Design Space of Moderate Size

In this case one consider more variables in the design space such that the assumption about the smoothness can no longer be true. That is, some design variables may have a smooth effect on the solution, but since there are sufficiently many design variables, the space of changes in the solution is also large and hence must contain higher frequency that cannot be represented on the coarsest level. Some intermediate levels, (still much coarse than the finest level) have to be included in the optimization process. This case was developed by Ta'asan, Salas and Kuruvila in [10],[11].

One approach to achieve this is to change the basis representing the design space such that one is using orthonormal decomposition. For the shape design problem it means that we use a basis of shape functions $g_{m,l}$ where the representation of the shape is

$$\alpha (x) = \sum _{m=1}^{m_l} \sum _{l=1}^{n_d} \beta_{m,l} g_{m,l} (x)$$ (46)

and where the index $l$ stands for the multigrid level, and the changes of state and adjoint variables corresponding to changes in $\beta _{m,l}, m=1,\dots,m_l$ can be approximated on this level $l$.

The idea is that these new shape functions, $g_{m,l}$ are going to be of increasing oscillations, due to their orthogonality. This implies that their effect on the solution will also be of increasing oscillations. Thus, we achieve a decomposition of the design space into subspaces of different smoothness as far as its effect on the solution. The implication of this is that a natural preconditioner of the optimization process is done. Since each level of those coarse grids on which optimization is to be done works independently of the other, at the relaxation step, we have removed the bad conditioning from the problem using this decomposition. The relaxation of the design variables is done now for the $\beta _{m,l}$ instead of the original $\alpha _j$. On level $l$ we update $m_l$ of the total design variables, $\beta _{m,l}, m=1,\dots,m_l$.

Relaxation: Level $l$


$\qquad$ (1) Relax State Equation
$\qquad$ (2) Relax Adjoint Equation
$\qquad$ (3) Update design variables $\beta _{m,l}, m=1,\dots,m_l$