Numerical modeling of flexoelectricity: Status, opportunities and challenges

Flexoelectricity — the induction of electric polarization in response to a mechanical strain gradient — is a universal electromechanical phenomenon that has gained increasing attention for its relevance in nanoscale sensors, actuators, and energy harvesting systems. Unlike piezoelectricity, flexoelectricity is not constrained by material symmetry and thus occurs in all dielectric materials, making it particularly significant at reduced dimensions where strain gradients are amplified. This review presents a comprehensive examination of flexoelectric behavior across one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) material systems, including nanostructures, polymers, ceramics, and functional composites. We highlight recent advances in theoretical frameworks and computational methodologies that have deepened the understanding of flexoelectric coupling mechanisms. Atomistic simulations using Density Functional Theory (DFT) and Molecular Dynamics (MD) have provided crucial insights into polarization response and electronic structure under strain gradients. Finite Element Methods (FEM) enable modeling of flexoelectric effects at the continuum level, linking microscopic and macroscopic responses. Emerging machine learning (ML) techniques and high-throughput screening have further accelerated the discovery and optimization of flexoelectric materials, offering data-driven strategies to predict flexoelectric coefficients and design application-specific materials. The review also discusses key theoretical developments, including tensorial representations, direct and converse effects, and symmetry considerations. By integrating insights from multiscale modeling, computational simulation, and data science, this paper offers a holistic perspective on the current state and future directions of flexoelectricity research. It serves as a critical resource for advancing the modeling and design of next-generation materials and devices based on flexoelectric principles.