A discontinuous Galerkin method based isogeometric analysis framework for flexoelectricity in micro-architected dielectric solids

TL;DR

The paper addresses the challenge of solving fourth-order flexoelectric equations in multi-patch architected dielectric solids by coupling a discontinuous Galerkin (DG) method with isogeometric analysis (IGA). It enforces $C^1$ continuity weakly across patch boundaries via an interior-penalty term, while exploiting intra-patch $C^1$ smoothness of NURBS to reduce interfacial complexity. The approach is validated against analytical results and applied to 2D truss lattices to reveal enhanced flexoelectric responses and favorable size-scaling behavior compared with solid geometries, under both direct and converse effects. The findings suggest that architected micro-structures can scale flexoelectricity toward meso- and macro-scale dielectric materials, enabling universal electromechanical responses in engineered solids.

Abstract

Flexoelectricity - the generation of electric field in response to a strain gradient - is a universal electromechanical coupling, dominant only at small scales due to its requirement of high strain gradients. This phenomenon is governed by a set of coupled fourth-order partial differential equations (PDEs), which require $C^1$ continuity of the basis in finite element methods for the numerical solution. While Isogeometric analysis (IGA) has been proven to meet this continuity requirement due to its higher-order B-spline basis functions, it is limited to simple geometries that can be discretized with a single IGA patch. For the domains, e.g., architected materials, requiring more than one patch for discretization IGA faces the challenge of $C^0$ continuity across the patch boundaries. Here we present a discontinuous Galerkin method-based isogeometric analysis framework, capable of solving fourth-order PDEs of flexoelectricity in the domain of truss-based architected materials. An interior penalty-based stabilization is implemented to ensure the stability of the solution. The present formulation is advantageous over the analogous finite element methods since it only requires the computation of interior boundary contributions on the boundaries of patches. As each strut can be modeled with only two trapezoid patches, the number of $C^0$ continuous boundaries is largely reduced. Further, we consider four unique unit cells to construct the truss lattices and analyze their flexoelectric response. The truss lattices show a higher magnitude of flexoelectricity compared to the solid beam, as well as retain this superior electromechanical response with the increasing size of the structure. These results indicate the potential of architected materials to scale up the flexoelectricity to larger scales, towards achieving universal electromechanical response in meso/macro scale dielectric materials.

Figures (9)

• Figure 1: Schematic of weak enforcement of $C^1$ continuity in a two-patch domain IGA. (a) The physical domains of an L-shaped beam modeled with two patches whose boundary is highlighted as a dashed line. The mapping of the parametric domain to the physical domain, defined by the knot vectors $\Xi_1=\Xi_2=\left[0, 0, 0, 0, 1/2, 1, 1, 1, 1\right]$, is shown in the inset. (b) NURBS basis function for the two patches, exhibiting $C^0$ continuity at the boundary, and (c) the derivatives of NURBS basis functions, which are discontinuous at the patch boundaries. (d) Representation of different domains, boundaries, and normal vectors used in the implementation of interior penalty-based discontinuous Galerkin method.
• Figure 2: Weak enforcement of continuity of gradients in a two-patch cantilever example. (a) The geometry and boundary conditions of the cantilever beam, discretized with two patches, (b) variation of axial strain $\epsilon_{11}$ along the length of the beam computed with usual $C^0$ continuous patch interface, (c) variation of $\epsilon_{11}$ with continuity of gradients enforced by interior penalty based DG method with a stabilization parameter $\tau=10^{10}$, and (d) variation of normalized jump with increasing value of $\tau$ on a logarithmic scale, where $\llbracket\epsilon_{11}\rrbracket^n = \llbracket \epsilon_{11} \rrbracket / \epsilon_{11}^{\mathrm{max}}$
• Figure 3: Mesh convergence study for a cantilever beam discretized with (a) two, and (b) four patches, with converged meshes for both the cases shown in the inset.
• Figure 4: Validation of the present model against analytical calculations of electromechanical coupling factor. (a) Variation of electromechanical coupling factor with normalized thickness under unit load and open circuit boundary condition, showing great match with the results for pure flexoelectric and piezoelectric-flexoelectric couplings. (b) variation of electric field across the thickness of the beam, under an applied voltage $\phi=20 \mathrm{V}$ and closed circuit boundary conditions.
• Figure 5: Normalized electric potential $(\phi^\prime=\phi/b \ \mathrm{V/m})$ distribution in the truss lattices based on four selected unit cells under compressive load and open circuit boundary conditions. Single unit cells under compression, termed as UC1 - UC4 (a, c, e, and f, respectively). All the UCs are electrically ground at the bottom and the top surface is subjected to equipotential electrical constraint. High magnitudes of electric potential at the corners can be observed due to high strain gradients. (b,d,f,h) show the normalized electric potential distribution in $5\times5$ tessellations of the UC1-UC4. Due to strain gradients at the intersections of the struts, the electric potential is sustained at the larger sizes.