A 'Surface-based' Geometrical Acoustic formulation within a Galerkin Boundary Element framework

Emthyas, A

Abstract

As sound propagates within a room, it experiences high-order reflection, diffraction, and scattering. This causes the reflection density to increase over time, such that the sound field becomes diffuse and chaotic. Under these conditions, there is little benefit in running a computationally costly full wave solver – if it is even feasible – so methods based on Geometrical models of acoustic propagation prevail. Raytracing is currently the de-facto method for this late-time, high-frequency regime in room acoustic modelling. It samples the propagating distributions of acoustic intensity by launching a set of rays and individually tracing their trajectories. This is computationally efficient when only specular reflections are present, but accurate inclusion of scattering or diffraction requires ‘ray-splitting’ to be introduced, causing an exponential increase in computational cost with reflection order, crippling the algorithm. Hence only crude Monte Carlo implantations of these processes are tractable with Raytracing.
An emerging solution for modelling late-reflections is “Surface-Based” Geometrical Acoustics. These formulations map a distribution of rays arriving at a boundary onto a pre- defined ‘approximation space’ of basis functions spanning position and angle, so the sound field is represented by a vector of boundary coefficients. Re-radiation of subsequent reflections is thus reduced to a matrix multiplication, with the steady-state solvable via a Neumann series. As rays only propagate one reflection order before being collected, the multiple ‘child’ rays that would be produced by scattering or diffraction of a ‘parent’ ray at the boundary are absorbed into the ‘approximation’ space at each reflection order. This maintains a fixed number of degrees of freedom and a linear computational cost with reflection order. This thesis presents a Surface-Based Geometrical Acoustic formulation cast in a Galerkin Boundary Element framework.
This thesis presents and implements the formulation in two dimensions and validates it against an Image Source Model for a rectangular room. The Galerkin Boundary Element scheme expediates comparison of different approximation schemes and their effect on convergence and accuracy can be easily studied. Examination of the resulting power distributions on the boundary for early reflections show power being smudged over a range of reflection angles, indicating approximation in the scheme. But this is perceptually appropriate for late-time diffuse fields as individual reflections will no longer be distinguishable, and late time energy decay rates are shown to be correct. Receiver responses for early reflections show very good agreement also, so long as angular resolution is set sufficiently high. The formulation is shown to converge with the number of angular degrees of freedom as well as smaller element sizes. The results show a high degree of accuracy and identical convergence trends when using continuous orthogonal polynomials, such as Legendre, Chebyshev or Lobatto, as angular basis. In contrast, other functions, such as continuous piecewise-linear, or discontinuous piecewise- constant, exhibit a significant degree of approximation for higher interpolation orders due to their discontinuous or non-smooth nature. Solutions in Geometrical Acoustics can be discontinuous. The ultimate ambition in formulating the model presented in this thesis is to include diffraction, and solutions when it is included will be continuous. This capability still remains as work for the future, but the choices made in this thesis were informed by that end goal.

Citation

Emthyas, A. A 'Surface-based' Geometrical Acoustic formulation within a Galerkin Boundary Element framework. (Dissertation). University of Salford

• Thesis Type: Dissertation
• Deposit Date: Apr 12, 2023
• Publicly Available Date: Apr 12, 2023
• Award Date: Sep 23, 2022