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.

翻译：挠曲电效应——应变梯度诱导电场产生的现象——是一种普适的机电耦合机制，因其对高应变梯度的依赖性，仅在微小尺度占据主导地位。该现象由一组耦合的四阶偏微分方程组控制，数值求解时要求有限元方法中的基函数具有$C^1$连续性。等几何分析虽能通过高阶B样条基函数满足这一连续性要求，但其应用局限于可通过单一IGA补丁离散的简单几何构型。对于需多个补丁离散的域（如架构材料），等几何分析面临补丁边界处$C^0$连续性的挑战。本文提出基于间断伽辽金法的等几何分析框架，可求解桁架架构材料域中的挠曲电四阶偏微分方程。通过引入基于内惩罚项的正则化技术确保解的稳定性。该公式相较于同类有限元方法具有显著优势：仅需计算补丁边界上的内边界贡献。由于每根杆件仅需两个梯形补丁建模，$C^0$连续边界的数量大幅减少。进一步，我们构建了四种特征单元胞组成的桁架点阵结构，并分析其挠曲电响应。结果表明，相较于实体梁，桁架点阵结构展现出更强的挠曲电效应，且随结构尺寸增大仍能维持这一优越的机电响应。这些发现揭示了架构材料在将挠曲电效应扩展至更大尺度的潜力，有望实现介观/宏观尺度介电材料中的普适机电响应。