PhD Defense: Numerical Geometric Acoustics

Physically modelled sound propagation is a viable means of simulating realistic architectural and environmental acoustics. Point-to-point room impulse responses (RIRs) are computed for many source and listener locations and convolved online to give a listener a realistic impression of the space. Existing methods solve the wave or Helmholtz equations, which is infeasible if the wavenumber is too small. Geometric acoustics, most commonly in the form of raytracing, works for high frequencies but becomes complicated in the offline setting. Motivated by the limitations of these approaches, we model geometric acoustic wave propagation by solving a pair of partial differential equations, the eikonal equation and transport equation. Our focus is offline sound propagation in the high-frequency regime where directly solving the wave or Helmholtz equations is infeasible.To this end, we conduct a survey of semi-Lagrangian eikonal solvers, and develop new, efficient, first order solvers for the eikonal equation in 3D called ordered line integral methods (OLIMs). Motivated by the requirements of sound propagation simulations, we introduce higher order semi-Lagrangian eikonal solvers which we term jet marching methods (JMMs). JMMs additionally transport higher order derivative information causally, allowing the eikonal equation to be solved to high order using compact stencils. To solve the transport equation, we apply paraxial raytracing to propagate the amplitude along each local ray. We first develop a JMM that handles a smoothly varying speed of sound on a regular grid in 2D. To conform to the requirements of industrial room acoustics applications, we develop a second order JMM for computing the eikonal on a tetrahedron mesh with a constant speed of sound. We compute multiple arrivals by reinitializing the eikonal equation on reflecting walls and diffracting edges. We use dynamic programming to enforce reflection and diffraction boundary conditions for these scattered fields in the semi-Lagrangian setting, requiring the use of the uniform theory of diffraction. Along the way, we carry out numerical analysis of our solvers, and conduct extensive tests.Examining Committee:

Chair: Dr. Maria K. Cameron Dean's rep: Dr. P.S. Krishnaprasad Members: Dr. Ramani Duraiswami Dr. Howard Elman
Dr. Ming Lin