Dynamics, singularities and asymptotics of higher order PDEs

高阶偏微分方程的动力学、奇异性和渐进性

基本信息

  • 批准号:
    1516753
  • 负责人:
  • 金额:
    $ 16.5万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2015
  • 资助国家:
    美国
  • 起止时间:
    2015-08-01 至 2019-07-31
  • 项目状态:
    已结题

项目摘要

This award supports the research program of the Principal Investigator (PI) related to the applied science field of Micro-Electro-Mechanical Systems (MEMS), an important technology for microscopic sensors and actuators (with applications in the automotive industry, consumer electronics, medical and health technologies, and a number of other areas). The ability to manipulate and predict the behavior of dynamic processes on microscopic scales is a necessity for the design of modern nanotechnology. This can be challenging because intuition of processes on the macro scale does not necessarily translate to very small spatial scales. Mathematical modeling is an essential tool for bridging this gap; however, the relevant equations are highly complex. In this project, the PI and his students will develop analytical and computational tools for studying these complex mathematical models. The goal is to establish a set of mathematical methodologies for practitioners that can inform the engineering of these important technologies. These methodologies will also be useful to many mathematical researchers studying problems from a variety of related fields, such as Fluid Mechanics, Pattern Formation, and Mathematical Biology.Mathematical modeling of nanotechnology requires a coupling of multiple physical theories, in particular those of elasticity and electrostatics. This gives rise to models that feature coupled systems of non-linear and higher order (greater than two) partial differential equations. The complex nature of these equations means that many existing tools, in particular those based on maximum and comparison principles, are not applicable. Without such constraining principles, these systems can exhibit rich and unexpected patterning behaviors. To study these behaviors and their implications for the engineering of micro-devices, the PI will develop novel singular perturbation methods and computational techniques. The particular focus of this project is studying singular solutions of these systems, mainly in the form of finite-time singularities and sharp interfaces. The PI will focus on obtaining both formal and rigorous results regarding the location, multiplicity, and local dynamics of these singular solution structures. The analytical arm of this project will be coupled to highly accurate and adaptive numerical simulations developed by the PI and his students.
该奖项支持首席研究员(PI)与微电子机械系统(MEMS)应用科学领域相关的研究计划,这是一项重要的微型传感器和执行器技术(在汽车工业、消费电子、医疗和健康技术以及许多其他领域中的应用)。能够在微观尺度上操纵和预测动态过程的行为是现代纳米技术设计所必需的。这可能是具有挑战性的,因为对宏观尺度上的过程的直觉并不一定转化为非常小的空间尺度。数学建模是弥合这一差距的重要工具;然而,相关的方程式非常复杂。在这个项目中,PI和他的学生将开发分析和计算工具来研究这些复杂的数学模型。目标是为从业者建立一套数学方法,以便向工程人员提供这些重要技术的信息。这些方法也将有助于许多数学研究人员研究来自不同相关领域的问题,如流体力学、图案形成和数学生物学。纳米技术的数学建模需要多种物理理论的耦合,特别是弹性理论和静电学理论。这导致了以非线性和高阶(大于两个)偏微分方程组的耦合系统为特征的模型的产生。这些方程的复杂性意味着许多现有的工具,特别是那些基于最大值和比较原则的工具,并不适用。如果没有这样的约束原则,这些系统可能会表现出丰富和意想不到的图案化行为。为了研究这些行为及其对微器件工程的影响,PI将开发新的奇异摄动方法和计算技术。这个项目的重点是研究这些系统的奇异解,主要是以有限时间奇性和尖锐界面的形式。PI将专注于获得关于这些奇异解结构的位置、多重性和局部动力学的形式和严格的结果。这个项目的分析臂将与PI和他的学生开发的高精度和自适应的数值模拟相结合。

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

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Alan Lindsay其他文献

Stochastic Simulation of Close-Contact Dynamics in Immune Recognition
  • DOI:
    10.1016/j.bpj.2019.11.1726
  • 发表时间:
    2020-02-07
  • 期刊:
  • 影响因子:
  • 作者:
    Jonathan M. Morgan;Alan Lindsay;Omer Dushek;Johannes Pettmann
  • 通讯作者:
    Johannes Pettmann
The developing pattern of Australian tertiary education: An analysis and critique of three reports
  • DOI:
    10.1007/bf02192552
  • 发表时间:
    1981-03-01
  • 期刊:
  • 影响因子:
    3.200
  • 作者:
    Alan Lindsay
  • 通讯作者:
    Alan Lindsay
A Lightweight Artificial Cognition Model for Socio-Affective Human-Robot Interaction
用于社会情感人机交互的轻量级人工认知模型
Assessing institutional performance in higher education: a managerial perspective
  • DOI:
    10.1007/bf01676865
  • 发表时间:
    1981-11-01
  • 期刊:
  • 影响因子:
    4.600
  • 作者:
    Alan Lindsay
  • 通讯作者:
    Alan Lindsay
An Interactive Narrative Format for Clinical Guidelines
  • DOI:
    10.1007/s13218-015-0354-3
  • 发表时间:
    2015-02-20
  • 期刊:
  • 影响因子:
    3.600
  • 作者:
    Marc Cavazza;Fred Charles;Alan Lindsay;Jonathan Siddle;Gersende Georg
  • 通讯作者:
    Gersende Georg

Alan Lindsay的其他文献

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{{ truncateString('Alan Lindsay', 18)}}的其他基金

Collaborative Research: MODULUS: Data-Driven Discovery for Mechanisms of Nuclear Dynamics and Scaling
合作研究:MODULUS:数据驱动的核动力学和尺度机制发现
  • 批准号:
    2052636
  • 财政年份:
    2021
  • 资助金额:
    $ 16.5万
  • 项目类别:
    Standard Grant
New Trends in Localized Patterns in Partial Differential Equations: Mathematical Theory and Applications to Physics, Biology, and the Social Sciences
偏微分方程定域模式的新趋势:数学理论及其在物理、生物学和社会科学中的应用
  • 批准号:
    2013192
  • 财政年份:
    2020
  • 资助金额:
    $ 16.5万
  • 项目类别:
    Standard Grant
Modeling and Stochastic Simulation of Close-Contact Dynamics in Immune Recognition
免疫识别中近距离接触动力学的建模和随机模拟
  • 批准号:
    1815216
  • 财政年份:
    2018
  • 资助金额:
    $ 16.5万
  • 项目类别:
    Standard Grant

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进化偏微分方程的渐近性和奇点
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