This paper describes a new method for variable-topology shape optimization. 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. Formulations that penalize perforated material, either explicitly or through a partial relaxation, can generate manufacturable designs. However, the same illposedness that motivates relaxed formulations reappears as the penalty against perforated material is strengthened. The practical consequence is that numerical implementations of the penalized formulations fail to converge with grid refinement. The new approach uses a control on the design perimeter to effectively exclude 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. Numerical examples demonstrate that manufacturable designs can be obtained in a single, automatic operation. Grid refinement improves geometric resolution without altering the design topology. The new method also provides a means to control the number and the length scale of holes in the optimal design.