Topology optimization with an implicit function and parameterized cutting surface

This paper introduces a new implicit function based method for topology optimization that can: obtain solutions with smooth boundaries, be solved using standard mathematical programming methods and reduce the number of design variables. Using implicit functions for topology optimization is attractive because the solutions have clearly defined, smooth boundaries. Most current methods use the zero level-set of the implicit function to define the boundary. The implicit function is then modified during optimization to move the boundary location and connectivity. The new approach proposed in this paper abandons the zero level-set idea and instead uses a fixed signed-distance implicit function. The definition of the boundary from the fixed implicit function is then modified during optimization. This is achieved by using a cutting surface and defining the boundary as the intersection of the cutting surface and signed-distance function. The cutting surface is parameterized and the parameters become the design variables during optimization. Thus, the optimization problem can be solved using mathematical programming and the number of parameters used to define the cutting surface is less than the number of elements in the analysis mesh. The new method is demonstrated using minimization of compliance, minimization of volume and complaint mechanism problems. The results show that the method can obtain good solutions to well-known problems with smooth, clearly defined boundaries and that this can be achieved using significantly fewer design variables compared with element-based methods.