Topology Optimization for Wave Propagation Problems

Abstract: This thesis considers topology optimization methods for wave propagation problems. These methods make no a priori assumptions on topological properties such as the number of bodies involved in the design. The performed studies address problems from two different areas, acoustic wave propagation and microwave tomography. The final study discusses implementation aspects concerning the efficient solution of large scale material distribution problems.Acoustic horns may be viewed as impedance transformers between the feeding waveguide and the surrounding air. Modifying the shape of an acoustic horn changes the quality of the impedance match as well as the angular distribution of the radiated waves in the far field (the directivity). This thesis presents strategies to optimize acoustic devices with respect to efficiency and directivity simultaneously. The resulting devices exhibit desired far field properties and high efficiency throughout wide frequency ranges.In microwave tomography, microwaves illuminate an object, and measurements of the scattered electrical field are used to depict the object's conductive and dielectric properties. Microwave tomography has unique features for medical applications. However, the reconstruction problem is difficult due to strongly diffracting waves in combination with large dielectric contrasts. This thesis demonstrates a new method to perform the reconstruction using techniques originally developed for topology optimization of linearly elastic structures. Numerical experiments illustrate the method and produce good estimates of dielectric properties corresponding to biological objects.Material distribution problems are typically cast as large (for high resolutions) nonlinear programming problems over coefficients in partial differential equations. Here, the computational power of a modern graphics processing unit (GPU) efficiently solves a pixel based material distribution problem with over 4 million unknowns using a gradient based optimality criteria method.