An efficient method for modelling flexoelectricity in soft material toward uncertainty quantification and reliability analysis

本项目的主要目标是开发和实现一个强大的计算不确定性量化方法来研究柔性材料中的挠曲电效应。软材料可以在广泛的现代技术中找到。除其他外，它们可用作：结构材料、泡沫和粘合剂、胶体和颗粒材料、润滑剂和燃料添加剂以及轮胎橡胶。此外，许多生物材料（例如血液、肌肉、牛奶）也可归类为软物质。 由于电极化和应变梯度的耦合作用，在软介质中产生挠曲电。这可以用应变梯度下离子的非均匀位移来解释。由于纳米尺度的软材料可以承受更大的变形，因此预计由于耦合引起的能量转换将大大增强。在这方面，由于挠曲电的机电耦合的数值建模固有地受到各种来源的不确定性，这导致从期望的性能的显著波动。在数值逼近中，不确定性通常与数值方法及其输入参数有关。前者是由物理行为的假设和简化所导致的偏差误差。而后者则与输入参数的随机方差有关。输入的不确定性越高，预测的不确定性就越大。我们将提供一个有效的工具，以支持在大变形的软材料的挠曲电的表征，分析和设计优化。所提出的方法将捕获相关的机电耦合效应的全部统计特性。它还允许在概率空间内预选最佳材料和设计几何形状。将通过计算预期应用中感兴趣的功能量得到（或不得到）满足的概率来评估可靠的性能。 为了解决挠曲电的控制方程，我们将设计一个基于NURBS的伊加制定使用多级离散化的层次结构。与等几何公式，复杂的几何形状可以更容易地建模，而多级层次方案将能够减少计算成本的数值实验中，而不损失的精度。这样，我们将在一组基准测试示例中虚拟测试各种负载条件下的性能。

The main objective of this project is to develop and implement a robust computational uncertainty quantification method to study the effect of flexoelectricity in soft materials. Soft materials can be found in a wide range of the modern technology. Among others, they can be used as: structural materials, foams and adhesives, colloids and granular materials, lubricants and fuel additives, and rubber in tires. In addition, a number of biological materials (e.g. blood, muscle, milk) are also classifiable as soft matter. Flexoelectricity is generated in soft dielectrics as a result of the coupling between electric polarization and strain gradient. It can be explained by the non-uniform displacement of ions under a strain gradient. The energy conversion due to the coupling is expected to largely enhanced since the soft material at nanoscale can sustain larger deformation. In this regards, numerical modeling of the electromechanical coupling due to flexoelectricity is subjected inherently to various source of uncertainty which leads to significant fluctuations from the desired performance. In the numerical approximation, the uncertainties can be generally related to the numerical method and its input parameters. The former is the bias error resulting from the assumptions and the simplifications of the physical behavior. Meanwhile, the latter concerns to the stochastic variance of the input parameters. The higher the uncertainties in the inputs, the more uncertain the predictions will be. We will provide an effective tool to support the characterization, analysis, and design optimization of flexoelectricity in soft materials with large deformation. The proposed method will capture the full statistical properties of the related electromechanical coupling effect. It will allow also the preselection of optimal materials and design geometries within the probability space. Reliable performance will be evaluated by calculating the probability that the functional quantity of interest in the intended application is (or is not) met. To solve the governing equations of flexoelectricity we will devise a NURBS-based IGA formulation using a hierarchy of multilevel discretizations. With the isogeometric formulations, complex geometries can be modeled more easily, while the multilevel hierarchy scheme will enable reducing the computation cost in the numerical experiments without loss of the accuracy. Doing so, we will virtually test the performance under various loading conditions in a set of benchmark examples.