CAREER: Numerical Analysis for Meshfree and Particle Methods via Nonlocal Models
职业:通过非局部模型进行无网格和粒子方法的数值分析
基本信息
- 批准号:2240180
- 负责人:
- 金额:$ 46.71万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2023
- 资助国家:美国
- 起止时间:2023-07-01 至 2028-06-30
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
Meshfree and particle methods are widely used in the computational studies of partial differential equations (PDEs) with applications to many mechanical, hydrodynamical and biophysical processes. They offer many advantages over traditional mesh-based methods, particularly for problems with complex or moving geometries, large deformations of materials, or other singular behaviors of solutions. The goal of this project is to investigate an innovative approach for designing and analyzing meshfree and particle methods through the study of continuum nonlocal models and their robust discretizations. A central idea is to design "asymptotically compatible schemes" with respect to the nonlocal/integral relaxation scale of PDEs so as to advance the stability, accuracy and efficiency of meshfree and particle methods. The proposed research will advance the theoretical understanding of meshfree and particle methods and enhance their practical performance and functionality. Central to the project is an integrated plan of educational goals, and these include promoting the engagement and retention of female and underrepresented groups in math and science, enhancing applied and computational mathematics curricula, and disseminating new scientific discoveries. The PI proposes a variety of activities that aim at promoting computational thinking in middle/high school students and teachers, inspiring and preparing undergraduates for early research experience, and contributing to a more connected and inclusive intellectual environment. The project primarily focuses on three objectives. The first is the development of monotone meshfree methods for elliptic equations via nonlocal relaxation. This includes the study of linear and nonlinear elliptic equations in Euclidean space and on manifolds. The central theme is to preserve the monotonicity in the discretization while keeping a compact selection of nearby points. The second subject is meshfree methods for systems of equations via the study of nonlocal vector calculus. The study of well-posed nonlocal models through nonlocal vector calculus is crucial for designing of stable numerical discretizations, without which instabilities are often observed (e.g., in the SPH method and peridynamics correspondence model). In the last subproject, we focus on improving the accuracy and efficiency of Lagrange type particle methods for nonlinear time-dependent PDEs through nonlocal analysis. The discussion includes aggregation equation, degenerate diffusion, and convection-diffusion equation. The study of Wasserstein gradient flows will also be needed for rigorous convergence studies.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.
无网格方法和粒子方法在偏微分方程的计算研究中得到了广泛的应用,并被应用于许多力学、流体动力学和生物物理过程。与传统的基于网格的方法相比,它们具有许多优点,特别是对于具有复杂或移动几何形状,材料大变形或其他奇异行为的问题。本项目的目标是通过研究连续非局部模型及其鲁棒离散化,研究设计和分析无网格和粒子方法的创新方法。一个中心思想是设计“渐近相容的计划”的非局部/积分松弛尺度的偏微分方程,以提高稳定性,精度和效率的无网格和粒子方法。该研究将促进对无网格和粒子方法的理论理解,并增强其实际性能和功能。该项目的核心是一项教育目标综合计划,其中包括促进女性和代表性不足的群体参与和保留数学和科学,加强应用和计算数学课程,以及传播新的科学发现。PI提出了各种活动,旨在促进初中/高中学生和教师的计算思维,激励和准备本科生的早期研究经验,并有助于建立一个更加连接和包容的知识环境。该项目主要侧重于三个目标。第一个是通过非局部松弛发展椭圆型方程的单调无网格方法。这包括研究线性和非线性椭圆方程在欧几里德空间和流形上。中心主题是保持离散化的单调性,同时保持附近点的紧凑选择。第二个主题是通过研究非局部向量微积分的方程组的无网格方法。通过非局部向量演算研究适定的非局部模型对于设计稳定的数值离散化是至关重要的,如果没有这些离散化,经常会观察到不稳定性(例如,在SPH方法和周波对应模型中)。在最后一个子计划中,我们致力于通过非局部分析来提高非线性时变偏微分方程的拉格朗日型粒子方法的精度和效率。讨论了凝聚方程、退化扩散方程和对流扩散方程。Wasserstein梯度流的研究也将需要严格的收敛研究。该奖项反映了NSF的法定使命,并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Xiaochuan Tian其他文献
A Class of High Order Nonlocal Operators
一类高阶非局部算子
- DOI:
10.1007/s00205-016-1025-8 - 发表时间:
2016 - 期刊:
- 影响因子:2.5
- 作者:
Xiaochuan Tian;Q. Du - 通讯作者:
Q. Du
Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation
- DOI:
10.1016/j.cma.2020.113264 - 发表时间:
2020-10-01 - 期刊:
- 影响因子:
- 作者:
Yu Leng;Xiaochuan Tian;Nathaniel A. Trask;John T. Foster - 通讯作者:
John T. Foster
$ L^{p} $ compactness criteria with an application to variational convergence of some nonlocal energy functionals
$ L^{p} $ 紧致性准则在某些非局部能量泛函的变分收敛中的应用
- DOI:
- 发表时间:
2018 - 期刊:
- 影响因子:1
- 作者:
Q. Du;T. Mengesha;Xiaochuan Tian - 通讯作者:
Xiaochuan Tian
The double adaptivity paradigm: (How to circumvent the discrete inf-sup conditions of Babuška and Brezzi)
双自适应范式:(如何规避 Babuška 和 Brezzi 的离散 inf-sup 条件)
- DOI:
- 发表时间:
2020 - 期刊:
- 影响因子:2.9
- 作者:
L. Demkowicz;T. Führer;N. Heuer;Xiaochuan Tian - 通讯作者:
Xiaochuan Tian
The Conceptual Metaphor of Governance in The Governance of China
中国治理中的治理概念隐喻
- DOI:
- 发表时间:
2021 - 期刊:
- 影响因子:0
- 作者:
Xiaochuan Tian;Yunhao Ba;Xinyu Zhang - 通讯作者:
Xinyu Zhang
Xiaochuan Tian的其他文献
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{{ truncateString('Xiaochuan Tian', 18)}}的其他基金
Numerical Methods for Nonlocal Models with Applications to Multiscale and Nonlinear Systems
非局部模型的数值方法及其在多尺度和非线性系统中的应用
- 批准号:
2111608 - 财政年份:2021
- 资助金额:
$ 46.71万 - 项目类别:
Standard Grant
Mathematical Analysis and Numerical Methods for Peridynamics and Nonlocal Models
近场动力学和非局部模型的数学分析和数值方法
- 批准号:
2044945 - 财政年份:2020
- 资助金额:
$ 46.71万 - 项目类别:
Standard Grant
Mathematical Analysis and Numerical Methods for Peridynamics and Nonlocal Models
近场动力学和非局部模型的数学分析和数值方法
- 批准号:
1819233 - 财政年份:2018
- 资助金额:
$ 46.71万 - 项目类别:
Standard Grant
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