Adaptive Finite Element Methods for Multiscale Problems Governed by Geometric PDE

NochettoDMS-0807811 对表现出不同时空尺度的连续介质现象进行充分的数值处理是一项艰巨的数学和计算挑战。 现代算法应该能够解析某些物理量的精细尺度，而不会过度解析其他物理量，从而优化计算工作并使真实的 3D 模拟变得可行。 生物物理学（例如流体和凝胶状态的生物膜）、材料科学（例如晶体表面形态和双层执行器）和形状优化（包括流体和固体力学）中的数学模型是本项目中讨论的典型但又非常独特的示例。 它们的多尺度结构体现在时间上的快速瞬变和慢速运动状态，以及空间上的点和线奇点（界面）和薄层。 大域变形可能会导致拓扑变化，这是另一个有趣的特征。 该项目的目标是为这些多尺度问题设计、测试和分析可靠且高效的自适应有限元方法，这些问题大多由几何偏微分方程控制，并具有基于后验误差估计的时空误差控制。 该项目建立在先前 NSF Grant DMS-0505454 的工作之上，实际上是扩展和增强的。 它由许多小但相互关联的主题组成，从数值分析和计算问题到应用，贯穿始终的几何和适应性的作用。 该项目融合了纳米技术、材料和制造、生物技术和高性能计算等几个具有战略重要性的研究领域中相当微妙的分析、计算和建模问题。设计智能机电设备以及了解生物体中脂质膜在纳米和微米尺度上的关键功能，需要能够处理大形状变化的强大而灵活的计算算法。 该项目开发了此类工具。 这是一个跨学科的合作项目，涉及美国和国外的许多科学家（其中四名工程师和物理学家）以及一些学生和博士后。 由于大力投入教育和人力资源开发，大部分资源都用于部分支持学生和博士后。

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.