  De-homogenization is becoming an effective method to significantly expedite
the design of high-resolution multiscale structures, but existing methods have
thus far been confined to simple static compliance minimization. There are two
critical challenges to be addressed in accommodating general cases: enabling
the design of unit-cell orientation and using free-form microstructures. In
this paper, we propose a data-driven de-homogenization method that allows
effective design of the unit-cell orientation angles and conformal mapping of
spatially varying, complex microstructures. We devise a parameterized
microstructure composed of rods in different directions to provide more
diversity in stiffness while retaining geometrical simplicity. The
microstructural geometry-property relationship is then surrogated by a neural
network to avoid costly homogenization. A Cartesian representation of the
unit-cell orientation is incorporated into homogenization-based optimization to
design the angles. Corresponding high-resolution multiscale structures are
obtained from the homogenization-based designs through a conformal mapping
constructed with sawtooth function fields. This allows us to assemble complex
microstructures with an oriented and compatible tiling pattern, while
preserving the local homogenized properties. To demonstrate our method with a
specific application, we optimize the frequency response of structures under
harmonic excitations within a given frequency range. It is the first time that
a sawtooth function is applied in a de-homogenization framework for complex
design scenarios beyond static compliance minimization. The examples illustrate
that multiscale structures can be generated with high efficiency and much
better dynamic performance compared with the macroscale-only optimization.
Beyond frequency response design, our proposed framework can be applied to
other general problems.
