Design Variables

Topology optimization methods operate on a computational mesh. In structural optimization, this mesh is made up of finite elements. In CFD topology optimization, a CFD mesh is used. The smallest geometrical quantity is a single finite element or a single CFD control volume (cell). A cell is manipulated by modifying a certain physical property of it. With this approach, this cell property becomes a design variable of the optimization problem. Hence, the cells are called design cells of the optimization problem. All design cells make up the so-called design space of the optimization problem. This cell-based approach has two important consequences:

1. The optimization problem is posed in such a way that a large geometrical freedom is provided: No parameterized geometry is required, no shape functions must be predefined. The optimizer operates on a cell-by-cell level, which implies a large design freedom.
2. The number of design variables, namely the number of design cells, is likely to be very large: A CFD model for topology optimization might contain hundreds of thousands or even millions of design cells.

The first point leads to the potential of the topology optimization approach:

• The modeling effort to set up an optimization problem is low
• No complex geometrical model is required (for example, parametrized CAD model)
• The large freedom within the problem allows completely new, unexpected, and innovate solutions
• A "pure" topology optimization result is basically a set of cells in the first place and must be treated as a so-called design proposal

The second point implies that only specific optimization methods can be used for topology optimization because of the large number of design variables.