As CFD methods become an essential contributor to the virtual design process, the application of automatic optimisation methods offers significant benefits. Topology and shape optimisation by means of adjoint methods is the most promising way of tackling complex flow-geometry problems.

Topology optimisation can be seen as a generalisation of shape optimisation, which not only deals with manipulation of a provided form, but also allows the derivation of an optimal component directly from the design space.

The general form of an optimisation problem also applies naturally to topology and shape optimisation:

" Minimise a defined cost function by varying a number of defined design variables, obeying a set of fixed constraints " The major advantage of employing adjoint methods (a special way to calculate the sensitivity of the cost function to the design variables) is that the effort of finding the sensitivities for a required cost function is independent of the number of defined variables.

Transferred into the CFD world this can result in the following example setup:

• Inlet and outlet positions are defined, fixed and used as constraints.
• The available space is volume meshed independent of any expected final solution. This differs from conventional CFD engineering practise. Thus the extent of the meshed volume acts as a de facto constraint on the topology to be extracted.
• Each volume cell in the available space is then assigned an individual momentum sink. Consequently, the number of mesh cells equals the number of design variables.

Using adjoint methods the sensitivity of the cost function (e.g. total pressure loss) is calculated for each cell. In this way “bad” and “good” cells can be identified and “bad” cells can be “solidified”. The result is that the “bad” cells are effectively removed from the domain, leaving a volume shaped to give an optimised cost as defined by a particular set of boundary conditions. This shape can then be extracted using appropriate quantities and used as a guideline or directly employed for CAD preparation.

The topology optimisation method described above is limited in resolution by the cell size used to mesh the design space. As a result, the accuracy of its surface definition is limited to the local cell dimensions. However, in a second stage the newly created geometry can be re-meshed and a similar adjoint method used to find the sensitivity of the objective function to the motion of surface. These sensitivities can then be fed into an iterative mesh motion routine to arrive at a highly accurate optimal shape.