The peridynamic equation consists in an integro‐differential equation of the second order in time which has been proposed for modeling fractures and damages in the context of nonlocal continuum mechanics. In this article, we study numerical methods for the one‐dimension nonlinear peridynamic problems. In particular we consider spectral Fourier techniques for the spatial domain while we will use the Störmer–Verlet method for the time discretization. In order to overcome the limitation of working on periodic domains due to the spectral techniques we will employ a volume penalization method. The performance of our approach is validated with the study of the convergence with respect to the spatial discretization and the volume penalization. Several tests have been performed to investigate the properties of the solutions.

中文翻译：

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非线性扰动模型的带体积罚分的谱方法

周动力方程包含一个时间上的二阶积分微分方程，该方程已被提出用于在非局部连续介质力学的背景下模拟断裂和损伤。在本文中，我们研究一维非线性周动力问题的数值方法。特别地，我们将频谱傅里叶技术用于空间域，而我们将使用Störmer-Verlet方法进行时间离散化。为了克服由于频谱技术而在周期域上工作的局限性，我们将采用体积罚分法。我们关于空间离散化和体积罚分的收敛性研究验证了我们方法的性能。已经进行了一些测试来研究溶液的性质。