Theory on Vertex Morphing

Vertex morphing is based on the discretization of the design field s(ξ t) as s_h, with ξ the field parameters and t the time. In this case, this discretization is generally made through a set of shape functions N(ξ):

Each of the discrete design parameters s_h influence several points of the discretized geometry z_h, therefore a filter function A is applied. Integrating over the surface, we obtain

The key aspect of vertex morphing resides in this filter functions A. The discrete gradient design of the objective function can be approximated by:

With vertex morphing, the discrete design controls s_j are identified to be the design handles or vertices of the design model. When the geometry and design field discretizations are the same, only the choice of a filter is to be added to the analysis model. This filter may be constant or adaptative.

References

• Bletzinger, K.-U. (2017). Shape Optimization. In Encyclopedia of Computational Mechanics Second Edition (eds E. Stein, R. Borst and T.J.R. Hughes). https://doi.org/10.1002/9781119176817.ecm2109