Variable-topology shape optimization with a control on perimeter

A method for variable-topology shape optimization is described. The method addresses certain problems that arise in relaxed formulations, i.e., homogenization design methods. For example, a complete relaxation generates optimal designs containing material with perforated microstructures that may be difficult or expensive to manufacture. This approach uses a control on the design perimeter to exclude effectively chattering designs, which have an infinite perimeter, from the feasible solution space. This achieves a well-posed shape design problem without the introduction of microstructure.