二维层流非线性周波模型的时空离散
A space-time discretization of a nonlinear peridynamic model on a 2D lamina
摘要：
周波理论是一种用于动态断裂分析的非局部理论，它包含一个时间上的二阶偏积分微分方程。本文考虑二维空间域中的一个周期运动的非线性模型。我们实现了一个谱方法的基础上的解决方案的傅立叶展开的空间离散化，而我们考虑的Newmark- $β$方法的时间推进。这种计算方法利用了周波算子的卷积形式和离散傅里叶变换的使用。我们证明了全离散逼近的收敛性，并研究了该方法应用于线性周期模型的稳定性。最后，我们进行了几个数值测试和比较，以验证我们的结果，并提供模拟实现体积惩罚技术，以避免由于频谱方法的周期边界条件的限制。
英文摘要：
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.
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