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2015-11

An optimal shape design problem for plates

by Klaus Deckelnick, Michael Hinze, Tobias Jordan.


Series: 2015-11, Preprints

MSC:
49J20 Optimal control problems involving partial differential equations
65N12 Stability and convergence of numerical methods

Abstract:
We consider an optimal shape design problem for the plate equation, where the variable
thickness of the plate is the design function. This problem can be formulated as a control
in the coefficient PDE-constrained optimal control problem with additional control and
state constraints. The state constraints are treated with Moreau-Yosida regularization of a
dual problem. Variational discretization is employed for discrete approximation of the
optimal control problem. For discretization of the state in the mixed formulation we compare the standard continuous piecewise linear ansatz with a piecewise constant one
based on the lowest-order Raviart-Thomas mixed finite element. We derive bounds for the
discretization and regularization errors and also address the coupling of the
regularization parameter and the finite element grid size. The numerical solution of the
optimal control problem is realized with a semismooth Newton algorithm. Numerical examples
show the performance of the method.

Keywords:
elliptic optimal control problem, optimal shape design, pointwise state constraints, Moreau-Yosida regularization, error estimates