Topology Optimization and Effective Properties of Composites

9704813 格拉博夫斯基 本研究的目的是探讨均匀化、复合材料及其有效性质的三个问题。第一个问题将结合联合收割机均匀化理论、复合材料有效模量的界和Young测度技术来研究具有弱间断目标泛函的不适定优化问题的松弛。该方法产生的一类二次泛函的答案。 第二个计画将使用结合偏微分方程理论与变分原理的新技术，寻求多晶复合材料有效行为的精确关系。该方法在至少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和渗流阈值附近的随机电阻网络之间的相变之间的类比，另一方面利用最近关于远离临界点的高对比度连续复合材料的结果。 复合材料是看起来均匀的介质，但实际上在显微镜下观察时具有复杂的结构。这些材料正在进入我们的日常生活，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。复合材料的思想是将其简单但不完美的成分的有益特性联合收割机结合起来。这里的困难在于，复合材料的性质（称为有效性质）不仅取决于你混合的是什么，还取决于你如何混合它，换句话说，它们取决于你在显微镜下看到的几何形状（称为微观几何形状）。因此，我们的目标是指定哪些微几何形状可以产生最好的复合材料，前提是您使用相同的材料，并按规定的比例。一般来说，这是一个不适定问题。也就是说，它可能无法通过最强大的计算机以原始形式解决。这个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（成为适定问题）。该建议的第二部分将寻找和研究多晶复合材料的某些有效性质不依赖于微观几何形状的非常特殊的情况。 这项工作可能有应用的水下传感器的设计基于压电多晶体。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，组成复合物的两种成分具有极其不同的性质（如空气和岩石，或盐水和冰）。 对这种复合材料的性质进行建模是一个艰巨的数学问题。这方面的进展将加强我们的理解，使我们能够更好地控制各种问题， 从多孔岩石中提取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).