Adaptive Finite Element Methods for Multiscale Problems Governed by Geometric PDE

几何偏微分方程控制多尺度问题的自适应有限元方法

• 批准号: 0807811
• 负责人: Ricardo Nochetto
• 金额: $ 51.01万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807811&HistoricalAwards=false
• 关键词: Adaptive; Finite; Element; Methods; Multiscale

NochettoDMS-0807811 The adequate numerical treatment of continuum phenomenaexhibiting disparate space-time scales is a formidablemathematical and computational challenge. Modern algorithmsshould be able to resolve fine scales for certain physicalquantities without overresolving others, thereby optimizing thecomputational effort and making realistic 3d simulationsfeasible. Mathematical models in biophysics (such asbiomembranes in both fluid and gel state), in materials science(such as crystal surface morphologies and bilayer actuators), andin shape optimization (including both fluid and solid mechanics)are typical yet quite distinct examples addressed in thisproject. Their multiscale structure manifests in rapidtransients and slow motion regimes in time, as well as point andline singularities (interfaces) and thin layers in space. Largedomain deformations, perhaps leading to topology changes, isanother intriguing feature. The goal of this project is todesign, test, and analyze reliable and efficient adaptive finiteelement methods for these multiscale problems, most governed bygeometric partial differential equations, with space-time errorcontrol based on a posteriori error estimation. This projectbuilds upon, and in fact extends and enhances, the work of theprior NSF Grant DMS-0505454. It is organized in a number ofsmall but interrelated topics permeated by the roles of geometryand adaptivity throughout, from issues in numerical analysis andcomputation to applications. This project blends quite delicate analytical,computational, and modeling issues in several areas of researchof strategic importance such as nanotechnology, materials andmanufacturing, biotechnology, and high performance computing. Designing smart electro-mechanical devices as well asunderstanding key functions of lipid membranes in livingorganisms, both at the nano and microscales, require powerful andflexible computational algorithms capable of dealing with largeshape variations. This project develops such tools. It is acollaborative and interdisciplinary project involving a number ofscientists in the US (four of them engineers and physicists) andabroad, as well as several students and postdocs. Because asubstantial effort is devoted to education and human resourcedevelopment, most resources are used to partially supportstudents and postdocs.

NOCHETTODMS-0807811连续现象的足够数值处理不同的时空尺度是一种正式的数学数学和计算挑战。 现代算法应该能够解决某些物理水平的精细尺度，而不会过度解决其他物理，从而优化了计算的努力并制作现实的3D模拟。 材料科学（例如晶体表面形态和双层执行器）中的生物物理学（例如流体和凝胶状态的生物膜）中的数学模型是典型但在TheProjection中所解决的典型例子。 它们的多尺度结构在时间上以快速变形和慢动作状态表现出来，以及空间中的点和线奇点（接口）和薄层。 largedomain变形，可能导致拓扑变化，是其他有趣的功能。 该项目的目的是针对这些多尺度问题的可靠和有效的自适应有限元方法，大多数控制的副局部差分方程，并根据后验误差估计，可靠，有效的适应性有限元方法。 该项目建设在Prior NSF Grant DMS-0505454上的工作扩展和增强。 它是在许多小小的组织中组织的，但相互关联的主题渗透到整个几何和适应性的作用，从数值分析和计算的问题到应用程序。 该项目在战略重要性的几个领域（例如纳米技术，材料和制造，生物技术和高性能计算）中融合了非常精致的分析，计算和建模问题。设计智能的电力机械设备，以及在纳米和显微镜的Livingorshism中脂质膜的关键功能，都需要能够处理大量变化的功能强大且富裕的计算算法。 该项目开发了这样的工具。 它是学科和跨学科的项目，涉及美国的许多科学家（其中四个工程师和物理学家）和多个学生和博士后。 由于Asubstantial努力致力于教育和人力资源开发，因此大多数资源都用于部分支持学生和博士后。