A shape function based interpolation approach for nodal design variable based structural optimization

In this article, a novel density interpolation scheme for topological optimization based on nodal density variables is presented. The method relies on the definition of an independent interpolation mesh for the description of the density field. This mesh is used for the interpolation of the density field at the Gauss points of the design domain mesh. Provided that the interpolation mesh is realized by linear elements, the proposed scheme is range restricted and assures physically meaningful values of the interpolated density. Explicit analytical sensitivity expressions can be derived. The utilization of an independent interpolation mesh allows for the utilization of any element type and geometry on the body for the computation of the displacements. It is also shown that the chosen interpolation scheme has interesting properties of volume and material volume conservation with respect to the mesh utilized on the body and the domain of the interpolation mesh can differ from the domain of the mesh for the displacements computation. This allows for the realization of very regular interpolation meshes even in case of very complex shapes of the domain of the body to be optimized. Examples are provided showing the performance of the proposed interpolation scheme.