We present a new method for solving the transport equations for complex multi-component structures based on a boundary element formulation of the stationary Liouville equation. The method is an improved version of the Dynamical Energy Analysis technique introduced recently by the authors. It 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.