Structural Optimisation Using Parametrised Level Set Methods and Extended Finite Element Method

In the framework of level‐set‐based optimisation, several approaches to solve bi‐material and topological problems have been developed. In this study a combined concept of different approaches is introduced, joining together a parametrised geometry description with superellipses or b‐splines and a modified approach for the extended finite element method within an optimisation setup of topological and shape iterations introduced by Allaire [1]. The parametrisation with geometric figures or splines allows us to reduce the number of design variables to a minimum whereas we hold up a sufficient precision in the geometry description. Moreover it simplifies the shape derivatives as we gain an implicit description for the moving interfaces. In order to solve the boundary value problem of structural mechanics, we introduce a modified extended finite element approach which uses a sub‐meshing technique that enables us to keep existing strategies from homogeneous structures and transform them onto a discontinuous material using enriched shape functions provided by the standard extended finite element method. Shape sensitivities are evaluated on the sub‐elements and extrapolated to the element nodes. The topological derivative as developed by Novotny [4] is evaluated and prescribes the initiation of new discontinuities as inclusions or holes.