Geometric shape optimization is concerned with novel discrete surface energies and computational geometry algorithms for processing and optimizing polyhedral meshes in industrial applications such as computer aided design (CAD) and computer graphics. The main focus of this project addresses the development of efficient mesh processing algorithms and robust modeling tools. The mathematical tools are novel discrete differential and curvature operators.
This project addresses the strong needs for multilevel algorithms in industrial applications and in computer graphics where large surface meshes must be efficiently processed. Typical applications are solutions of PDEs on surfaces, surface optimization, and automatic mesh parametrization. Funding is provided by the DFG Research Center MATHEON - "Mathematics for key technologies".
For large, nonlinear, and time dependent PDE constrained optimization problems with 3D spatial domain, reduced methods are a viable algorithmic approach. The computation of reduced gradients by adjoint methods requires the storage of 4D data, which can be quite expensive from both a capacity and bandwith point of view. This project investigates lossy compression schemes for storing the state trajectory, based on hierarchical interpolation in adaptively refined meshes as a general predictor.