弯电纳米能量采集器无网格法模拟与水平集拓扑优化

Micro-/nano electromechanical systems (MEMS/NEMS) have demonstrated many important applicaitions in areopsace engineering, medicine, military engineering and communications. The competition in the development of MEMS/NEMS is intense between many countries. Flexoelectricity is the generation of electric polarization under strain gradient, or the mechanical deformation subject to electric gradient (converse-flexo). Flexoelectric nano energy harvester has recently attracted intense research interests. Flexoelectricity is related to the conventional piezoelectricity, but fundamentally different. Flexoelectricity is dependent on the gradient of strain while the latter is only related to the magnitude of strain. Flexoelectricity grows dominantly in energy density when scale reduces to submicro or nano, meaning the promise of enabling self-powered nano device such as body implant and small-scale wireless sensor. Among the emerging applications of flexoelectricity, energy harvester is the basic front device in nano-system of wide technological impact. Despite the advantages offered by flexoelectricity, research in this field is still in germination. Laboratory tests are limited in measuring, explaining and quantifying the phenomenon. It is of upmost importance to develop numerical modelling methods for that can be used to explain the mechanism complementarily. Since the governing equations of flexoelectricty are fourth-order partial PDEs which require second (spatial) derivatives in the weak form and hence a numerical method of C1 discretization is required. Meshless methods are particularly suitable for modelling such type of problem. In the proposed project, meshless methods will be used to study the surface piezoelectricity and surface elasticity of flexoelectric materials. The level set methods will be used to for the topological optimization of flexoelectric nano energy harvester. The project is expected to have great impact in both theory development of meshless methods.

微机电器件在航天、医学、军事和通讯等应用领域具有极其重要的地位，各大国在此领域竞争激烈，其中微机电器件的自供电是目前发展瓶颈问题之一。弯电(flexoelectric)纳米能量采集器因具有解决微机电器件自供电问题的潜力，近年来引起广泛研究关注。与传统压电有本质区别，弯电是材料在应变梯度作用下产生电极化，而压电仅与应变有关。在纳米尺度，弯电可产生极高的能量密度，远大于压电效应。目前弯电研究尚处于起步，主要集中在实验研究。然而现有的实验结果在解释物理现象、量测和定量分析纳观物理量上存在困难，因此亟待发展数值模拟理论和方法以揭示机理、互补研究。弯电问题最终求解四阶偏微分方程，需采用高阶连续C1数值方法，无网格法在此方面独具优势。本项目采用无网格法研究表面弯电能和表面弹性能尺度效应，探索弯电能量采集结构拓扑优化过程动边界问题的水平集-无网格分析基础理论，具有重要理论价值和应用前景。

本项目均按计划实施，已经完成研究目标。项目在实施过程中取得了一系列理论与计算方法上的创新。首先，已经将无网格应用到纳米压电能量收集装置的力电耦合问题分析中，研究了结构形状优化和拓扑优化对压电结构机电性能的影响。第二，对弯曲电效应问题和大变形分析的基本理论和计算公式进行推导；使用基于梯度的形状优化方法和基于结构敏感度的拓扑优化方法实现了纳米压电结构的优化设计。第三，项目组成员提出了一种基于紧密支撑径向基函数（CSRBF）的无网格方法。传统的有限元方法（FEM）无法用于分析柔性电结构，因为控制柔性电的四阶偏微分方程需要位移场的C1连续性，分析考虑了表面效应的大变形挠曲电梁。首次实现考虑表面效应与大变形分析的挠曲电纳米结构无网格法工作。与复杂的混合有限元法相比，具有更高阶连续性的无网格形状函数更具优势，主要是因为无网格公式只需要离散位移和电势场，这种基于CSRBF的无网格公式可以解决柔电结构中的几何非线性问题。第四，项目对三维挠曲电程序的IGA等几何分析方法的实现进行了开源和论文发表，见项目成果附录。项目执行期间，进展顺利，共发4篇项目资助论文，取得了丰富的研究成果，培养博士生2名，硕士生1名。