  Topological materials (TMs) are well-known for their topological protected
properties. Phononic system has the advantage of direct observation and
engineering of topological phenomena on the macroscopic scale. For the inverse
design of 3D TMs in continuum medium, however, it would be extremely difficult
to classify the topological properties, tackle the computational complexity,
and search solutions in an infinite parameter space. This work proposed a
systematic design framework for the 3D mechanical higher-order topological
insulators (HOTIs) by combining the symmetry indicators (SI) method and the
moving morphable components (MMC) method. The 3D unit cells are described by
the MMC method with only tens of design variables. By evaluating the inherent
singularity properties in the 3D mechanical system, the classic formulas of
topological invariants are modified accordingly for elastic waves. Then a
mathematical formulation is proposed for designing the helical multipole
topological insulators (MTIs) featured corner states and helical energy fluxes,
by constraining the corresponding topological invariants and maximizing the
width of band gap. Mechanical helical HOTIs with different symmetries are
obtained by this method and verified by full wave simulations. This design
paradigm can be further extended to design 3D TMs among different symmetry
classes and space groups, and different physical systems.
