Peridynamics is a nonlocal theory for dynamic fracture analysis consisting in a second order in time partial integro-differential equation. In this paper, we consider a nonlinear model of peridynamics in a two-dimensional spatial domain. We implement a spectral method for the space discretization based on the Fourier expansion of the solution while we consider the Newmark-β method for the time marching. This computational approach takes advantages from the convolutional form of the peridynamic operator and from the use of the discrete Fourier transform. We show a convergence result for the fully discrete approximation and study the stability of the method applied to the linear peridynamic model. Finally, we perform several numerical tests and comparisons to validate our results and provide simulations implementing a volume penalization technique to avoid the limitation of periodic boundary conditions due to the spectral approach.

近场动力学是动态断裂分析的非局部理论，包括时间上的二阶偏积分微分方程。在本文中，我们考虑了二维空间域中近场动力学的非线性模型。我们基于解的傅立叶展开实现空间离散化的谱方法，同时考虑 Newmark- β时间前进的方法。这种计算方法利用了近场动力学算子的卷积形式和离散傅立叶变换的使用。我们展示了完全离散近似的收敛结果，并研究了应用于线性近场动力学模型的方法的稳定性。最后，我们进行了几次数值测试和比较以验证我们的结果，并提供了实施体积惩罚技术的模拟，以避免由于光谱方法而导致的周期性边界条件的限制。