Topology optimization design of stretchable metamaterials with Bézier skeleton explicit density (BSED) representation algorithm

Abstract

A new density field representation technique called the Bézier skeleton explicit density (BSED) representation scheme for topology optimization of stretchable metamaterials under finite deformation is proposed for the first time. The proposed approach overcomes a key deficiency in existing density-based optimization methods that typically yield designs that do not have smooth surfaces but have large number of small intricate features, which are difficult to manufacture even by additive manufacturing. In the proposed approach, Bézier curves are utilized to describe the skeleton of the design being optimized where the description of the entire design is realized by assigning thickness along the curves. This geometric representation technique ensures that the optimized design is smooth and concise and can easily be tuned to be manufacturable by additive manufacturing. In the optimization method, the density field is described by the Heaviside function defined on the Bézier curves. Compared to NURBS or B-spline based models, Bézier curves have fewer control parameters and hence are easier to manipulate for sensitivity derivation, especially for distance sensitivities. Due to its powerful curve fitting ability, using Bézier curve to represent density field allows exploring design space effectively and generating concise structures without any intricate small features at the borders. Furthermore, this density representation method is mesh independent and design variables are reduced significantly so that optimization problem can be solved efficiently using small-scale optimization algorithms such as sequential quadratic programming. Numerical optimization results in three typical tests, uniaxial tension, biaxial tension and shear tests are presented by imposing symmetry on a unit cell for lattice structure design. Numerical examples illustrate that the optimized metamaterials have better stretchability and higher stiffness, and hence the proposed method has great potential of impacting the design of future applications that exploit stretchability of structures.