A Numerical Framework for Nonlinear Peridynamics on Two-Dimensional Manifolds Based on Implicit P-(EC)[math] Schemes

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-03-01 DOI:10.1137/22m1498942

Alessandro Coclite, Giuseppe M. Coclite, Francesco Maddalena, Tiziano Politi

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引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 622-645, April 2024.
Abstract. In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non-parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by reinterpreting the triangular computational mesh as a non-oriented graph, thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC)[math] formulation of the implicit [math]-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a 2D sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials.

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基于隐式 P-(EC)[math]方案的二维平面上非线性周流体力学数值框架

SIAM 数值分析期刊》第 62 卷第 2 期第 622-645 页，2024 年 4 月。摘要本手稿提出了一种在任意形状的二维（2D）封闭流形上进行非线性周动力学计算的原创数值程序。在离散尺度上处理非参数化二维流形时，会出现计算两个非相邻点之间大地距离的问题。在这里，通过将三角形计算网格重新解释为无定向图，实现了计算大地测量距离的路由程序，从而返回了一种合适的通用方法。此外，围动力学方程的时间积分要求采用隐式[math]-Newmark 方案的 P-(EC)[math]表述。对所提出的整个程序的收敛性进行了质疑和严格证明。通过模拟二维球体的演变，分析了其能力和局限性。进行数值研究的主要动机是与演化问题中的奇异点相关的问题。所获得的结果揭示了积分微分方程的非局部性在导致实际材料中奇点自发形成的复杂过程中所起的作用。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.