Topology, Shape, and Size Optimization of Additively Manufactured Lattice Structures Based on the Superformula

This paper investigates the application of Superformula for structural synthesis. The focus is set on the lightweight design of parts that can be realized using discrete lattice structures. While the design domain will be obtained using the Superformula, a tetrahedral meshing technique will be applied to this domain to generate the topology of the lattice structure. The motivation for this investigation stems from the property of the Superformula to easily represent complex biological shapes, which opens a possibility to directly link a structural synthesis to a biomimetic design. Currently, numerous results are being reported showing the development of a wide range of design methods and tools that first study and then utilize the solutions and principles from the nature to solve technical problems. However, none of these methods and tools quantitatively utilizes these principles in the form of nature inspired shapes that can be controlled parametrically. The motivation for this work is also in part due to the mathematical formulation of the Superformula as a generalization of a superellipse, which, in contrast to the normal surface modeling offers a very compact and easy way to handle set of rich shape variants with promising applications in structural synthesis. The structural synthesis approach is organized as a volume minimization using Simulated Annealing (SA) to search over the topology and shape of the lattice structure. The fitness of each of candidate solutions generated by SA is determined based on the outcome of lattice member sizing for which an Interior Point based method is applied. The approach is validated with a case study involving inline skate wheel spokes.