The Eikonal Equation & Acoustic Cloaking
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The Eikonal Equation & Acoustic Cloaking

Status
Past
Created
Jul 12, 2023 06:47 AM
Areas
numerical methods
optimization
math
Date
Sep 1, 2022 → Aug 31, 2023
📜 Publications, Posters & Talks
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Introduction

The propagation of the sound wave in a domain can be described by the Eikonal equation. If the speed of sound is anisotropic (it has different speeds along different directions), then we need an anisotropic Eikonal equation.
An anisotropic solution to the Eikonal equation.
An anisotropic solution to the Eikonal equation.
An isotropic solution to the Eikonal equation.
An isotropic solution to the Eikonal equation.
Let and let . If certain compatibility conditions on are satisfied, then:
is an anisotropic Eikonal equation. The function is referred to as the (anisotropic) eikonal. If for all , then reduces to an isotropic eikonal equation.
Domain of Dependence of a given grid point (brown).
Domain of Dependence of a given grid point (brown).
Now consider an acoustic cloaking material: even though it has holes inside, observers from the outside will only detect the sound waves as if they were plane waves. In other words, the inside of the material cannot be detected by sound waves. It has been shown that such material can be designed with a finite density and stiffness if only the sound speed is anisotropic in this material. Our goal is to use numerical methods, such as FEM and optimization, to design such material for a given domain.