Multigrid Methods for Optimal Shape Design Problems

                                  Eyal Arian
                                 ICASE, Mail
                                  Stop 132C,
                        NASA Langley Research Center,
                              Hampton, VA 23681

                                     and

                                Shlomo Ta'asan
                            Dept. of Mathematics,
                         Carnegie-Mellon University,
                             Pittsburgh, PA 15213


We present a multigrid algorithm to solve optimal shape design problems
governed by elliptic PDEs.  The computational complexity of the proposed
algorithm is O(N), independent of the number of design parameters, where N is
the number of grid points on the finest level.

The necessary conditions for a minimum are obtained using Lagrange multipliers
and are given as a set of three equations:  state, costate and design.  These
equations are represented on coarser levels using the full approximation
scheme (FAS).  On each level the residuals of the design equation are used in
the optimization step to up date the design parameters (which determine the
shape position).

The optimization step is performed in a local region neighboring the boundary,
using the elliptic characteristics of the problem where a high frequency
perturbation on the boundary has an exponential decaying effect on the
solution in the interior of the domain.  Thus a crucial p oint in the method
is to construct an optimization step which reduces effectively the
high-frequency errors in the design parameters.

Numerical results include shape optimization of a 2D nozzle geometry using
Dirichlet and Neumann boundary conditions.