标题： 高对比系数应力位移公式中线性弹性的无锁定多尺度方法 显示英文标题

标题： A locking free multiscale method for linear elasticity in stress-displacement formulation with high contrast coefficients

摘要： 在高对比度介质的线性弹性问题中实现强对称应力逼近提出了显著的计算挑战。传统方法通常由于自由度过多而面临过高的计算成本，限制了它们的实际适用性。为了解决这个挑战，我们引入了一种高效的多尺度模型降阶方法和一种计算成本低廉的粗网格模拟技术，用于高度异质、高对比度介质中的线性弹性方程。我们首先利用稳定的应力-位移混合有限元方法离散化线性弹性问题，然后介绍位移和应力的多尺度基函数的构建。混合公式提供了几个优势，例如无需后处理即可直接计算应力、局部动量守恒（确保物理一致性）以及对几乎不可压缩材料的鲁棒性，即使对于接近不可压缩的材料也是如此。理论分析证实，我们的方法是inf-sup稳定的且无锁定效应，并且相对于粗网格大小具有一阶收敛性。值得注意的是，随着扩大过采样区域，收敛率与对比度无关。数值实验验证了该方法的有效性，在极端对比条件下也展示了其优越性能。 显示英文摘要

摘要： Achieving strongly symmetric stress approximations for linear elasticity problems in high-contrast media poses a significant computational challenge. Conventional methods often struggle with prohibitively high computational costs due to excessive degrees of freedom, limiting their practical applicability. To overcome this challenge, we introduce an efficient multiscale model reduction method and a computationally inexpensive coarse-grid simulation technique for linear elasticity equations in highly heterogeneous, high-contrast media. We first utilize a stable stress-displacement mixed finite element method to discretize the linear elasticity problem and then present the construction of multiscale basis functions for the displacement and the stress. The mixed formulation offers several advantages such as direct stress computation without post-processing, local momentum conservation (ensuring physical consistency), and robustness against locking effects, even for nearly incompressible materials. Theoretical analysis confirms that our method is inf-sup stable and locking-free, with first-order convergence relative to the coarse mesh size. Notably, the convergence remains independent of contrast ratios as enlarging oversampling regions. Numerical experiments validate the method's effectiveness, demonstrating its superior performance even under extreme contrast conditions.