Alternative Concepts for Preserving Regularity in Free Surface Shape Optimization
A common problem in shape optimization with free node parameterizations, i.e. the situation where the surface itself is used as design unknown without a smooth parameterization, is the loss of regularity. The usual remedy here is the use of gradient smoothing, sometimes also called Sobolev-Gradient method. Here, a smoother shape update is obtained by solving a Laplace-Type PDE to obtain the H^1-Gradient over the L_2-Update.
By considering shape Hessians, one can see that this approach should give Newton-like performance when then objective is perimeter driven. The main focus of the talk are alternative gradient smoothing strategies, for problems, where less regularity of the surface is desirable, in particular surfaces of bounded variation. Such problems arise naturally in the post-processing of 3D scans of objects with kinks and corners or in inverse problems, where the object to be determined is non-smooth.
To this end, we blend algorithms for bounded variation L^1-Regularization stemming from mathematical imaging to processing the surfaces textures of scanned objects and finally to regularizing the actual surface.