CAREER: Multiscale Modeling, Analysis and Simulation for High-Contrast Media

许多天然和人造材料（例如多孔介质和生物组织，以及技术中使用的复合材料）在其大多数性能上表现出巨大的空间变异性，例如水力和电导率以及介电渗透率。这些被称为高对比度介质。由于这些材料在地下流动、生物工程、悬浮液、医学和地球物理成像等广泛的应用领域中遇到，近年来它们在数学界受到越来越多的关注。研究者研究了具有几何性质的高对比度复合材料，即异质畴要么具有高度集中的不均匀性，要么包含多个尺度（即，与材料的某些长度特征相比，其不均匀性非常小）。研究生和本科生都包括在该项目的工作中。在数学上，高对比度介质是用偏微分方程来描述的，偏微分方程的系数在域内的取值范围非常大。研究者研究的这类问题也具有快速变化的系数，这意味着它们在比复合材料所占据的域的尺寸小得多的长度尺度上波动。这些问题需要在建立设计多尺度模型的策略方面做出特别的努力。该领域的最新进展导致了许多方法的发展，这些方法主要是为问题参数中有界振荡的问题设计的。然而，系数具有高纵横比的问题对解析和数值求解方法的发展提出了额外的挑战。因此，人们寻求对问题参数具有鲁棒性的技术。研究者开发了高效的分析和数值工具，用于描述材料特性和复杂几何形状变化较大的介质中的强非均质流动。

GorbDMS-1350248 Many natural and man-made materials (e.g. porous media and biological tissues on the one hand, and composites used in technology on the other) exhibit vast spatial variability in most of their properties, such as hydraulic and electrical conductivities and dielectric permeability. These are called high contrast media. Because these materials are encountered in a wide range of application areas, such as subsurface flows, bioengineering, suspensions, and medical and geophysical imaging, they have received increased attention within the mathematical community in recent years. The investigator studies high contrast composite materials that exhibit an additional feature, which is of geometric nature, namely, the heterogeneous domain either has highly concentrated inhomogeneities or contains multiple scales (that is, its inhomogeneities are very small compared to some length characteristic of the material). Graduate and undergraduate students are included in the work of the project. Mathematically, high contrast media are described by partial differential equations with coefficients that take on an extremely large range of values in the domain. The investigator studies problems of this type that also have rapidly varying coefficients, meaning that they fluctuate on a length scale that is much smaller than the size of the domain occupied by the composite. These problems require special efforts in establishing strategies for designing multiscale models. Recent advances in the field have led to the development of many approaches that were designed mostly for problems with bounded oscillations in problem parameters. However, problems whose coefficients have high aspect ratio pose additional challenges for developing both analytical and numerical solution methods. One therefore seeks techniques that are robust with respect to problem parameters. The investigator develops efficient analytical and numerical tools for describing strongly heterogeneous flows in media with large variations in material properties and complex geometry.