A subdomain optimization problem is defined and formulated as a new category of shape optimal designs. Several subdomains, which may have different features such as in material properties, geometric properties, external force conditions and boundary conditions, are to be optimized in their configurations and various characteristics. The design change of a subdomain is described by defining suitable design velocity fields, usually over local regions. This description is found to be very versatile allowing many practical applications possible.
For the design sensitivity analysis, general approaches based on either the variational formulation or the boundary integral equation formulation are possible. But the former is better suited to the subdomain optimization because most of our problems by nature involve various nonhomogeneous properties. Therefore, in this thesis, the variational approach is taken for all the derivation of the sensitivity formulas.
Several examples are presented to show typical applications from different fields. Specialized sensitivity formulas are derived for each individual area. Numerical design sensitivities are calculated and compared with those by finite differencing. Numerical optimizations are the performed and the results discussed.
The first motivating problem is for the layout of chips on a printed circuit board, where the maximum temperature rise is to be minimized. The result is very reasonable and shown practically applicable. For tests and understanding of the performance of structural problems, a set of beams composed of different materials are considered. As more complex examples, the eigenvalue problems for a square membrane with added masses, and rectangular or quarter annular plates with different line supports or stiffeners are treated. The fundamental or the second natural frequency, of which some are repeated, is to be maximized with respect to the location of the masses, supports or stiffeners. Selection and description of design velocity fields are studied and illustrated for rotational as well as translational change in configurations.
In the numerical study, different finite elements and meshes are tried. Also different velocity fields are tested. It is shown that the sensitivities are insensitive to the selection of local design velocity field as should be theoretically. However, the sensitivities are rather senstivie to the accuracy of finite element analysis. They may be in good agreement, as seen in most cases, with those by finite differencing, but often far from the exact or accurate solutions. In some cases, the error due to mesh distortion or remodeling during optimization iteration obscures the effect of design changes and the sensitivity results can be grossly wrong. A detailed study in this direction is required before any meaningful complex problems are tried.
