Superomniphobic surfaces display both superoleophobic and superhydrophobic properties, having a contact angle greater than 150 for both water and oil droplets. In this work, lattice Boltzmann simulations On droplets impacting the surface textures of various topologies are performed to understand the mechanism of how the superomniphobic properties can be achieved by optimizing the geometry of re-entrant surfaces and the inherent hydrophobicity of substrates. Detailed kinetics for droplet impinging is analyzed for both liquid impalement and emptying, showing distinct dependences on geometrical details of re-entrant surfaces. The origins of the enhanced stability of Cassie states are ascribed to (i) the barrier of the Cassie-to-Wenzel transition for the impalement process, (ii) the driving force for liquid receding in the emptying process, and (iii) the contact line pinning from the entrance effect and the edge effect. Finally, we check the strategies proposed here by designing a new re-entrance structure that possesses an excellent property in maintaining the droplet Cassie state.