WIAS Preprint No. 335, (1997)

A duality approach in the optimization of beams and plates


  • Sprekels, Jürgen
    ORCID: juergen.sprekels@wias-berlin.de
  • Tiba, Dan

2010 Mathematics Subject Classification

  • 49D05


  • Optimal design, nonconvex duality




We introduce a class of nonlinear transformations called "resizing rules" which associate to optimal shape design problems certain equivalent distributed control problems, while preserving the state of the system. This puts into evidence the duality principle that the class of system states that can be achieved, under a prescribed force, via modifications of the structure (shape) of the system can be as well obtained via the modifications of the force action, under a prescribed structure.

We apply such transformations to the optimization of beams and plates and, in the simply supported or in the cantilevered cases, the obtained control problems are even convex. In all cases, we establish existence theorems for optimal pairs, by assuming only boundedness conditions. Moreover, in the simply supported case, we also prove the uniqueness of the global minimizer. A general algorithm that iterates between the original problem and the transformed one is introduced and studied. The applications also include the case of variational inequalities.

Appeared in

  • SIAM J. Control Optimiz., 37 (1998), pp. 486-501

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