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RUI: Multiscale Methods, Singular Limits, and Spectral Problems Related to Materials Science

RUI：与材料科学相关的多尺度方法、奇异极限和谱问题

基本信息

批准号：
2110036
负责人：
Silvia Jimenez Bolanos
金额：
$ 20.65万
依托单位：
Colgate University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2110036&HistoricalAwards=false
关键词：
RUI Multiscale Methods Singular Limits

项目摘要

The overarching theme of this Research in Undergraduate Institutions (RUI) project is the development of mathematical tools to study models related to applications in engineering, materials science, and mechanics. The investigator and her collaborators address questions pertaining to elasticity, photonics, and multiscale analysis of heterogeneous media. The considered applications span a wide range of problems in biology, physics, and engineering, these include growing tissues, engineered swelling or shrinking gels, and high-speed computing. The project provides opportunities for undergraduate research experiences. The investigator is committed to the training of undergraduate students, particularly from underrepresented groups in mathematics, through hands-on research, high-quality teaching, and external educational and outreach events. The project comprises four principal research topics. Prestrained Elasticity, where the investigator tackles questions on the derivation of dimensionally reduced models for thin prestrained films, via methods of calculus of variations, in connection with the analysis of existence and regularity of isometric immersions of Riemannian metrics. Bloch Waves in Three-Dimensional High-Contrast Photonic Crystals, where the objective is to develop analysis for the electromagnetic wave propagation inside three-dimensional photonic crystals made from high contrast inclusions, and to investigate the propagation band structure. Multiscale Analysis of a Coupled Problem of Stokes Flow with Magnetic Janus Particles, where a multiscale approach is developed to describe the behavior of Janus particles in a viscous non-conducting fluid in presence of an externally applied magnetic field, and to understand the crucial role of the characteristic features of these two-phase particles. Homogenization for a Variational Problem with a Slip Boundary Condition, where the investigator and her collaborators use the two-scale convergence to obtain the homogenized model for a periodic mixture of an elastic solid with a slightly viscous fluid, under the Navier slip condition posed on their interface.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

本科院校研究（RUI）项目的总体主题是开发数学工具来研究与工程、材料科学和力学应用相关的模型。研究者和她的合作者解决了有关弹性、光子学和异质介质多尺度分析的问题。考虑的应用范围涉及生物、物理和工程领域的广泛问题，包括组织生长、工程膨胀或收缩凝胶以及高速计算。该项目为本科生提供了研究经历的机会。研究者致力于通过实践研究、高质量教学以及外部教育和推广活动，培训本科生，特别是来自数学领域代表性不足的群体的本科生。该项目包括四个主要研究课题。预紧弹性，研究者通过变分法，在分析黎曼度量的等距浸入的存在性和规律性方面，解决了关于预紧薄膜的维数缩减模型的推导问题。三维高对比度光子晶体中的布洛赫波，其目的是对由高对比度内含物制成的三维光子晶体中的电磁波传播进行分析，并研究其传播带结构。Stokes流与磁性Janus粒子耦合问题的多尺度分析，其中开发了一种多尺度方法来描述存在外部外加磁场的粘性非导电流体中Janus粒子的行为，并了解这些两相粒子的特征特征的关键作用。具有滑移边界条件的变分问题的均匀化，研究者和她的合作者使用双尺度收敛来获得弹性固体与微粘性流体在其界面上的Navier滑移条件下的周期性混合物的均匀化模型。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。