AI 中文总结
针对含裂缝的混合维多孔弹性模型，提出一种无锁解耦方法，通过引入总压力重述模型，建立能量耗散定律，开发时间解耦格式，经空间离散构建全离散格式，推导其能量稳定性和误差估计，并用数值实验验证。
AI 中文摘要
我们为含裂缝的混合维多孔弹性模型提出了一种无锁解耦方法。通过引入总压力，将含裂缝的比奥系统重新表述为一个包含位移、总压力、基质压力和裂缝压力的四场公式。我们为连续模型建立了能量耗散定律，表明其与热力学第二定律的一致性。基于此公式，开发了一种时间解耦格式。在初始时间步采用全耦合格式，而在后续时间步，先求解流动问题，再求解力学问题。在力学方程中加入稳定项以减少稳定性分析中对模型参数的限制。建立了半离散格式的能量稳定性。对于空间离散化，位移和总压力用泰勒 - 胡德单元近似，而基质压力和裂缝压力用拉格朗日有限元。然后构建了全离散解耦格式。推导了全离散格式的能量稳定性和误差估计，且该方法无锁。给出了数值实验以支持理论结果。
英文摘要
We propose a locking-free decoupling method for a mixed-dimensional poroelasticity model with fractures. By introducing the total pressure, the fractured Biot system is reformulated as a four-field formulation involving the displacement, total pressure, matrix pressure, and fracture pressure. We establish an energy dissipation law for the continuous model, which shows its consistency with the second law of thermodynamics. Based on this formulation, a time-decoupled scheme is developed. At the initial time step, a fully coupled scheme is employed, while for subsequent time steps, the flow problem is solved first, followed by the mechanics problem. A stabilization term is incorporated into the mechanical equation to help reduce the restrictions imposed on the model parameters in the stability analysis. Energy stability is established for the semi-discrete scheme. For the spatial discretization, the displacement and total pressure are approximated by the Taylor--Hood element, while Lagrange finite elements are used for the matrix and fracture pressures. A fully discrete decoupled scheme is then constructed. Energy stability and error estimates are derived for the fully discrete scheme, and the method is shown to be locking-free. Numerical experiments are presented to support the theoretical results.
