Optimal shapes of mechanically motivated surfaces

Subject of this contribution is form finding of “optimal” structural shapes with regard to the load carrying behaviour of surface structures under certain load cases. In general, those optimal shapes prefer a membrane state of stress to transfer loading. Bending is omitted as much as possible. It will be focused on two different disciplines and related numerical approaches which deal with solutions of the mentioned task: form finding of prestressed membranes and general shape optimization. As design is an inverse problem both approaches share similar problematic properties as e.g. indeterminate in-plane location of surface discretization or necessary regularization and filtering of sensitivity and other data. As it will turn out, those remedies found for the very special methods of membrane design can be abstracted and transferred to general optimization procedures. That merges into elegant, numerical shape optimal design techniques which combine advantages of both approaches and allow for effective and efficient shape optimization of free formed surfaces, directly on the finite element mesh and for a large number of variables. Typical applications are, for example, membrane design, free form architecture and structural engineering, and metal sheet design.