Design Space of a Small Dimension

This is the simplest of all cases and was developed in Ta'asan [7]. The assumption is that the design space is of a small dimension and moreover, it is such that its effect on the solution is smooth. That is, a change in the design variables produces a relatively smooth change in the solution, that can be approximated using the coarsest grid. If the design variables describe the shape of a boundary to be designed, we may have a representation of the form

$$\alpha (x) = \sum \alpha _j f_j(x),$$ (45)

where the functions $f_j$ can be represented on the coarsest grid.

This suggests to update the design variables on the coarse grid only. The question is how to perform the optimization of the design variables on coarse levels but to have convergence toward the finest grid solution? The answer is in our coarse grid optimization problem. The coarse grid functional is such that, at convergence of the solution is that of the fine grid problem, due to the FAS formulation we have mentioned.

On all grids we perform relaxation to smooth the errors in the state and costate variables. On the coarsest grid only, we solve the optimization problems which depends, through the FAS transfers, on the finer level optimization problem. On that level we may perform any optimizer of out choice, for example, a quasi-Newton method such as BFGS. The coarse grid iterations are done until convergence of its optimization problem.

RELAXATION: Level l


(1) Relax State Equation
(2) Relax Adjoint Equation
(3) If level = 1 Update Design Variables by BFGS