A smoothed natural neighbour Galerkin method for flexoelectric solids

In this paper, a smoothed natural neighbour Galerkin method is developed for modeling flexoelectricity in dielectric solids. The domain integrals in the weak form are implemented on the background Delaunay triangle meshes. Each Delaunay triangle is divided into four sub-domains. In each sub-domain, by introducing the gradient smoothing technique, the rotation gradients, and the electric field gradients can be represented as the first-order gradients of the displacement and the electric potential, respectively. Thus, the continuity requirement for the field variables is reduced from C1 to C0, and the integrals within the sub-domains are converted to the line integrals on the boundary. Then, the field variables are approximated via the non-Sibsonian partition of unity scheme, which enables the direct imposition of the essential boundary conditions. The proposed method is validated through examples with analytical solutions. Results show that the numerical solutions agree well with the analytical solutions.