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

GORBDMS-1350248许多天然和人造材料(一方面是多孔介质和生物组织，另一方面是用于技术的复合材料)在大多数性能上表现出巨大的空间变异性，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、医学和地球物理成像等广泛的应用领域都有应用，近年来它们在数学界受到了越来越多的关注。研究人员研究了具有几何性质的高对比度复合材料，即非均质区域要么具有高度集中的非均质性，要么包含多个尺度(即，与材料的某些长度特性相比，其非均质性非常小)。研究生和本科生被包括在该项目的工作中。在数学上，高对比度介质由偏微分方程式描述，其系数在域中具有极大的值范围。研究人员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占据的区域大小小得多的长度尺度上波动。这些问题需要特别努力来制定设计多尺度模型的战略。该领域的最新进展导致了许多方法的发展，这些方法主要是针对问题参数有界振荡的问题而设计的。然而，系数具有大长宽比的问题给发展解析和数值解方法带来了额外的挑战。因此，人们寻求关于问题参数的健壮技术。研究人员开发了有效的分析和数值工具，用于描述材料性质变化很大和几何形状复杂的介质中的强烈非均质流动。

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.