# Doctoral Dissertation Defense: Chen Lai

## Advisors: Drs. Florian Potra and Susan Minkoff (UT Dallas)

Thursday, December 14, 2017

3:00 PM - 5:00 PM

3:00 PM - 5:00 PM

Mathematics/Psychology : 401

**Title:**

*Wave propagation: Theory and experiment*

**Abstract**

Wave propagation through air and ground is studied for two different phenomena: (i) Nearly Perfectly Matched Layer Boundary Conditions for Operator Upscaling of the Acoustic Wave Equation and (ii) Modeling of Air Platform Detection Methodology Based on the Human Auditory System.

(i) Acoustic imaging and sensor modeling are processes that require repeated solution of the acoustic wave equation. Solution of the wave equation can be computationally expensive and memory intensive for large simulation domains. One scheme for speeding up solution of the wave equation is the operator-based upscaling method. The algorithm proceeds in two steps. First, the wave equation is solved for fine grid unknowns internal to coarse blocks assuming the coarse blocks do not need to communicate with neighboring blocks in parallel. Second, these fine grid solutions are used to form a new problem which is solved on the coarse grid. Accurate and efficient wave propagation schemes also must avoid artificial reflections off of the computational domain edges. One popular method for preventing artificial reflections is the Nearly Perfectly Matched Layer (NPML) method. In this paper we discuss applying NPML to operator upscaling for the wave equation. We show that although we only apply NPML to the first step of this two step algorithm (directly affecting the fine grid unknowns only), we still see a significant reduction of reflections back into the domain. We describe three numerical experiments (one homogeneous medium experiment and two heterogeneous media examples) in which we validate that the solution of the wave equation exponentially decays in the NPML regions. Numerical experiments of acoustic wave propagation in two dimensions with a reasonable absorbing layer thickness resulted in a maximum pressure reflection of 3–8%. While the coarse grid acceleration is not explicitly damped in our algorithm, the tight coupling between the two steps of the algorithm results in only 0.1–1% of acceleration reflecting back into the computational domain.

(ii) We develop a general framework for the aural detection of air platforms. We consider air platform noise and ambient sound information in our formulation as well as physiological characteristics of the human auditory system to provide the probability of acquisition with respect to range. The primary focus is sound propagation in a heterogeneous domain with realistic atmospheric conditions, which can be prohibitively expensive in computation. In order to minimize the computational cost, we formulate a mixed numerical method in a modified finite difference scheme with the NPML method for the Helmholtz equation. We adapt the idea of using Green’s function in the Helmholtz equation formulation in order to implement a domain decomposition scheme. We describe the subgrid unknowns only based on the coarse block unknowns by using the Green’s functions. We formulate a linear system of the coarse block unknowns to solve. In the coarse block problem, we incorporate the forcing function or specific air platform signal and solve for the subgrid unknowns. We can easily update the forcing function for various acoustic signatures in the coarse block problem. The numerical case studies and application examples demonstrate the efficient performance and long time stability of our formulation for unbounded domain problems. Numerical experiments resulted in a maximum relative error in amplitude of 2–4%. Ollerhead’s aural detection model serves as the foundation of aural detection in our study. A collection of air platform signatures in various environmental conditions has been produced. This study also provides an algorithm scheme that can be integrated with Defense combat simulations.