The flexoelectric effect, coupling polarization and strain gradient as well as strain and electric field gradients, is universal to dielectrics, but, as compared to piezoelectricity, it is more difficult to harness as it requires field gradients and it is a small-scale effect. These drawbacks can be overcome by suitably designing metamaterials made of a non-piezoelectric base material but exhibiting apparent piezoelectricity. We develop a theoretical and computational framework to perform topology optimization of the representative volume element of such metamaterials by accurately modeling the governing equations of flexoelectricity using a Cartesian B-spline method, describing geometry with a level set, and resorting to genetic algorithms for optimization. We consider a multi-objective optimization problem where area fraction competes with four fundamental piezoelectric functionalities (stress/strain sensor/ actuator). We computationally obtain Pareto fronts, and discuss the different geometries depending on the apparent piezoelectric coefficient being optimized. In general, we find competitive estimations of apparent piezoelectricity as compared to reference materials such as quartz and PZT ceramics. This opens the possibility to design devices for sensing, actuation and energy harvesting from a much wider, cheaper and effective class of materials.

翻译：柔性电效应耦合了极化与应变梯度以及应变与电场梯度，是介电材料的普遍特性，但相较于压电效应，其更难被利用，因为需要场梯度且属于小尺度效应。这些缺点可通过设计由非压电基材构成但展现表观压电性的超材料来克服。我们建立了一个理论与计算框架，通过使用笛卡尔B样条方法精确模拟柔性电的控制方程、用水平集描述几何形状并采用遗传算法进行优化，实现了这类超材料代表性体积单元的拓扑优化。我们考虑了多目标优化问题，其中面积分数与四种基本压电功能（应力/应变传感器/执行器）相互竞争。我们计算得到了帕累托前沿，并根据优化的表观压电系数讨论了不同的几何构型。总体而言，与石英和PZT陶瓷等参考材料相比，我们获得了具有竞争力的表观压电性估计。这为利用更广泛、更廉价且高效的材料设计传感、驱动和能量收集器件提供了可能性。