保结构增广拉格朗日有限元方法用于雷诺和斯托克斯流中的空化
Structure-Preserving Augmented Lagrangian Finite Element Methods for Cavitation in Reynolds and Stokes Flows
AI总结：
提出增广拉格朗日有限元方法求解雷诺和斯托克斯润滑模型中的空化问题，离散互补条件精确成立，证明一阶误差估计，数值实验表明压力相近不保证空腔预测相近。
AI中文摘要：
本文针对润滑的雷诺和斯托克斯模型中空化问题，提出了增广拉格朗日有限元方法，其中离散互补条件精确成立，且增广参数从方法中消去。对于雷诺方程，这源于混合分片线性方法中的节点求积，我们证明了经典的一阶误差估计。我们还确定了无乘子稳定化替代方案的可计算稳定性阈值。对于斯托克斯流，我们表明当使用零体积粘度的偏应力时，约束标量是机械压力，但使用常规不可压缩应力时则不然。我们采用跳跃稳定化的Crouzeix-Raviart单元和分片常数压力进行离散，并证明了适定性、二维和三维稳定性以及一阶误差估计。我们给出了数值算例，验证了这些性质，并表明两种模型中相近的压力分布并不意味相近的空腔预测，后者还对计算域端部条件敏感。
英文摘要：
In this paper we propose augmented Lagrangian finite element methods for cavitation in the Reynolds and Stokes models of lubrication, for which the discrete complementarity conditions hold exactly and the augmentation parameter drops out of the method. For the Reynolds equation this follows from nodal quadrature in a mixed piecewise linear method, for which we prove the classical first-order error estimate. We also determine the computable stability threshold of a multiplier-free stabilised alternative. For Stokes flow we show that the constrained scalar is the mechanical pressure when the deviatoric stress, with zero bulk viscosity, is used, but not with the customary incompressible stress. We discretise with a jump-stabilised Crouzeix-Raviart element and piecewise constant pressure, and prove well-posedness, stability in two and three dimensions, and a first-order error estimate. We present numerical examples which verify these properties and show that close pressure profiles in the two models do not imply close cavity predictions, the latter also being sensitive to the end conditions of the computational domain.
