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

职业：高对比度媒体的多尺度建模、分析和仿真

• 批准号: 1350248
• 负责人: Jiwen He
• 金额: $ 42万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1350248&HistoricalAwards=false
• 关键词: CAREER; Multiscale; Modeling; Analysis; Simulation

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.

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）在其大多数属性中表现出巨大的空间变异性，例如水力和电导率以及介电导率。 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、悬浮液以及医学和地球物理成像等广泛的应用领域中遇到，因此近年来它们在数学界受到越来越多的关注。 研究人员研究的高对比度复合材料表现出额外的特征，即几何性质，即异质域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含在该项目的工作中。 在数学上，高对比度介质由偏微分方程描述，其系数在域中具有极大的值范围。 研究人员研究的此类问题也具有快速变化的系数，这意味着它们的波动长度远小于复合材料占据的域的大小。 这些问题需要在制定多尺度模型设计策略方面付出特别努力。 该领域的最新进展导致了许多方法的发展，这些方法主要是针对问题参数有界振荡的问题而设计的。 然而，系数具有高纵横比的问题对开发解析和数值求解方法提出了额外的挑战。 因此，人们寻求对于问题参数而言稳健的技术。 研究人员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异质流。