Dual strategies for solving the Stokes problem with stick–slip boundary conditions in 3D

Abstract

The paper deals with the numerical realization of the 3D Stokes flow subject to threshold slip boundary conditions. The weak velocity–pressure formulation leads to an inequality type problem that is approximated by a mixed finite element method. The resulting algebraic system is non-smooth. Besides the pressure, three additional Lagrange multipliers are introduced: the discrete normal stress releasing the impermeability condition and two discrete shear stresses regularizing the non-smooth slip term. Eliminating the discrete velocity component we obtain the minimization problem for the smooth functional, expressed in terms of the pressure, the normal, and the shear stresses. This problem is solved either by a path following variant of the interior point method or by the semi-smooth Newton method. Numerical scalability is illustrated by computational experiments.

Introduction

The no-slip condition is the standard boundary condition in fluid flow models. It characterizes the stick of a fluid on a solid wall, i.e., the tangential velocity on the boundary of a computational domain vanishes. However, a slip of a fluid is observed along a wall in many real situations (e.g. water flow along hydrophobic surfaces). The simplest stick–slip condition is the Navier one [17]:

$${\mathit{\sigma }}_t=-\kappa {\mathbf{u}}_t,$$

where ${\mathit{\sigma }}_t$ is the shear stress, ${\mathbf{u}}_t$ is the tangential component of the velocity $\mathbf{u}$, and $\kappa \ge 0$ is an adhesive coefficient. One can see from (1) that a slip appears whenever ${\mathit{\sigma }}_t\ne \mathbf{0}$. The stick–slip conditions introduced by Fujita [6] consider their threshold character using the slip bound function $g\ge 0$. They read as follows:

$$\left.\begin{array}{ccc}\Vert {\mathit{\sigma }}_t\Vert \le g,\hfill & \hfill \hfill & \hfill \\ \Vert {\mathit{\sigma }}_t\Vert <g\hfill & \hfill \Rightarrow \hfill & {\mathbf{u}}_t=\mathbf{0},\hfill \\ \Vert {\mathit{\sigma }}_t\Vert =g\hfill & \hfill \Rightarrow \hfill & \exists c\ge 0:{\mathbf{u}}_t=-c{\mathit{\sigma }}_t.\hfill \end{array}\right\}$$

Hence, the slip may occur only if the bound $g$ is attained and the tangential velocity and the shear stress have the opposite direction. This condition is well-known in contact problems of solid mechanics as the Tresca friction law [7].

In this paper we combine (1), (2) in one boundary condition formulated for the Stokes flow model. For the mathematical analysis of this problem we refer to [8], [19]. Computational experiments based on an alternating direction method can be found in [4]. Another way of solving non-smooth problems using the primal–dual set strategy which is associated to a limit case of the semi-smooth Newton (SSN) method has been introduced and analysed in [9]. Our aim is to develop efficient algorithms for solving large-scale algebraic systems arising from an appropriate finite element discretization of 3D problems that were successful in 2D cases. In particular, we extend the path-following (PF) variant of the interior point method [13] and the SSN method [16]. Note that the extensions from 2D to 3D are not straightforward, since the norm appearing in (2) is represented by the absolute value in 2D while by the Euclidean norm in ${\mathbb{R}}^2$ for 3D problems. This fact changes the structure of the resulting (saddle-point or dual) algebraic problems since the simple bounds in 2D are replaced by the separable spherical constraints in 3D. The PF algorithm is an appropriate modification of that one proposed in [14], while the implementation of the SSN method is based on similar ideas as in [15]. Note that original versions of these algorithms were developed and successfully tested in context of 3D contact problems of solid mechanics. In contrast to [15], it is not necessary to symmetrize the (generalized) Jacobian matrices in the SSN algorithm if the adhesive coefficient $\kappa$ is positive. A considerable difference between problems of solid and flow mechanics is the presence of the incompressibility and impermeability conditions in the latter. Consequently, the dual algebraic formulations involve a relatively small number of constrained unknowns which excludes the efficient use of some types of algorithms. For this reason we do not use the active-set minimization algorithms [12]. It has been observed that for our type of 2D problems [13] they are less efficient, especially for large-scale computations. This property is much more significant in 3D.

The rest of the paper is organized as follows. In Section 2 we introduce different formulations of our problem. Section 3 deals with the finite element approximation based on the P1-bubble/P1 finite element pair. The respective algebraic problem is expressed in the saddle-point and the dual form. In Section 4 the algorithms are presented. The PF algorithm is assembled as the solver of the dual problem with the separable spherical constraints. The SSN algorithm solves the projective form of the saddle-point problem in which the constraints are expressed by the projections on circles in ${\mathbb{R}}^2$. Section 5 summarizes results of numerical experiments. Finally, Section 6 gives several concluding remarks.