Journal of Non-Newtonian Fluid Mechanics, 39 (1991) 269-290 Elsevier Science Publishers B.V., Amsterdam

269

Numerical integration of differential viscoelastic models

Joseph Rosenberg * Center for Advanced Materials, Lawrence Berkeley Laboratory and Department of Chemical

Engineering, University of California, Berkeley, CA 94720 (U.S.A.)

Roland Keunings Unite de Mecanique AppliquPe, UniversitP Catholique de Louvain, Place du Levant 2, B-1348

Louvain-la-Neuve (Belgium)

(Received July 23, 1990; in revised form November 26, 1990)

Abstract

Various numerical techniques are compared in the integration of the upper-convected Maxwell constitutive model on the basis of given kine- matics in the planar stick-slip problem. The techniques include three differ- ent finite element methods (Galerkin, streamline-upwind, and streamline- upwind Petrov-Galerkin), and a streamline integration scheme. The results illustrate the failure of Galerkin’s method in flows possessing steep solution gradients or singularities. The two upwinding techniques are shown to be superior to Gale&in’s method but they remain inaccurate near singularities and in flow regions where the solution gradients are transverse to the streamlines. The streamline integration scheme, which in essence is a method of characteristics applied to the purely hyperbolic constitutive model, leads to stable and accurate stress predictions even close to singularities. In view of these results, we have developed a Picard iterative method based on streamline integration to produce solutions to the full set of governing equations for an Oldroyd-B fluid. Results for the stick-slip flow problem could only be obtained for modestly low values of the Weissenberg number with a rather coarse discretization of the flow domain. The reason for the poor convergence properties of the iterative scheme is the large sensitivity of the computed extra-stress to minute changes in kinematics in flow regions of high stress gradients. Our stress integrator based on streamline integration

* Present address: POLYFLOW, Place de I’UniversitC 16, B-1348 Louvain-la-Neuve, Bel- gium.

0377-0257/91/$03.50 Q 1991 - Elsevier Science Publishers B.V.

270

increases the (iterative) convergence difficulties since it captures those stress gradients with great accuracy. The successful use of streamline integration in viscoelastic calculations requires further work towards decoupled iterative schemes capable of handling large stress sensitivity.

Keywords: finite element techniques; numerical simulation; streamline integration; streamline upwinding; viscoelastic flow

1. Introduction

Until recently, Galerkin finite element methods have been the techniques of choice for the numerical modeling of viscoelastic flow with differential constitutive equations. While some problems endowed with smooth exact solutions have been solved successfully with this approach, it is known that Gale&in techniques often produce unstable results in flows possessing stress singularities or boundary layers. A source for the numerical instabilities is undoubtedly the hyperbolic nature of differential viscoelastic constitutive models (see [1,2] for detailed reviews of viscoelastic flows).

Alternative numerical methods that take into account the hyperbolic nature of viscoelastic flows have been developed recently by several authors [3,4]. The goal of the present paper is to compare the stability and accuracy of some of these new techniques in a flow problem possessing a stress singularity and large solution gradients. Included in the comparison are three finite element methods (Galerkin, streamline-upwind and streamline- upwind Petrov-Galerkin) and a method of characteristics known as stream- line integration. Results are discussed for the prediction of viscoelastic extra-stress using the upper-convected Maxwell model and given kinematics corresponding to the Newtonian, planar stick-slip problem. As expected, Gale&in’s method behaves very poorly in these numerical experiments. The two upwinding techniques are markedly superior to Galerkin’s method, except in the neighborhood of the singularity (i.e. the exit section) where they do not converge with mesh refinement. Streamline integration, however, produced results that are stable and accurate, and which converge with increasing levels of discretization even near the singularity. The simulation results predict a steep stress boundary layer transverse to the main flow direction and emanating from the exit section. The two upwinding tech- niques, which introduce stabilizing artificial stress diffusion in the streamline direction only, are unable to capture the boundary layer without spurious oscillations.

In view of these results, we have developed a decoupled technique based on streamline integration for the prediction of both kinematics and visco-

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elastic stress for an Oldroyd-B fluid in stick-slip flow. Galerkin’s method is used to discretize the conservation equations while streamline integration is applied to the constitutive model: a simple substitution (Picard) scheme updates the nonlinear iterations. It is found that small changes in kinematics computed near the singularity produce very large variations in the stress field. This high sensitivity of stress on kinematics, which is approximated very accurately by the streamline integration scheme, is likely to pose serious difficulties for any iterative scheme. It does indeed limit the convergence of the present Picard iterative scheme to modestly low values of the Weissen- berg number. Improved decoupled iterative schemes are clearly needed to exploit the appreciable numerical properties of streamline integration to their full extent. This work also shows the need for carefully defined convergence criteria for decoupled methods. We find indeed in our numeri- cal experiments that the generally accepted, single criterion based on the relative difference between successive updates does not allow for a complete assessment of convergence of Picard-type iterative techniques.

2. Governing equations

Differential viscoelastic models used currently in numerical simulations have the general form

IJ= -PI+ TN+ TV, (1)

TN = 2pL,D, (2)

aT, A(T,; X).Tv+hst=2p,D,

where u is the Cauchy stress, p is the pressure, Z is the unit tensor, and TN, TV are the Newtonian and viscoelastic contributions to the extra-stress, respectively. The differential equation (3) is written here for a single pair of relaxation time X and viscosity coefficient pV. The Newtonian stress in- volves a viscosity coefficient pN and the rate of strain tensor D = l/2( vv + vuT), where u is the velocity vector. The symbol A in (3) denotes a model-dependent tensor function which is equal to the unit tensor for a vanishing relaxation time. Finally, the operator 6/6t is an objective time derivative defined as a linear combination of lower and upper-convected derivatives. We have

ST L=“;“+(l-a)z” 6t

O<a<l,

A V

where TV and TV are respectively the lower and upper-convected derivatives

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of the extra-stress defined by

A aT,

Tv=~+v~vT,+T;v~~ + vu. TV, (5)

~~=~+~~~T,-T;vv-vv~~T,. (6)

In steady-state problems, and for a given velocity field, the generic differen- tial viscoelastic model (3) can be cast in the form

0. VT, = +(T,, VV), (7)

where B is a model-dependent tensor function. It is easily shown that (7) constitutes a set of first-order hyperbolic equations for the components of TV whose characteristic curves are the streamlines [5]. The various numerical techniques that we used to integrate (7) are described in Section 3; numeri- cal results are discussed in Section 4.

The complete formulation of viscoelastic flow problems includes the constitutive equations (l-3), the conservation laws and suitable initial and boundary conditions. In the case of steady-state, incompressible creeping flows under isothermal conditions, the conservation laws read

v .(-PI+ TV+ ~,)=0, (8)

v.v=o. (9)

We present a decoupled method for the solution of (l-3) and (8-9) in Section 5; the corresponding numerical results are discussed in Section 6.

3. Numerical techniques for stress computation

Our objective is to solve the hyperbolic constitutive model (7) on the basis of a given steady-state, two-dimensional velocity field, using three different finite element techniques and the method of characteristics.

The finite element techniques rest upon the following approximation of the unknown stress field:

TV* = 5 T,'&, (W i=l

where the c#B;‘s are given finite element shape functions, while the symbols T;' denote unknown nodal values of the viscoelastic stress.

In the standard Galerkin finite element technique (GFEM), the ap- proximation (10) is substituted into the constitutive equation (7), and the resulting residuals are made orthogonal to the set of basis functions,

(&; v. VT,* - ~B(T"*, VU)) =o. (11)

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Here, the brackets (;) denote the L2 inner product over the flow domain fit, i.e.

(f; g> = l,fR dQ- 02)

It is an acknowledged challenge to design finite element techniques that enjoy high-order accuracy and good stability properties in the solution of hyperbolic problems. The mathematical analysis given in [6] for linear first-order hyperbolic systems shows that GFEMs are formally accurate but unstable, i.e. they tend to produce oscillatory results unless the exact solution is globally smooth. One way of stabilizing the results is to use upwind schemes. Instead of (7), upwind schemes solve the modified problem

u-vT,=+(T,. vv)+v -(K.vT,), (13)

by means of the Galerkin principle. The symbol K in (13) denotes an artificial diffusivity tensor whose magnitude is of the order of the character- istic finite element size h. Modifying the original problem in this manner has the consequence of limiting the convergence rate to at most first order, whatever the degree of the polynomial basis functions c#+ As shown in [7], upwind finite element techniques based on isotropic artificial diffusivity tensors produce smooth but inaccurate solutions; typically, the numerical results are excessively smoothed out in the direction transverse to the main flow. The problem of crosswind diffusion is solved in the streamline upwind (SU) method [7], where the artificial diffusivity acts in the streamwise direction only. The SU artificial diffusivity tensor thus reads

K=k$-, (14)

where k is a scalar of order h [7]. SU methods have improved stability properties relative to Galerkin’s techniques, but they cannot be more than first-order accurate [7].

Finite element methods exhibiting the stability properties of SU schemes while being of higher-order accuracy have been developed by Hughes and co-workers [7,8]. These techniques, known collectively as streamline-upwind Petrov-Galerkin (SUPG) methods, are not based on a modification of the original problem. In the present context, the SUPG discretization of (7) reads

(w;; v.vT,* - $(I;*, vu)) =O,

where the weight functions are given by

W; = +, +ku* v+,.

(15)

(16)

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The symbol k again denotes a scalar of order h [7,8]. The specific choice (16) of weight functions results in artificial diffusivity being added in the streamwise direction only. In contrast to SU methods, however, the ensuing stability improvement relative to Gale&in techniques is not accompanied by a restriction to first-order accuracy. Indeed, it can be shown that Galerkin methods as applied to linear hyperbolic problems are formally less accurate than their SUPG counterparts [6].

The finite element techniques described above are global methods applied to the purely hyperbolic problem (7). Hyperbolic equations are most natu- rally solved by the method of characteristics, a local method whereby the original set of partial differential equations is transformed into a set of ordinary differential equations to be integrated along individual characteris- tic curves. In the present context, we can write the constitutive model (7) as

where s is the arc length along a streamline. Streamline integration (SI) is a method whereby the constitutive law (17) is numerically integrated along individual streamlines [3]. Integration starts at an upstream location where TV is known, and is continued until a desired point in space is reached. Formally, SI amounts to the following computational task

where the integral sign denotes a streamline integration operator, and sO and s, refer respectively to upstream and downstream locations along the streamline.

We identify the streamlines in two-dimensional steady flows as contour lines of the streamfunction 4. The latter is related to the flow kinematics through the Poisson equation

(19)

where u, and uu are the velocity components in a Cartesian (x, v) coordi- nate system. We solve (19) by means of a standard Gale&in finite element method. The construction of individual streamlines based upon the stream- function computed over an arbitrary finite element mesh is conceptually simple, but very tedious to implement in a computer code. Details of our procedure are given [9]. We use a second-order accurate predictor/corrector scheme to integrate (17) along the constructed streamlines. The algorithm automatically adjusts the step size along the streamline such that a measure of the local discretization error is kept below a user-defined tolerance. The complete strategy is also described [9].

275

4. Viscoelastic stresses in stick-slip flow: Newtonian kinematics

The planar stick-slip flow problem [2] was chosen to compare the numeri- cal accuracy and stability of GFEM, SU, SUPG and SI methods. The flow geometry and boundary conditions are depicted in Fig. 1. This problem is very challenging in view of the stress singularity at the exit point [lo]. For the sake of illustration, we shall consider numerical results obtained with the upper-convected Maxwell (UCM) fluid, for which we have

A(T,; A) =I, a = 0,

B(T,,vv)=h(T;vv+v~~.T,)-T,+2~~D,

in (3,4) and (7).

(20)

Newtonian kinematics were computed by means of the finite element mesh shown in Fig. 1 and the classical velocity-pressure Galerkin method [2] with P2-Co interpolation for velocity and P’-Co interpolation for pressure. The same finite element mesh was used to compute the viscoelastic stresses with GFEM, SU and SUPG. Here, we selected a P’-Co interpola- tion for T,, resulting in 576 stress nodal values. Two successive mesh refinement steps were carried out by dividing each element of the original mesh into four and sixteen sub-elements, leading to 2079 and 7875 stress unknowns, respectively. The stress computations on all three meshes are based on the Newtonian kinematics computed with the original mesh of Fig. 1. SI results were obtained on the two particular streamlines shown in Fig. 1. In order to construct those streamlines, we solved the Poisson equation (19) using a P2-Co finite element interpolation of the streamfunction and the 4 X 4 subdivision of the original mesh. Computing viscoelastic stresses for the UCM fluid based on given kinematics is a linear mathematical problem.

X

t Y

Fig. 1. Planar stick-slip flow. Partial view of the finite element mesh used in the calculations based on Newtonian kinematics. Also shown are the streamlines $ = 0.432 and J, = 0.056; the streamfunction equals 0 on the stick-slip boundary and 1 on the plane of symmetry. Fully-developed Poiseuille flow conditions are specified 10 half-channel widths upstream of the exit plane; uniform flow conditions are specified 20 half-channel widths downstream of the exit plane.

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0.8

0.6

-10 -5 0 5 10 15 20

axial coordinate

’ - (b)

0.8-

0.6- P

0.4 1

0.21

0 / , .

I I I I I /

-10 -5 0 5 10 15 20

axial coordinate

Fig. 2. Newtonian kinematics along streamlines (a) I/J = 0.432 and (b) I+L = 0.056. The x- and y- components of velocity are shown in dashed and solid lines, respectively.

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Results can thus be obtained whatever the value of the Weissenberg number We. In the present paper, we discuss numerical predictions for We = 3, with We defined by

~(?=xv H' (21)

where V is the average axial velocity and H is the half-channel width. Figure 2 shows the computed Newtonian kinematics along the two

particular streamlines 4 = 0.056 and 0.432 illustrated in Fig. 1. Important streamwise velocity gradients occur in the exit section. It should be noted that the predicted Newtonian flow field is incompressible in the Galerkin weighted residual sense only. Along the streamline closest to the wall (i.e. 4 = 0.056) the dimensionless divergence of the velocity v . (Hu/V) attains a maximum of about l/10 at the exit section. The corresponding maximum value for the streamline further away from the wall (i.e. + = 0.432) is one order of magnitude smaller. We wish to emphasize that the ‘bump’ in the y-component of the velocity field along 1c, = 0.056 (Fig. 2b) is not the result of spatial, node-to-node oscillations; indeed, the scale over which it occurs is much larger than the elements size in that region of the flow field.

SI results for the UCM extra-stress at We = 3 are shown in Fig. 3. They have been obtained with a dimensionless step size h/H automatically varying between 5 X lop2 and 5 X lop6 along a given streamline. The smallest values of the step size are used in the exit region where the mathematical problem becomes quite stiff. It should be pointed out that the integration step sizes used here with SI are orders of magnitude smaller than the characteristic dimension of the finite elements used in our GFEM, SU, and SUPG calculations. Careful numerical experiments [9] with various ranges of step sizes and alternative streamline integration rules reveal that the results shown in Fig. 3 constitute highly stable and accurate predictions of the UCM extra-stress based on Newtonian kinematics. We can state safely that these results are fully-converged as far as discretization errors are concerned. In view of their accuracy, we shall use the SI predictions as a basis for comparison.

For the same flow conditions, GFEM, SU, and SUPG were used to predict the UCM extra-stress with 1 x 1, 2 x 2, and 4 x 4 sub-divisions of the finite element mesh shown in Fig. 1. Results are compared to the SI predictions in Figs. 4-6, in terms of the yy-component of the extra-stress along the two streamlines 11/ = 0.056 and 0.432. Significant differences are seen between Gale&in and SI results. The discrepancy is largest near the singularity, but it is also noticeable over most of the computational domain. Actually, Galerkin’s results diverge from their SI counterparts with in- creased mesh refinement. The intrinsic instability of Galerkin’s method

278

lo i(a)

8-

6-

0

:

.,.. & ;‘:, ,_,---

: -- .______________.

-2 I I I I I I I -10 -5 0 5 10 15 20

axial distance

8o 1 (b)

-10 I I I I I I I

-10 -5 0 5 10 15 20 axial coordinate

Fig. 3. Streamline integration predictions of UCM extra-stress along streamlines (a) # = 0.432 and (b) J, = 0.056. The XX-, xy-, and yy- components of TV are shown in dotted, dashed, and solid lines, respectively.

279

30

20

10

0

-10

-20

-30

1 l

i

+ -l(

\J

I I I I / I

I -5 0 5 10 15 20

axial distance

-100 / I I I I I I

-10 -5 0 5 10 15 20

axial distance

Fig. 4. Gale&in versus streamline integration predictions for the yy-component of UCM extra-stress along streamlines (a) 4 = 0.432 and (b) J, = 0.056. SI prediction is shown in solid line; Gale&in results using 1 x 1, 2 x 2, and 4 x 4 sub-divisions of the original mesh (Fig. 1) are shown in long-dashed, short-dashed, and dash-dotted lines, respectively.

280

2o $4

1

15-4

axial distance

250 l(l I

I 200 4

I 150 4

I

1 100

50 _I

0

I -50 1

1 ,

W

\

\+_

/I7 I

. /

-100 ~ I I / I / I

-10 -5 0 5 10 15 20

axial distance

Fig. 5. Streamline-upwind versus streamline integration predictions (see legend of Fig. 4).

281

i-_ _ . 8

. 6-1 Ifi

41

-2 ~ I / I -10 -5 0 5 10 15 20

axial distance

4oo 104

300 J p

1

1 \,

200

c

\

100

1

I’ \

4 0’

1 I

I I \’

-100

\

/

-200 / I I I I I -10 -5 0 5 10 15 20

axial distance

Fig. 6. Streamline-upwind Petrov-Galerkin versus streamline integration predictions (see legend of Fig. 4).

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applied to purely hyperbolic problems allows the propagation over a large part of the flow domain of spurious oscillations emanating from the singu- larity.

Results obtained with both SU and SUPG are significantly superior to the Galerkin predictions along the streamline of Fig. 1 that is further from the wall (Figs. 5(a) and 6(a)). Convergence to the SI results is observed with increased mesh refinement. In this particular flow problem, a 4 X 4 sub-divi- sion of the mesh used for predicting the kinematics is required to obtain accurate extra-stress values along that streamline. Inspection of Figs. 5(b) and 6(b) reveals, however, that SU and SUPG results are quite inaccurate along the streamline of Fig. 1 that is closer to the wall; the inaccuracy persists over several channel widths downstream of the exit section. The numerical results obviously do not converge with mesh refinement there. By contrast, SI enjoys stability and accuracy even near the singularity. (Stability properties of the integration rules used in SI are well documented, see e.g. ref. 11).

Additional insight into the behavior of the three finite element techniques can be gained from inspection of Fig. 7. Here we show the yy-component of the UCM extra-stress computed at the cross-section located one half-chan- nel width downstream of the exit. The dimensionless abscissa x equals 0 at the plane of symmetry and 1 at the slip boundary. The results obtained with the three finite element meshes reveal the presence of a steep stress boundary layer located at the slip boundary and transverse to the flow direction. The Galerkin method cannot cope with such high stress gradients. It produces spurious oscillations that propagate far away from the slip boundary. Mesh refinement does not solve the problem but it rather increases the amplitude of the oscillations. The SU and SUPG predictions also show spurious oscillations, but these tend to be confined near the slip boundary. Conver- gence with mesh refinement is actually observed in most of the cross-section. The SU and SUPG techniques do not, however, behave very well in flow regions where the solution gradient is not aligned with the streamlines, as is the case here near the slip boundary.

The above results show how delicate it is to compute viscoelastic stresses on the basis of Newtonian kinematics in the stick-slip flow problem. The stress singularity, typical of many flow problems of practical significance, and the transverse stress boundary layer downstream of the exit are mainly responsible for those numerical difficulties. In a modified problem, the domain containing the row of elements adjacent to the stick/slip boundary of the 1 X 1 mesh was truncated, thus removing the singularity and the transverse boundary layer from the computational domain. The kinematics were left unaltered over the reduced flow field. Of course, this operation does not change the SI results along streamlines present in the reduced

283

‘Ooo (a) 800 -

600 -

400 -

200 -

0

-lOO- \ \ \

-300 - \ \ !

-500 1 I I I

i 14 3000

2000

1000

5oo (b) \ 300 -

0.0 0.2 0.4 0.6 0.8 1 .o

X

Fig. 7. (a) Gale&in, (b) SU, and (c) SUPG predictions of the yy-component of the UCM extra-stress at the cross-section located one half-channel width downstream of the exit. Dashed, dotted, and solid lines correspond to 1 x 1, 2 X 2, and 4~ 4 sub-divisions of the original mesh, respectively.

284

0.0 0.2 0.4 0.6 0.8 1.0

X Fig. 8. Gale&in predictions of the yy-component of the UCM extra-stress at the cross-section located one half-channel width downstream of the exit. The solid line is for a 1 X 1 mesh and the modified problem; the dashed line corresponds to a 4 X 4 mesh and the original problem.

domain. GFEM, SU, and SUPG methods, however, are expected to produce results different from those of the original problem, since they are global techniques. As detailed in [9], all finite element schemes performed very well on the modified problem; good agreement with SI results and global convergence with mesh refinement was observed. A typical result is shown in Fig. 8, where we consider again the yy-component of the UCM extra-stress in the cross-section located one half-channel width downstream of the exit. We see that the Galerkin results for the truncated problem obtained with the 1 X 1 mesh are by far superior to their counterparts for the original problem and a 4 x 4 mesh. The conclusion to be drawn from this test is that finite element schemes can behave very well in smooth hyperbolic problems, even if they are based on Galerkin’s principle.

5. A decoupled technique based on streamline integration

The above results for the computation of viscoelastic extra-stress on the basis of given kinematics reveal the superiority of the streamline integration approach over global finite element schemes, in terms of both numerical accuracy and stability. Solving the constitutive equation (7) using SI simulta- neoud with the conservation laws (8,9) is a formidable task, however. Indeed, the viscoelastic extra-stress depends upon the velocity along given streamlines through (7); the streamlines themselves are unknown a priori and they depend upon the stress and velocity fields in a non-trivial manner.

285

The situation is akin to viscoelastic computations with integral constitutive models. The implementation of a coupled technique based on streamline integration is not feasible on current supercomputers, unless one is ready to perform the streamline integration process very crudely (i.e. with large integration steps). Available schemes based on SI are decoupled methods where the computation of the viscoelastic extra-stress is performed sep- arately from that of the kinematics. From known kinematics, one calculates the viscoelastic stress using SI; the kinematics are then updated by solving the conservation equations (8,9), and the procedure is repeated. The update scheme is usually of the Picard type.

We discuss in the remainder of the present paper the (iterative) conver- gence behavior of a decoupled method based on SI for the prediction of Oldroyd-B kinematics and stresses in stick-slip flow. No discussion will be offered on the qualitative behavior of the computed solutions (such as the stress behavior near the singularity), in view of the difficulties met in obtaining convergent iterates and the inordinate cost of serious mesh refine- ment experiments.

The method implemented in our work is similar to that of Luo and Tanner [3]. If n denotes the iteration index, the basic steps of the procedure are as follows:

Step I Compute by SI the viscoelastic stress T,” based on current velocity nn and current finite element mesh; stress values are needed at each Gauss point of the finite element mesh used in the kinematics computation (Step

2). Step 2 Update kinematics by solving the perturbed Newtonian problem

v .(-p”+‘I+2(~N+~A)Dn+1)= -v .(T;-2/.@“),

v . ,n+’ = 0. C-22)

Equations (22) are solved by means of the standard Galerkin finite element technique. Step 3 Compute the streamlines corresponding to updated kinematics v”+‘. Step 4 Adapt the finite element mesh so that elements conform with streamlines. Step 5 Check for convergence. Return to Step 1 if necessary.

Step 1 can be very expensive for an arbitrary finite element mesh. For example, the SI evaluation of TV at the nine Gauss points of all the elements of the mesh shown in Fig. 1 required of the order of 30 CPU minutes on a CRAY X-MP single processor computer! The reason is simply that Gauss points generally lie on different streamlines. Luo and Tanner [3] use streum- line elements to avoid the difficulty. Streamline elements are quadrilateral

286

elements which have a pair of opposing edges that remains aligned with a pair of streamlines during the non-linear iterations. Streamlines are thus conveniently approximated in each element as lines of constant local coordi- nate (in the parent element). In addition, families of Gauss points lie on common streamlines. For the simulations described in the present section, we use bi-linear quadrilateral streamline elements; streamlines are thus approximated by straight lines over each element.

Following Luo and Tanner, we have introduced in both sides of (22a) an arbitrary Newtonian stress component 2pELAD. Specific values of r_1* do not affect the final converged solution, but they may have an impact on the speed of convergence of the iterative procedure. In effect, this arbitrary viscous term is equivalent to the addition of an artificial time derivative in the momentum equation; indeed, a simple backward difference approxima- tion of the time derivative of D becomes

- 0”) = 2pA(Dnt1 -D”), (23)

where the artificial time step At, is identified with the inverse of 2~~. A time derivative of the rate of deformation tensor has no physical meaning in the momentum equation; it is simply an artificial term which can aid convergence, as we shall see.

The check for convergence of decoupled iterations (Step 5) is not at all obvious. In our opinion, it has not been treated in the literature with the care it deserves. Decoupled Picard schemes enjoy first-order convergence at best. As a result, convergence is much more difficult to assess than with the second-order, Newton technique. Convergence of decoupled techniques is usually checked on the basis of maximum relative changes (MRCs). These quantities are defined for viscoelastic stress, velocity, pressure, streamfunc- tion, and nodal positions in the following way:

max 1s:” -S: 1 MRC(S) = ;

i

(24) max max 1 S:+l

i IL1).

Here, S represents a generic field variable, the superscript n is the iteration index, and the subscript i refers to either Gauss points or nodal values. The MRCs represent the maximum relative change in computed field variables between consecutive iterations. We have found in our numerical experiments that decreasing or even very small MRCs (e.g. of order 10-4) do not guarantee convergence of the Picard scheme. MRCs can be arbitrarily small while the solution is slowly but constantly diverging. This can be checked by monitoring distances for each field of interest, defined by

DIST(S, So) = x(S;+‘- SF)‘, (25)

287

where i and n have the same meaning as in (24); the superscript 0 refers to a particular fixed iterate. Distances are a measure of the movement of the solution during the iterations with respect to a fixed point in solution space; they become constant at convergence. In our numerical experiments, we have found cases where all MRCs had attained very low levels, i.e. of order 10v4, while distances were constantly growing, indicating that convergence was not achieved. It should however be noted that monitoring MRCs and distances is not sufficient to assess convergence of decoupled iterations unambiguously. Indeed, The fact that distances reach constant values does not necessarily imply convergence. For lack of a truly satisfactory set of criteria, we say that the solution has converged when all MRCs remain below 5 X lop2 over prolonged iterations (typically 10 to 20), while all distances do not grow.

6. Stick-slip flow of an Oldroyd-B fluid

We have used the above procedure to compute planar stick-slip flow of an Oldroyd-B fluid. The constitutive equation is that of the UCM fluid for the viscoelastic stress (i.e. eqn. (20)), while the viscous component in eqn. (2) is selected such that pN/pL, = l/8. Figure 9 shows the finite element mesh used in these calculations. Three sets of computations were made using differing SI step size parameters. Computations started at We = 0 and were continued with increments in We of 0.1 until convergence was lost (see Table 1). Quoted execution times are for a CRAY X-MP single processor machine.

We found that the choice of the viscosity pA greatly affects the conver- gence of the iterations. Values of pLA were chosen by trial and error for each value of We. If pLA was chosen too small, iterations tended to diverge wildly. If pLA was chosen too high, the computed MRCs were deceptively low, but the solution nevertheless diverged. Convergence, when it did occur, was typically linear and slow. MRCs for the viscoelastic stress were usually an order of magnitude larger than those for the velocity, which themselves were generally an order of magnitude larger than those for pressure and stream- function. MRCs for the nodal locations were typically several orders of magnitude less than those for pressure and streamfunction. Loss of conver- gence was almost always caused by an MRC in TV which exceeded our convergence criterion of 5 X 10p2.

With a coarse streamline integration (Case I of Table l), convergence was lost for We above 0.2. More refined streamline integrations (Cases II and III) produced results up to We = 0.5. As discussed in ref. 2, divergence of a decoupled method is not necessarily indicative of the existence of irregular points in either the continuous or discrete solution families.

288

Fig. 9. Finite element mesh used for computing stick-slip flow of an Oldroyd-B fluid,

Mesh refinement experiments are prohibitively expensive with the SI decoupled method. We could not afford to use a mesh more refined than that of Fig. 9. We did however perform a final experiment which emulates the 4 X 4 sub-division for the stress computation used in Section 4. Nine Gauss points were defined on each quarter of a given element, thus provid- ing a more accurate computation of the integral involving T,” in the weak formulation of the momentum equation. The streamline integration parame- ters of Case III were used (see Table 1). Refining the stress calculation in this way had the adverse effect of reducing the attainable We down to 0.1. Actually, the 4 x 4 sub-division requires prediction of TV much closer to the singularity than does the 1 X 1 counterpart. Small changes in the velocity field there can produce large changes in stress, leading sometimes to MRCs for velocity and stress that differ by two orders of magnitude. This high sensitivity of stress to changes in kinematics, which indeed is reproduced

TABLE 1

Simulation statistics for stick-slip flow of an Oldroyd-B fluid. Case I: e = 10e3, h/H E (5 x lo-*, 5 x 10m3), Case II: c =10e4, h/H E (5 x lo-*, 5 x 10m3), Case III: e =10m4, h/H E (5 x lo-*, 5~ 1O-5), where e is the tolerance for local truncation error in the streamline integration, and h/H is the dimensionless streamline integration step.

We Total number Total CPU time of iterations (s)

Case I: coarse streamline integration 0.1 35 0.2 55

Case II: medium streamline integration 0.1 15 0.2 60 0.3 90 0.4 120 0.5 180

Case III: refined streamline integration 0.1 20 0.2 40 0.3 80 0.4 140 0.5 200

45 10 60 10

30 90

150 200 300

45 10 90 10

180 10 330 20 510 40

10 20 20 20 30

289

very accurately by the streamline integration procedure, is likely to pose serious convergence problems for most decoupled iterative schemes.

7. Conclusions

Our work shows that the computation of viscoelastic stresses based upon given kinematics is a delicate task in view of the hyperbolic nature of the constitutive equations and the presence of sharp gradients or singularities in the stress field. Finite element upwinding techniques constitute definite improvements over the standard Gale&in method, in terms of both accuracy and stability. Upwinding techniques fail, however, close to singularities and in flow regions where stress gradients occur transverse to the flow direction. The streamline integration method can be made both accurate and stable even near singularities. For problems devoid of large stress gradients, all numerical techniques considered in this paper produce adequate results.

The implementation of streamline integration within a decoupled iterative technique for computing stick-slip flow of an Oldroyd-B fluid has been only partially successful. The range of attainable values of the Weissenberg number is indeed limited by the large sensitivity of stress on kinematics in the exit region. The significant degree of accuracy afforded by the streamline integration technique becomes, in that regard, a source of convergence problems; it allows the method to capture better stress boundary layers or singularities, and thus increases stress sensitivity to changes in kinematics. The development of decoupled iterative schemes that can handle large stress sensitivity remains a challenge.

Acknowledgments

This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Science Division of the U.S. Department of Energy under Contract no DE-AC03-76SF00098. The numerical simulations described in this paper have been conducted on a CRAY X-MP supercomputer of the National Magnetic Fusion Energy Computer Center, Lawrence Livermore National Laboratory. Some of the work was done while Joseph Rosenberg was a visitor in the Unite de Mecanique Appliquee, Universite Catholique de Louvain.

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11

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