  The design of specified nonlinear mechanical responses into a structure or
material is a highly sought after capability, which would have a significant
impact in areas such as wave tailoring in metamaterials, impact mitigation,
soft robotics, and biomedicine. Here, we present a topology optimization
approach to design structures for desired polynomial nonlinear behavior,
wherein we formulate the problem in such a way as to decouple the nonlinear
response from the stiffness. We show results across qualitatively different
polynomial behaviors while achieving a high degree of precision, creating a
path toward analytically tractable nonlinear dynamical systems. The approach
enables access to previously difficult to design for, or hitherto unachieved,
nonlinear behavior via optimized structures, which can furthermore be
incorporated as unit cells of designer materials with tailored nonlinear
properties.
