Pattern Selection: Growth, Fronts, and Defects

模式选择：生长、前沿和缺陷

基本信息

批准号：
1612441
负责人：
Arnd Scheel
金额：
$ 30.2万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-07-01 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612441&HistoricalAwards=false
关键词：
Pattern Selection Growth Fronts Defects

项目摘要

This research project aims to mathematically describe physical and biological growth processes through direct modeling and analysis of coherent structures. Particular model examples create regular patterns such as stripes or hexagonal lattices of spots. The goal of this research is to find mathematical characterizations of patterns that explain the ubiquity of such regular patterns across nature and, at the same time, to provide recipes for the self-organized manufacturing of microstructure. The project includes research activities that will train graduate and undergraduate students. The project focuses on perturbation methods that are applicable to systems without a technically viable spatial dynamics interpretation, including nonlocal and multi-dimensional problems. It will explore the interaction of the domain growth and pattern-formation and mechanisms of the speed-selection for traveling fronts. In particular, the research aims to develop techniques for the study of perturbation and bifurcation problems in infinite dimensional dynamical systems in the presence of essential spectrum using rigorous core/far-field decompositions. Beyond local perturbation analysis, these strategies will be implemented in numerical continuation algorithms, with a priori and a posteriori error bounds. Particular goals set forth are wavenumber and pattern predictions in growing domains, characterization of invasion speeds in pattern-forming systems, and depinning asymptotics in heterogeneous media.

该研究项目旨在通过对相干结构的直接建模和分析，以数学方式描述物理和生物生长过程。特定的模型示例创建规则图案，例如条纹或六角形斑点格子。这项研究的目标是找到图案的数学特征，以解释这种规则图案在自然界中的普遍存在，同时为微结构的自组织制造提供配方。该项目包括培训研究生和本科生的研究活动。该项目重点关注适用于没有技术上可行的空间动力学解释的系统的扰动方法，包括非局部和多维问题。它将探索域增长和模式形成的相互作用以及行进前沿的速度选择机制。特别是，该研究旨在开发使用严格的核心/远场分解在存在基本谱的情况下研究无限维动力系统中的扰动和分岔问题的技术。除了局部扰动分析之外，这些策略还将在具有先验和后验误差界限的数值连续算法中实现。提出的具体目标是生长域中的波数和模式预测、模式形成系统中入侵速度的表征以及异质介质中的渐进脱钉。