The classical level set method, which represents the boundary of the unknown geometry as the zero-level set of a function, has been shown to be very effective in solving shape optimization problems. The present work addresses the issue of using a level set representation when there are simple geometrical and topological constraints. We propose a logarithmic barrier penalty which acts to enforce the constraints, leading to an approximate solution to shape design problems.
Bibliographical noteFunding Information:
We thank Grant Erdmann, whose suggestion that a logarithmic barrier method could be used to preserve constraints lead to the penalty functional we employ in this paper. We are grateful to the anonymous referees who made very helpful suggestions to the original manuscript, many of which we have incorporated in the present version. This work is supported in part by the National Science Foundation.
- Level set method
- Steepest descent method
- Topology preservation