Communications in Mathematical Sciences

Volume 16 (2018)

Number 6

Variant of optimality criteria method for multiple state optimal design problems

Pages: 1597 – 1614



Krešimir Burazin (Department of Mathematics, J.J. Strossmayer University of Osijek, Croatia)

Ivana Crnjac (Department of Mathematics, J.J. Strossmayer University of Osijek, Croatia)

Marko Vrdoljak (Department of Mathematics, Faculty of Science, University of Zagreb, Croatia)


We consider multiple state optimal design problems, aiming to find the best arrangement of two given isotropic materials, such that the obtained body has some optimal properties regarding $m$ different right-hand sides. Using the homogenization method as the relaxation tool, the standard variational techniques lead to necessary conditions of optimality. These conditions are the basis for the optimality criteria method, a commonly used numerical (iterative) method for optimal design problems. In Vrdoljak (2010), one variant of this method is presented, which is suitable for the energy maximization problems. We study another variant of the method, which works well for energy minimization problems. The explicit calculation of the design update is presented, which makes the implementation simple and similar to the case of single state equation. The method is tested on examples, showing that exact solutions are well approximated with the obtained numerical solutions.


stationary diffusion, optimal design, homogenization, optimality criteria method

2010 Mathematics Subject Classification

49J20, 49K35, 49M05, 80M40

Received 26 September 2017

Accepted 21 April 2018

Published 7 February 2019