This article proposes an efficient concurrent coupling of two different material scales—a macroscale and a microscale—in a direct solution scheme based on explicit time integration. Both scales may be discretized with different element sizes and the microdomain may exhibit a heterogeneous structure. A surface coupling is described, which imposes the macrovelocities at the interfaces on the microscale. Using an averaged stress state of several elements on the microscale within a bounded volume, forces are derived which transfer the micromaterial response back to the macroscale. Whereas established surface couplings based on Lagrange multipliers achieve an exact solution of the interface problem, the proposed coupling is based on a weak staggered scheme. The advantage is that no common global system of equations has to be solved and the approach preserves the efficiency of direct solution schemes almost completely. It is therefore well applicable to the simulation of wave propagation phenomena in heterogeneous materials with complex constitutive models and suitable for massive parallelization. Example simulations demonstrate the capabilities and current limitations.

中文翻译：

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使用具有显式时间积分的直接求解方案的波传播的并发两尺度耦合

本文在基于显式时间积分的直接求解方案中提出了两种不同材料尺度（宏观尺度和微观尺度）的有效并发耦合。两个尺度都可以用不同的元素尺寸离散化，并且微域可能表现出异质结构。描述了一种表面耦合，它在微观尺度的界面上施加了宏观速度。使用有限体积内微观尺度上几个元素的平均应力状态，推导出将微观材料响应转移回宏观尺度的力。虽然基于拉格朗日乘子建立的表面耦合实现了界面问题的精确解，但所提出的耦合基于弱交错方案。优点是不必求解通用的全局方程组，并且该方法几乎完全保留了直接求解方案的效率。因此，它非常适用于具有复杂本构模型和适合大规模并行化的异质材料中的波传播现象的模拟。示例模拟展示了功能和当前限制。