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Flexoelectricity,which is a novel electromechanical coupling effect between the strain gradient and electric polarization,can have a significant impact on the functional properties of nanostructures and is thus of considerable interest for applications.This electromechanical coupling effect exists in any insulators,unlike piezoelectricity,which arises only in noncentrosymmetric materials.We developed a general and unified first-principles theory of flexoelectricity and a practical supercell-based method,which allow us to compute this effect directly in the context of density-functional calculations and design flexoelectric devices.A macroscopic theory is also developed to investigate the flexoelectric effect on the materials properties.