Topology optimization with supershapes

Abstract

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.

Generalized Bezier components and successive component refinement using moving morphable components

“…The geometric primitives were modeled using a topology description function (TDF). Guo et al (2014) developed this to construct a TDF based on a form of superellipse (Norato 2018) that modeled structural components directly. Rectangular shaped components were modeled using the component mid-point, length, width, and angular orientation.…”

A well connected, locally-oriented and efficient multi-scale topology optimization (EMTO) strategy

“…This research field now comprises a vast variety of methods: homogenization (Bendse and Kikuchi 1988), Solid Isotropic Material with Penalization (SIMP) (Bendse 1989), evolutionary methods (Xie and Steven 1993), level set methods (Wang et al 2003), moving morphable components (MMC) (Guo et al 2014), generalized geometry projection (GGP) (Coniglio et al 2019), among others. Those methods are described in more detail in Xia et al (2018) and Norato (2018).…”

A MATLAB code for topology optimization using the geometry projection method

“…The foregoing formulation can be used for geometric components of any shape that are made of a single, isotropic material, so long as a signed distance and its derivatives with respect to the geometric parameters can be computed. For instance, this scheme has been employed to design structures made of bars (Norato et al 2015), flat plates (Zhang et al 2016a), curved plates bent along a circular arc (Zhang et al 2018), and supershapes (Norato 2018), which are a generalization of hyperellipses with variable symmetry. The case of multi-material structures, where components can be made of one out of a given set of isotropic materials, and where the optimizer simultaneously determines the optimal components layout and the best material for each component, requires a different strategy to combine the geometric primitives and to interpolate the properties of the different materials (cf.…”