Topology Optimization and Effective Properties of Composites

本课题旨在研究与均质化、复合材料及其有效性能相关的三个问题。第一个问题将结合齐次化理论、复合函数有效模的界和杨氏测量技术来研究具有弱不连续目标泛函的不适定优化问题的松弛性。该方法给出了一类二次泛函的答案。第二个项目将使用一种结合PDE理论和变分原理的新技术来寻找多晶复合材料有效行为的确切关系。该方法至少在一个这样的问题上是成功的。第三个项目将研究高对比度复合材料在渗流阈值附近的临界行为，一方面利用伊辛铁磁体的相变与渗流阈值附近的随机电阻网络之间的类比，另一方面利用远离临界点的高对比度连续复合材料的最新结果。复合材料是一种看起来均匀但实际上在显微镜下具有复杂结构的介质。这些材料正在进入我们的日常生活中，如滑雪板、高尔夫球杆、汽车、飞机、计算机、水泥、建筑物和桥梁的建筑部件、传感器等等。复合材料的概念是将其简单但不完美的成分的有益特性结合起来。这里的困难在于复合材料的特性（称为有效特性）不仅取决于混合的是什么，还取决于混合的方式。换句话说，它们取决于你在显微镜下看到的几何形状（称为微观几何）。因此，我们的目标是在规定比例使用相同材料的情况下，确定哪种微几何形状能产生最佳的复合材料。一般来说，这是一个不适定问题。也就是说，最强大的计算机可能无法以原始形式解决这个问题。该建议的第一部分旨在将问题重新表述为适合在计算机上解决的形式（变成一个适定问题）。本提案的第二部分将寻找和研究非常特殊的情况下，多晶复合材料的一些有效性能不依赖于微观几何结构。本研究对基于压电多晶的水下传感器的设计具有一定的应用价值。该提案的最后一部分构成了一个长期项目，研究“自然的复合材料”，如多孔岩石或海冰（由冰和盐水包裹体组成）。这里的问题是，组成混合物的两种成分具有截然不同的性质（比如空气和岩石，或者盐水和冰）。对这种复合材料的性能进行建模是一个艰巨的数学问题。这方面的进展将增强我们的理解，并使我们能够更好地控制各种各样的问题，从从多孔岩石中开采石油到全球气候变化（这在很大程度上取决于北极和南极海冰的变化）。

9704813 Grabovsky This proposal aims to investigate three problems related to homogenization, composite materials and their effective properties. The first problem will combine homogenization theory, bounds on the effective moduli of composites and Young measure techniques in order to study the relaxation of the ill-posed optimization problems with weakly discontinuous objective functionals. The method yields an answer for a class of quadratic functionals. The second project will seek out the exact relations for the effective behavior of polycrystalline composites using a new technique that combines PDE theory and variational principles. The method is successful in at least one such problem. The third project will study the critical behavior of high contrast composites near the percolation threshold by utilizing the analogy between the phase transitions in Ising ferromagnets and the random resistor networks near the percolation threshold on the one hand, and recent results on the high contrast continuum composites away from the critical point on the other. Composite materials are media that look homogeneous but in fact have complex structure when viewed under a microscope. These materials are finding their way into our everyday lives in objects such as skis, golf clubs, automobiles, aircraft, computers, cement, construction components of buildings and bridges, sensors and many many more. The idea of composite materials is to combine beneficial properties of its simple but imperfect constituents. The difficulty here is that the properties of composites (called effective properties) depend not only on what you mix but on how you mix it. In other words, they depend on the geometry of what you see under a microscope (called microgeometry). Therefore the goal is to specify which microgeometries produce the best composites provided you use the same materials in prescribed proportions. In general this is an ill-posed problem. That is, it may not be solved in its raw form by the most powerful computers. The first part of this proposal aims to reformulate the problem into a form suitable for solution on a computer (into a well-posed problem). The second part of this proposal will seek out and investigate the very special cases when some effective properties of polycrystalline composites do not depend on the microgeometry. This work may have applications for the design of underwater sensors based on piezoelectric polycrystals. The last part of this proposal constitutes a long term project on studying "nature's composites" like porous rocks or sea ice (consisting of ice with brine inclusions). The problem here is that the two constituents comprising the composite have extremely different properties (like air and rock, or brine and ice). Modeling the properties of such composites presents a formidable mathematical problem. The progress here would enhance our understanding and give us more control of a diverse array of problems ranging from extraction of oil from porous rocks to global climate change (that depends to a great extent on what is happening to Arctic and Antarctic sea ice).