Math 2.0

Calculus of variations and optimal control; optimization Optimal control: formulation and existence Problem

Shape optimization

Builds on Optimal control of partial differential equationsPartial differential equationFunctionalShape derivative

Problem

Let D⊆ℝdD \subseteq \R^d be a bounded open set, called the design region. Let 𝒪\mathcal{O} be a class of open subsets Ω⊆D\Omega \subseteq D, the admissible shapes, for example those with a given volume. For each Ω∈𝒪\Omega \in \mathcal{O} let yΩy_\Omega be the solution of a boundary value problem for a partial differential equation on Ω\Omega. Let J(Ω)=j(Ω,yΩ)J(\Omega) = j(\Omega, y_\Omega) be a real-valued shape functional. The shape optimization problem is to find Ω∗∈𝒪\Omega^* \in \mathcal{O} with J(Ω∗)≤J(Ω)J(\Omega^*) \le J(\Omega) for every Ω∈𝒪\Omega \in \mathcal{O}.

Remarks

It is a form of optimal control of partial differential equations in which the control is the domain itself. Example: minimize the first Dirichlet eigenvalue λ1(Ω)\lambda_1(\Omega) of the Laplacian among open sets Ω⊆ℝd\Omega \subseteq \R^d of a given volume. The Rayleigh–Faber–Krahn inequality solves it: the minimizers are the balls. In general minimizers need not exist in a given class of shapes, since minimizing sequences can develop holes or oscillating boundaries. One then restricts the class, for example by a uniform cone condition or a perimeter bound, or relaxes the problem; relaxation leads to topology optimization. Gradient methods use the shape derivative of JJ, often computed with an adjoint equation.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Shape_optimization
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AboutOptimal control of partial differential equations
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