A Level-Set Framework for Shape Optimisation

For my PhD in applied mathematics, I considered the application of the level-set method to shape optimisation. This is not an entirely new connection, and a lot of literature exists that uses level sets in applications. There exists also a lot of theory around level-set equations and time evolution of level-set geometries, but mostly in a quite abstract setting.

My main contribution, therefore, is the formulation of a level-set framework that is particularly tailored towards shape-sensitivity analysis and other questions necessary for shape optimisation. This framework is based on a Hopf-Lax formula for the time evolution of level-set shapes, similar in spirit to the classical Fast Marching Method. (But note that my focus is not mainly on numerical algorithms but also on theoretical conclusions from this formulation.)

A lot of the program code developed during my research has been published as free software in the level-set package for GNU Octave.

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