  Utilizing synthetic dimensions generated by spatial or temporal modulation,
topological pumping enables the exploration of higher-dimensional topological
phenomena through lower-dimensional physical systems. In this letter, we
propose a rational design paradigm of fast topological pumping based on 1D and
2D time-modulated discrete elastic lattices for the first time. Firstly, the
realization of topological pumping is ensured by introducing quantitative
indicators to drive a transition of the edge or corner state in the lattice
spectrum. Meanwhile, with the help of limiting speed for adiabaticity to
calculate the modulation time, a mathematical formulation of designing
topological pumping with the fastest modulation speed is presented. By applying
the proposed design paradigm, topological edge-bulk-edge and corner-bulk-corner
energy transport are successfully achieved, with 11.2 and 4.0 times of
improvement in modulation speed compared to classical pumping systems in the
literature. In addition, applying to 1D and 2D space-modulated systems, the
optimized modulation schemes can reduce the number of stacks to 5.3% and 26.8%
of the classical systems while ensuring highly concentrated energy transport.
This design paradigm is expected to be extended to the rational design of fast
topological pumping in other physical fields.
