The objective of this article is to summarise the work that we have been doing as a group in the context of stabilised finite element formulations for viscoelastic fluid flows. Viscoelastic fluids are complex non-Newtonian fluids, characterised by having an irreducible constitutive equation that needs to be solved coupled with the momentum and continuity equations. The finite element approximation of this kind of fluids presents several numerical difficulties. It inherits obviously the problems associated with the approximation of the incompressible Navier–Stokes equations. But, on top of that, now the constitutive equation is highly non-linear, with an advective term that may lead to both global and local oscillations in the numerical approximation. Moreover, even in the case of smooth solutions, it is necessary to meet some additional compatibility conditions between the velocity and the stress interpolation in order to ensure control over velocity gradients. The stabilised methods detailed in this work allow one to use equal order or even arbitrary interpolation for the problem unknowns (\({\varvec{\sigma }}\)-\({\varvec{u}}\)-p) (elastic deviatoric stress-velocity-pressure) and to stabilise dominant convective terms, and all of them can be framed in the context of variational multi-scale methods. Some additional numerical ingredients that are introduced in this article are the treatment of the non-linearities associated with the problem and the possibility to introduce a discontinuity-capturing technique to prevent local oscillations. Concerning the constitutive equation, both the standard as the logarithmic conformation reformulation are discussed for stationary and time-dependent problems, and different versions of stabilised finite element formulations are presented in both cases.

本文的目的是总结在粘弹性流体流动的稳定有限元公式化的背景下我们作为一个整体所做的工作。粘弹性流体是复杂的非牛顿流体，其特征在于具有不可解的本构方程，需要与动量和连续性方程一起求解。这种流体的有限元逼近存在几个数值上的困难。显然，它继承了与不可压缩的Navier-Stokes方程的逼近有关的问题。但是，最重要的是，本构方程现在是高度非线性的，带有对流项，可能导致数值逼近中的整体和局部振动。而且，即使在解决方案顺利的情况下，为了确保对速度梯度的控制，有必要满足速度和应力插值之间的一些其他兼容性条件。在这项工作中详细介绍的稳定化方法允许对问题未知数使用等阶甚至任意插值（\（{\ varvec {\ sigma}} \） - \（{\ varvec {u}} \） - p）（弹性偏应力-速度-压力）并稳定主导对流项，所有这些项都可以框架化在变分多尺度方法的背景下。本文介绍的一些其他数字成分是对与该问题相关的非线性的处理，以及引入不连续性捕获技术以防止局部振荡的可能性。关于本构方程，讨论了平稳和与时间有关的问题的标准，即对数构象重构的标准，并且在两种情况下均提出了不同版本的稳定有限元公式。