渗透流体和变形耦合的异质多孔弹性介质中Voigt和Reuss界限与解耦约束之间的对应关系
The correspondence between Voigt and Reuss bounds and the decoupling constraint in a two-grid staggered solution algorithm to coupled flow and deformation in heterogeneous poroelastic media
AI总结：
本文研究了在两网格 staggered 解决方案算法中，Voigt 和 Reuss 界限与解耦约束之间的对应关系，通过分析两网格方案的收敛性，构建了限制和扩展算子，并推导了粗尺度弹性质。
摘要：
我们对用于建模异质多孔弹性介质中耦合流体和变形的Biot系统两网格 staggered 解决方案算法进行了收敛性分析。该算法首先在细网格上使用混合有限元方法求解流体子问题（通过冻结某一平均应力度量），然后在粗网格上使用符合 Galerkin 方法求解流体力学子问题。限制算子将细尺度流体解映射到粗尺度流体力学网格，扩展算子将粗尺度流体力学解映射到细尺度流体网格。耦合迭代重复进行直到收敛，并采用Backward Euler进行时间推进。分析基于研究迭代差值所满足的方程，以证明在某些条件下两网格方案是一个收缩映射。这些条件用于构建限制和扩展算子，并以细尺度数据推导粗尺度弹性质。我们证明了平均应力度量中的可调参数与计算均质化多相复合材料中经常遇到的Voigt和Reuss界限相关。
英文摘要：
We perform a convergence analysis of a two-grid staggered solution algorithm for the Biot system modeling coupled flow and deformation in heterogeneous poroelastic media. The algorithm first solves the flow subproblem on a fine grid using a mixed finite element method (by freezing a certain measure of the mean stress) followed by the poromechanics subproblem on a coarse grid using a conforming Galerkin method. Restriction operators map the fine scale flow solution to the coarse scale poromechanical grid and prolongation operators map the coarse scale poromechanical solution to the fine scale flow grid. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The analysis is based on studying the equations satisfied by the difference of iterates to show that the two-grid scheme is a contraction map under certain conditions. Those conditions are used to construct the restriction and prolongation operators as well as arrive at coarse scale elastic properties in terms of the fine scale data. We show that the adjustable parameter in the measure of the mean stress is linked to the Voigt and Reuss bounds frequently encountered in computational homogenization of multiphase composites.
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