A generalized optimization problem in which the design space is also a design variable to be determined is defined and a numerical implementation method is proposed. In conventional optimization, only some of the structural parameters are designated as design variables while the remaining parameters related to the design space are often taken for granted. The number of design variables along with the layout or configuration specifies a design space. To solve this type of design space problem, a simple initial design space is selected and gradually improved while the usual design variables are being optimized. To make the design space evolve into a better one, the designer may increase the number of design variables, but in this transition, there are discontinuities in the objective and constraint functions. Accordingly, the sensitivity analysis methods based on continuity cannot be used in this discontinuous stage. To overcome this difficulty, a numerical continuation scheme is proposed. It is based on a new concept of a pivot phase design space. This is applicable only for the cases when new design variables are added.

To find the set of new design cells to be added, first a set of potential cells to be possibly added are specified, for example, by including all the adjacent cells to the existing active cells. The directional derivatives based on the introduction of pivot phase space are calculated and then a portion of the set are selected using the ranking of the sensitivities. The selection ratio is chose by the user. It is noted that only one analysis is necessary for the sensitivity information, and this makes this process very efficient.

In order to reduce computational cost and improve efficiency, however, it is necessary to include a process of removing design variables when the values remain constant over iterations, because in this case the design variables are not used for optimization. In the topology optimization of density approach, a cell whose density is zero is not used efficiently for optimization, and also there is little probability of regaining its density. A numerical routine of changing the design space by reducing the number of design variables is implemented. The new optimization formulation and its numerical scheme enable one to change the design space automatically either by increasing or by decreasing the number of design variables depending on the characteristics of a given problem.

Two new categones of structural optimization problems, topology optimization and plate thickness optimization, are formulated. For the topology optimization, a short cantilever and a bridge are optimized for various cases:​ according to the loading condition, initial domain size, mesh size and selection ratio for new design variable addition. The results show that a high selection ratio is required to obtain a good optimum design under bending loads, and a domain of too small an initial size brings in checkerboards and eventually fails to converge to an optimum.

It is seen that the new proposed process of adding new design variables works very well and efficient. In the examples with compliance as the objective function, different solutions are obtained for the same design space depending on whether a fixed initial design space is used or an evolutionary approach by addition is used. Also there is a bigger chance of falling into a local minimum when the resolution in relation to the cell size is higher. In most of the cases tested, the evolutionary approach has generated a better optimum than the fixed domain cases.

With the addition of a routine for reducing the design variables, the total number of design variables is found to decrease gradually because, in many cases, the reduced design variables outnumber those added. This integrated routine of design variable addition and reduction reduces computational cost.

The second category of applications is plate thickness design, where the layout of design patches is adaptively optimized and the optimum thickness distribution elaborated. The optimum layout problem of design patches is taken to minimize the compliance objective function of plate problems. The advantages of the proposed approach are that (1) the effect of a patch refinement on the objective and constraint functions is calculated quantitatively by the proposed sensitivity analysis, and (2) the refinement is performed adaptively on the basis of the obtained sensitivities. Again only one analysis is necessary to obtain the sensitivity information of the refined cells.

In conclusion, a mathematically sound approach of obtaining the sensitivity of added design variables is proposed by introducing a pivot phase design space. It is a directional derivative and can be calculated efficiently by the adjoint variable method. With the new capability of increasing design cells and the existing method of removing design cells, it is new possible to obtain the design truly in an evolutionary manner. As shown numerically, there are many local solutions involved. The convergence property and obtaining global solutions are different and yet need future studies.