Weakly-imposed Dirichlet boundary conditions for non-Newtonian ...

Weakly-imposed Dirichlet boundary conditions for non-Newtonian fluid flow M.G.H.M. Baltussena , Y.J. Choia,b , M.A. Hulsena,∗, P.D. Andersona a

Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands b Dutch Polymer Institute (DPI), P.O. Box 902, 5600 AX Eindhoven, The Netherlands

Abstract We propose a new formulation for weakly imposing Dirichlet boundary condition in non-Newtonian fluid flow. It is based on the Gerstenberger-Wall formulation for Newtonian fluids [1], but extended to non-Newtonian fluids. It uses a stabilization term in the weak form that is independent from the actual fluid model used, except for an adjustable parameter κ, having the physical dimension of a viscosity. The new formulation is tested, combined with an extended finite element method, for the flow past a cylinder between two walls using both a generalized Newtonian and a viscoelastic fluid. It is shown that the convergence is optimal for the generalized Newtonian fluid by comparing with a converged boundary-fitted solution using traditional strong boundary conditions. Also the solution of the viscoelastic fluid compares very well with a traditional solution using a boundary-fitted mesh and strong Dirichlet boundary conditions. For both fluid models we also test ∗

Corresponding author. Email addresses: [email protected] (M.G.H.M. Baltussen), [email protected] (Y.J. Choi), [email protected] (M.A. Hulsen), [email protected] (P.D. Anderson)

Preprint submitted to Journal of Non-Newtonian Fluid Mechanics

May 19, 2011

various values of the κ parameter and it turns out that a value equal to the zero-shear-viscosity gives good results. But, it is also shown that a wide range of κ values can be chosen without sacrificing accuracy. Keywords: Weakly-imposed Dirichlet boundary conditions, XFEM, generalized Newtonian fluid, viscoelastic fluid 1. Introduction In recent years, the focus in numerical simulation of non-Newtonian fluid flows has been shifted towards heterogeneous systems, such as particle-filled flows [2, 3, 4, 5] and two-phase flows [6, 7]. In these flows, the interface between different materials is sharp and moving. When using the finite element method, the method of choice for limited motion of the interface has been the ALE scheme [8, 2]. In this scheme the mesh moves independently from the fluid motion. The results are usually very accurate since the mesh movement is such that the interface is always aligned with element edges. For large movement of the interface, the mesh can become highly distorted and frequent remeshing is required. Automatic remeshing is not easy, especially in 3D, and requires very specialized and expensive techniques. Therefore, the fictitious domain (FD) method [9, 10] has been quite popular in recent years for simulation of particle filled viscoelastic fluid flows [11, 5]. The FD method is relatively easy to use, since it requires a Eulerian mesh for the fluid only. However, the accuracy near the interface of velocity gradients, stresses and pressures is not good, since the interface is not aligned with element edges and intersects the interior of the elements. There is evidence that in very thin gaps between two solid surfaces, 2

the velocity solution is not even close to the exact solution [12]. In order to combine the accuracy of ALE with the ease of use of FD, the eXtended Finite Element Method (XFEM) is increasingly being used to model moving sharp interfaces. See Belytschko et al. [13] for a review of XFEM in material modelling and Choi et al. [4] and the references therein for application of XFEM to fluids. However, imposing the interface conditions, such as continuity of the velocity, is not straightforward in XFEM, since the interface, like in the FD method, does not coincide with element edges. Weakly imposing the interface conditions is an attractive possibility to overcome this problem. To evaluate the stability and accuracy of such an approach, we focus our attention to the more simple problem of weakly-imposed Dirichlet boundary conditions for the velocity vector. A quite general method for weakly imposing a Dirichlet boundary condition is the Lagrangian multiplier method, as used in the FD method. The Lagrangian multiplier space is defined on a separate mesh of the interface, where the elements are typically somewhat larger than the fluid elements. This has been used by Choi et al. [4] for viscoelastic fluid flows and works rather well. One of drawbacks of this approach is that the separate mesh for the Lagrangian multiplier needs to be adapted to the internal fluid mesh, which can be quite complicated if the latter is locally refined, especially in 3D. Also standard iterative solvers are difficult to apply to the saddle-point structure of the system matrix. Another approach is Nitsche’s method [14] for imposing Dirichlet conditions in a weak way at arbitrary surfaces. The method has been developed for diffusion-type problems in mind and requires several boundary terms, includ3

ing a penalizing term on the interface to stabilize the system. The penalizing factor depends on the mesh and problem being solved and it is not easy to obtain the optimal value. The method is increasingly being used within the XFEM community, see for example Dolbow & Harari [15] for diffusion-type problems. It is not obvious how to extend this method to non-Newtonian fluids. A method that does not require a stabilizing penalty is the BaumannOden approach for discontinuous Galerkin methods [16]. It retains the other boundary terms in Nitsche’s method but with a change of sign for one term. Apparently, the sign change has some negative effect on convergence [17]. Also here, it is not obvious how to extend this method to non-Newtonian fluids. Recently, Gerstenberger & Wall [1] proposed a new formulation for imposing weak Dirichlet boundary conditions, that seems to give optimal convergence rates for Newtonian fluids. It is based on an additional stress field variable defined on the fluid mesh that replaces the Lagrangian multiplier. This seems to be an attractive approach, where the separate mesh for the Lagrangian multiplier is not needed anymore. Furthermore the additional stress field can be eliminated on element level. Also, there are no adjustable parameters that need to be optimized for the problem, in contrast to Nitsche’s method. Application of the Gerstenberger-Wall formulation to non-Newtonian fluids does not seem to be straightforward, since it requires an inversion of the constitutive model. In this paper, we propose a new formulation for weakly-imposed Dirichlet boundary condition for non-Newtonian fluids. It is based on the Gerstenberger4

ΓD Γ = ΓD ∪ ΓN

n Ω ΓN

Figure 1: Fluid domain Ω with Dirichlet boundary ΓD , Neumann boundary ΓN and outwardly directed unit normal vector n.

Wall formulation for Newtonian fluids [1], but extended to non-Newtonian fluids by using a stabilization term in the weak form that is independent from the actual fluid model used. It also contains an adjustable parameter κ, having the physical dimension of a viscosity, similar in spirit to the DEVSS formulation [18]. For Newtonian fluids and κ equal to the viscosity of the fluid, the Gerstenberger-Wall formulation is retained. For non-Newtonian fluids the optimal value of κ is near the zero-shear-rate viscosity of the fluid, but it is shown that a wide range of values gives good results. 2. Problem formulation We consider a flow problem on a domain Ω with a stationary boundary Γ, that consists of a Neumann boundary part ΓN and a Dirichlet boundary part ΓD (see Fig. 1). The governing equations are the momentum and mass balance and the constitutive model for the fluid. Assuming that the flow is incompressible and inertia can be neglected, the momentum and mass 5

balance are given by: −∇ · (2µD + τ ) + ∇p = f

in Ω,

(1)

∇·u=0

in Ω,

(2)

where u is the velocity, p is the pressure, f is a possible body force per unit volume, µ is a constant (solvent) viscosity, D = (∇u + ∇uT )/2 and τ is the extra-stress tensor. Neglecting inertia is only for simplicity of the presentation and the removed terms can always be added at a later stage. For the constitutive model we choose either: a generalized Newtonian model: τ = 2η(γ)D ˙

in Ω,

(3)

or a Giesekus model: τ = G(c − I),

λc + c − I + α(c − I)2 = 0 5

In these equations, η(γ) ˙ is a viscosity function with γ˙ =

in Ω. √

(4)

2D : D, c is the

conformation tensor, G is the modulus, λ is the relaxation time, α is the mobility parameter and the triangle (5) denotes the upper-convected time derivative, as defined by 5

c=

∂c + u · ∇c − (∇u)T · c − c · (∇u). ∂t

(5)

The chosen constitutive models are just examples and the approach described in this paper can be applied to a much wider class of models. The boundary conditions for the momentum balance are given by ¯ u=u

on ΓD ,

(6)

(2µD + τ − pI) · n = ¯t

on ΓN ,

(7)

6

where n is the outwardly directed unit normal vector on boundary Γ. The ¯ and traction ¯t can be functions of both position and time. specified velocity u In case of the Giesekus fluid we have an additional boundary condition ¯ c=c

on Γin ,

(8)

where Γin is the part of Γ where u · n < 0 (inflow boundary). The specified ¯ can be a function of both position and time. Finally, conformation tensor c the value of the conformation tensor needs to specified initially: c(x, 0) = c0 (x),

for all x in Ω.

(9)

The weak formulation of the momentum and mass balance, Eqs. (1) and (2), is obtained after multiplying with test functions v and q, integrating over the domain Ω, partially integration of the divergence term, and replacing the traction on ΓN with ¯t (as given by Eq. (7)). This results in:

(∇v)T , 2µD + τ − (∇ · v, p) − v, n · (2µD+τ ) − pn Γ

D

= (v, f ) + (v, ¯t)ΓN , (10) −(q, ∇ · u) = 0,

(11)

for any test function v and q. We have defined proper inner products on Ω, ΓD and ΓN with obvious notation. We have also changed the sign of the mass balance to make it symmetric with respect to the corresponding pressure term in the momentum balance. In case of the Giesekus model we need to derive a weak form of the constitutive model Eq. (4) also, but this will be postponed to the section on the viscoelastic fluid flow problem. The most straightforward way of imposing the Dirichlet boundary condition Eq. (6) is imposing this condition on u directly and setting v = 0 on 7

ΓD . In this way the boundary integral on ΓD in Eq. (10) vanishes. In general, however, this is difficult to implement if the boundary ΓD does not coincide with the boundary of the computational domain. A weak formulation of the Dirichlet boundary conditions is an attractive alternative in that case. In practice, the direct (strong) approach will also be used for some parts of ΓD , but for convenience of the formulation we will assume in the following that ΓD is fully imposed in a weak way. We will also assume that if the inflow boundary Γin is present, it coincides with the boundary of the computational domain and can easily be handled in the traditional (strong) way. Various ways of imposing Eq. (6) in a weak way can be considered. A rather general approach is the Lagrangian multiplier method, which imposes a (weak) constraint on the space for u:
(∇v)T , 2µD + τ − (∇ · v, p) + (v, λ)ΓD = (v, f ) + (v, ¯t)ΓN ,

(12)

−(q, ∇ · u) = 0,

(13)

¯ )ΓD = 0, (µ, u − u

(14)

where µ is the test function for the Lagrangian multiplier λ on ΓD . Comparing Eq. (12) with Eq. (10) shows that we can identify λ with (minus) the traction on the boundary ΓD : λ = −n · (2µD + τ ) + pn.

(15)

The Lagrangian multiplier method has been used successfully in the fictitious domain method [9, 10] and also in the XFEM approach [19, 20, 21, 4]. Disadvantages of the Lagrangian multiplier method are: • The system has additional unknowns (Lagrangian multipliers), which are located in nodes on the boundary mesh and not in the nodes of 8

the regular mesh. An exception is the work of Mo¨ es et al. [20], where the Lagrangian multiplier is discretized on the nodes of the regular mesh. • The required interpolation of the Lagrangian multipliers is unclear and needs to satisfy the LBB condition for stability [20]. Finding an appropriate discretization of the Lagrangian multiplier is not straightforward, but possible, at least in 2D [20, 21]. • The resulting saddle point problem gives rise to a zero diagonal block which is not very well suited for fast iterative solvers. Therefore, we will consider other more attractive methods for imposing weak Dirichlet boundary conditions. In the next section we will first discuss weak Dirichlet boundary conditions for the special case of the Stokes equation, since this will be the basis for our proposed method for non-Newtonian flows. 3. Weak Dirichlet boundary conditions for the Stokes equation 3.1. Theory We obtain the Stokes system by setting τ = 0 in Eqs. (10)–(11), which reduces to:
(∇v)T , µ(∇u + ∇uT ) − (∇ · v, p)
− v, µn · (∇u+∇uT ) − pn Γ = (v, f ) + (v, ¯t)ΓN , D

(16) −(q, ∇ · u) = 0.

9

(17)

In order to impose the Dirichlet condition Eq. (6) in a weak sense we need the velocity u as a “flux” on the boundary, similar to the traction for the Neumann boundary. Therefore we partially integrate the first term in Eq. (16) and get
− µ∇ · (∇v + ∇v T ), u − (∇ · v, p)

− v, µn · (∇u + ∇uT ) − pn Γ + µn · (∇v + ∇v T ), u Γ D

= (v, f ) + (v, ¯t)ΓN ,

(18)

−(q, ∇ · u) = 0.

(19)

Now we set the “flux” u on the boundary ΓD to be the Dirichlet condition ¯ and leave it unspecified at ΓN . Finally, we partially integrate the value u first term once again and obtain the following weak form:
(∇v)T , µ(∇u + ∇uT ) − (∇ · v, p)

¯ Γ − v, µn · (∇u + ∇uT ) − pn Γ − µn · (∇v + ∇v T ), u − u D

D

= (v, f ) + (v, ¯t)ΓN , −(q, ∇ · u) = 0.

(20) (21)

By applying the same procedure to Eq. (21), i.e. partial integration and ¯, setting the “flux” u on the boundary ΓD to the Dirichlet condition value u we can replace Eq. (21) with ¯ )ΓD = 0, −(q, ∇ · u) + (qn, u − u

(22)

which makes the system symmetric. In practice it hardly affects the results, but it is attractive to have a symmetric system from an efficiency point of view. 10

The weak form obtained (Eqs. (20)– (21)) is consistent and the Dirichlet condition is imposed weakly. Unfortunately, this weak form is unstable. However, Nitsche [14] showed that adding a consistent penalizing term to the left-hand side of Eq. (20) stabilizes the system (in [14] a scalar diffusion problem is analyzed):
µ ¯ Γ = ..., · · · + K (v, u − u D h

(23)

where K is a sufficiently large positive constant and h is a characteristic size of the elements. Choosing K too low results in an unstable system and choosing it too high leads to an inaccurate solution, hence an optimal value for K has to be found where the system is both stable and accurate. Unfortunately, the optimal K depends on the physical problem and the discretization. By requiring coercivity of the appropriate bilinear form, an estimate for the optimal value for K/h can be found [15, 22]. Dolbow & Harari [15] give analytical expressions for linear second-order problems using linear triangular elements and Embar et al. [22] find an estimate for the optimal value when using spline-based finite elements by solving an eigenvalue problem. In both cases, the optimum value is local and varies along the boundary. Another method of stabilization is based on the Baumann-Oden approach for discontinuous Galerkin methods [16], where the sign of the last boundary term in the left-hand side Eq. (20) has been reversed: ¯ )ΓD = . . . · · · + µn · (∇v + ∇v T ), u − u

(24)

The weak form is still consistent, but no longer symmetric, which has some negative effect on convergence [17] and solving the system will be less efficient. 11

Gerstenberger and Wall [1] recently introduced a new method for weakly imposing Dirichlet conditions. It is based on the Lagrangian multiplier form Eqs. (12)–(14), however the “Lagrangian multiplier” is now defined on the domain instead of on the boundary only. For this, they introduce a symmetric tensor field σ and corresponding test function θ that relates to λ and µ on the boundary ΓD as follows: λ = −n · σ,

µ = −n · θ

on ΓD .

(25)

Considering the interpretation Eq. (15) of λ it seems quite natural to define the field σ in the domain as σ = µ(∇u + ∇uT ) − pI,

(26)

which fulfills the requirement Eq. (25) on ΓD . The next step is to “invert” the constitutive equation Eq. (26): 1 1 (σ + pI) = (∇u + ∇uT ). 2µ 2

(27)

This relation between σ and the rate-of-deformation tensor is imposed weakly and combined with the constraint equation (Eq. (14)) into a single weak form using the test function θ. The final weak form is given by
(∇v)T , µ(∇u + ∇uT ) − (∇ · v, p) − (v, n · σ)ΓD = (v, f ) + (v, ¯t)ΓN , (28)

θ,

−(q, ∇ · u) = 0,

(29)


1 1 ¯ )ΓD = 0. (σ + pI) − (∇u + ∇uT ) + (n · θ, u − u 2µ 2

(30)

This scheme basically fixes most of the disadvantages of the Lagrangian multiplier approach: 12

• The additional field σ is defined on the domain (in the nodes of the regular mesh) and if interpolated piecewise discontinuously, it can be eliminated on element level [1]. Also σ can be restricted to the elements intersected by the boundary ΓD . • The choice of interpolation of σ is not critical and the same order interpolation as u, but piecewise discontinuous, seems to work well. • The typical structure of a mixed method (large zero block) is gone and iterative solvers can more easily be applied. However, it also introduces some drawbacks: • The system is inherently non-symmetric, although the original Stokes problem is symmetric. • The “inversion” of the constitutive model can be difficult, if not impossible, to perform for non-Newtonian fluid models. Our objective is to apply a similar scheme to non-Newtonian fluids, but avoid the last drawback. Therefore, we rewrite the scheme using the symmetric tensor field E =

1 (σ + pI) 2µ

and abandon the stress tensor as a field variable:


(∇v)T , µ(∇u + ∇uT ) − (∇ · v, p) − (v, 2µn · E − pn)ΓD = (v, f ) + (v, ¯t)ΓN , (31) −(q, ∇ · u) = 0,

(32)

¯ )ΓD = 0, (H, E) − (H, ∇u) + (H · n, u − u

(33)

13

where the symmetric tensor field H is a test function for the field E. The symmetric tensor field E can be interpreted as a weak representation of the rate-of-deformation tensor D = 12 (∇u + ∇uT ). If the test function space H (and θ) fully includes the space for the pressure p, which is the case for all practical purposes, both formulations (for the Stokes equation) are identical. In Appendix A we present a derivation of the scheme using a mixed scheme and “boundary fluxes” as is common in discontinuous Galerkin methods [23, 17]. Furthermore, in Appendix B we derive the corresponding “primal” formulation, by eliminating E altogether. The resulting scheme is basically adding a symmetric and positive definite term to the weak momentum balance in its original form Eq. (10) (for τ = 0):

(∇v)T , µ(∇u + ∇uT ) − (∇ · v, p)− v, µn · (∇u + ∇uT ) − pn Γ D
−1 ¯ ) + (u − u ¯ )n Γ +µMkm (vn, φk )ΓD : φm ,n(u − u D

= (v, f ) + (v, ¯t)ΓN , −(q, ∇ · u) = 0,

(34) (35)

where φk are the shape functions for interpolation of E and Mkm = (φk , φm ). The shape functions φk will be taken equal order with the velocity shape functions, but discontinuous across element boundaries. In this way, the additional term can be computed on element level and only contributes to elements cut by the boundary ΓD . Note, that the additional term works similar to the Nitsche term Eq. (23) by adding a positive penalizing term, however in contrast to the Nitsche term, it does not have any adjustable parameters.

14

ΓD

Ω ΓN

Ωm

Figure 2: Fluid domain Ω with Dirichlet boundary ΓD and Neumann boundary ΓN as part of a bigger mesh domain Ωm .

3.2. Numerical methods We evaluate the various weak Dirichlet boundary conditions for use with the extended finite element method (XFEM). The mesh region Ωm fully includes the fluid region Ω and the boundary Γ intersects the elements (see Fig. 2). The discretization of the velocity u and pressure p is given by uh (x) =

X

ph (x) =

X

νk (x)uk

x ∈ Ω,

(36)

ψk (x)pk

x ∈ Ω.

(37)

k

k

The test functions v and q are similarly discretized. Note, that the shape functions νk and ψk are only evaluated/defined inside the domain Ω and on the boundary Γ. However, the nodal values uk and pk are also defined in the nodes outside the domain Ω for elements intersected by the boundary. For the shape functions νk and ψk we will use the Taylor-Hood Q2 Q1 element, which satisfies the LBB condition [24]. 15

In case of the Gerstenberger-Wall scheme, the symmetric tensor for E is discretized by E h (x) =

X

φk (x)E k

x ∈ Ω.

(38)

k

The test function H is similarly discretized. The shape function φk is taken identical to νk on element level, i.e. Q2 -interpolation, however φk is discontinuous across element edges, whereas νk is a continuous function. Note, that only degrees of freedom in nodes that are either in Ω or belong to elements intersected by the boundary Γ end up in the system vector. The weak form has been defined on Ω only, and therefore volume integration is performed on Ω only. This requires special integration techniques to integrate only on part of an element. We use the technique depicted in Fig. 3 and described in [4], which subdivides the element domain into small quadrilaterals and triangles. The (discretized) boundary Γ consists of small linear segments that do not cross element boundaries (see Fig. 3). Therefore, the boundary integrals can be evaluated on element level. For the integration on quadrilateral and triangular subdomains we use 3 × 3 and six-point Gauss integration, respectively. For integration on the line segments of the boundary we use three-point Gauss integration. Note, that the matrices Mkm , as used in the primal formulation (34), also need to be computed by integrating on the part of an element that is inside the domain Ω. 3.3. Results In order to study the convergence behavior of the various methods, we solve a Stokes flow problem around a stationary cylinder with a known exact

16

Γ Ω

Figure 3: Subdivision of quadrilateral elements (solid lines) into quadrilateral subdomains using a quadtree. The quadrilaterals at the smallest level that are intersected by the boundary are further subdivided into triangular subdomains on the part that belongs to Ω. The discrete interface consists of linear segments that do not cross element boundaries, making it possible to perform boundary integration on element level. In the actual computations, the number of levels in the quadtree is five.

17

solution, given in [25]: (R2 − r2 ) cos2 θ + r2 ln(r/R) + (1/2)(r2 − R2 ) , r2 (R2 − r2 ) cos θ sin θ u˜y = , r2 2 cos θ + 10, p˜ = − r

u˜x =

(39) (40) (41)

where R is the radius of the cylinder and (r, θ) are polar coordinates. The problem is solved on a square domain of side length 2, with a cylinder of radius 0.2 centered at the origin, as proposed in [25]. We apply the exact value of the velocity on the boundary of the domain as a strong Dirichlet boundary condition, and on the cylinder surface as a weak Dirichlet boundary condition. We specify the pressure in a single point of the domain. The relative L2 -errors in the velocity and pressure are defined as
1/2 R 2 ˜ ku − u k dx h L2 ,u = Ω R ,
2 dx 1/2 k˜ u k Ω
1/2 R (p − p˜)2 dx Ω h L2 ,p = ,
R 2 dx 1/2 p ˜ Ω

(42) (43)

where the subscript ‘h’ indicates the discretized field variables. The mesh used is a regular uniform mesh with square elements (width and height of the elements is denoted by h). The relative L2 -errors for the methods discussed in Sec. 3.1 are plotted against the element size h in Fig. 4. Basically optimal convergence rates are obtained for all methods: third order in the velocity and second order in the pressure. Some remarks: • The Gerstenberger-Wall scheme has the lowest error for both velocity and pressure. It also gives very smooth convergence curves. 18

10-2

relative L2 -error

10-3

10-4

10-5

Nitsche K=30 velocity Nitsche K=30 pressure Baumann-Oden velocity Baumann-Oden pressure Gerstenberger-Wall velocity Gerstenberger-Wall pressure

10-6

10-7

10-2

10-1

element size h

Figure 4: Convergence in relative L2 -norm with respect to element size h.

Results

are shown for the Nitsche scheme with K = 30, the Baumann-Oden scheme and the Gerstenberger-Wall scheme.

19

• The error in velocity is the largest for the Baumann-Oden and convergence is not smooth. The pressure error is also somewhat larger than the best scheme, but less so than for the velocity. • The value of K = 30 for the Nitsche scheme has been found by minimizing the error for a single value h. Apparently this is not optimal for other values of h. Higher values of K will rectify this. This behavior is typical for Nitsche scheme: the value of K needs to be big enough to obtain stable behavior. A too low value of K leads to unstable behavior and large errors. The conclusion is that the Gerstenberger-Wall scheme is the best performer: accurate, robust and (at least for Stokes) without any adjustable parameters. 4. An approach for non-Newtonian fluids 4.1. Theory Although the Gerstenberger-Wall scheme performs very well for Stokes flow, straightforward extension to non-Newtonian fluids of the original scheme using a stress tensor σ is difficult. It would require an inversion of the constitutive model, which is only easy for a few restricted cases, like a power-law model for η(γ) ˙ in the generalized Newtonian model. For a viscoelastic model it is not possible at all. The primal formulation Eq. (34), however, suggests an easy extension to general constitutive models by adding the same positive stabilization term to the original weak form Eq. (10), but now with an

20

adjustable parameter κ instead of the viscosity µ:

(∇v)T , 2µD + τ − (∇ · v, p)− v, n · (2µD + τ ) − pn Γ D
−1 ¯ ) + (u − u ¯ )n Γ +κMkm (vn, φk )ΓD : φm ,n(u − u D

= (v, f ) + (v, ¯t)ΓN , −(q, ∇ · u) = 0.

(44) (45)

The reason we need an adjustable parameter κ 6= µ is, that the “effective viscosity” of the non-Newtonian model is not determined by only the constant µ, but also by the flow-dependent behavior of τ . For melts even a value of µ = 0 is quite common. In order to justify the proposed approach using an adjustable parameter κ, we first test it for the Stokes equation (τ = 0) with κ 6= µ. We investigate the effect of the κ parameter on the relative L2 -errors by fixing the size of the fluid element to h = 0.0377 and solving the problem of Sec. 3.3 with different values for κ/µ. In Fig. 5, the results are shown and it is clear that indeed the error is minimal near κ = µ. However, more importantly this minimum is very shallow and over a wide range of κ/µ values the errors are only slightly above the minimum. In practice a value of 0.1 < κ/µ < 200 gives smooth solutions and low errors. Beyond this range (both larger and smaller values) errors start to increase and clearly unstable/wiggly solutions near the cylinder surface start to appear. Hence, once we avoid very small or very large values of κ, relative to the viscosity, the actual value of κ is not critical. This behaviour is very promising for simulating non-Newtonian flows, where the “effective” viscosity of the fluid is not constant. By choosing a suitable value for κ we will show in the following sections that indeed the 21

relative L2 -error

10-2

10

-3

10-4

10

velocity pressure

-5

10-2

10-1

100

101

102

103

104

105

106

κ/µ

Figure 5: Relative L2 -errors as a function of κ/µ for the Stokes problem of Sec. 3.3.

scheme as proposed in Eqs. (44)–(45) is suitable for non-Newtonian models (generalized Newtonian and viscoelastic). As a final remark, we note that there is a resemblance between the Nitsche stabilization term (Eq. (23)) and our stabilization term in Eq. (44). This suggests that we can replace the Nitsche term by our stabilization term, obtaining a symmetric version for the Stokes equation. This seems to work just fine for κ = µ and from an efficiency point of view this seems an attractive approach for the Stokes equation. However, it turns out that around κ = 0.5µ the method becomes unstable and big errors appear similarly to the unstable behavior of the Nitsche scheme if K is too low. This makes the symmetric version less suitable for non-Newtonian flows. So, basically the transposed boundary term destabilizes the system for small values of κ or K. We can

22

R

H = 4R

L = 4R Figure 6: The problem domain for the flow around a confined cylinder.

verify this by using the Nitsche stabilization in our non-symmetric system Eq. (44). It turns out (not shown) that indeed now the Nitsche stabilization has similar properties for low values of K as shown for our scheme in Fig. 5. However for values of K above the minimum the errors are for the velocity up to 1.5 times larger for those of our proposed method. For the pressure the errors are even up to three times larger. Also the other disadvantages of finding the optimal value of K and the dependence on the mesh size (h) of the front factor remain. 4.2. Results for a generalized Newtonian fluid We model the flow of a generalized Newtonian fluid around a cylinder confined between two parallel plates, see Fig. 6. The radius of the cylinder is R, the channel has height H = 4R and length L = 4R. The actual computations are performed using R = 0.25 and thus L = H = 1. The generalized Newtonian fluid has a shear-rate dependent viscosity which is

23

100

η 10-1 η0

10-2 10-1

100

101

102

103

ˆ γ˙ λ

104

105

106

107

108

Figure 7: The viscosity versus the shear rate.

modelled with the Carreau equation (µ = 0): η(γ) ˙ = η0 +

η∞ − η0 , ˆ γ) (1 + (λ ˙ 2 )(1−n)/2

(46)

where η0 is the viscosity at zero shear rate, η∞ is the viscosity at infinitely high ˆ is the timescale at which the viscosity becomes rate dependent shear rate, λ and n is the parameter determining the slope of the shear rate dependence. ˆ = 1 and n = 0.5. In this test problem we will use η0 = 1, η∞ = 0.01, λ The function η(γ) ˙ for these parameters is given in Fig. 7. No-slip boundary conditions are applied at the top and bottom wall, as well as on the cylinder wall. Parabolic in- and outflow profiles are applied at the left and right boundaries (ux = 6y(H − y)/H, with y = 0 at the lower wall, giving an average velocity of H). The zero velocity on the cylinder surface is imposed as a weak Dirichlet boundary condition, whereas the other conditions are 24

100

η η0

10-1

0

0.2

0.4

0.6

0.8

1

relative position on cylinder wall

Figure 8: The relative viscosity on the wall of the cylinder. The relative position is 0 at the top of the cylinder wall.

imposed as a strong Dirichlet boundary condition. The pressure is set to zero in the lower left corner of the domain. We first show the viscosity on the wall of the cylinder in Fig. 8, in order to verify that we indeed are in the shear-thinning range of the constitutive equation. This figure shows that the viscosity varies about one order of magnitude. It is highest at the front and back stagnation point of the cylinder and lowest at the top and bottom. In order to perform convergence analysis we compute the same problem on a dense boundary-fitted mesh with 40000 elements. A coarser version of this mesh can be seen in Fig. 9. The error is defined by the L2 norm of the difference between the solution when using the weak Dirichlet boundary con-

25

Figure 9: The boundary-fitted mesh used for convergence analysis. The actual mesh is ten times more refined in both co-ordinate directions and consists of 40000 elements.

dition Eq. (44) and the boundary-fitted solution for the velocity and pressure in the following way: L2 ,u L2 ,p

R ( Ω kuh − ubf k2 dx)1/2 R = , ( Ω kubf k2 dx)1/2 R ( Ω (ph − pbf )2 dx)1/2 R , = ( Ω p2bf dx)1/2

(47) (48)

where the subscript ‘h’ indicates the discretized field variable and the subscript ‘bf’ the solution from the boundary-fitted simulation. The weak boundary solutions are computed on a regular uniform mesh with square elements. In Fig. 10 results are shown for different values of κ/η0 as a function of the element size h. The results shows that the convergence is optimal for κ/η0 ≥ 1, whereas for κ/η0 = 0.1 convergence problems are visible, as can be expected from the results of Sec. 4.1. Taking κ = η0 seems a good choice for this problem, but higher values can be taken without a problem, as can be seen in Fig. 11, where the error versus κ/η0 is given for h = 0.0196. Choosing 0.1 < κ/η0 < 105 results in a low error for both velocity and pressure. 26

100

relative L2 -error

10-1

10-2

10-3

velocity, κ/η0 = 0.1 pressure, κ/η0 = 0.1 velocity, κ/η0 = 1 pressure, κ/η0 = 1 velocity, κ/η0 = 100 pressure, κ/η0 = 100

10-4

10-5

10-2

10-1

element size h

Figure 10: The relative L2 -error for the velocity (bottom) and pressure (top) solution for different value of κ/η0 as a function of the element size h in the problem with a generalized Newtonian fluid. The pressure error has been multiplied by a factor of 10 to separate the data from the velocity data.

27

relative L2 -error

10-2

10-3

10

velocity pressure

-4

10-2

10-1

100

101

102

103

104

105

106

107

κ/η0

Figure 11: The relative L2 -error for velocity and pressure versus κ/η0 in the problem with a generalized Newtonian fluid.

4.3. Results for a viscoelastic fluid 4.3.1. Weak form and numerical discretization For the discretization of the momentum balance we employ the DEVSSG method. (For more information on the DEVSS-G method, see the review paper by Baaijens [26] and references therein.) If we combine the primal form of Eqs. (44) and (45) with the DEVSS-G method, the weak formulation

28

of the momentum and mass balance can be stated as follows:

(∇v)T , 2µD + θ(∇u − GT ) + τ − (∇ · v, p)− v, n · (2µD + τ ) − pn Γ D
−1 ¯ ) + (u − u ¯ )n Γ +κMkm (vn, φk )ΓD : φm ,n(u − u D

= (v, f ) + (v, ¯t)ΓN , −(q, ∇ · u) = 0,
F , −∇u + GT = 0,

(49) (50) (51)

where F is a test function for the projected velocity gradient G. In Eq. (49), the DEVSS-G parameter θ is chosen equal to the polymer viscosity, θ = Gλ. We decouple the momentum and mass balance equations from the constitutive equation as proposed by D’Avino and Hulsen [27], in which the stress tensor τ is replaced by a time-discretized but space-continuous form of the constitutive equation: τ (cn+1 ) = G∆t − un+1 · ∇cn + (∇un+1 )T · cn + cn · ∇un+1




+ Gh(cn , ∆t) − GI, (52) where ∆t is the computational time step and h(cn , ∆t) = cn −


∆t n c − I + α(cn − I)2 , λ

(53)

for the Giesekus model. By substituting Eq. (52) into Eq. (49), we get a Stokes-like problem for (un+1 , pn+1 , Gn+1 ) depending on cn . Then the value of cn+1 can be found by solving the constitutive equation with known values of (un+1 , pn+1 , Gn+1 ). For the time discretization of the constitutive equation, we use a secondorder time-integration scheme, based on a Gear scheme, proposed by D’Avino 29

and Hulsen [27]. We incorporate the log-conformation representation [28] and the SUPG stabilization technique [29], as well: d+τ un+1 ·∇d,

3 n+1 s 2


− 2sn + 21 sn−1 +un+1 ·∇sn+1 −g(Gn+1 , sˆn+1 ) = 0, ∆t (54)

where s = log c, sˆn+1 = 2sn − sn−1 and d is a test function for the logconformation s. The function g(G, s) comes from the matrix-logarithm of the evolution equation of the conformation tensor c. The detailed derivation and exact expression for the evolution equation of the log-conformation s can be found in [28]. ¯ ¯ In Eq. (54), the SUPG parameter τ is given by τ = βh/2U c , where β is a non-dimensional parameter, h is a typical size of an element in the direction of velocity and Uc is a characteristic velocity magnitude for the flow problem. We choose the characteristic velocity Uc as the magnitude of the velocity in the integration points: Uc = kuj k,

j = 1, . . . , J

(55)

where J is the number of integration points in the element. To prevent very small characteristic velocity values near the Dirichlet boundary where the velocity is zero, a β¯ parameter is defined as a function of the Courant number C = Uc ∆t/h: β¯ = Note that τ=

  +1 if C ≥ 1,  C

if C < 1.

  h/2Uc

if C ≥ 1,

 ∆t/2

if C < 1.

30

(56)

(57)

As the characteristic velocity becomes smaller, β¯ imposes a lesser weight on the convection in Eq. (54), which leads to stability improvements. The tensor fields G and s, and corresponding test functions F and d, are discretized by continuous Q1 interpolation. 4.3.2. Problem description We consider a planar flow past a stationary cylinder of radius R, confined between two parallel plates, see Fig. 12. The length of the channel L = 30R

y H

x R

L Figure 12: Geometry for the flow around a cylinder, confined between two parallel plates.

and height of the channel H = 4R. The flow is generated by specifying a flow rate Q that is constant in time. Thus the average velocity of the fluid is U = Q/H = Q/4R. We assume no-slip boundary conditions on the cylinder and on the channel walls. The zero velocity on the cylinder surface is imposed as a weak Dirichlet boundary condition, whereas the other conditions are imposed as a strong Dirichlet boundary condition. We also assume the flow to be periodic which means that we extend the flow domain periodically in x-direction such that cylinders are positioned 30R apart. Since the flow is 31

Figure 13: Base mesh M0 from which the other meshes are derived.

periodic, an inflow boundary condition for the conformation tensor is not necessary. We present two test problems using the Oldroyd-B, i.e. α = 0 in Eq. (4), and the Giesekus constitutive model. The zero-shear-rate viscosity is given by η0 = µ + Gλ. The dimensionless parameters governing the problem are the Weissenberg number Wi = λU/R, the viscosity ratio β = µ/η0 and the mobility parameter α for the Giesekus model. The parameters used for simulations are given in Table 1. Case I is a well-known benchmark problem for the simulation of viscoelastic flows around a rigid body (see [28] and the references therein). Note that β = 0 implies µ = 0 for Case II. Otherwise stated, we use κ = η0 . The effect of κ parameter will be investigated in Sec. 4.3.5. We use four meshes M1, M2, M3 and M4 summarized in Table 2. The base mesh M0 is shown in Fig. 13.

We also use a boundary-fitted mesh

Table 1: Two test problems.

model

Wi

β

α

Case I

Oldroyd-B

0.3

0.59

-

Case II

Giesekus

0.3

0.0

0.1

(BFM) with 195584 elements for comparison as a reference solution. In 32

the BFM, the cylinder surface is treated as a standard Dirichlet boundary. The boundary-fitted mesh has more refined mesh resolutions than the mesh M4 along the radial direction on the cylinder by refinements around the cylinder, as demonstrated in Fig. 14. We will only show steady-state results,

Figure 14: Boundary-fitted mesh. Note that the actual mesh used in the computations is much finer than the mesh shown here.

which are obtained by time-stepping. In order to judge whether we have obtained a steady-state, we monitor the values of the velocity u and the logconformation s in the domain, and the drag on the cylinder as a function of time. We solve the problem until these values do not change within a time step up to an order of O(10−5 ). 4.3.3. Drag coefficient results The drag Fx on the rigid body is defined by: Z Z Fx = σ · n ds · ex = (−pI + 2µD + τ ) · n ds · ex , γ

(58)

γ

where n is the outwardly directed unit normal vector on the boundary γ of the rigid body and ex is the unit vector in x-direction. In Table 3, we give the values for the steady-state dimensionless drag coefficient KD : KD =

Fx . η0 U R

33

(59)

Table 2: Meshes used for simulations.

M0

M1

M2

M3

M4

Number of elements

1000

11388

23256

84000

110390

Element size near the interface

0.2

0.0548

0.0351

0.0191

0.0151

The drag coefficient obtained by BFM for Case I shows good agreement with the results of Hulsen et al. [28]. The values of KD obtained by weak boundary conditions using the mesh M4 are slightly smaller than those of BFM, both for Case I & II, but the difference seems negligible. 4.3.4. Stress profiles Let’s consider the stresses on the upper cylinder surface and along the centerline in the wake. The curved coordinate ξ along the cylinder surface and the centerline is shown in Fig. 15; ξ = 0 at the front stagnation point, ξ = π at the back stagnation point and ξ = [π, 3π] along the centerline. In Fig. 16, we have plotted the non-dimensional polymer stress component

ξ=0

ξ=π

Figure 15: The curved coordinate ξ.

34

ξ = 3π

∗ τxx = τxx /(η0 U/R) along the coordinate ξ for Case II. For a better comparison

of the stress profiles using different meshes, we show the details of the stress profile on the top of the cylinder in Fig. 17. By refining mesh resolutions, we obtain smoother profiles of the stress component and good agreements with the result of BFM. 50

M1 M2 M3 M4 BFM

45 40 35

∗ τxx

30 25 20 15 10 5 0 -5 0

1

2

3

4

5

6

7

8

9

10

ξ ∗ as a function of curved Figure 16: The non-dimensional polymer stress component τxx

coordinate ξ for Case II.

Table 3: Dimensionless steady-state drag coefficient KD

M1

M2

M3

M4

BFM

Case I

123.064 123.119 123.172 123.187 123.192

Case II

78.845

78.826

78.820

35

78.816

78.824

Ref. [28] 123.193 -

50

M1 M2 M3 M4 BFM

45

∗ τxx

40

35

30

25 1.2

1.4

1.6

1.8

2

2.2

2.4

ξ Figure 17: The magnified view of Fig. 16 on the top of the cylinder.

4.3.5. Effect of the κ parameter Now we investigate the effect of the κ parameter on the stress component ∗ . We solve the test problem Case II for various κ values using the mesh τxx

M3, since the result of M3 is quite similar to that of M4 as shown in Fig. 17. ∗ Fig. 18 shows the stress component τxx on the top of the cylinder for various

κ/η0 values. For κ/η0 > 1, we can see a gradual increase of oscillations on the stress component as the κ parameter increases. But these oscillations are quite small and seem to be negligible. Even κ/η0 = 1000 produces a reasonably good solution for this problem. For 0.1 ≤ κ/η0 < 1 we can also see a gradual increase of oscillations on the stress component as κ decreases (κ/η0 = 0.1 is shown in Fig. 18). However, if κ/η0 < 0.1, these oscillations increase sharply as κ value decreases, as can be expected from the results of

36

Sec. 4.1. The value κ/η0 = 0.05 produces noticeable oscillations on the stress component (not shown here); if we use κ/η0 ≤ 0.01, the simulation becomes unstable and the solution diverges. Once we avoid very small (κ/η0 < 0.1) or very large (κ/η0 > 1000) values of κ, the actual value of κ is not critical. Taking κ = η0 seems a good choice even for shear-thinning viscoelastic flows. 50

κ/η0 κ/η0 κ/η0 κ/η0 κ/η0

45

BFM

= 0.1 =1 = 10 = 100 = 1000

∗ τxx

40

35

30

25 1.2

1.4

1.6

1.8

2

2.2

2.4

ξ ∗ Figure 18: The stress component τxx on the top of the cylinder for various κ/η0 values.

5. Conclusion In this paper we have introduced a new formulation for imposing Dirichlet conditions in a weak way for non-Newtonian fluids flows. Although the formulation is general, it is particularly useful when combined with the extended finite element method. The formulation reduces to the Gerstenberger-Wall 37

[1] formulation in case of a Newtonian fluid model and the κ parameter is taken equal to the viscosity of the fluid. We have shown that the formulation is accurate and stable by performing simulations for a generalized Newtonian and a viscoelastic fluid model. It turns out that a value of κ equal to the zero-shear-viscosity gives good results. However, it is also shown that a wide range of κ values can be chosen without sacrificing accuracy. Acknowledgments M.G.H.M. Baltussen thanks the support of the EU within the frame work six project ARTIC. The work of one of the authors (Y.J. Choi) is part of the Research Programme of the the Dutch Polymer Institute (DPI), Eindhoven, The Netherlands, project no. #616. The authors thank Wolfgang Wall and his group members for inspiring discussions during a visit of some of the authors to the Technische Universit¨at M¨ unchen. References [1] A. Gerstenberger and W.A. Wall. An embedded Dirichlet formulation for 3D continua. Int. J. Numer. Meth. Engng., 82:537–563, 2010. [2] G.

d’Avino,

P.L.

Maffettone,

F.

Greco,

and

M.A.

Hulsen.

Viscoelasticity-induced migration of a rigid sphere in confined shear flow. J. Non-Newtonian Fluid Mech., 165:466–474, 2010.

38

[3] Ahamadi M. and Harlen O.G. Numerical study of the rheology of rigid fillers suspended in long-chain branched polymer under planar extensional flow. J. Non-Newtonian Fluid Mech., 165:281–291, 2010. [4] Y.J. Choi, M.A. Hulsen, and H.E.H. Meijer. An extended finite element method for the simulation of particulate viscoelastic flows. J. NonNewtonian Fluid Mech., 165:607–624, 2010. [5] W.R. Hwang and M.A. Hulsen. Direct numerical simulations of hard particle supsensions in planar elongational flow. J. Non-Newtonian Fluid Mech., 136:167–178, 2006. [6] C. Chung, M.A. Hulsen, J.M. Kim, K.H. Ahn, and S.J. Lee. Numerical study on the effect of viscoelasticity on drop deformation in simple shear and 5:1:5 planar contraction / expansion microchannel. J. NonNewtonian Fluid Mech., 155:8093, 2008. [7] S. Afkhami, P. Yue, and Y. Renardy. A comparison of viscoelastic stress wakes for 2D and 3D Newtonian drop deformation in a viscoelastic matrix under shear. Phys. Fluids, 21:072106, 2009. [8] H.H. Hu, N.A. Patankar, and M.Y. Zhu. Direct numerical simulations of fluidsolid systems using the Arbitrary LagrangianEulerian technique. J. Comput. Phys., 169:427–462, 2001. [9] R. Glowinski and T.-W. Pan and J. Periaux. A fictitious domain method for Dirichlet problem and applications. Comp. Meth. Appl. Mech. Eng., 111:283–303, 1994.

39

[10] R. Glowinski and T.-W. Pan and T.I. Hesla and D.D. Joseph. A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow, 25:755–794, 1999. [11] W.R. Hwang and M.A. Hulsen and H.E.H. Meijer. Direct simulations of particle suspensions in a viscoelastic fluid in sliding bi-periodic frames. J. Non-Newtonian Fluid Mech., 121:15–33, 2004. [12] A. Sarhangi Fard, M.A. Hulsen, N.M. Famili, H.E.H. Meijer, and P.D. Anderson. Adaptive non-conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries. Int. J. Numer. Meth. Fluids, 2011. In press. [13] T. Belytschko, R. Gracie, and G. Ventura.

A review of ex-

tended/generalized finite element methods for material modeling. Modelling and Simulation in Materials Science and Engineering, 17:043001, 2009. [14] J. Nitsche.

¨ Uber ein Variationsprinzip zur L¨osung von Dirichlet-

Problemen bei Verwendung von Teilr¨aumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg, 36:9–15, 1971. [15] J. Dolbow and I. Harari. An efficient finite element method for embedded interface problems. Int. J. Numer. Meth. Engng., 78:229–252, 2009. [16] C.E. Baumann and J.T. Oden.

A discontinuous hp finite element

method for convection-diffusion problems. Comp. Meth. Appl. Mech. Eng., 175:311–341, 1999. 40

[17] D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for ellliptic problems. SIAM J. Numer. Anal., 39:1749–1779, 2002. [18] R. Gu´enette and M. Fortin. A new mixed finite element method for computing viscoelastic flows. J. Non-Newtonian Fluid Mech., 60:27–52, 1995. [19] H. Ji and J.E. Dolbow. On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. Int. J. Numer. Meth. Engng., 61:2508–2535, 2004. ´ B´echet and M. Tourbier. Imposing Dirichlet boundary [20] N. Mo¨es and E. conditions in the extended finite element method. Int. J. Numer. Meth. Engng., 67:1641–1669, 2006. ´ B´echet, N. Mo¨es, and B. Wohlmuth. A stable Lagrange multiplier [21] E. space for stiff interface conditions within the extended finite element method. Int. J. Numer. Meth. Engng., 78:931–954, 2009. [22] A. Embar, J. Dolbow, and I. Harari. Imposing dirichlet boundary conditions with Nitsche´s method and spline-based finite elements. Int. J. Numer. Meth. Engng., 83:877–898, 2010. [23] F. Bassi and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible NavierStokes equations. J. Comput. Phys., 131:267–279, 1997. [24] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer Verlag, New York, 1991. 41

[25] G.J. Wagner and N. Mo¨es and W.K. Liu and T. Belytschko. The extended finite element method for rigid particles in Stokes flow. Int. J. Numer. Meth. Engng., 51:293–313, 2001. [26] F.P.T. Baaijens. Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newtonian Fluid Mech., 79:361–385, 1998. [27] G. D’Avino and M.A. Hulsen. Decoupled second-order transient schemes for the flow of viscoelastic fluids without a viscous solvent contribution. J. Non-Newtonian Fluid Mech., 165:1602–1612, 2010. [28] M.A. Hulsen and R. Fattal and R. Kupferman. Flow of viscoelastic fluids past a cylinder at high Weissenberg number: Stabilized simulations using matrix logarithms. J. Non-Newtonian Fluid Mech., 127:27–39, 2005. [29] A.N. Brooks and T.J.R. Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comp. Meth. Appl. Mech. Eng., 32:199–259, 1982. Appendix A. Deriving the Gerstenberger-Wall scheme from a DG formulation We will start from the mixed formulation as used in deriving discontinuous Galerkin methods for diffusion problems [17]. First, write the Stokes equation

42

(Eqs. (1)–(2) with τ = 0) as a first-order mixed system: −∇ · (2µE) + ∇p = f

in Ω,

(A.1)

∇·u=0

in Ω,

(A.2)

1 E − (∇u + ∇uT ) = 0 2

in Ω,

(A.3)

where E is a symmetric new tensor field variable. After multiplication with test functions v, q and H, integration over the domain Ω and partial integration, we find

(∇v)T , 2µE) − (∇ · v, p) − v, 2µn · E − pn Γ = (v, f ),

(A.4)

−(q, ∇ · u) = 0,

(A.5)

(H, E) + (∇ · H T , u) − (H · n, u)Γ = 0.

(A.6)

Now we have to set the boundary “fluxes”, for which we choose 2µn · E − pn = ¯t

on ΓN ,

(A.7)

¯ u=u

on ΓD ,

(A.8)

and otherwise we assume the “internal” values. This “natural” choice has been introduced by Bassi & Rebay [23]. After partially integrating back the third equation (equation for E) we arrive at

(∇v)T , 2µE) − (∇ · v, p) − v, 2µn · E − pn Γ = (v, f ) + (v, ¯t)ΓN , D

(A.9) −(q, ∇ · u) = 0,

(A.10)

¯ )ΓD = 0. (H, E) − (H, ∇u) + (H · n, u − u

(A.11)

43

As a final step, we replace 2E in the volume integral of Eq. (A.9) with ∇u + ∇uT :

(∇v)T , µ(∇u + ∇uT ) − (∇ · v, p) − v, 2µn · E − pn Γ

D

= (v, f ) + (v, ¯t)ΓN , (A.12) −(q, ∇ · u) = 0,

(A.13)

¯ )ΓD = 0. (H, E) − (H, ∇u) + (H · n, u − u

(A.14)

Note, that the last step cannot really be motivated, but it is the same as
adding (∇v)T , µ(∇u + ∇uT ) − 2µE to the equation, similar to the DEVSS stabilization in viscoelastic fluid flows [18]. Appendix B. The primal formulation It is common to eliminate the gradient variable to obtain the primal formulation [17]. For this, we introduce the shape functions φk for the interpolation of E and H on the mesh used: E = E k φk ,

H = H k φk ,

(B.1)

where we use summation convention with k running over all nodes in the mesh. Furthermore, we introduce the L2 -projection fˆ of a function f on the space spanned by the shape functions φk : −1 (φm , f ), fˆ = φk Mkm

Mkm = (φk , φm ).

(B.2)

If f belongs to the approximation space spanned by φk , of course fˆ = f . 44

If we substitute Eq. (B.1) into Eq. (A.14), and let the result be valid for any H k , we get
1 −1
1 −1 ¯ ) + (u − u ¯ )n Γ . (B.3) φm , n(u − u E k = Mkm φk , (∇u + ∇uT ) − Mkm D 2 2 Substituting this result into Eq. (B.1) and using the definition of the L2 projection, we find the following expression for E:
1 ˆ ˆ T ) − 1 φk M −1 φm , n(u − u ¯ ) + (u − u ¯ )n Γ . E = (∇u + ∇u km D 2 2

(B.4)

Substituting this into Eq. (A.12) we get the following system:

ˆ + ∇u ˆ T ) − pn (∇v)T , µ(∇u + ∇uT ) − (∇ · v, p)− v, µn · (∇u ΓD
−1 ¯ ) + (u − u ¯ )n Γ +µMkm (vn, φk )ΓD : φm ,n(u − u D

= (v, f ) + (v, ¯t)ΓN , −(q, ∇ · u) = 0.

(B.5) (B.6)

For all practical purposes we will take φk equal to the velocity shape function (equal order), but piecewise discontinuous. This means that the projected ˆ can safely be replaced by ∇u. For “non-curved” elements they gradient ∇u are even identical. Some remarks: • Comparing Eq. (B.5) with the original weak form Eq. (10) shows that a single symmetric and positive-definite term has been added, explaining the stability of the method. • Due to the discontinuous shape functions φk , the mass matrix Mkm is block-diagonal. Therefore the additional term can be computed on element level and standard finite element assembly can be used. 45

• The mass matrix Mkm is positive definite and the matrix multiplication −1 with Mkm can easily be computed (on element level) with a Cholesky

decomposition. • Only elements that are intersected by the boundary ΓD have a contribution to the additional term. As a final remark, we note that by using the expression for E in Eq. (B.4), the DEVSS stabilization mentioned in Appendix A can be written as

¯ Γ , (∇v)T , µ(∇u + ∇uT ) − 2µE = µn · (∇v + ∇v T ), u − u D

(B.7)

ˆ has been replaced by ∇u. This means, that the DEVSS stabiwhere ∇u lization is identical to adding a Baumann-Oden [16] stabilization term (see Eq. (24)).

46