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Flexoelectricity is an electromechanical coupling between the electric field and the mechanical strain gradient, as well as between the mechanical strains and the electric field gradient, observed in all dielectric materials, including those with centrosymmetry. Flexoelectricity demands C1-continuity for straightforward numerical implementation as the governing equations in the gradient theory are fourth-order partial differential equations. In this work, an alternative collocation-based mixed finite element method for direct flexoelectricity is used, for which a newly developed quadratic element with a high capability of capturing gradients is introduced. In the collocation method, mechanical strains and electric field through independently assumed polynomials are collocated with the mechanical strains and electric field derived from the mechanical displacements and electric potential at collocation points inside a finite element. The mechanical strain gradient and electric field are obtained by taking the directional derivative of the independent mechanical strain and electric field gradients. However, an earlier proposed linear element is unable to capture all mechanical strain gradient components and, thus, simulate flexoelectricity correctly. This problem is solved in the present work by using quadratic shape functions for the mechanical displacements and electric potential with fewer degrees of freedom than the traditional mixed finite element method. A Fortran user-element code is developed by the authors: first, for the linear and, after that, for the quadratic element. After verifying the linear element with numerical results from the literature, both linear and quadratic elements’ behaviors are tested for different problems. It is shown that the proposed second-order collocation-based mixed FEM can capture the flexoelectric behavior better compared to the existing linear formulations.

Direct flexoelectricity is a size-dependent phenomenon, very prominent at smaller scales, that connects the strain gradients and the electric field. The very existence of strain gradients enhances noncentrosymmetry and heightens the interaction between piezoelectricity and flexoelectricity, demanding fully coupled higher-order electromechanical formulations. The numerical instability of the existing finite elements used to model flexoelectricity alone is revealed due to their reliance on the stabilization parameter. Thus, two new finite elements Qu2s2p2l0 (QL0-4) and Qu2s2p2l1 (QL1-16) are proposed for mixed FEM that are numerically robust without any need of such stabilization parameters. Additionally, the existing finite element Qu2s1p2l0 [Q47 in (Deng et al. in J Appl Mech 84:081004, 2017)], is implemented from scratch to replicate known results and benchmark the performance of newly proposed finite elements. To verify the robustness of these elements, various benchmark problems for flexoelectricity in dielectric solids, such as a thick cylinder and truncated pyramid are simulated. The great agreement of the numerical results with the existing ones reflects the foundational nature of the proposed elements. Furthermore, the proposed mixed finite elements were used to successfully analyze cantilever beam and truncated pyramid problems where piezoelectric effects were taken into account for the first time. Current results are intrumental in simulating flexoelectricity and piezoelectricity together to highlight their interactions using newly proposed numerically robust finite elements.

Polycrystal microstructures, with their distinct physical, chemical, structural and topological entities, play an important role in determining the effective properties of materials. Particularly for computational studies, the well-known Voronoi tessellation technique is regularly used for obtaining microstructures. Standard Voronoi tessellations, however, exhibit statistics that are generally far removed from those in real microstructures. Nevertheless, such tessellations can be optimized to obtain certain key features and statistics seen in real microstructures. In this work, we develop the open-source software package OptiMic that enables the generation of optimized microstructures for both finite element as well as atomistic simulations. OptiMic allows for both monodispersive grains as well as irregular grains obtained currently via Voronoi tessellations. These initial microstructures can then be optimized to reflect desired statistical features. A key feature of the tool is that it gives the user extensive control on the optimization process via customizable cost functions. The software currently performs tessellations with the Voronoi method and can be easily extended to include other methods like grain-growth, phase-field etc.