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.

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. ==Definition== Mathematically, shape optimization can be posed as the problem of finding a bounded set \Omega , minimizing a functional \mathcal{F}(\Omega) , possibly subject to a constraint of the form \mathcal{G}(\Omega)=0. Usually we are interested in sets \Omega which are Lipschitz or C 1 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. ==Examples== ==Techniques== 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. x(0) = x_0 \in \Omega_0, \quad x'(s) = V(x(s)), \quad T_s(x_0) = x(s), \quad s \geq 0 , and denote \Omega_0 \mapsto T_s(\Omega_0) = \Omega_s. Then the Gâteaux or shape derivative of \mathcal{F}(\Omega) at \Omega_0 with respect to the shape is the limit of d\mathcal{F}(\Omega_0;V) = \lim_{s \to 0}\frac{\mathcal{F}(\Omega_s) - \mathcal{F}(\Omega_0)}{s} if this limit exists. If in addition the derivative is linear with respect to V , there is a unique element of \nabla \mathcal{F} \in L^2(\partial \Omega_0) and d\mathcal{F}(\Omega_0;V) = \langle \nabla \mathcal{F}, V \rangle_{\partial \Omega_0} where \nabla \mathcal{F} is called the shape gradient. This gives a natural idea of gradient descent, where the boundary \partial \Omega is evolved in the direction of negative shape gradient in order to reduce the value of the cost functional. Higher order derivatives can be similarly defined, leading to Newtonlike methods. Typically, gradient descent is preferred, even if requires a large number of iterations, because, it can be hard to compute the second-order derivative (that is, the Hessian) of the objective functional \mathcal{F} . If the shape optimization problem has constraints, that is, the functional \mathcal{G} is present, one has to find ways to convert the constrained problem into an unconstrained one. Sometimes ideas based on Lagrange multipliers, like the adjoint state method, can work. ==See also== SU2 code Topological derivative Topology optimization ==External links== Optopo Group — Simulations and bibliography of the optopo group at Ecole Polytechnique (France). Homogenization method and level set method.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1374909487 under CC BY-SA 4.0; modifications have been made.