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.

这项研究在本科机构（RUI）项目中的总体主题是开发与工程，材料科学和力学应用相关模型的数学工具。研究者及其合作者介绍了与异质媒体的弹性，光子学和多尺度分析有关的问题。所考虑的应用在生物学，物理和工程方面涉及广泛的问题，其中包括生长的组织，工程性肿胀或收缩凝胶以及高速计算。该项目为本科研究经验提供了机会。调查人员致力于通过动手研究，高质量的教学以及外部教育和外展活动的数学群体，特别是来自代表性不足的小组的培训。该项目包括四个主要研究主题。在研究人员通过变异的计算方法来解决有关薄膜的尺寸降低模型的推导的问题，该问题与分析Riemannian指标的存在和规则性有关。三维高对比度光子晶体中的Bloch波，其目的是为由高对比度夹杂物制成的三维光子晶体内部的电磁波传播进行分析，并研究传播带结构。对磁性janus颗粒流动的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在存在外部施加磁场的情况下，在存在粘性的非传导流体中的janus颗粒的行为，并了解这些两阶段粒子的特征性特征的关键作用。均质问题的同质化，具有滑动边界条件，研究人员和她的合作者使用二尺度的收敛来获得同质化模型，以定期将弹性固体与略带粘性流体的液体结合在一起，在其界面上带来的Navier Slip条件下，这一奖项反映了NSF的法定任务和审查的范围。