Variational Inequalities and their Applications in the Predictive Modeling of Heterogeneous Media

本项目的研究目标是建立一个基本的认识和方法来预测包括沉淀固体、多相多功能复合材料和自组装晶体在内的非均质介质的性质，并将这些知识用于非均质介质的新模型和复合材料的优化设计。复合材料在日常生活中被广泛使用，从塑料袋和混凝土(用作建筑材料)到波音787梦幻客机(其机身由碳复合材料制成)。今天研究人员面临的挑战包括将这一成功扩展到智能/多功能复合材料。这些复合体具有多种功能，可以受到多种外部刺激的刺激并对其做出反应。多功能复合材料的性能主要由组成相的微观结构决定。本研究系统地研究了微结构如何影响多功能复合材料的性能，以及对于期望的性能，什么是最优的微结构，例如磁电耦合系数。这些预测建模和优化设计问题是通过一种逆方法解决的：微结构是被设计为寻找微结构和材料性能之间的定量关系的“先验”。此外，还将开发一个数值工具包，以计算各种应用的最佳微结构。电池和涡轮发动机等许多技术的瓶颈在于新型多功能材料的开发。这项研究的结果可用于为这些技术开发新的多功能复合材料和新的制造工艺，这些技术有可能缓解当前能源危机的某些方面。在教育方面，将推出一个互动可视化演示项目，以提高学生学习工程学的兴趣和责任感。

The research objective of this project is to create a fundamental understanding and methods for predicting the properties of heterogeneous media including precipitated solids, multiphase multifunctional composites and self-assembled crystals, and to exploit such knowledge for new models for heterogeneous media and optimal design of composite materials. Composites are broadly used in daily life ranging from plastic bags and concrete (used as a construction material) to the Boeing 787 Dreamliner whose fuselage is made of carbon composites. The challenge facing researchers today consists of extending this success to smart/multifunctional composites. These are composites that serve several functions and can be excited by and respond to multiple external stimuli. The properties of multifunctional composites are predominantly determined by the microstructure of the constituent phases. The present research systematically addresses how microstructures influence the properties of multifunctional composites and what are the optimal microstructures for a desired property, e.g., the magnetoelectric coupling coefficient. These predictive modeling and optimal design problems are addressed by an inverse method: microstructures are "a priori" designed to find the quantitative relation between microstructures and material properties. Also, a numerical toolset will be developed to compute optimal microstructures for various applications. The bottleneck of many technologies, e.g., batteries and turbine engines, hinges on the development of new multifunctional materials. The results of this research can be used to develop new multifunctional composites and new manufacturing processes for these technologies, which have the potential to mitigate some aspects of the current energy crisis. Educationally, an interactive visualization demonstrations project will be launched to promote students' interest and responsibility to learn engineering.