Customized broadband pentamode metamaterials by topology optimization

•A systematic inverse-design strategy for pentamode metamaterials (PMs) is proposed.•Broadband isotropic/anisotropic PMs with customized properties are constructed.•Mechanisms for broadband single-mode range of exclusive longitudinal waves are revealed.•An inverse-design strategy for functional underwater devices using PMs is developed.•Pentamode metasurfaces are achieved to convert broadband propagating wave to evanescent surface mode for noise shielding.