Shape optimization

Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints. In many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain. Topology optimization is, in addition, concerned with the number of connected components/boundaries belonging to the domain. Such methods are needed since typically shape optimization methods work in a subset of allowable shapes which have fixed topological properties, such as having a fixed number of holes in them. Topological optimization techniques can then help work around the limitations of pure shape optimization. Mathematically, shape optimization can be posed as the problem of finding a bounded set , minimizing a functional possibly subject to a constraint of the form Usually we are interested in sets which are Lipschitz or C1 boundary and consist of finitely many components, which is a way of saying that we would like to find a rather pleasing shape as a solution, not some jumble of rough bits and pieces. Sometimes additional constraints need to be imposed to that end to ensure well-posedness of the problem and uniqueness of the solution. Shape optimization is an infinite-dimensional optimization problem. Furthermore, the space of allowable shapes over which the optimization is performed does not admit a vector space structure, making application of traditional optimization methods more difficult. Shape optimization problems are usually solved numerically, by using iterative methods. That is, one starts with an initial guess for a shape, and then gradually evolves it, until it morphs into the optimal shape. To solve a shape optimization problem, one needs to find a way to represent a shape in the computer memory, and follow its evolution. Several approaches are usually used. One approach is to follow the boundary of the shape.