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.

中文翻译：

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基于隐式 P-(EC)[数学]方案的二维流形非线性近场动力学数值框架

SIAM 数值分析杂志，第 62 卷，第 2 期，第 622-645 页，2024 年 4 月。
摘要。在这份手稿中，提出了任意形状的二维（2D）闭流形上的非线性近场动力学的原始数值程序。当处理离散尺度的非参数化二维流形时，会出现计算两个非相邻点之间的测地距离的问题。这里，通过将三角形计算网格重新解释为无向图来实现用于计算测地距离的路由过程，从而返回合适且通用的方法。此外，近场动力学方程的时间积分要求为隐式[math]-Newmark格式的P-(EC)[math]公式。整个提议程序的收敛性受到质疑并得到严格证明。通过模拟二维球体的演化来分析其能力和局限性。所进行的数值研究主要是出于与进化问题中奇点的兴起相关的问题。所获得的结果返回了一幅有趣的图景，展示了积分微分方程的非局部特征在导致实际材料中奇点自发形成的复杂过程中所发挥的作用。