Definition
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.
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