CAREER: Numerical Analysis for Meshfree and Particle Methods via Nonlocal Models

无网格方法和质点方法广泛应用于偏微分方程组的计算研究，在许多力学、流体力学和生物物理过程中都有应用。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的几何形状、材料的大变形或解的其他奇异行为的问题。本项目的目标是通过对连续介质非局部模型及其稳健离散化的研究，探索一种设计和分析无网格和质点方法的创新方法。其核心思想是针对偏微分方程组的非局部/积分松弛尺度设计“渐近相容格式”，以提高无网格法和质点法的稳定性、精度和效率。本文的研究将加深对无网格法和质点法的理论理解，提高它们的实用性能和功能。该项目的核心是一项教育目标的综合计划，其中包括促进女性和代表性不足的群体参与和留住数学和科学，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，启发和准备本科生的早期研究经验，并为建立一个更相互联系和包容的智力环境做出贡献。该项目主要侧重于三个目标。一是通过非局部松弛发展了求解椭圆型方程的单调无网格方法。这包括研究欧氏空间和流形上的线性和非线性椭圆型方程。中心主题是保持离散化中的单调性，同时保持对邻近点的紧凑选择。第二个主题是通过研究非局部矢量微积分来求解方程组的无网格法。利用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是至关重要的，否则经常会观察到不稳定性(例如，在SPH方法和动力学对应模型中)。在最后一个子项目中，我们致力于通过非局部分析来提高拉格朗日型粒子方法求解非线性时变偏微分方程组的精度和效率。讨论包括聚集方程、简并扩散方程和对流扩散方程。沃瑟斯坦梯度流的研究也将被用于严格的收敛研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.