This paper presents an extension of Direct \(\hbox {FE}^{2}\) method for the study of dynamic problems in heterogeneous materials. The proposed method can be formulated based on either the Hill–Mandel principle or the extended Hill–Mandel principle, the latter of which enforces the energy contributed by the internal force and the inertial force consistent at two scales. Unlike the traditional \(\hbox {FE}^{2}\) method, it is not necessary to conduct two levels of finite element simulations linked by extensive information interchange. Instead, we reformulate the macroscopic variational statement with the microscopic contributions, leading to only a single coupled boundary value problem. The classical microscopic boundary condition used in the traditional \(\hbox {FE}^{2}\) method can be employed but it is implemented through kinematical constraints between the macro nodes and the micro nodes in the Direct \(\hbox {FE}^{2}\) method. The proposed method is illustrated by two numerical examples including a fiber-reinforced composite and an acoustic metamaterial. The results are verified by direct numerical simulations and it is shown that micro-inertia effects are not important in modeling low-velocity impact behavior of the composite but they are essential in capturing wave attenuation performance of locally resonant acoustic metamaterials.

本文提出了Direct \（\ hbox {FE} ^ {2} \）方法的扩展，用于研究异质材料中的动力学问题。可以基于希尔-曼德尔原理或扩展的希尔-曼德尔原理来制定所提出的方法，后者的原理是通过内力和惯性力贡献的能量在两个尺度上保持一致。与传统的\（\ hbox {FE} ^ {2} \）方法不同，不需要进行通过大量信息交换链接的两个级别的有限元模拟。取而代之的是，我们用微观贡献来重新构造宏观变分表述，从而仅导致单个耦合的边值问题。传统\（\ hbox {FE} ^ {2} \）中使用的经典微观边界条件可以采用这种方法，但是可以通过Direct \（\ hbox {FE} ^ {2} \）方法中的宏节点和微节点之间的运动学约束来实现。通过两个数值示例说明了所提出的方法，包括纤维增强复合材料和声学超材料。通过直接数值模拟验证了结果，结果表明，微惯性效应对于模拟复合材料的低速冲击行为并不重要，但对于捕获局部共振声超材料的波衰减性能至关重要。