Developement of multigrid methods for convergence

acceleration of solvers for Navier-Stokes equations on

non-structured meshes

Ales Janka

December 14, 2003

1 Objective of the study

The objective of the study is to develop an algorithm for automatic directional coarsening ofanisotropic meshes based on the finest level mesh to provide coarser meshes for geometric multigrid.Characteristic parameters of the meshes are high aspect ratio (about 1:30.000 near the profile)and variation parameter 1.2. The meshes are assumed to have a layer structure (cf. Fig. 1) inthe boundary-layer zone. Moreover, the boundary layer is considered as structured, with someirregularities which might occur, such as column or layer condensations (cf. Fig. 2). The objectiveof the study is to handle primarily the boundary-layer zone in such a way, that the sequence ofcoarser meshes, coarsened by a coarsening factor 2, keep the structured nature. Secondly, theexterior of the boundary zone can be handled in a different way, usually by an unstructuredcoarsening algorithm [2, 3].

Figure 1: Layer structure of the mesh

1.1 General strategy

The major goal of the coarsening process is to coarsen structured areas of the fine mesh, whilepreserving the structures. Therefore, we chose to proceed with the coarsening process by layers,starting at the profile boundary. The whole algorithm could be summed-up as follows:

Algorithm 1.1. 1. Generate on each node of the finest mesh an initial nodal metric cor-responding to the given finest level mesh.

2. Atribute to each node in the boundary-layer zone a layer index, layer 1 is formed by thenodes at the profile boundary, two adjacent layers have adjacent indices.

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Figure 2: Layer and column condensations

3. Modify the initial metric to introduce the isotropic or anisotropic coarsening. For anisotropic(directional) coarsening, the prefered (local) direction can be given eg. by a normal vectorto the surface of a layer.

4. Fix each node in an odd layer (layer=1,3,. . . ) and remesh the finest mesh (while preservingthe fixed nodes) by an existing mesh-generator MTC. This mesher proceeds in a local way,to improve the mesh based on the specified metric.

1.2 Description of MTC

The topological 2D and 3D mesh-generation tool MTC is being developed by Thierry Coupez atEcole des Mines de Paris, Centre de Mise en Forme des Materiaux, Sophia Antipolis. It is based onan idea to improve, iteratively, an initial unsatisfactory mesh by local improvements. Its generalalgorithm can be expressed as follows, for further details see [5].

Algorithm 1.2 (MTC iteration) Let us denote Th, Eh, Nh, respectively, the sets of allmesh triangles, edges and nodes. Repeat for different cavities C0 ⊂ Th composed by a group ofadjacent elements obtained as nearest neighbourhood of a node n ∈ Nh or of an edge connecting2 nodes (n1, n2) ∈ Eh, n1, n2 ∈ Nh.

1. Denote N0 ⊂ Nh and E0 ⊂ Eh respectively, the set of all nodes and all edges of the cavityC0.

2. Denote ∂E0 ⊂ E0 the set of all edges belonging to the border of the cavity C0.

3. Evaluate the quality Q0 of the set C0 of elements T ∈ C0 by a given function, eg. (1) and(2).

4. For each n ∈ N0 do:

(a) Connect the node n to each edge e ∈ ∂E0, n 6∈ e to get a set of elements Cn = Te, Te =(e, n), attempting to retriangulate the cavity C0.

(b) Evaluate the quality Qn of the set Cn by a given function, eg. (1).

(c) If Q0 < Qn (in the sense of Definition 1.3) then set C0 ← Cn, Q0 ← Qn.

Until stagnation of the changes perpetuated to the mesh.

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Definition 1.3 (Quality function) Let us have a cavity Cn ⊂ Th and let us define (element-wise) a measure of quality in a given element metric M ∈ Rd×d by

Qn = q(T ), T ∈ Cn.(1)

The quality q(T ) of one element T of the cavity, measured in the metric M , is defined by

q(T ) = min(

1hM

, hM

)d

· |T |Mhd

M

,(2)

where |T |M denotes the volume of T in the Riemann metric space, d is the space dimension andhM is the average of lengths of edges of T measured by the metric M ,

hM =

2d(d + 1)

∑(i,j)∈T

(M(~xj − ~xi), (~xj − ~xi))l2

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,

~xi ∈ Rd are the coordinates of the node i.Let us equally define an ordering relation ”<” for each two qualities Qi, Qj of cavities Ci and

Cj . Set Ci = Ci and Cj = Cj . We say that Qi < Qj when for the worst-quality elements Ti ∈ Ci

and Tj ∈ Cj , q(Ti) = min(Qi), q(Tj) = min(Qj), there is q(Ti) < q(Tj). If q(Ti) = q(Tj), setCi ← Ci \ Ti, Cj ← Cj \ Tj and repeat the comparison.

We need to remark, that the discrete metric field M(Ω) : Ω 7→ R(d×d) is stored by nodes forimplementational convenience. For the element quality evaluation MTC use its interpolations,constant at the element T or even at the whole cavity Cn.

1.3 Preliminary numerical experiments

Let us show how Algorithm 1.1 works on a simple 2D model example. Let us have a 2D segment(cf. Fig 3) with the initial metric defined on each node x of its finest level mesh,

M0(x) = RT · Λ ·R,(3)

with

R =(

cos(α) sin(α)− sin(α) cos(α)

)Λ =

(1

(hα)21

2hαhr1

2hαhr

1(hr)2

).

The parameter hα is the characteristic meshsize in the tangential direction to the layer in thenode x, hr is the radial characteristic meshsize (ie. layer thickness) and α is the angle, see Fig. 3.For this simple, structured case, these parameters are easy to define. Note, that contrary to theclassical case, the matrix Λ is not diagonal. The extradiagonal element has been chosen so thatthe quality q(T ) is judged maximal for a right-angle element T of the desired dimensions, unlikein the standard case, where isoceles triangles are prefered.

Figure 3: 2D segment: parameters

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The initial metric M0 is then coarsened in the radial direction to get the desired metric fieldM .

M(x) = RT ·DT · (R ·M0(x) ·RT ) ·D ·R,

with R and M0 as above and

D =(

1 00 1

2

).

The original and the coarsened mesh can be seen in Fig. 4.

Figure 4: 2D segment: Initial (left) and coarsened (right) mesh

Of course, in this simple case, we could have defined the desired metric straight away, withoutM0. For general geometries, the step via M0 seems natural.

2 Subproblems to solve

As we already hinted in the example above, one of the corner-stones of using MTC is the choiceof the desired metric, defined in a discrete way on each node. This is why, we elaborate this topicin this section.

2.1 Metric and interpolation of metric

First, we should say, that a metric M(x) defines at each point not only the characteristic meshsizesin each direction, but also to some extent the angles [7].

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Also, to each element T there exist only one x-independent metric M such that its elementquality q(T ) of (2) is maximal, measured in the metric M . The inverse does not hold - let us havetwo triangles T1 = (0, 0); (3, 0); (0, 1) and T2 = (0, 0); (3, 0); (1.5, 1), their respective maximumquality metrics are

M1 =( 1

(3)21

2·31

2·3 1

)M2 =

( 1(3)2 00 1

),

but for these metrics, there exist an infinite number of triangles for which q(T ) attains its maxi-mum, q(T ) = 1, cf. Fig. 5.

Figure 5: some elements T for which q(T ) = 1 for given metrics M1 (left) and M2 (right)

Even though the metric M(x) takes account of angles of the reference (ideal) triangle, thequality function q(T ) with this metric M is not able to distinguish among infinitely many trianglesthe ideal (right-angle) one. Hence, it follows, that another constraint is needed, eg. in thedefinition of the quality function q(T ), but not necessarily only there, to preferenciate the right-angle triangles.

During our presentation, we do not use an additional constraint in the function q(T ), but werely on the fixed nodes of the odd layers as the needed constraint.

2.2 Definition of the initial nodal metric

In the simple model example of Section 1.3, we are able to construct the initial metric M0 layerby layer, based on the layer thickness parameter hr and the tangential mesh-size hα. This mightnot be the case for more general geometries, where one could have between nodes of layers L andL + 1 more than one layer of elements, cf. Fig 6.

Figure 6: Non-constant layer thickness

Therefore, we prefer to define the initial metric independently of layers.

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The aim is to define in every node of the initial mesh a metric, such that the quality Qh of theentire mesh,

Qh = q(T ), T ∈ Th,

would not be too bad (we are observing the values of the minimum min(Qh), of the algebraicaverage ave(Qh) and of the maximum max(Qh). Of course, the quality Qh depends not only onthe defined initial nodal metric field M0(x), but also on the way the nodal metrics are interpolatedon elements. To keep Qh as close as possible to the measure used in MTC to evaluate the meshquality, we should use the same metric interpolation as in MTC, ie. the algebraic average of nodalmetrics.

2.2.1 Mapping the neighbourhood onto the unit sphere

One of the ways to obtain a reasonable initial nodal metric on a node xi is to try to map its nearestneighbour nodes xj , j ∈ N (i) onto a unit sphere. Actually, we solve a simple minimization problemwith constraints:

f(M) =∑

j∈N (i)\i

(‖xj − xi‖2M − 1

)2 → min,

M = MT ,

M is positive definite.

As f(M) is quadratic in the coefficients Mij of the matrix M , the Jacobian is linear in Mij whichsimplifies the problem to solve. Moreover, the constraint M = MT can be directly substituted tof(M). Only the last constraint on positiveness of M is not straightforward to apply, it containsone sided constraints on the spectrum of M . In some cases, the geometrical configuration of theneighbourhood of xi results in non-invertibility of the Hessien ∂2f(M)/∂M2

ij . Therefore, we addartificial nodes xj′ , the centres of edges (xj1, xj2) ∈ Eh connecting two neighbours xj1, xx2 of xi,j1, j2 ∈ N (i).

2.2.2 Quality optimization

Another way to obtain a reasonable nodal metric field is expressing M(xi) in terms of a combina-tion of some basis Mk. In our case, we use the element-metrics on elements Tk ∈ Th, adjacentto the node xi, xi ∈ Tk. The element-metrics are uniquely defined; for each element Tk, xi ∈ Tk,we construct a metric Mk in which the element Tk is an equilateral simplex with edges of length1.

The combination of the basis elements Mk could be non-linear, we can use, eg.

M(xi) =∑

k

ckMk, or M(xi) =

[∑k

ckM−1k

]−1

, or M(xi) =

[∑k

ckM− 1

2k

]−2

,

with∑

k ck = 1. The choice from the above formulas depends on the used metric interpolationstrategy. Following the arguments of precision and simplicity in [1], we use the formula in themiddle, M(xi) =

[∑k ckM−1

k

]−1, when generating the initial metric, while MTC uses an algebraic

average of metrics for interpolating the initial nodal metrics to elements.The advantage of the Quality optimization approach over the previous one is that it does not

contain a constraint on the spectrum of M(xi); when Mk are SPD and ck > 0, we are assured ofpositivity of the resulting metrics M(xi). The coefficients ck can be locally specified for each xi

to maximize the quality Qi of the elements adjacent to xi, Qi = q(T ), xi ∈ T, T ∈ Th, ie.

min(Qi)→ max,

ck > 0.

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Figure 7: RAE2822: Quality of the initial metric: mapping onto unit sphere

Note, that the calculation of q(T ) takes into account the value of M(xi) through the interpolationof nodal metrics M(xj) to an element metric M on T , xj ∈ T .

The basis Mk is constructed for each node i in such a way, that it should be able to generatean ideal metric on the node i. Theoretically, we are not constrained in the choice of Mk toelement-metrics of the adjacent elements to the node i.

2.2.3 Some numerical experiments

Let us demonstrate the method on a 2D RAE2822 mesh geometry. As a convenient measure forcomparing the quality of the initial metric fields, we take the mesh quality parameter q(T ) definedby (2) for each element of the initial mesh.

Moderately stretched RAE mesh Let us first consider a mesh as in Fig. 9 which is onlymoderately stretched, with the aspect ratio 100. Figs. 7 and 8 show the respective qualities ofthe mesh measured by the same quality function q(T ) based on the two different fields of nodalmetrics.

We can see, that the Quality optimization approach leads to a metric field which tends to givebetter qualities of elements then in the case of Mapping to the unit sphere. Of course, our interestis to have the initial nodal metric which gives the best quality possible.

The corresponding coarsened meshes are to be seen in Figs. 10 and 11. We see, that there is nota substantial difference between the two approaches of generating the nodal metric-field. However,the coarsened mesh with the “less precise” metric of Section 2.2.1 respects less the coarsening andstructures in some areas, mainly where the metric field changes more abruptly.

Stretched RAE mesh There are cases yet to investigate, where neither of the two methodsfor generating the initial nodal metric fields work. As an example we can take another mesh of

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Figure 8: RAE2822: Quality of the initial metric: quality optimization

the same profile RAE2822 (courtesy of Guillaume Caillot, Dassault Aviation, cf. Fig. 12). Thismesh is a stretched one with the aspect ratio of order 1000. From the Figs. 13 and 14 we seeregions behing the profile with a dangerously low quality of initial metrics obtained by the samealgorithms as above. This fact might be related to the problem of interpolation/intersection ofmetrics, as described in [1]. Further study is yet to be done.

Although the initial metric in the chosen measure q(T ) of (2) does not seem satisfactory, thecoarsened meshes are still very good. Fig. 15 shows a section of an RAE mesh behind the wingprofile, together with the numbering of layers of nodes. Fig. 16 contains on the left the originalmesh (with axes of the plot stretched to better visualize the stretched zone). The plot in themiddle of Fig. 16 shows the coarsened mesh in the case of initial nodal metric by Mapping tocircle, while the left plot shows the case of metric by Quality optimization. Note, that in the caseof Mapping to circle, diagonals of the mesh can be swapped more often than in the case of Qualityoptimization. There, it occurs just on places, where the quality of the initial metric is very low(cf. Fig. 14).

3 Conclusion

Coarsening of the RAE meshes by the developed algorithm has given quite satisfactory coarsemeshes, even for meshes with higher aspect ratio 1000.

However, there persist some difficulties with the definition of the initial nodal metric. True,they did not influence too much the coarsening process, but this came only from the fact, thatnodes in every other layer of nodes have been fixed. Thus, even with a bad metric, the resultingmesh was satisfactory. However, once we would like to manipulate and coarsen an anisotropicmesh not possessing the layer structure, the bad initial metric might cause some problems.

A natural continuation of the work in 2D is the generalization for 3D boundary layer meshes.Also, we would like to introduce an additional constraint into the quality function q(T ) to stress

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out the importance of right-angle elements. We haven’t yet treated initial meshes with layer-condensation phenomenon (cf. Fig. 2). To be able to keep a favorable structure of layers ofelements, we would like to review the attribution of nodes to layers. At present, this is done basedon topological connectivity, which would result in this case into a bad configuration of layers whichwould not permit to preserve the structured parts of the mesh.

References

[1] Borouchaki, H.; George, P.L.; Hecht, F.;Laug, P. et Saltel, E.: Mailleur bidimensionnel deDelaunay gouverne par une carte de metriques. Parie I.: Algorithmes, Rapport de RechercheINRIA no. 2741, 1995.

[2] Carre, G.; Carte, G.; Guillard, H. and Lanteri, S.: Multigrid strategies for CFD problems onnon-structured meshes, Multigrid Methods VI (Gent, 1999), 1–10, Lect. Notes Comput. Sci.Eng. 14, Springer, Berlin, 2000.

[3] Carte, G.; Coupez, T.; Guillard, H. and Lanteri, S.: Coarsening techniques in multigrid appli-cations on unstructured meshes, European Congress on Computational Methods in AppliedSciences and Engineering, ECCOMAS 2000, Barcelona, 2000.

[4] Castro-Dıaz, M.J.; Hecht, F.; Mohammadi, F.; Pironneau, O.: Anisotropic unstructured meshadaptation for flow simulations, Int. J. Numer. Meth. Fluids 25 (1997), 475–491.

[5] Coupez, T: Generation de maillage et adaptation de maillage par optimisation locale, Revueeuropeene des elements finis 9 (2002), no. 4, 403–423.

[6] George, P.L. and Borouchaki, H.: Triangulation de Dalaunay et maillage, applications auxelements finis, Hermes, Paris 1997.

[7] Knupp, P.M.: Algebraic mesh quality metrics, SIAM J. Sci. Comput 23 (2001), no. 1, 193–218.

[8] Labbe, P.; Dompierre, J.; Vallet, M.G.; Guibault, F.; Trepanier,J.Y.: A measure of theconformity of a Mesh to an Anisotropic Metric, CERCA Research Report, Ecole polytechniquede Montreal 2001, 10th Int. Meshing Roundtable 2001.

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Figure 9: RAE2822: mesh geometry, moderately stretched mesh

Figure 10: RAE2822: Mapping to circle, coarsened mesh (top) and mesh coarsened up to 8thlayer (bottom)

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Figure 11: RAE2822: Quality optimization, coarsened mesh (top) and mesh coarsened up to 8thlayer (bottom)

Figure 12: RAE Dassault geometry: zoom on the back of the profile

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Figure 13: RAE Dassault: Quality of the initial metric: mapping onto unit sphere

Figure 14: RAE Dassault: Quality of the initial metric: quality optimization

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Figure 15: RAE Dassault: the finest mesh - detail of the zone behind the wing (top), numberingof nodes in layers (bottom)

Figure 16: RAE Dassault: The finest mesh (left), and coarse meshes with metric by mapping tocircle (middle) and by quality optimization (right) (NB. the axes do not have the same scale)

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