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In this paper, a finite element method based on the ABAQUS code and user subroutines is presented to evaluate the propagation of elastic waves in a phononic crystal slab with Archimedean-like tilings. Three systems composed of cylinders embedded in host materials in square, ladybug and bathroom lattices are considered. Complete and accurate band structures and transmission spectra are obtained to identify the eigenmodes and band gaps. We find that Archimedean-like structures can have some advantages over the traditional square lattices regarding the completeness of the band gaps and their position and width. Also, due to the same square primitive unit cell, the two Archimedean-like lattices have similar wave responses at lower frequencies. The results indicate that the finite element method is efficient and suitable for the band structure computation of the phononic crystal slabs with complex lattice structures.