We introduce a second-order computational homogenization procedure designed to address heterogeneous poromechanical media. Our approach relies on the method of multiscale virtual power, a variational multiscale method that extends the Hill–Mandel principle of macro-homogeneity. Constraints on displacement and pore pressure fields are managed using periodic and second-order minimally constrained fluctuating spaces. Numerical comparisons reveal that first-order models fail to accurately represent nonzero net fluid flow and volume changes at the micro-scale. In contrast, our second-order approach effectively captures nonuniform fluid flow across representative volume element boundaries, in agreement with results from direct numerical simulations. Our findings indicate that the classical first-order expansion of the pressure field is inadequate for poromechanical homogenization in cases involving micro-scale volume changes, such as swelling or contraction. The proposed second-order approach not only overcomes these limitations but also proves effective in cases where the principle of separation of scales is not strictly upheld.

中文翻译：

---

大变形下桥接多孔机械尺度的二阶计算均质化


我们引入了一种二阶计算均质程序，旨在解决异质多孔机械介质。我们的方法依赖于多尺度虚拟功率方法，这是一种扩展宏观同质性的 Hill-Mandel 原理的变分多尺度方法。位移和孔隙压力场的约束使用周期性和二阶最小约束的波动空间进行管理。数值比较表明，一阶模型无法准确表示微尺度上的非零净流体流量和体积变化。相比之下，我们的二阶方法有效地捕获了穿过代表性体积单元边界的不均匀流体流动，这与直接数值模拟的结果一致。我们的研究结果表明，在涉及微尺度体积变化（例如膨胀或收缩）的情况下，压力场的经典一阶膨胀不足以进行多孔机械均质化。所提出的二阶方法不仅克服了这些限制，而且在不严格遵守刻度分离原则的情况下也被证明是有效的。