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

GorbDMS-1350248 许多天然和人造材料（例如一方面是多孔介质和生物组织，另一方面是技术中使用的复合材料）在其大多数特性中表现出巨大的空间变化性，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。 由于这些材料在广泛的应用领域，如地下流动，生物工程，悬浮液，医学和地球物理成像，他们在数学界近年来受到越来越多的关注。 研究者研究高对比度复合材料，其表现出额外的特征，其具有几何性质，即，异质域具有高度集中的不均匀性或包含多个尺度（即，其不均匀性与材料的某些长度特征相比非常小）。 研究生和本科生都参加了该项目的工作。 在数学上，高对比度介质由偏微分方程描述，该偏微分方程的系数在域中具有极大范围的值。 研究人员研究的这类问题也具有快速变化的系数，这意味着它们在一个长度尺度上波动，该尺度远小于复合材料所占据的域的大小。 这些问题需要特别的努力，在建立多尺度模型的设计策略。 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中存在有界振荡的问题而设计的。 然而，问题的系数具有高的纵横比提出了额外的挑战，发展的分析和数值求解方法。 因此，寻求技术，是鲁棒的问题参数。 研究人员开发了有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烈非均匀流动。

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.