Existence proofs for a class of plate optimization problems.

*(English)*Zbl 0542.49002
System modelling and optimization, Proc. 11th IFIP Conf., Copenhagen 1983, Lect. Notes Control Inf. Sci. 59, 773-779 (1984).

Summary: [For the entire collection see Zbl 0538.00033.]

In this paper we shall discuss optimization of thin, solid elastic plates of given domain and elastic properties. The design variable is the variable thickness of the plate and the problem is thus a distributed parameter optimization problem.

Numerical experiments have shown that one cannot guarantee existence of solutions to, for example, the minimum compliance (maximum stiffness) problem if the admissible designs considered correspond to thickness functions in a bounded, closed and convex set of the bounded, measurable functions on the plate domain. This means that this type of optimization problem has to be regularized and in this paper we suggest doing this by imposing an extra (physically reasonable) constraint on the admissable designs in the form of a bound on the average of the gradient of the thickness function, i.e. we consider thickness functions in a bounded subset of the Sobolev space \(H^ 1\). For this type of admissable designs we obtain the existence for not only the traditional static optimization problem such as the minimum compliance problem, but also for dynamic problems.

In this paper we shall discuss optimization of thin, solid elastic plates of given domain and elastic properties. The design variable is the variable thickness of the plate and the problem is thus a distributed parameter optimization problem.

Numerical experiments have shown that one cannot guarantee existence of solutions to, for example, the minimum compliance (maximum stiffness) problem if the admissible designs considered correspond to thickness functions in a bounded, closed and convex set of the bounded, measurable functions on the plate domain. This means that this type of optimization problem has to be regularized and in this paper we suggest doing this by imposing an extra (physically reasonable) constraint on the admissable designs in the form of a bound on the average of the gradient of the thickness function, i.e. we consider thickness functions in a bounded subset of the Sobolev space \(H^ 1\). For this type of admissable designs we obtain the existence for not only the traditional static optimization problem such as the minimum compliance problem, but also for dynamic problems.

##### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

74G60 | Bifurcation and buckling |

74H45 | Vibrations in dynamical problems in solid mechanics |

49R50 | Variational methods for eigenvalues of operators (MSC2000) |

93C20 | Control/observation systems governed by partial differential equations |