Martin Berggren
Department of Computing Science
Umeå University
Umeå
Boundary shape optimization problems for systems governed by partial differential equations involve a calculus of variation with respect to boundary modifications. As typically presented in the literature, the first-order necessary conditions of optimality are derived in a quite different manner for the problems before and after discretization, and the final directional-derivative expressions look very different. However, a systematic use of the material-derivative concept allows a unified treatment of the cases before and after discretization. The final expression when performing such a derivation (which appears to be new) includes the classical before-discretization (``continuous'') expression, which contains objects solely restricted to the design boundary, plus a number of ``correction'' terms that involve field variables inside the domain. Some or all of the correction terms vanish when the associated state and adjoint variables are smooth enough.