Adaptive non-conformal mesh reﬁnement and extended ﬁnite … · 2012. 2. 12. · Adaptive...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2012; 68:1031–1052Published online 11 May 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2595

Adaptive non-conformal mesh refinement and extended finiteelement method for viscous flow inside complex

moving geometries

A. Sarhangi Fard1,2, M. A. Hulsen1, H. E. H. Meijer1, N. M. H. Famili2 andP. D. Anderson1,*,†

1Materials Technology Institute, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, TheNetherlands

2Polymer Engineering Group, Tarbiat Modares University, PO Box 14115-111, Tehran, Iran

SUMMARY

Two different techniques to analyze non-Newtonian viscous flow in complex geometries with internal mov-ing parts and narrow gaps are compared. The first technique is a non-conforming mesh refinement approachbased on the fictitious domain method (FDM), and the second one is the extended finite element method(XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non-conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in thehigh shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEMis applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous fieldvariables are captured by using virtual degrees of freedom that serve as enrichment and by applying specialintegration over the intersected elements. The accuracy of the two methods is demonstrated by direct com-parison with results of a boundary-fitted mesh applied to a two-dimensional cross section of a twin-screwextruder. Compared with non-conforming FDM, XFEM shows a considerable improvement in accuracyaround the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.

Received 24 June 2010; Revised 25 January 2011; Accepted 18 March 2011

KEY WORDS: flow field simulation; fictitious domain method; mesh refinement; extended finite elementmethod; twin-screw extruder

1. INTRODUCTION

Numerical simulation of flow in complex geometries with internal moving rigid bodies has showngreat progress in the past decades. The modeling of moving discontinuities with the classical finiteelement method (or finite volume method) is cumbersome because of the need to update the meshto match the geometry of moving rigid bodies. The remeshing of complex geometries such as atwin-screw extruder (TSE) is a tedious and time-consuming process and becomes even worse orimpossible in case of three-dimensional structured meshes. To overcome these problems, severalnumerical techniques were developed, and we can roughly classify these methods in two categories:moving mesh and fixed mesh methods.

One of the important moving mesh methods is the Arbitrary Lagrangian–Eulerian (ALE) tech-nique, which is frequently used to simulate the interaction of flows with a flexible structural domain[1, 2]. This method is based on using a deforming grid for the fluid. The grid deforms with the flex-ible structure at the interface, and then the grid deformation is extended into the fluid field. Becausethe boundary of the fluid mesh completely aligns with the boundary of the rigid body, the quality

*Correspondence to: P. D. Anderson, Materials Technology, Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands.

†E-mail: [email protected]

Copyright © 2011 John Wiley & Sons, Ltd.

1032 A. S. FARD ET AL.

of the deformed mesh is preserved, and the accuracy of the ALE method is higher than any fixedmesh method. ALE methods have some limitations if the internal moving part has a rigid com-plex geometry because the fluid mesh highly deforms and becomes distorted. Sometimes remeshinghelps to overcome this problem, but in case of three-dimensional problems, it is really complicatedor impossible to deform the finite element mesh with flow advection (e.g., the three-dimensionalgeometry of a TSE). On the other hand, remeshing of a deformed mesh is, from a computational timeview, expensive.

Fixed mesh methods were introduced to circumvent deforming mesh problems and were basedon a reference mesh and immersing boundaries of moving parts through the fixed mesh. The orig-inal immersed boundary (IB) method was introduced by Peskin [3] and used a fixed mesh for thecomputational domain that is intersected by the fluid/solid interface. Using a level set function, theadditional interface nodes are added to the computational domain with additional degrees of freedom(DOF). A special treatment is then applied to the solid region and the interface DOF [4]. Recently,Ilinca and Hétu [5] applied the IB method to simulate the flow inside single-screw extruders andTSEs. Avalosse [6] presented a mesh superposition technique (MST) applied to a TSE. With MST,the whole computational domain is covered with one fixed mesh without considering the internalmoving rigid bodies and one separate dynamic mesh for the moving parts. At each time, the posi-tion of the moving part mesh is updated. With superposition of these two meshes, the kinematics ofthe dynamic mesh are imposed on the static mesh using penalty constraints. This method is easy toimplement, but to avoid diffusion errors, precise mesh refinement at the interface of fluid-rigid bodyis required [7]. Zhang et al. [8] used an MST for flow simulation inside TSEs.

Recently, a distributed Lagrangian multiplier/fictitious domain method (FDM) technique wasdeveloped with many similarities to MST or IB methods [9, 10]. With FDM, a fixed mesh for thecomputational domain, including internal moving parts, is created. The kinematics of the rigid-bodymotion is enforced with a constraint on the conservation equations. This constraint is imposed viaLagrangian multipliers. The rigid-body boundary has been discretized through collocation points.The rigid-body motion is imposed on (evenly distributed) points on the solid boundary (a line in twodimensions or a surface in three dimensions). D’Avino and Hulsen [11] used weak constraints forthe rigid-body boundary discretization. The particle boundary is divided into elements, where therigid-body motion is imposed in a weak sense. Here, the weak constraint approach is used.

In this work, two different techniques to analyze non-Newtonian viscous flow in complex geome-tries with internal moving parts and narrow gaps are compared. The first technique is a non-conforming mesh refinement approach based on the FDM, and the second one is the extendedfinite element method (XFEM). First, we present a new strategy for a non-conforming mesh refine-ment method based on FDM for rectangular Taylor–Hood elements and its adaptation for complexgeometries with internal moving parts and narrow gap regions. The local mesh refinement techniquepresented is based on one fixed reference mesh. At each time step, the size of the intersected ele-ments containing the boundary of the moving parts is decreased by splitting elements. To ensurecontinuity of non-conforming refined elements with coarse elements, we use extra constraints,implemented by using a Lagrangian multiplier. The quality of the refined mesh is the same as that inthe reference mesh. A two-dimensional cross section of a TSE with the typical narrow gap regionsbetween screws and the barrel is chosen as a test case. The flow inside this geometry is the result ofa combined drag and pressure flow, with high shear rates in the gap regions.

Secondly, we employ the XFEM enrichment scheme using virtual DOF. The XFEM was origi-nally introduced for crack problems to predict the discontinuity of field variables such as displace-ments and stress in solids [12, 13]. Recently, XFEM is further developed to include applications influid mechanics. For example, Wagner et al. [14] used XFEM to simulate problems of rigid parti-cles in Stokes flows using analytic solutions as the partition of unity enrichment, and Gerstenbergerand Wall [15] applied XFEM to fluid-structure interaction problems in Newtonian fluids using aHeaviside function as the enrichment. Choi et al. interpreted the XFEM enrichment scheme usingvirtual DOF and applied the method to particulate viscoelastic flows [16]. Chessa and Belytschko[17] applied XFEM to two-phase fluids. In this manuscript, we quantitatively compare the results ofthe non-conforming mesh refinement using FDM with XFEM to explore the effect of discontinuitiesof physical variables in complex geometries with internal moving parts and narrow gap regions.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2012; 68:1031–1052DOI: 10.1002/fld

FDM AND XFEM FOR STOKES FLOW INSIDE COMPLEX GEOMETRIES 1033

2. MATHEMATICAL FORMULATION

The polymer melt inside the flow domain � with boundaries � (Figure 1) is modeled as an isother-mal and incompressible fluid. The Reynolds number is low, and inertia will be neglected. Also,viscoelastic effects are ignored, and the fluid is described using a generalized Newtonian constitutiveequation. The flow is governed by the momentum and the continuity equations:

�r � � Crp D 0 in �, (1)

r � vD 0 in �, (2)where v is the velocity vector, and p is the pressure. The relation between the extra stress tensor �and the symmetric part of the velocity gradient D is given by

� D 2�.IID/D, (3)where D D 1

2ŒrvC .rv/T � is the rate of deformation tensor and IID is the second invariant of D.

The Carreau model is used:

�. P�/D �1C .�0 � �1/.1C .� P�/2/.n�1/=2, (4)where �1, �0, �, and n are parameters of the Carreau model [18], and the shear rate P� is related tothe second invariant of D:

P� Dr1

2IID. (5)

Dirichlet boundary conditions for the velocity are imposed on � :

vD vd on �d , (6)vD vs on �s , (7)

where vd and vs are the velocities on moving (�d ) and stationary (�s) boundaries, respectively.

3. NUMERICAL METHODS

The governing equations are partial differential equations that can be solved with the finite elementdiscretization method [19]. The weak form of the conservation equations is obtained by multi-plying Equations (1) and (2) with test functions � and q, respectively, and partially integrate themomentum balance:

..r�/T , �/� .r � �,p/D 0 8� 2 .H 10 .�//2, (8).q,r � v/D 0 8q 2 L2.�/, (9)

where .f, g/DR�

f W g dA.

Ω

dΓ2D cross sectionsΓ

Figure 1. Two-dimensional cross section of a self-wiping co-rotating twin-screw extruder with narrow gapsbetween screw–screw and screw–barrel.



The computational domain is discretized using a finite element mesh. The field variables, v andp, are discretized as follows:

vh.x/DnvXiD1

�i .x/vi 8x 2�, (10)

ph.x/DnpXiD1

i .x/pi 8x 2�, (11)

where vh and ph are the interpolated functions of the velocity and pressure in �, nv is the numberof velocity nodal points, np is the number of pressure nodal points, �i .x/ and i .x/ are the shapefunctions of velocity and pressure in�, and vi and pi are the nodal values of velocity and pressure,respectively. Rectangular Taylor–Hood elements are used; nine nodes (corners, mid-sides, and cen-ter) for velocity with biquadratic shape functions and four nodes (corners) for pressure with bilinearshape functions. Because the viscosity is shear rate dependent, the nonlinear part of the equationsis linearized using a Picard iteration. The numerical procedure is described in 2D; generalization to3D is straightforward.

In the case of geometries with moving internal parts, like for example TSEs (Figure 2a–2c), theflow domain changes. To avoid the need of regeneration of the mesh for each position of the movingpart, the FDM is applied [9, 10]. In FDM, the kinematic condition of the moving parts is enforcedusing constraints that are implemented with a Lagrangian multiplier (Figure 2d):

..r�/T ,�/� .r � �,p/C .�,�/�d D 0, (12).q,r � v/D 0, (13)

.�, v� vd /�d D 0, (14)

where � is a test function for the Lagrangian multiplier �, and .., ./�d denotes the proper innerproducts on the interface of the rigid body.

The main challenge of FDM is prescribing precisely the enforced velocity of the moving rigidbody on the fluid flow. The accuracy becomes critical in high shear rate regions. For example,in a TSE, there are narrow gaps (screw–screw and screw–barrel) where the shear rate is ordersof magnitude higher than in the other parts of the flow regions. The accuracy of FDM in highshear rate regions can be increased by locally decreasing the element size of the fixed mesh (meshrefinement technique) across the interface of rigid body-fluid. Bertrand et al. used standard and

(a) (b)

(c) (d)

Figure 2. Boundary-fitted finite element mesh for two-dimensional cross section of twin-screw extruder atthree screw positions: (a) 0ı, (b) 45ı, (c) 90ı, and (d) reference mesh for fictitious domain method with

constraint mesh of screws for 0ı (ı) and 40.5ı (G) screw positions.



nonstandard mesh refinement techniques for the quadratic Crouzeix–Raviart triangular element,PC2 � P1 for a two-dimensional twin-screw problem based on the FDM [20], and later extendedtheir method to three-dimensional tetrahedron elements [21]. The challenge of local mesh refine-ment is to impose continuity between refined elements and coarse elements. Bertrand et al. modifiedthe shape functions associated with the non-conforming nodes. This method is easy to implement asit only requires a few modifications of the shape functions and the assembling procedure. Anotherrefinement approach is using mortar element techniques. This method is introduced by Bernardiet al. [22, 23]. The application of this method is also developed for spectral elements [24]. Mortarelement methods solve the coupling of the new refined elements with coarse elements via addingextra constraint equations to the stiffness matrix. Belgacem [25] used a Lagrangian multiplier forcoupling the non-conforming regions with the rest of the computational domain. This method pre-serves the quality of the reference mesh. Its main drawback is the increasing number of constraintsthat may lead to an ill-conditioned stiffness matrix. In all FDM-based methods, there is continuityof field variables across the interface between fluid and internal rigid body because the fixed gridcovers both the fluid and the rigid body domain, and there is no guarantee that the boundary offluid elements aligns with the boundary of the rigid body. Therefore, it is not possible to accuratelypredict discontinuities or jumps from fluid to solid along their interface for field variables such asvelocity or pressure. Capturing of discontinuous field variables such as pressure has a crucial effecton accuracy in the gap region where the pressure gradient is high and dominates the flow.

3.1. Lagrangian multiplier

Two approaches exist to impose the kinematics of a moving object. The first is using collocationpoints (control points) to depict the geometry of the moving part (Figure 3b). In this approach, theconstraint is imposed for each collocation point, therefore Equation (14) becomes

�k � .vk � vkd /D 0, k D 1, � � � ,J , 8�k , (15)

where J is the number of collocation points, �k is an arbitrary value (test value) at collocationpoint k, vk is the interpolated velocity at the coordinate of collocation point k, and vk

dis the veloc-

ity imposed on the moving object at the collocation point k. The number of collocation points shouldbe chosen such that it allows to cover all elements that are intersected with the moving object. Toavoid an over-constrained system, the maximum number of collocation points per element shouldnot be more than two [21]. The approach is easy to implement, but when applying local mesh refine-ment, J should also dynamically change, dependent on newly intersected elements. The approachenforces constraints only in discrete points (collocation points), and there is no continuity betweenvalues of imposed constraints along the object-fluid interface. This becomes more significant if ahigher-order integration scheme is used (e.g., XFEM). The other issue is that there is no guaranteeto have collocation points for those elements, which have a very small overlap with the rigid body.

W

HR

Γ

Ω dV

(a) (b) (c)

Figure 3. (a) Schematic of a cavity (H DW ) with a rotating disk in the center (R=W D 2.25=8) and sta-tionary walls. Typical mesh for cavity, that rotation object is considered as a constraint on boundary �d : (b)

with collocation points and (c) with weak constraint.



To overcome these problems, we used an alternative approach: a weak constraint on the object-fluid interface, instead of on the individual collocation points (Figure 3c). Based on Equation (14),the constraints are enforced in a weak way by integration over the entire length of the interfacemesh [11]: Z

�d

� � .v� vd /ds D 0 8� 2 L2.�d /. (16)

With this method, the object-fluid interface is described with a mesh that possesses a topology. Weuse a linear shape function to interpolate the Lagrangian multiplier over the discretized interface:

�h.x/Dn�XiD1

Qi .x/�i . (17)

Here, x represents the coordinates of the point on the boundary of the rigid body (�d ) on whichwe wish to find an approximate (interpolated) value of the Lagrangian multiplier �h.x/, i is thenode index, n� is the total number of nodes of the interface mesh, �i are the nodal values of theLagrangian multiplier, and Qi .x/ is the shape function associated with node i in �d .

In the weak constraint approach, the constraints are continuously distributed over the interface ofrigid body. Therefore, a guarantee exists that with a sufficient number of integration points (suchthat an ‘exact’ integration is performed), the intersected elements are influenced by a constraint ofthe moving object. Similar to the collocation points approach, the number of elements of the inter-face mesh should be balanced with the number of intersected elements of finite element mesh, butit is much less critical here. D’Avino and Hulsen [11] compared the weak constraints method withcollocation, and their results show that in the weak constraint approach, the Lagrangian multipliers,as well as the number of boundary elements �d , do not influence significantly the accuracy of thesolution. Weak constraint approach compared with collocation points showed less error around �d .

In this work for all weak constraints, we used 30 integration points for each element of �d . Theintegration points are generated using subintervals through a midpoint integration rule. Also, theboundary of the object �d is discretized into linear elements, where the constraints are imposedusing the linear Lagrangian multiplier functions.

3.2. Non-conformal local mesh refinement

Moving internal boundaries and narrow gaps are two important issues in the simulation of the veloc-ity field inside complicated geometries. In the previous section, it was described that to overcomeregeneration of the mesh for geometries with internal moving parts, the velocity at the boundary ofthe moving parts can be enforced using a Lagrangian multiplier �. To implement this method, theboundaries of the moving parts are described with a surface mesh, and the accuracy of the FDMdepends on the number of elements intersected with moving boundaries and the degree of approxi-mation of �. The resolution depends on the number of elements used in the moving boundary mesh(or the number of collocation points), but with an increasing number of constraints, the number ofintersected elements of the main domain should also be increased. In geometries with narrow gaps,high velocity gradients occur, and their positions change in time. To obtain adequate accuracy inthese regions, the size of the elements there should be small. Local mesh refinement is efficient,but it leads to a non-conformal mesh. Here, we use the mortar element method [25], and an auto-matic mesh refinement process is designed, which relies on one fixed reference mesh. First, criteriahave to be defined to detect target elements of the reference mesh for refinement. In the FDM, analgorithm is used to find all elements that intersect the internal moving boundary �d . Those ele-ments are located in the narrow gap regions, thus they are refinement targets. Next, we subdividethe target elements into small elements. For example, in Figure 4 for quadrilateral nine-node ele-ments, the splitting procedure is shown for 2 � 2 (Figure 4b) and 3 � 3 (Figure 4c) subdivisions.All sub-elements and target elements have the same order, although it is possible to generate loweror higher order sub-elements. The challenge of non-conformal refinement is to solve issues relatedto the discontinuity of the field variables, such as velocity or pressure between the refined elements



Ω

Γ

Γ

Γ

ΓΩ Ω

Ω Ω

Γ

Γ

Γ

Γ

Γ

Γ

Γ

Γ

(a) (b) (c)

Figure 4. Non-conformal subdivision of a two-dimensional quadrilateral element: (a) target element�e; (b)2� 2 subdivision; and (c) 3� 3 subdivision. Coupled nodes are shown by� and discontinuous nodes by �.

and the coarse mesh, because the new nodes of sub-elements do not belong to coarse elements. Forexample, if the element e is divided into four sub-elements (refinement order 2), then

�e D�e1 [�e2 [�e3 [�e4 . (18)

All non-coupled new nodes are located at the boundary of �e (�ei , i D ¹1, 2, 3, 4º), and these non-conforming boundaries should be coupled to the reference mesh. Because the refinement of elementsis processed one by one, it is more efficient from a computational point of view to connect all newsub-elements that have a same order of refinement (Figure 5a).

To ensure the continuity of field variables between refined elements with neighboring coarseelements, we use a weak constraint implemented with a Lagrangian multiplier over the non-conforming interface. In a general geometrically non-conforming domain (Figure 5b), the mortarsare �i 2 ¹�1,2,�1,3,�2,3º. We have to minimize the difference in function values (e.g., velocity)across each mortar (�i ) using a Lagrangian multiplier (�) and test function � in a weak constraint:Z

�1,2

� � .v1 � v2/ds D 0 (19)Z�1,3

� � .v1 � v3/ds D 0 (20)Z�2,3

� � .v2 � v3/ds D 0, (21)

1Γ

2Γ

3Γ

4Γ

5Γ

6Γ

7Γ

(a) (b)

Figure 5. (a) Connecting of side-by-side sub-elements with the same order of refinement; (b) Mortardecomposition in a geometrically non-conforming case, three mortars.



and the corresponding (symmetric) contribution to the momentum balance. Discretization of � isdefined on the boundaries of the refined elements.

This mesh refinement strategy is first applied to simulate a fluid flow inside a cavity with station-ary boundaries and one rotating circular object (speed 1 rev/s) in its center. The geometry is shownin Figure 3a. The presence of the object is described by Lagrangian multipliers, and the problem issolved with weak constraints. The reference mesh consists of 15 � 15 quadrilateral elements withnine nodes (Figure 6a). Different mesh refinements are applied. In the first one, we subdivide thoseelements that intersect with the object to 2� 2 sub-elements (Figure 6b). Next is subdividing inter-sected elements to 3� 3 and their neighboring elements to 2� 2 sub-elements (Figure 6c). The lastone is subdividing intersected elements to 4 � 4 sub-elements and neighboring elements to 3 � 3and 2� 2 sub-elements (Figure 6d). The computed shear rate is used to investigate the performanceof these different mesh refinements (Figure 6). With increasing the refinement steps, the calculatedmean shear rate inside the object decreases to zero.

3.3. The extended finite element method

In the FDM, the fluid flow problem is solved with a finite element mesh, which includes also thesolid moving part. This means that the boundary of the solid body is not aligned with fluid elementboundaries; intersected elements exist, which are partly inside the fluid and partly inside the rigidbody. As a result, there is a non-physical continuity for field variables such as a velocity and pres-sure across the interface of the fluid-rigid body along these intersected elements. To include jumpsfor field variables along the interface, we use the XFEM. For a one-sided discontinuity problem(Figure 7), the intent is solving the fluid flow (�C) only.

The variation of each unknown continuous field variable, such as f , over � can be interpolatedusing shape functions:

f .x/�nfXiD1

Ni .x/fi , (22)

(a) (b) (c) (d)

Figure 6. Local mesh refinement technique is applied for a cavity with rotating rigid body in the center.Accuracy of non-conforming mesh refinement technique is represented with calculated average shear rateinside the rigid body; (a) no refinement NP� D 0.66 s�1, (b) 2� 2 subdividing NP� D 0.15 s�1, (c) 3� 3 - 2� 2

subdividing NP� D 0.03 s�1, (d) 4� 4 - 3� 3 - 2� 2 subdividing NP� D 0.015 s�1.



× × ×

×× ×

× ×+Ω

−Ω

e1

e2e3

e4

(a) (b)

Figure 7. (a) One side discontinuity problem between fluid domain (inside)�C and rigid body domain (out-side) ��. Non-intersected elements fully inside �C are fully integrated, intersected elements (gray color)are integrated on the inside part (�C), and the elements (�) fully outside (��) are ignored. Virtual degreesof freedom in nodes outside but belonging to intersected elements are shown with �. (b) Subdivision of ele-ments: no subdivision for fully inside element e1, two-level quadtree subdivision for element e2, three-levelquadtree subdivision for element e3, and four-level quadtree subdivision for element e4. Final quadrilateral

sub-elements are decomposed to small triangles to describe the interface.

where x represents the coordinates of the point in � on which we wish to find an approximate(interpolated) value for the function f , i is the node index, nf is the total number of nodes in �,fi are the nodal values of the function f , and Ni .x/ is the shape function associated with node i in�. To impose the jump for field variables such as f , the approximation function of f in � can beredefined as follows:

f .x/�nfXiD1

Ni .x/H.s/fi , (23)

where H.s/ is the Heaviside function defined with a scalar level set function s:

H.s/D²C1 if s > 00 if s < 0,

(24)

s.x/

8<:> 0 if x 2�C< 0 if x 2��D 0 if x is on the interface.

(25)

The level set function s.x/ is also used to determine whether elements are fully inside, fully out-side, or intersected with the fluid-rigid body interface. The definition of the level set function isbased on the shape of the rigid-body boundary. For regular geometries with boundaries that canbe described with simple mathematical equations such as for a cylinder, it is straightforward todefine s.x/ analytically. For non-regular or complicated geometries, s.x/ is defined approximatelyor numerically.

If we have the situation like in Figure 7a, elements (�) fully outside (��) are ignored, elementsfully inside (�C) are fully integrated as before, but elements (shaded elements) that are partiallyinside and outside need only to be integrated on the inside part (Figure 7b). The DOF in the insidenodes (�) have the same meaning as before; they are standard DOF. The nodes that are connected tofully outside elements are removed from the solution, and fully outside elements are not assembledin the final solution matrix; field variables in their nodal points are set to zero as Dirichlet boundarycondition. The DOF in nodes outside of (�C) but belonging to intersected elements are virtual DOF(�). They do not define the field variable at that point but define the spatial distribution of the fieldvariable in the part of the element that is inside the body. This is similar to virtual or ghost gridpoints in the finite volume method. The value in the ghost nodes is just considered for interpolating



values at the fluid domain part of an intersected element. Using the XFEM method, the velocity andpressure are redefined as follows:

vh.x/DnvXiD1

�i .x/H.s/vi (26)

ph.x/DnpXiD1

i .x/H.s/pi (27)

An important part of XFEM is the numerical integration scheme. The intersected elements aresubdivided into subdomains, and integration is performed over these subdomains. The integration onsubdomains of an element determines the robustness and accuracy of XFEM. The XFEM integrationscheme for quadrilateral elements is shown in Figure 7b. The first step is to create a quadtree; usingthe level set function, the intersected elements are determined. Then the quad-intersected elementsare subdivided to small quad elements (1-level subdividing), newly intersected sub-elements aredetermined, and further subdividing them to small sub-elements (2-level subdividing) follows. Withincreasing the level of quad-subdivision, the integration area (or volume) is much better defined bythe interface geometry, which increases the accuracy of integration. It also increases the CPU time,so finding the optimum number for the subdividing level is important. In this work, a 5-level subdi-vision is used. After creating the quadtree, the smallest quad-intersected subdomains are subdividedinto small triangles. The algorithm of triangulation using a level set function is well described in Minand Gibou[26]. Now after domain subdividing, the flow part aligns with the interface, and numericalintegration can be done only on subdomains (triangles or quadrilateral subdomains) that are fullyinside. We used a 3 � 3 Gauss integration rule for the quadrilateral elements and sub-elements anda 6-point Gauss integration for triangles.

Also an internal Dirichlet boundary condition on �d is imposed using a weak constraintimplemented with a Lagrangian multiplier, similar to FDM.

In Figure 8a, an intersected element is shown. If the integration area (a1) is very small comparedwith the element area, the final solution matrix becomes ill conditioned. This situation becomesworse when iterations on the solution are needed, for example, for nonlinear problems, and a diver-gence of the solution can occur. To avoid this problem, we used a method introduced by Choiet al. [16] that moves the mesh inside the domain slightly along a direction normal to the inter-face of the fluid-rigid body (Figure 8b), moving nodes inside the fluid, while nodes inside the

+Ω

−Ω

1a2a +Ω

−Ω

n̂

(a) (b)

Figure 8. (a) An intersected element: a1 is the area of the inside part of the element, and a2 is the area ofthe outside part of the element; (b) Moving mesh (shown with dashed line) in the direction normal to the

screw surface.



rigid body are fixed. By solving the Poisson equation, the velocity of nodes un can be obtainedas follows:

r2un D 0 in � (28)un D 0 on �s (29)un D ˛ebn on �d (30)

wherebn is the outwardly directed unit normal vector on the rigid body surface, and ˛e is a coeffi-cient to control the value ofbn in element e with a default value of unity. Each mesh point xn movesaccording to the following advection equations:

dxndtD²

un if xn 2�C0 if xn 2�� (31)

xn.tn D 0/D xn,0 (32)with xn,0 as the original position of the unmoved mesh nodes. A fourth-order Adams–Bashforthmethod is used to solve the system (31). The nodal points outside the rigid body are moveduntil each integration area of intersected elements is larger than 3% of the entire element area(a1=.a1 C a2/ > 0.03). Moving the mesh may distort some elements. Displacing of the mesh iscontrolled by the time step t of Adams–Bashforth method and value of ˛e . To avoid large localdistortions of elements, the following procedure is applied:

� A node in-between element vertices is set to the midpoint of vertex nodes after eachAdams–Bashforth time step.� The coefficient ˛e for elements located in the gap regions is set to less than unity (0.5 < ˛e <0.9) and in the other regions, is set to unity. Sometimes if the gap is extremely narrow, a negativevalue (e.g., ˛e D�0.05) is used, moving the mesh in opposite direction.� We use small values of t , t D 0.001.Extended finite element method is also applied to the cavity problem with a rotating rigid body

in the center (Figure 3a). The resulting shear rates are compared with a boundary-fitted (BF) meshand non-conforming FDM (Figure 9). The XFEM results, even on a coarse mesh, demonstrate highaccuracy close to the BF result. In contrast, the FDM results near the boundary of the object revealhigh errors.

4. RESULTS

4.1. Geometry

To validate the numerical methods introduced, the cross section of a TSE is selected as a testgeometry (Figure 10). It consists of two moving internal parts. Narrow gaps are present betweenscrew–screw and screw–barrel. For a given screw radius (Rs), centerline distance (Cl ), screw–screwgap distance (ıs), barrel–screw gap distance (ıb), and number of parallel channels (e), it has a uniqueshape because of the requirement that one screw must, in any position, wipe the other one. A numberof different descriptions for the screw geometry of co-rotating TSEs exist [27–29]. We use a slightlyadapted version of the description of a self-wiping profile provided by Booy [27]:

l./D

8̂̂<ˆ̂:Rs.�c � 1/ 06 6 ˛2Rs

�q�2c � sin2. � ˛2 / � cos. �

˛2/

�˛26 6 ˛

2C 2ˇ

Rs˛2C 2ˇ 6 6 �

e,

(33)

where l./ is the distance from the center to the edge of each corresponding screw. The geometricalrelations of the parameters are as follows:

�c DCl � ısRs

, (34)



(a) (b) (c) (d)

Figure 9. Contour color plot of shear rate for a cavity with rotating rigid body in the center: (a) boundary-fitted mesh, (b) non-conforming fictitious domain method 4�4 - 3�3 - 2�2 subdividing, (c) extended finite

element method with a coarse mesh, (d) extended finite element method with a fine mesh.

x

yC

R

( )l

*( )r P

O′

Figure 10. Geometric parameters of self-wiping twin-screw extruder.

Table I. Screw geometry parameters.

Number of flights (e) 2.0Screw radius (Rs) 15.275mmCenter distance of screws (Cl ) 26.2mmClearance between screws (ıs) 0.20mmClearance screw–barrel (ıb) 0.15mm

Table II. Rheological parameters for the Carreau model.

�0 1290 Pa�s�1 0.0 Pa�sn 0.559 –� 0.112 s

ˇ D arccos��c2

�, (35)

˛ D �e� 2ˇ. (36)

The values of the parameters are listed in Table I.



(a) 0°

(d) 90°

(b) 22.5°

(e) 112.5°

(c) 45°

(f) 157.5°

Figure 11. Typical boundary-fitted meshes with quadrilateral nine-node elements for different screworientation.

Figure 12. Typical boundary-fitted mesh of twin-screw extruder with 98,431 nodes and mesh quality in thegap regions.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

5

10

15

20

25

BF1 = 62,521 nodes

BF2 = 98,431 nodes

BF = 118,677 nodes

BF3 = 171,077 nodes

−5 −4 −3 −2 −1 0 1 2 3 4 5 x 10

24.75

24.8

24.85

24.9

y/Rs

V/V

p

BF1BF2

BFBF3x = 10:8246 mm

−3

= 0

Figure 13. Comparison of non-dimensional tangential velocity VVp.Vp D 2�Rs!/ along the inter-meshing

line (xD10.8246 mm) for different size of boundary-fitted (BF) mesh. With increasing number of nodes theresults converge to the same values.



Based on Equation (33), a level set function s.x/ is defined to determine whether the points areinside .s.x/ < 0/, outside .s.x/ > 0/, or on the screw surface .s.x/D 0/:

s.x/D r./D jOP j � l./ 8P 2�, (37)where r./ is a linear distance function of point P with coordinate x from the center of the screws(O or KO in Figure 10). If the element intersects with one screw (e.g., left screw) in the XFEM inte-gration scheme, the related reference point for that screw (e.g.,O) is considered. For those elementsthat intersect with both screws (frequently in the intermeshing region), elements are subdivided intosmall elements until their subdivided elements intersect with just one screw. Note, s.x/ is not the

Table III. Meshes used for simulations of finite element method and mesh refinement.

nelem1 nnodes2 RS3 nelem14 nnodes14 S6 CPU time (s)

F1 29,791 120,921 – – – 0.138 123F2 29,791 120,921 2 33,019 140,883 0.069 426F3 29,791 120,921 3–2 45,103 201,285 0.049 1745F4 29,791 120,921 4–2 53,398 236,815 0.034 2220F5 52,726 213,257 – – – 0.103 238F6 52,726 213,257 2 57,478 241,769 0.051 774F7 52,726 213,257 3–2 73,249 321,075 0.034 4469F8 52,726 213,257 4–2 84,337 368,595 0.026 5602

Note: 1Number of elements for reference mesh.2Number of nodes for reference mesh.3Refinement scheme.4Number of element after refinement.5Number of nodes after refinement.6Size of intersected elements, S=(circumference of screw)/(number of intersectedelements), circumference of screw� 81.67 m.

(a) (b) 99°

(d) 54°(c) 310.5°

Figure 14. Example of a mesh F5 used for fictitious domain method and mesh refinement; (a)the referencemesh F5, (b)2� 2 subdividing for intersected element F6, (c)3� 3 subdividing for intersected elements and2 � 2 for their neighboring elements F7, (d) 4 � 4 subdividing for intersected elements and 2 � 2 for their

neighboring elements F8.



real signed distance function for the screw surface, and it defines a linear signed distance (shortestdistance) from the reference point (O or KO).

We use a shear-thinning fluid to show the differences in numerical methods proposed because inthe intermeshing and gaps regions, shear rates are high and strongly influencing the viscosity. Thevalues of rheological parameters for the Carreau model are listed in Table II.

4.2. Boundary-fitted results

To compare results, we used a classical finite element method with a single boundary fitted meshthat precisely takes into account the correct interface conditions. Because the geometry is changingwith time, we just consider a few orientations. Figure 11 shows typical BF meshes for selected ori-entations ¹ D 0ı, 22.5ı, 90ı, 112.5ı, and 157.5ıº. Figure 12 shows the details of this mesh. The

−1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.70

0.2

0.4

0.6

0.8

1

x/Rb

V/V

p

y = 0

F1

F2

F5

F3F4

0.68

0.72

0.76

0.8

0.84

F6F7

F8

BF

= 0

Figure 15. The non-dimensional tangential velocity . VVp/ as a function of x

Rbover the line y D 0 mm at

the left side screw for the screw position D0ı, using different meshes and different refinement scheme andcomparison with boundary-fitted result (BF).

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

5

10

15

20

25

x = 15.177 mm

y/Rb

V/V

p

BFF8F7

F6F4

F3

F1

F2

F5

0.13 0.14 0.15 0.16 0.17 0.18 0.19

0.5

1

1.5

2

2.5

3

3.5

= 112.5

Figure 16. The non-dimensional tangential velocity ( VVp

) as a function of yRb

over the line xD15.177 mmfor the screw position D112.5ı, using different meshes and different refinement scheme and comparison

with boundary-fitted result (BF).



velocity values are calculated for D 0ı and for screw rotation speed ! D 1 rev/s along the inter-meshing position x D 10.8246mm (the origin of Cartesian coordinates is located in the center ofleft screw), using different number of nodes (Figure 13). With decreasing element size, the values ofthe tangential velocity converge. A BF mesh with 118,877 nodes was selected as a reference resultto make the comparison between all numerical methods introduced.

4.3. Finite element method results

Non-conforming local mesh refinement was described in the previous section to increase the accu-racy of the FDM by reducing the size of intersected elements. We apply non-conforming FDM ontwo reference meshes F1 and F5. For each reference mesh, three refinement strategies are used: 2�2subdividing for intersected element, 3�3 subdividing for intersected elements plus 2�2 subdividingfor their neighboring elements, and 4�4 subdividing for intersected elements plus 2�2 subdividingfor their neighboring elements. Mesh information is summarized in Table III. As a demonstration,a simple quadrilateral-structured mesh, represented in Figure 14, is refined with the three schemesusing the non-conforming mesh refinement technique, and the shafts of the screws are consideredhollow to reduce the number of elements (Figure 14).

For one screw revolution, increments of 4.5ı are chosen for the time steps between two con-secutive screw configurations, corresponding to a time interval of 0.0125 s for a counterclockwise

Table IV. Meshes used for simulations of extended finite element method.

nelem1 nnodes2 S3 CPU time (s)

X1 1544 6527 0.689 46X2 5348 22,089 0.345 97X3 13,482 55,103 0.209 195X4 29,791 120,921 0.138 328X5 52,726 213,257 0.103 634

Note: 1Number of elements for reference mesh.2Number of nodes for reference mesh.3Size of intersected elements, S=(circumference of screw)/(number of intersectedelements), circumference of screw� 81.67mm.

−1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.70

0.2

0.4

0.6

0.8

1y = 0

x/Rb

V/V

p

F8

0.984

0.988

0.992

0.996

1

1.004

X2

X1

BF

X3X4X5

X2

= 0

Figure 17. The non-dimensional tangential velocity . VVp/ as a function of x

Rbover the line yD0 mm at the

left side screw for the screw position D 0ı, using different extended finite element method meshes andcomparison with boundary-fitted result (BF) and fictitious domain method with high-order refinement (F8).



rotation speed of 1 rev/s. Because the geometry of the two-lobe screw elements is symmetric, onlyone-half of the complete cycle is considered in the simulations.

Figure 15 shows the non-dimensional tangential velocity . VVp/ as function of x

Rbover the line

y D 0, x


SD0.069, but the number of computational nodes for F5 is 1.5 times higher than for F2. Anotherexample is between two refined meshes F4 and F7; the refinement scheme for F7 is 3�3 subdividingand for F4, it is 4 � 4 subdividing. The size of the smallest intersected element for these meshes isthe same SD0.034, but the total number of nodes after refinement for F7 is 1.5 times higher thanfor F4, whereas the average difference between the velocity values for these meshes with 1% erroris close.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

F

X

y/Rb

x = 15.177 mm

p

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1FX

V/V

p

y/Rb

x = 15.177 mm

(b)(a)

= 112.5= 112.5

Figure 20. Comparison of XFEM (X) and refined FDM (F) meshes for large gap screws for the screw posi-tion D112.5ı over the line xD15.177mm. (a) The non-dimensional tangential velocity . V

Vp/ as a function

of yRb

. (b) The relative pressure p (N/mm2) as a function of y=Rb .

(a) (b)

(c)

Figure 21. Contour color plot of velocity at x direction for screw orientation D 0ı: (a) BF, (b) X5, and(c) F8.



We also sampled the velocity values over the line x D 15.177 mm that crosses the intermesh-ing region at a screw orientation of D 112.5ı (Figure 16). The trend is similar as in Figure 15,except for some oscillations found and especially for the large difference with the BF result in thescrew–screw gap region. In the channel, the error is about 6%–10 %, but in the intermeshing gapregion, the error is large. In this region, the drag flow (in y direction) from each screw is in oppositedirection, and the pressure driven flow dominates the resulting velocity. Apparently, this pressureflow is miscalculated in the FDM approach.

The total number of nodes can be a good criterion for the size of the memory needed for thesecomputations. With regards to CPU time, in all cases, the refined FDM is more expensive than thenon-refined FDM. For example, for mesh F5, the CPU time with 213,257 nodes is five times smallerthan that for the refined solution F3 with a total number of 201,285 nodes. The reason for this is thatthe refinement procedure takes time.

4.4. Extended finite element method results

We solved the same problem with XFEM, using five meshes X1, X2, X3, X4, and X5 summarizedin Table IV. The convergence of the tangential velocity over the line yD0, x


To analyze the large difference between XFEM and FDM results, mainly in the gap region, thepressure values over the intermeshing crossing line are shown for F8, X5, and BF in Figure 19. Ahigh-pressure gradient is present in the gap, and it causes a local flux. As can be seen in Figure 19,this pressure gradient is for the BF and XFEM about three times higher than for F8. To verify theeffect of the gap region on the accuracy, we increase the gap size between the screws and betweenscrew and barrel by increasing the radius of the barrel to 17 mm and the centerline distance to 28mm. The results for pressure and velocity in the gap region over the line xD15.177 mm are shownin Figure 20. Indeed, the differences between XFEM and FDM become much smaller now.

Figures 21, 22, and 23 finally show the contour plots of the velocity in x direction, of the velocityin y direction, and the pressure, respectively. Good qualitative agreement found between the XFEMand the BF simulations results. As expected, FDM shows larger deviations from the BF results nearthe high shear rate regions (gap regions) because the standard finite element integration on inter-sected elements by the rigid body cannot predict the jumps in field variables. For the TSE, capturingthe jump in the pressure over the gap regions is important to predict the high local value of thevelocities and stresses there.

5. CONCLUSIONS

The objective of this paper was to compare modified techniques, based on mortar elements, theFDM, and the XFEM, to simulate the fluid flow inside complex geometries such as those in TSEs.Using non-conforming mesh refinement, we simulated geometries with moving internal boundarieswith just one structured reference mesh and refined this mesh locally according to the position of themoving parts. Ensuring continuity for field variables in the non-conforming elements was enforcedby using a Lagrangian multiplier. The results show that local mesh refinement increases the accuracyof the velocity field compared with the classical FDM. Compared with the classical finite element

(a) (b)

(c)

Figure 23. Contour color plot of pressure for screw orientation D 0.0ı: (a) BF, (b) X5, and (c) F8.



FDM, local mesh refinement is a much more efficient method. However, in more complex geome-tries, the accuracy of non-conforming mesh refinement is not sufficient anymore. To increase theaccuracy of the computed velocity field in ,for example, the gap regions of TSEs, we applied XFEMwith some modifications regarding to the geometry complexity and to the fluid properties. In XFEM,the finite element shape functions are extended by using virtual DOF for the description of the dis-continuities around the interface. Then, intersected elements are integrated only on the fluid domainpart, and DOF associated to elements fully inside the rigid body are removed from the system. Theno-slip boundary condition is imposed by using constraints implemented with Lagrangian multipli-ers. The accuracy and convergence of this scheme have been verified by comparing its results withthose of BF mesh problems. The results are also compared with those based on local mesh refine-ment in FDMs. The XFEM method was much more accurate especially in the narrow gap regions,where FDM proves to be inaccurate.

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