  A new density field representation technique called the Bezier 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, Bezier 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 Bezier
curves. Compared to NURBS or B-spline based models, Bezier 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 Bezier 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.
