Collaborative Research: Efficient, Accurate, and Structure-Preserving Numerical Methods for Phase Fields-Type Models with Applications

合作研究：高效、准确、结构保持的相场型模型数值方法及其应用

• 批准号: 2012269
• 负责人: Cheng Wang
• 金额: $ 15万
• 依托单位: University of Massachusetts, Dartmouth
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012269&HistoricalAwards=false
• 关键词: Collaborative; Research; Efficient; Accurate; Structure

The project will develop computational methods for simulations of phase transformations in materials at the atomic and nanometer scales, aiming at understanding behavior at large time scales. With these simulations, the project will contribute to the understanding of processes such as complex biological growth and cancer, multi-phase active-particle and ionic fluids relevant in biological growth and development, and in the study of other complex phenomena in physics, and material engineering. The focus will be on a particular class of models: gradient flow equations with singular energy potentials. The project will develop theory and software; the codes developed in this project will be scaled up to conduct real-world three-dimensional simulations. In addition, some numerical algorithms to be developed could impact the field of deep learning. This project will provide interdisciplinary applied mathematics and scientific computing training and research experiences for both graduate and undergraduate students at the two institutions involved. In the proposed gradient flow models, a singularity is involved in the energy potential, so that the positivity-preserving property becomes a crucial feature to make the numerical approximation well-defined. In addition, energy stability and optimal rate convergence analysis will be considered for these gradient model with singular energy potential, such as the doubly degenerate Cahn-Hilliard model describing surface diffusion, a new phase field crystal model with heat transport for simulating solidification, a new quasi-incompressible Cahn-Hilliard-Navier-Stokes model for two-phase density mismatched flow, the Poisson-Nernst-Plank model for ionic mixtures, and multi-phase magneto-hydrodynamics equations. Novel finite difference, mixed finite element, and/or Fourier pseudo-spectral spatial approximations will be utilized. Convergence analysis up to the third order temporal accuracy will be investigated in details, which will be the first such work for gradient flows with singular potential. Moreover, numerical solvers for these highly nonlinear schemes will be designed and analyzed, based on the preconditioned steepest decent and Nesterov accelerated methods. Highly efficient adaptive nonlinear multigrid methods based will also be tested and studied in details.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将开发用于模拟原子和纳米尺度材料相变的计算方法，旨在了解大时间尺度下的行为。通过这些模拟，该项目将有助于理解复杂的生物生长和癌症，与生物生长和发育相关的多相活性粒子和离子流体等过程，以及物理学和材料工程中其他复杂现象的研究。重点将放在一类特殊的模型：具有奇异能量势的梯度流方程。该项目将开发理论和软件;该项目中开发的代码将扩大规模，以进行真实世界的三维模拟。此外，一些有待开发的数值算法可能会影响深度学习领域。该项目将为两个有关机构的研究生和本科生提供跨学科的应用数学和科学计算培训和研究经验。在所提出的梯度流模型中，奇异性涉及的能量势，使保正属性成为一个至关重要的功能，使数值近似定义良好。此外，还将对描述表面扩散的双退化Cahn-Hilliard模型、模拟凝固过程的新的考虑热输运的相场晶体模型、两相密度不匹配流动的新的准不可压Cahn-Hilliard-Navier-Stokes模型、离子混合物的Poisson-Nernst-Plank模型、和多相磁流体动力学方程。新的有限差分，混合有限元，和/或傅立叶伪谱空间近似将被利用。收敛性分析的三阶时间精度将详细调查，这将是第一个这样的工作梯度流奇异位。此外，这些高度非线性格式的数值求解器的设计和分析，基于预条件最速下降和Nesterov加速方法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

An Energy Stable Finite Element Scheme for the Three-Component Cahn–Hilliard-Type Model for Macromolecular Microsphere Composite Hydrogels

高分子微球复合水凝胶三组分Cahn-Hilliard型模型的能量稳定有限元方案

• DOI: 10.1007/s10915-021-01508-w
• 发表时间: 2021-04
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Maoqin Yuan;Wenbin Chen;Cheng Wang;Steven M. Wise;Zhengru Zhang
• 通讯作者: Zhengru Zhang

Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshiz equation

Landau-Lifshiz方程二阶半隐式投影法的收敛性分析

• 发表时间: 2021
• 期刊: Applied numerical mathematics
• 影响因子: 2.8
• 作者: Chen, Jingrun;Wang, Cheng;Xie, Changjian
• 通讯作者: Xie, Changjian

A modified Crank-Nicolson scheme for the Flory-Huggins Cahn-Hilliard model

Flory-Huggins Cahn-Hilliard 模型的修正 Crank-Nicolson 方案

• 发表时间: 2022
• 期刊: Communications in computational physics
• 影响因子: 3.7
• 作者: Chen, W.;Jing, J.;Wang, C.;Wang, X.;Wise, S.
• 通讯作者: Wise, S.

A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system

• DOI: 10.1090/mcom/3642
• 发表时间: 2020-09
• 期刊: Math. Comput.
• 作者: Chun Liu;Cheng Wang;S. Wise;Xingye Yue;Shenggao Zhou

Convergence Analysis of Structure-Preserving Numerical Methods Based on Slotboom Transformation for the Poisson-Nernst-Planck Equations

• DOI: 10.4310/cms.2023.v21.n2.a7
• 发表时间: 2022-02
• 期刊: ArXiv
• 作者: Jie Ding;Cheng Wang;Shenggao Zhou