一类非线性微分方程的基于动态微分算子的非局部数值范式

IF 2.8 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Computational Particle Mechanics Pub Date : 2023-03-07 DOI:10.1007/s40571-023-00568-z
Xiaohu Yu, Airong Chen, Haocheng Chang
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引用次数: 0

摘要

本文利用周动力微分算子,给出了一类一般非线性常微分方程的一种新的非局部数值范式。利用无网格正交技术将微分控制方程和初始/边界条件从局部微分形式重新表述为非局部积分形式。将解域划分为有限个数的点,这些点的性质通过对相邻点的相应性质的加权求和得到。利用拉格朗日乘子法和变分原理,可以用牛顿-拉夫森迭代法求解具有初始/边界条件的非线性常微分方程。此外,通过对几个影响因子的比较,说明了该方法与其他方法的区别。通过求解Riccati方程、Poisson方程和流体流动方程三个基准,验证了所提数值方法的适用性和准确性,结果与文献数值结果一致。最后,将该方法应用于驰振问题,揭示了驰振机理。
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Peridynamic differential operator-based nonlocal numerical paradigm for a class of nonlinear differential equations

This paper presents a novel nonlocal numerical paradigm for a class of general nonlinear ordinary differential equations using the peridynamic differential operator. Differential governing equations and initial/boundary conditions are reformulated from the local differential form to the nonlocal integral form using a meshless orthogonal technique. The solution domain is partitioned into a finite number of points, of which the properties are obtained through weighted summation over the corresponding properties of neighboring points. Using the Lagrange multiplier method and the variational principle, nonlinear ordinary differential equations with initial/boundary conditions can be solved through the Newton–Raphson iteration method. Moreover, the differences between the proposed method and other methods are illustrated by comparing several impact factors. Furthermore, three benchmarks, including the Riccati equation, the Poisson equation, and the fluid flow equation, have been solved to show the applicability and accuracy of the proposed numerical method, and the results are consistent with the numerical results in the previous literature. Finally, the proposed method is applied to the galloping vibration problem to reveal the galloping mechanism.

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来源期刊
Computational Particle Mechanics
Computational Particle Mechanics Mathematics-Computational Mathematics
CiteScore
5.70
自引率
9.10%
发文量
75
期刊介绍: GENERAL OBJECTIVES: Computational Particle Mechanics (CPM) is a quarterly journal with the goal of publishing full-length original articles addressing the modeling and simulation of systems involving particles and particle methods. The goal is to enhance communication among researchers in the applied sciences who use "particles'''' in one form or another in their research. SPECIFIC OBJECTIVES: Particle-based materials and numerical methods have become wide-spread in the natural and applied sciences, engineering, biology. The term "particle methods/mechanics'''' has now come to imply several different things to researchers in the 21st century, including: (a) Particles as a physical unit in granular media, particulate flows, plasmas, swarms, etc., (b) Particles representing material phases in continua at the meso-, micro-and nano-scale and (c) Particles as a discretization unit in continua and discontinua in numerical methods such as Discrete Element Methods (DEM), Particle Finite Element Methods (PFEM), Molecular Dynamics (MD), and Smoothed Particle Hydrodynamics (SPH), to name a few.
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