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 re-interpreting 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)$^k$ formulation of the implicit $\beta$-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 two-dimensional 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.
翻译:在本手稿中,为任意形状的二维(2D)闭合的二维形的无线近地动力学提出了原始数字程序。在处理离散规模的非参数化 2D 元时,出现计算两个非相邻点之间的大地测量距离问题。这里,为计算大地测量距离采用了一种路由程序,将三角计算网格重新解读为非方向图形,从而返回了一种适当和一般的方法。此外,对远地动力学方程式的时间整合要求为隐含的$\beta$-Newmark 方案的P-(EC)$k$配方。总体拟议程序的趋同受到质疑和严格证明。通过模拟二维领域的演进,分析其能力和局限性。进行的数字调查主要受进化问题中奇异性变异性问题相关问题的驱动。获得的结果还令人感兴趣的是,在导致真实材料自发形成的复杂过程中,对非本地异方程式的非本地性方程式所起的作用作了描述。