Topology optimization with supershapes

This work presents a method for the continuum-based topology optimization of structures whereby the structure is represented by the union of supershapes. Supershapes are an extension of superellipses that can exhibit variable symmetry as well as asymmetry and that can describe through a single equation, the so-called superformula, a wide variety of shapes, including geometric primitives. As demonstrated by the author and his collaborators and by others in previous work, the availability of a feature-based description of the geometry opens the possibility to impose geometric constraints that are otherwise difficult to impose in density-based or level set-based approaches. Moreover, such description lends itself to direct translation to computer aided design systems. This work is an extension of the author's group previous work, where it was desired for the discrete geometric elements that describe the structure to have a fixed shape (but variable dimensions) in order to design structures made of stock material, such as bars and plates. The use of supershapes provides a more general geometry description that, using a single formulation, can render a structure made exclusively of the union of geometric primitives. It is also desirable to retain hallmark advantages of existing methods, namely the ability to employ a fixed grid for the analysis to circumvent re-meshing and the availability of sensitivities to use robust and efficient gradient-based optimization methods. The conduit between the geometric representation of the supershapes and the fixed analysis discretization is, as in previous work, a differentiable geometry projection that maps the supershapes parameters onto a density field. The proposed approach is demonstrated on classical problems of 2-dimensional compliance-based topology optimization.