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Abstract Topology optimization has been proved to be an efficient tool for structural design. In recent years, the focus of structural topology optimization has been shifting from single material continuum structures to multimaterial and multiscale structures. This paper aims at devising a numerical scheme for designing bionic structures by combining a two-stage parametric level set topology optimization with the conformal mapping method. At the first stage, the macro-structural topology and the effective material properties are optimized simultaneously. At the second stage, another structural topology optimization is carried out to identify the exact layout of the metamaterial at the mesoscale. The achieved structure and metamaterial designs are further synthesized to form a multiscale structure using conformal mapping, which mimics the bionic structures with “orderly chaos” features. In this research, a multi-control-point conformal mapping (MCM) based on Ricci flow is proposed. Compared with conventional conformal mapping with only four control points, the proposed MCM scheme can provide more flexibility and adaptivity in handling complex geometries. To make the effective mechanical properties of the metamaterials invariant after conformal mapping, a variable-thickness structure method is proposed. Three 2D numerical examples using MCM schemes are presented, and their results and performances are compared. The achieved multimaterial multiscale structure models are characterized by the “orderly chaos” features of bionic structures while possessing the desired performance. 

Topology optimization has been proved to be an automatic, efficient and powerful tool for structural designs. In recent years, the focus of structural topology optimization has evolved from mono-scale, single material structural designs to hierarchical multimaterial structural designs. In this research, the multi-material structural design is carried out in a concurrent parametric level set framework so that the structural topologies in the macroscale and the corresponding material properties in mesoscale can be optimized simultaneously. The constructed cardinal basis function (CBF) is utilized to parameterize the level set function. With CBF, the upper and lower bounds of the design variables can be identified explicitly, compared with the trial and error approach when the radial basis function (RBF) is used. In the macroscale, the ‘color’ level set is employed to model the multiple material phases, where different materials are represented using combined level set functions like mixing colors from primary colors. At the end of this optimization, the optimal material properties for different constructing materials will be identified. By using those optimal values as targets, a second structural topology optimization is carried out to determine the exact mesoscale metamaterial structural layout. In both the macroscale and the mesoscale structural topology optimization, an energy functional is utilized to regularize the level set function to be a distance-regularized level set function, where the level set function is maintained as a signed distance function along the design boundary and kept flat elsewhere. The signed distance slopes can ensure a steady and accurate material property interpolation from the level set model to the physical model. The flat surfaces can make it easier for the level set function to penetrate its zero level to create new holes. After obtaining both the macroscale structural layouts and the mesoscale metamaterial layouts, the hierarchical multimaterial structure is finalized via a local-shape-preserving conformal mapping to preserve the designed material properties. Unlike the conventional conformal mapping using the Ricci flow method where only four control points are utilized, in this research, a multi-control-point conformal mapping is utilized to be more flexible and adaptive in handling complex geometries. The conformally mapped multi-material hierarchical structure models can be directly used for additive manufacturing, concluding the entire process of designing, mapping, and manufacturing.