Coordination Funds

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Recent developments in the digitization of 3D bodies as well as in digital fabrication have triggered a tremendous growth in complexity of typical workpieces that can be handled in automated CAD/CAE/CAM workflows. The design is no longer a purely geometric task but needs to take the desired functionality and serviceability of the final workpiece into account. It implies the necessity to include structural analysis into the process. Therefore, a tedious and error-prone conversion of the underlying digital 3D models is an established workflow in industry. To avoid this, novel research areas have emerged like "Isogeometric Analysis" and "Geometry Processing" where NURBS or polytope meshes, respectively, are used as a common geometric reference model. Besides the advantages of relying on one model, there is a disadvantage that in each work step the reference model has to satisfy different requirements. These vary from topological constraints (e.g. self-intersection free) to structural analysis requirements (e.g. type of structural model) up to geometric requirements (e.g. convexity, aspect ratio of cells) and stem from the particular aims of the specific work steps. Geometry processing aims for an accurate representation with a small geometric approximation error while reducing the computational cost for graphics applications. A way to reach this goal is to cluster best-fitting regions resulting in a polygonal surface mesh. For the structural analysis computational mechanics employs typically finite element (FE) programs, which lead to a physical approximation error of the stress and strain state of the structure. One way to reduce this error is to introduce additional degrees of freedom by refining the mesh. The aim of computational mechanics is to minimize the physical approximation error while reducing the computational effort. While stand-alone solutions exist, there is a need to combine the features of digital 3D geometry descriptions with finite element techniques. We aim at polytope mesh generation methods that are tailored to the specific requirements of the simulation, as well as new FE analysis methods that exploit the flexibility of polytopic meshes. Compared to smooth NURBS surfaces or standard FE meshes, arbitrarily shaped elements, ranging from planar facets to freeform patches, offer more flexibility in mesh generation and adaptation. Therefore, it is necessary to combine the expertise from numerical mathematics, geometry processing and computational mechanics to develop novel methods that enable the usage of general polytope meshes for design and analysis simultaneously. Polytope meshes have the potential to provide a computationally efficient representation (with few cells) of the target geometry while adapting flexibly to local geometric features. Moreover, this flexibility opens up new possibilities to locally adjust and refine the mesh structure allowing for a tightly interleaved operation of mesh generation and simulation.