Collaborative Research: Efficient, Stable and Accurate Numerical Algorithms for a class of Gradient Flow Systems and their Applications

合作研究：一类梯度流系统高效、稳定、准确的数值算法及其应用

基本信息

批准号：
1720440
负责人：
Jie Shen
金额：
$ 13万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720440&HistoricalAwards=false
关键词：
Collaborative Research Efficient Stable Accurate

项目摘要

This project focuses on the development of efficient, stable and accurate numerical algorithms for gradient flow systems which are ubiquitous in modeling of real-world phenomena. It is expected that the research will not only lead to efficient numerical algorithms for a class of problems of current interests, but also contribute to better understandings of some fundamental issues in materials science, biotechnology, and other related fields through numerical simulations. This project will also provide opportunities for the involved graduate and undergraduate students to learn critical skills of computational and applied mathematics and to develop state-of-the-art numerical algorithms for science and engineering applications. The free energies of gradient flow systems usually consist of various nonlinear potentials formulated in diverse complex formats which present a major challenge in the construction of efficient and accurate time discretization schemes. The project aims at overcoming this challenge by using a flexible and robust IEQ approach that enables one to develop time discretization schemes for a large class of gradient flow systems. The goals of this proposal are three folds: (i) to develop a unified numerical framework of time-marching schemes for solving general gradient flow models with high nonlinearity; (ii) to develop efficient numerical schemes for a number of challenging gradient flow models of current interests (e.g., nonlinear coupled multivariable models, nonlocal models, anisotropic models, nonlinear coupled systems that follow various physical principles, tensor based liquid crystal models); (iii) to investigate some fundamental issues of viscoelastic drops on substrates and active liquid crystal droplets using the developed predictive numerical tools. The proposed schemes will lead to numerical predictive tools that extend the capability of mathematical and experimental analysis, and contribute to better understanding of some pressing science and engineering issues related to multi-phase complex fluids.

该项目着重于为梯度流量系统的有效，稳定和准确的数值算法的开发，这些梯度流量系统无处不在，这些算法无处不在。预计该研究不仅会导致一系列目前利益问题的有效数值算法，而且还可以通过数值模拟更好地理解材料科学，生物技术和其他相关领域的一些基本问题。该项目还将为参与的研究生和本科生提供机会，以学习计算和应用数学的关键技能，并为科学和工程应用开发最先进的数值算法。梯度流系统的自由能通常由各种非线性电势组成，这些非线性电势以各种复杂格式提出，在构建有效，准确的时间离散化方案方面构成了重大挑战。该项目旨在通过使用灵活且强大的IEQ方法来克服这一挑战，该方法使人们能够为大型梯度流系统开发时间离散化方案。该提案的目标是三个折叠：（i）开发一个统一的数值框架，以解决具有高非线性的一般梯度流模型；（ii）开发有效的数值方案，用于许多当前兴趣的许多具有挑战性的梯度流模型（例如，非线性耦合的多变量模型，非定位模型，各向异性模型，非线性耦合系统，遵循各种物理原理，基于量液态液晶模型）；（iii）使用开发的预测数值工具研究底物和活性液晶液滴上粘弹性滴的一些基本问题。所提出的方案将导致数值预测工具扩展数学和实验分析的能力，并有助于更好地理解与多相复杂流体相关的一些紧迫的科学和工程问题。