Nonlinear Multiscale Phenomena: Analysis, Control, and Computation

非线性多尺度现象：分析、控制和计算

• 批准号: 1411808
• 负责人: Ricardo Nochetto
• 金额: $ 99.41万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411808&HistoricalAwards=false
• 关键词: Nonlinear; Multiscale; Phenomena; Analysis; Control

Capturing the essential behavior of nonlinear phenomena with the simplest possible models is of paramount importance in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of devices, and the analysis of both models and algorithms. These crucial aspects of modern research are blended together in this research project, which deals with modeling, formulation, and numerical analysis of physical and biological phenomena at a scale where surface tension competes with bulk effects and could in principle be manipulated (or controlled) to produce scientifically interesting and practically useful dynamical behavior. Applications of the work include nano and microtechnology (such as the design and control of micro electro-mechanical systems (MEMS)), biotechnology (such as the study of biomembranes), and high performance computing (such as the design of novel efficient numerical methods). Results of the work will enhance modeling and prediction capabilities and help educate students and postdocs in exciting, mathematically and computationally challenging, and practically relevant areas of research. This project investigates models, such as biomembranes, ferrofluids, liquid crystals, and bilayer actuators, that are governed by nonlinear geometric partial differential equations defined on deformable domains that are unknown beforehand. Numerical approximation is carried out via adaptive finite element methods, with a posteriori error estimation and multilevel solvers, which allow for the resolution of problems with very disparate space-time scales with relatively modest computational resources. The project will advance understanding of adaptive approximation methods and the role of geometry in key questions concerning:1. Convergence and complexity of adaptive finite element methods (FEM) for elliptic PDE; study of fractional diffusion, hybridizable discontinuous Galerkin methods, hp-FEM and isogeometric methods, and the Laplace-Beltrami operator on parametric surfaces. 2. Design of high order arbitrary Lagrangian-Eulerian methods for parabolic PDE on deformable domains and surfaces. 3. Control of problems involving surface tension and magnetic effects, with or without free boundaries, relevant for device design in technology and biomedicine. 4. Computational modeling and analysis of ferrofluids and liquid crystals; these are technologically useful and mathematically intriguing complex fluids which can be actuated by magnetic and electric fields, and thus manipulated and controlled for specific purposes. 5. Novel FEM for geometric PDE: handling of large deformations with isometry constraints, typical of bilayer actuators, and dealing with fully nonlinear PDE.

用最简单的模型捕捉非线性现象的基本行为在科学和工程中至关重要。这有助于理解基本机制、设计和实施用于设备模拟和控制的高效数值方法以及分析模型和算法。现代研究的这些关键方面在这个研究项目中融合在一起，该研究项目涉及物理和生物现象的建模，制定和数值分析，其规模为表面张力与体积效应竞争，原则上可以操纵（或控制）以产生科学上有趣和实用的动力学行为。 这项工作的应用包括纳米和微技术（如微机电系统（MEMS）的设计和控制），生物技术（如生物膜的研究）和高性能计算（如新的高效数值方法的设计）。这项工作的结果将提高建模和预测能力，并有助于教育学生和博士后在令人兴奋的，数学和计算上具有挑战性的，实际相关的研究领域。本项目研究模型，如生物膜，铁磁流体，液晶和双层致动器，这些模型由预先未知的可变形域上定义的非线性几何偏微分方程控制。数值逼近是通过自适应有限元方法进行的，具有后验误差估计和多级求解器，这使得解决具有非常不同的时空尺度的问题具有相对适度的计算资源。该项目将促进对自适应逼近方法以及几何在以下关键问题中的作用的理解：1。椭圆型偏微分方程自适应有限元方法的收敛性和复杂性;研究分数扩散、可杂交的间断Galerkin方法、hp-FEM和等几何方法以及参数曲面上的Laplace-Beltrami算子。2.变形域和曲面上抛物型偏微分方程的高阶任意拉格朗日-欧拉方法设计。3.控制涉及表面张力和磁效应的问题，有或没有自由边界，与技术和生物医学中的设备设计相关。4.铁磁流体和液晶的计算建模和分析;这些都是技术上有用的和数学上有趣的复杂流体，可以通过磁场和电场驱动，从而操纵和控制特定的目的。5.几何PDE的新FEM：处理具有等距约束的大变形，典型的双层致动器，以及处理完全非线性PDE。