In this work, we report on the formulation and detailed stability analysis of a dynamic multi-scale scheme involving two different local computational strategies for modeling of elastic wave propagation. The coupled model involves the Local Interaction Simulation Approach and Cellular Automata for Elastodynamics, however the presented analysis approach is general and applies to other numerical techniques. This scheme is capable of coupling two numerical models with possibly dissimilar spatial discretization lengths and material properties, hence it is appealing for a multi-scale and/or multi-resolution analysis. The method developed in this paper employs an interface force–displacement coupling to yield the multi-scale model equations. It is shown that the governing equations contain a self-coupling term that affects the stability of the system, as it contributes to additional stiffness at the interface. Stability analysis is presented in terms of rotations of two vectors in $\beta$ space, where each vector represents individual model’s stability. Three model configurations of practical interest were investigated, analytical formulae derived and used to analyze stability. These analytical formulae were compared against results from numerical simulations and perfect agreement was observed.

在这项工作中，我们报告了动态多尺度方案的制定和详细的稳定性分析，该方案涉及用于模拟弹性波传播的两种不同的局部计算策略。耦合模型涉及弹性动力学的局部相互作用模拟方法和元胞自动机，但所提出的分析方法是通用的，适用于其他数值技术。该方案能够耦合两个具有可能不同的空间离散长度和材料特性的数值模型，因此它对多尺度和/或多分辨率分析很有吸引力。本文开发的方法采用界面力-位移耦合来产生多尺度模型方程。结果表明，控制方程包含一个影响系统稳定性的自耦合项，因为它有助于增加界面的刚度。稳定性分析是根据两个向量的旋转来表示的$\beta$空间，其中每个向量代表单个模型的稳定性。研究了三种具有实际意义的模型配置，推导出分析公式并用于分析稳定性。将这些分析公式与数值模拟的结果进行比较，并观察到完美的一致性。