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Primal hybrid formulation of an elliptic equation in smooth optimal shape problems. (English) Zbl 0831.65071

In optimal shape problems an admissible domain is controlled by a design function described by means of a suitable mapping \(F\) with values in \(C([0,1])\). The key-point of a numerical treatment of such problems is a sensitivity analysis. Two main approaches are possible: The first one calculates the derivative of a cost functional of a discretized optimal deisign problem, the second one starts with the continuous problem and then the derivative is approximated by means of a solution of a certain discretized state problem.
A second-order elliptic equation with the homogeneous Dirichlet boundary condition is taken as the state problem; three integral cost functionals are considered to be the cost functionals.
The primal hybrid formulation of the state and the adjoint problem is used. In examples, the optimized domains are controlled by Bézier curves. Convergence analysis is performed and existence is proved. Some numerical test results obtained on an HP 9000/710 workstation are presented.

MSC:

65K10 Numerical optimization and variational techniques
49M15 Newton-type methods
49J20 Existence theories for optimal control problems involving partial differential equations
49Q12 Sensitivity analysis for optimization problems on manifolds
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