A new finite element formulation for viscoelastic flows: Circumventing simultaneously the LBB condition and the high-Weissenberg number problem
A new finite element formulation for viscoelastic flows: Circumventing simultaneously the LBB condition and the high-Weissenberg number problem
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DOI：
10.1016/j.jnnfm.2019.04.003
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发表时间：
2019-05-01
影响因子：
3.1
通讯作者：
Tsamopoulos, J.
中科院分区：
文献类型：
--
作者：
Varchanis, S.;Syrakos, A.;Tsamopoulos, J.
本文提出了一种新的、完全一致的、高度稳定的粘弹性流动有限元模拟公式。在我们的方法中，我们为所有变量实现了等阶插值，并将经典有限元稳定技术（PSPG/DEVSS-TG/SUPG）与本构方程的对数构象表示相结合，使我们能够在高Weissenberg数下获得数值稳定的解。通过将本文方法的数值结果与文献中三个基准试验的数值结果进行比较，证明了所提出的有限元框架的有效性：粘弹性流体在方形盖驱动腔体中的二维流动和在通道中通过圆柱体，以及在立方体盖驱动腔体中的三维流动。我们考虑使用Oldroyd-B和线性PTT模型进行直接稳态和瞬态计算。在所有情况下，我们都可以达到（在某些情况下甚至超过）混合有限元方法所能达到的最大Weissenberg数值，但计算成本和编程工作量却大大降低。此外，我们进行了网格收敛测试，表明该方法具有收敛性，并且在空间上具有接近二阶精度。
In this paper, we propose a new, fully consistent and highly stable finite element formulation for the simulation of viscoelastic flows. In our method, we have implemented equal order interpolants for all variables and a combination of classical finite element stabilization techniques (PSPG/DEVSS-TG/SUPG) with the log-conformation representation of the constitutive equation that has allowed us to obtain numerically stable solutions at high Weissenberg numbers. The validity of the presented FEM framework is testified by comparing the numerical results of our method to those of the literature in three benchmark tests: the 2-dimensional flows of a viscoelastic fluid in a square lid-driven cavity and past a cylinder in a channel, and the 3-dimensional flow in a cubic lid-driven cavity. We consider both direct steady-state and transient calculations using the Oldroyd-B and linear PTT models. In all cases, we can reach, and in some cases surpass, the maximum Weissenberg number values attainable by mixed finite element methods, but at a considerably lower computational cost and programming effort. In addition, we perform mesh-convergence tests illustrating that the proposed method is convergent and features almost 2nd order accuracy in space.