Novel applications of local optimization semi-Cartesian grid for the complex band structure analysis of phononic crystals

•A method for complex band structure computation of phononic crystals is proposed.•Local optimization semi-Cartesian grid is proposed to improve the grid quality near the interface.•Floquet transform is applied to transform the elastic wave equation into quadratic eigenvalue equation.•Complex band structures of phononic crystals with sharp-edged scatterer shapes are computed and analyzed.