unstructured mesh motion using sliding cells and mapping...
TRANSCRIPT
Unstructured Mesh Motion Using Sliding Cells andMapping Domains
Sina Arabi , Ricardo Camarero and François Guibault
Départment de génie mécanique, École Polytechnique de Montréal, Campus de l’Université de Montréal, 2500chemin de Polytechnique, Montréal (Québec) Canada H3T 1J4
Email: [email protected]
ABSTRACT
This paper addresses the problem of generating un-structured meshes with fixed connectivity for largerigid body motion. The proposed approach consists ingenerating a mesh in computational space for a genericconfiguration of the moving body. The management ofbody and mesh motion is then carried out in compu-tational space using a sliding mesh paradigm. After-wards, the mesh in physical space is obtained throughthe Winslow equation to map the computational meshto the physical space.Two new discritization techniques are implemented,validated and compared for performing the Winslowoperator on unstructured grids. The first approachused a 9-point Cartesian stencil inside each patch ofthe computational mesh and discritizes the mappingoperators on that using conventional finite differenceschemes. The second approach used finite volume dis-critization technique by linearizing the system of map-ping equations. This methodology is applied to com-plex geometric configurations representative of engi-neering applications.
1 INTRODUCTION
The numerical modeling of unsteady problems withlarge amplitude or relative motion of bodies, requiresconsiderable care in the formulation of the domaindiscretisation specifically in devising schemes for themovement of the grids. These types of problems haveled to development of algorithms ([1]) for simulatingfluid physics, where dynamic grid generation for bothviscous and inviscid regions play a significant role.Despite the considerable efforts addressed at this prob-lem, efficiency and robustness remain critical issues.State of the art mesh motion methodologies applicableto unsteady problems can be divided into two major
categories, as illustrated in Fig. 1, namely, one groupis based on changing topology whereas the other main-tains the mesh topology.
Changingtopology
Fixedtopology
Morphing
LocalRemeshing
Remeshing
techniquesmesh motionUnstructured
Sliding Cells
Sliding Zones
Deformingmesh
Overset grids
Figure 1: Classification of Different moving meshMethodologies
The early attempts in dynamic grid generation are es-sentially remeshing techniques, where the entire gridis regenerated based on the new position of the bound-aries as presented in [2, 3] and [4]. This approach canproduce meshes of very high quality if the defined sizefunction is well behaved, for example by equidistribut-ing the error across the domain ([5]). However, it isvery expensive computationally, as in addition to theactual remeshing, the technique requires the interpola-tion of the solution at each time step.A major improvement of the efficiency of this ap-proach is to apply remeshing locally as determined bya mesh quality indicator, [6]. Based on this indicator,elements are removed, resulting in one or more voidsin the mesh which are then remeshed according to therequired distribution of mesh parameters provided bythe error indicating process and merged into the global
mesh. Although applied locally, compared to the com-plete remeshing, this method remains computationallyexpensive for transient problems. Another mesh adap-tation method, called morphing, applies local edge col-lapse operations for mesh coarsening, and incrementalpoint insertion algorithms for mesh refinement, [7, 8].Its robustness depends heavily on maintaining meshquality during each adaptation cycle.Mesh motion algorithms by fixed topology have beenpresented in the literature, with various approachesaccording to the amplitude of the body motion. Awidely used method, considers the mesh as a networkof springs and solves the static equilibrium equationsfor this network to determine the new location of thegrid points. However, the major disadvantage of thisapproach is that the grid smoothness and regularity arelost when the grid is subjected to large motion.Another promising method is based on the Radial Ba-sis Functions interpolation presented in [9] which canbe applied to mesh motion while preserving the cells’connectivity. This is an interpolation technique wherethe displacements of boundary nodes are propagatedonto the interior nodes. This has been applied success-fully for large relative motions in engineering applica-tions but since the cells remain attached to the bound-ary, it is not suitable for periodic rotary objects likemodelling the mixer blades or propellers.PDE operators, such as Laplace equation with vari-ous diffusivity coefficients and biharmonic equation,have been used as a mechanism to generate and tosmooth meshes. However, they still have limitationsfor the large linear and rotary motions in unsteady flowproblems. Another attempt to solve large mesh defor-mation has been proposed by [10] using linear-elasticsmoothing. One advantage of this approach is that ituses a variable elastic stiffness, inversely proportionalto the cell volume, in order to preserve the mesh qual-ity in viscous layers. In [11], an optimization proce-dure based on the adjoint method for linear elastic-ity mesh deformation technique is presented. Whilevery robust for several engineering applications, thismethod has the same limitations as the Laplace equa-tion and gives invalid cells for large motions, speciallyaround high curvature regions or sharp corner pointsof boundaries.Another technique is overset grids ([14]) that simplifythe mesh management by superposition of the staticand moving parts of the grids at the expense of the costof interpolation.
2 MESH MOVEMENT
The methods described so far, all share one major char-acteristic, which is that the mesh motion is carried
out in physical space. Generally, each method satis-fies a particular set of requirements at the expense oftheir important capabilities. For example, large mo-tions can be handled by local remeshing techniquesbut present difficulty with accuracy and complexity.Similarly, while the spring analogy is very efficientand easy to apply, it fails for large motions. Further-more, despite their significant benefits, the Radial Ba-sis Functions interpolation, linear elasticity techniquesand biharmonic equations are still not capable of han-dling large linear or periodic rotary motions encoun-tered in the engineering applications. This is due tothe constraint of the attachment of cells to the bound-aries and the difficulties of maintaining mesh topologyas the motion evolves.A new approach is presented in this paper, based ona flow analogy, where the grid cells in computationalspace are advected past moving boundaries by a ficti-tious potential-like fluid flow. Then, this is followedby mapping the computational mesh to physical space.In this approach, the cells are allowed to slide on theboundaries in order to release the constraints present inthe conventional mesh deformation techniques. In ad-dition, the cells connectivities will be maintained ex-cept at specific points on the boundary. The main ad-vantage of this approach is to preserve the mesh con-nectivity as time evolves, making it well suited for Ar-bitrary Lagrangian Eulerian (ALE) flow solvers. Thisapproach is shown with a bold line in the general framework for unstructured mesh motion technics, presentedin Fig. 1.The proposed procedure for this method consists inthree major steps. First, an unstructured mesh is gen-erated in computational space around a generic body.Then, the generic boundary is made to slide throughthe cells according to the defined trajectory for thephysical boundary, and finally, the mesh is mappedon the physical domain. The mapping to an arbi-trary domain in the physical space is carried out us-ing a procedure detailed in the following sections,where the shape is imposed by the body coordinatesthrough boundary conditions of the Winslow equa-tions. Fig. 2 shows the independent and dependentvariables in computational C and physical Ω domains,respectively, which appear in the governing equations.
3 MAPPING DOMAINS
In the present work, the Winslow operator, Eqns. 1, aresolved on a triangulated computational space to trans-form the mesh from the computational to the physicaldomain as motion evolves.
x
yη
ξC Ω
C Ω
Figure 2: Mapping of a computational (ξ,η) to a phys-ical (x,y) domain.
This system of equations are quasi-linear, coupled andelliptic in type.
g11xξξ −2g12xξη +g22xηη = 0 (1)g11yξξ −2g12yξη +g22yηη = 0
where
g11 = x2η + y2
η , g12 = xξxη + yξyη , g22 = x2ξ + y2
ξ
The most straightforward method to solve the systemof Eqns. 1 is using a finite difference scheme on astructured mesh. However, little attention has been ad-dressed to extend the method to unstructured meshes.The reason is that in contrast to Laplace and Poissonequations, the Winslow operator is in non-conservativeform, and therefore, the conventional discretizationschemes cannot be applied to unstructured meshes. In[12], a method based on Taylor series expansion tosolve this operator on equilateral triangles where allthe angles are equal to π/3 has been proposed. An-other method is presented in [13] based on generat-ing a virtual control volume in physical domain locallyaround each node in Ω as a local computational spacewith the same number of neighboring nodes in physi-cal space.Two new methods based on finite difference and finitevolume schemes are presented to solve Eqns. 1 on un-structured meshes.
4 NUMERICAL DISCRETIZATION
4.1 Finite difference scheme
In this work, a novel finite difference method on un-structured meshes is introduced, which is easier to ap-ply compared to the previous works and closer to theclassical schemes used on structured grids. This algo-rithm is based on a 9-point Cartesian stencil formed
inside each patch around a node (ξi,ηi) in computa-tional space, as shown in Fig. 3. The values at thestencil points can be obtained by interpolation or aleast-squares reconstruction method. From these, thefirst and second order derivatives are approximated tosecond order accuracy on the Cartesian stencil with anequal spacing ∆ξi = ∆ηi. In the present study, the val-ues of the dependent variables on the stencil nodes arelinearly interpolated from the nodal values of the ele-ments surrounding the node, and ∆ξi,∆ηi are chosenas a fixed fraction of the length of the shortest edgeconnected to node (ξi,ηi). The discrete operators are
Figure 3: Proposed 9 point stencil for the finite differ-ence discretization of the mapping operation for un-structured mesh
solved using an SOR-type iterative procedure. Since,almost all cells generated in the computational spacesatisfy the Delaunay criteria, the polygons are veryclose to the ideal regular polygon presented in [12].This property insures that all the information from thepolygon surrounding the node contributes to the so-lution process. In this method, if a triangle does notcontribute directly to the construction of the stencil’smodel, it is estimated that this would, heuristically, besmoothed out by the neighboring stencils.
4.2 Finite volume scheme
As mentioned in the previous section, the Winslow op-erator is in nonconservative form because the three co-efficients, g11,g12 and g22 are functions of the de-pendent variables in the computational space. Usinga linearization procedure ([13]), Eqns. 1 can be inte-grated over a control volume defined around each pointof the mesh in computational space. The integrationpath for the application of Green’s theorem is formedby joining the centroid of each triangular element tothe midpoints of its sides, as shown by the dashed linesin Fig. 4. The hashed region in this figure, indicates acontrol volume with a centroid node which is the stor-age location of all dependent variables.
This results in the integral form of the linearized
Figure 4: Computational mesh and a control volume
Eqns. 1
g11
∫ ∫xξξdΓ−2g12
∫ ∫xξηdΓ+g22
∫ ∫xηηdΓ= 0
(2)Applying the divergence theorem to the second orderderivative terms, for example for the first one, gives∫ ∫
xξξdΓ =∫ ∫
·FdΓ (3)
where the components of function F is F = (xξ,0).A similar procedure is applied to the xηη term. Onecritical step in the procedure is the calculation of thecross-derivatives which requires a special treatment.In [13], the authors proposed the use of augmentedcells around the control volumes. In the present work,it is proposed to solve all the operators in computa-tional domain and without considering the virtual con-trol volumes described in [13]. This method uses ac-tual control volume in computational space leading toa simpler arithmetic procedure. In addition, the cross-derivatives are found using the same control volumeand without considering auxiliary cells. Applying thisprocedure to the term xξη∫ ∫
xξηdΓ =∫ ∫
·QdΓ (4)
In this work, the same control volume shown in Fig. 4is used to compute the cross-derivative terms and byintegrating the terms in both ξ and η directions, thevalue of fluxes in these two directions will be obtained.The average value of these two fluxes gives the net fluxof that cross-derivative term,
Q =12(Q1 +Q2) (5)
where Q1 = (0,xη) and Q2 = (xξ,0).Integrating over the control volume and applying thedivergence theorem for each dependent variable, forexample ξ, gives∫ ∫
·FdΓ =∮
F.ndS (6)
The term under the RHS integral represents the net fluxthat passes through the surface of the volume and, forthe Winslow operator, can be evaluated as
g11
∮xξnxdS−2g12[
12(∮
xηnxdS+∮
xξnydS)] (7)
+g22
∮xηnydS = 0
It has been our specific experience that, taking onlyone component of the cross-derivative term after ap-plying the Green’s theorem, wrongly deforms the finalmesh.
5 SLIDING CELLS
The unstructured mesh management and mesh slid-ing technique require a computational mesh with aset of generic boundary definitions that match the el-ement topology of the physical mesh. These genericboundaries for the translation and rotational motionsare shown in Fig. 5.
ξ
η
x
y
ΩC
Direction of transition
ξ
η
x
y
ΩC
Direction of rotation
Figure 5: Mapping of the generic configurations fortranslation and rotational motions.
To manage the mesh for translation or rotational mo-tions, a computational mesh is created such that theboundary points lie on a slit of zero thickness or a cir-cle, respectively. These boundary points are displacedor relocated at each step in the computational domainso that they conform to the correct geometry in phys-ical space by applying mapping operators. Figure 6shows the link between boundary nodes (circles) andmesh nodes of the body (slit) (squares). The bound-ary conditions applied to the mesh nodes are the co-
ordinates of the corresponding body nodes. In Fig. 7,the moving body is shown a set of circles: single cir-cles at leading and trailing edges and pairs of circlesin between. At a given step, the motion consists ofchanging the topological connection of the mesh nodes(squares) to the body nodes (circles) as the body movesthrough the mesh. Translation motion, the mesh nodesin computational domain remain fixed and the bound-ary nodes split the cells that lie on the defined trajec-tory. Rotational motion is obtained by rotating aboutthe circle’s center which differs from the translationmotion. This rotary motion is realized by the genericconfiguration of a circle rotating inside a mesh in com-putational space, (ξ,η), as shown in Fig. 8. Then, themapping operator is solved at each time step, but withdifferent values of the boundary conditions.
Mesh point
Boundary point
Leading edge Trailing edge
Figure 6: Arrangement of the nodes on the slit
Mesh point
Boundary point
Leading edge Trailing edge
Figure 7: Sliding mesh nodes on the slit
6 RESULTS
Figures 9 and 10 illustrate the application of the trans-lation motion at three time steps. Tracking specificnodes, in both computational and physical spaces (forexample nodes 456 and 1366), clearly shows the meshdeformation resulting from the motion of the cylinder.It is necessary to mention that all the dimensions andlengths, except moving boundaries, in both computa-tional and physical domains are equal. This is not a re-quirement, but is used here to simplify the procedure.
The second example is the rotation of an airfoil shown
N=1
N=1
Nb=1
Nb=1
Figure 8: Rotating physical boundary and sliding com-putational boundary over the mesh points
1456911
1366 1821
2276
1456911
1366 1821
2276
1456911
1366 1821
2276
Figure 9: Computational mesh at first, intermediateand final steps of sliding a slit in a cavity
in Fig. 11. The results obtained by the Winslow’s op-erator are shown at three angular positions. The meshmotion can be understood by following the path ofnode 1 which represents a boundary node on the air-foil and cell number 530 which slides on the bound-ary. The mapping operator for this application is dis-cretized using the finite difference scheme explainedin the previous section.
1456911
13661821
2276
1
456
911
1366
1821
2276
1456
911
1366 1821
2276
Figure 10: Corresponding physical mesh at first, inter-mediate and final steps of sliding a circle in a cavity
530
2117
2646
3175
1
1588
530
2117
2646
3175
1
1588
530
2117
2646
3175
1
1588
530
2117
26463175
1
1588
530
2117
2646
3175
1
1588
530
2117
26463175
1
1588
Figure 11: Rotating and sliding the cells around anairfoil in computational and physical domains in threedifferent positions
The final example is the rotation of two four-petal con-figurations rotating in opposite directions as shownschematically in Fig. 12. Following the proposed pro-cedure, the moving boundaries are two circles in com-putational space, mapped to two four-petal roses inphysical space. For this case, the circles are discretizedwith the same number of nodes.
Figure 12: Boundaries in computational and physicalspaces
The objective for this case is to illustrate relative mo-tion with a wide range of amplitudes for the proximityof boundaries, while maintaining a fixed connectivity.This complex motion is shown in Figs. 13- 15 for threedifferent steps obtained using the Winslow equationssolved by the finite volume discretization scheme. Asis shown in these figures, the cells in the computa-tional domain at different time steps are fixed exceptfor the pointers on the boundaries. For example, track-ing boundary nodes 1 and 240 in computational spaceat different time steps shows how these two circlesslide inside the computational mesh. In addition, cellnumber 1, as a boundary cell number in the physicalspace, reveals how the boundary cells slide over thesurface.
7 CONCLUSION
In this paper, a novel method for large rigid body mo-tions based on the sliding cells and mapping domainswas implemented and applied to generic type of mo-tions. This method was found to overcome the dif-ficulties encountered in traditional mesh motion tech-niques such as using Laplace, Poisson or linear elas-ticity equations, biharmonic method or RBF where theboundary cells remain attached to the body as motionevolves.
This approach allows the cells connectivities to remainconstant therefore, avoids interpolation errors in mov-ing mesh problems which has a direct influence on theaccuracy and efficiency of the simulations.
It was found that using a computational mesh facili-
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Figure 13: Sliding the cells around the boundaries incomputational and physical domain at t1
tated the grid management, when the boundaries aresimplified for translation and rotational motions.
Moreover, since the cells slide over the boundaries asmuch as one boundary edge at each motion step, GCLcan be applied on the physical mesh correctly. In otherwords, the time rate of change of the total volume ofthe physical domain remains constant.
Finally, the decomposition of movement appliesequally well in three dimensional space, and the pro-posed approach could therefore be extended to 3D.However, the simplification process, whereby complexthree dimensional objects are represented by simplerobjects in computational space, becomes much moredifficult to handle in 3D. Work is currently underwayto generalize this aspect of the method to 3D.
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Figure 14: Sliding the cells around the boundaries incomputational and physical domain at t15
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