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：柔性电效应将极化与应变梯度以及应变与电场梯度耦合起来，是电介质的普遍特性，但相较于压电效应，由于需要场梯度且属于小尺度效应，因此更难驾驭。通过合理设计由非压电基材构成但具有表观压电性的超材料，可以克服这些缺陷。我们开发了一套理论与计算框架，对这类超材料的代表性体积单元进行拓扑优化：采用笛卡尔B样条方法精确建模柔性电性的控制方程，用水平集描述几何结构，并借助遗传算法进行优化。我们考虑了一个多目标优化问题，其中面积分数与四种基本压电功能（应力/应变传感器/致动器）相互竞争。通过计算获得帕累托前沿，并讨论了根据待优化的表观压电系数而不同的几何结构。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算值。这为利用更广泛、更廉价且更有效的材料类别设计传感、致动和能量收集器件开辟了可能性。