An augmented stress-based mixed finite element method for the steady state Navier-Stokes equations with nonlinear viscosity
An augmented stress-based mixed finite element method for the steady state Navier-Stokes equations with nonlinear viscosity
复制标题
非线性粘性稳态纳维-斯托克斯方程的增强应力混合有限元法
DOI：
10.1002/num.22166
复制
发表时间：
2017
影响因子：
3.9
通讯作者：
Camaño J
中科院分区：
文献类型：
--
作者：
Camaño J
本文提出并分析了一种新的基于应力的定常N-S方程的混合变分格式，该方程具有依赖于应变张量大小的定常密度和变粘度。我们的方法是一些作者在最近的一篇论文中应用于相同边值问题的技术的自然推广，但粘性非线性地依赖于速度的梯度而不是应变张量。在这种情况下，除了指出粘性与应变的关系产生了一个更具物理相关性的模型外，我们注意到，为了处理这种非线性，我们现在不仅需要将应变本身包括在内，还需要将涡度作为辅助未知数。此外，与前面的工作类似，为了处理速度的合适空间，在变分公式中增加了来自本构方程和平衡方程的Galerkin型项、定义两个附加未知量的关系以及Dirichlet边界条件。这样，由于所得到的增广格式可以重写为不动点算子方程，经典的Schauder和Banach定理以及单调算子理论被用来推导连续和相关离散格式的适定性。特别地，我们证明了后者可以利用任意的有限元子空间，然后我们得到了最优的先验误差估计和相应的收敛速度。其次，针对任意多边形和多面体区域，提出了一种可靠有效的基于残差的后验误差估计器。所使用的主要工具包括Raviart-Thomas和Clément插值算子，逆和离散不等式，以及基于三角气泡和边缘气泡函数的局部化技术。最后给出了几个数值算例，说明了该方法的良好性能，证实了后验误差估计器的可靠性和有效性，并展示了自适应算法的预期行为。©2017 Wiley期刊，Inc.数字方法偏差式33：1692-1725,2017
A new stress‐based mixed variational formulation for the stationary Navier‐Stokes equations with constant density and variable viscosity depending on the magnitude of the strain tensor, is proposed and analyzed in this work. Our approach is a natural extension of a technique applied in a recent paper by some of the authors to the same boundary value problem but with a viscosity that depends nonlinearly on the gradient of velocity instead of the strain tensor. In this case, and besides remarking that the strain‐dependence for the viscosity yields a more physically relevant model, we notice that to handle this nonlinearity we now need to incorporate not only the strain itself but also the vorticity as auxiliary unknowns. Furthermore, similarly as in that previous work, and aiming to deal with a suitable space for the velocity, the variational formulation is augmented with Galerkin‐type terms arising from the constitutive and equilibrium equations, the relations defining the two additional unknowns, and the Dirichlet boundary condition. In this way, and as the resulting augmented scheme can be rewritten as a fixed‐point operator equation, the classical Schauder and Banach theorems together with monotone operators theory are applied to derive the well‐posedness of the continuous and associated discrete schemes. In particular, we show that arbitrary finite element subspaces can be utilized for the latter, and then we derive optimal a priori error estimates along with the corresponding rates of convergence. Next, a reliable and efficient residual‐based a posteriori error estimator on arbitrary polygonal and polyhedral regions is proposed. The main tools used include Raviart‐Thomas and Clément interpolation operators, inverse and discrete inequalities, and the localization technique based on triangle‐bubble and edge‐bubble functions. Finally, several numerical essays illustrating the good performance of the method, confirming the reliability and efficiency of the a posteriori error estimator, and showing the desired behavior of the adaptive algorithm, are reported. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1692–1725, 2017