[ieee 4th international symposium on voronoi diagrams in science and engineering (isvd 2007) -...

Generating the Voronoı-Delaunay Dual Diagram for Co-Volume IntegrationSchemes

Igor Sazonov, Oubay Hassan, Ken Morgan and Nigel P. WeatherillCivil and Computational Engineering Centre,

Swansea University, Singleton park, Swansea, SA2 8PP, [email protected]

Abstract

Advantages of co-volume methods (based on the use of ahigh quality Voronoı diagram and the dual Delaunay mesh)for two- and three-dimensional computational electromag-netics are well known. The co-volume method is faster thantraditional methods for an unstructured mesh and needs lessmemory. The co-volume integration scheme preserves en-ergy, i.e. gives high accuracy of wave amplitude. It alsogives better accuracy if the scattering objects has sharp cor-ners or vertices. However, the co-volume method requiresuse of high quality unstructured dual Voronoı-Delaunay di-agrams which cannot be created by classical mesh gener-ation methods. For two-dimensional problems, a stitchingmethod gives the best mesh quality for a wide variety of do-mains. Generation of a three-dimensional dual mesh appro-priate for the use of a co-volume scheme is a much more dif-ficult issue. Here, an approach is being developed where themain ideas of the stitching method are exploited. Some ex-amples of three-dimensional meshes generated by this newmethod, as well as the results of the integration of Maxwell’sequations on those meshes, are presented.

1 Introduction

Co-volume methods exhibit a high degree of computa-tionally efficiency, compared to, for example, a finite ele-ment (FE) method. Yee’s scheme for the solution of theMaxwell equations [14] and the MAC algorithm for thesolution of the Navier–Stokes equations [1], are examplesof such co-volume solution techniques. The co-volumemethod for electromagnetic (EM) waves has additional ad-vantages. It preserves the energy and, hence, maintains theamplitude of a plane wave. It also better approximates thefield near boundary peculiarities (sharp edges, vertices, wirestructures) without requiring a reduction in the size of ele-ments.

Nevertheless, despite the fact that real progress has beenachieved in unstructured mesh generation methods, co-volume approaches for simulation involving complex shapedomains have not been widely applied. This is due to thedifficulties encountered when attempting to generate highquality dual diagrams, satisfying the requirements neces-sary for the use of the co-volume solution schemes.

In the present work, we list the requirements for themesh appropriate for the use with co-volume schemes. Wedemonstrate that current mesh generating methods cannotproduce the required mesh for general 3D geometries. Weconsider a new approach to mesh generation, based on theexperience obtained in generating 2D meshes. Some nu-merical examples of a propagating and scattered EM waveare described that demonstrate the effectiveness of the ap-proach for use with co-volume integration schemes.

2 Co-Volume method for Maxwell’s equa-tions

The co-volume algorithm for integration of Maxwell’sequations is based on two integral equations [12]: Ampere’slaw

∂

∂t

∫A

E dA = ε

∮∂A

H dl (1)

and Faraday’s law

∂

∂t

∫A

H dA = −µ

∮∂A

E dl (2)

applied to a surface A and its boundary ∂A. Here E andH are vectors of the electric and magnetic fields, respec-tively; dA is an element of the surface area directed normalto the surface, dl is an element of the contour length di-rected along tangent to the contour; ε and µ are the electricpermittivity and magnetic permeability.

To implement the method effectively, we must have twomutually orthogonal meshes built for the domain of inte-gration for the electric and magnetic fields [4]. The dual

4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007)0-7695-2869-4/07 $25.00 © 2007

a

b

Figure 1. Voronoı cell (pink) and Delaunay el-ements (blue) around a generator (Delaunaynode) (a) for a generic case and (b) for degen-erate elements: eight vertices of every ele-ment are on the same sphere.

Delaunay–Voronoı diagram is the obvious choice. In 2D,corresponding edges of the Voronoı and Delaunay meshesare mutually orthogonal. In 3D, every edge of the Voronoıdiagram is orthogonal to the corresponding face of Delau-nay triangulation and vice versa (see Figure 1).

If edges of Delaunay elements are used for the electricfield and edges of Voronoı cells are used for the magneticfield1, then equations (1)–(2) can be approximated as

Eni − En−1

i

∆tAV

i = ε

MVi∑

k=1

Hn+0.5ji,k

lVji,k, i = 1, . . . , ND

s

(3)

1We can reverse this choice but for typical boundary conditions it isappropriate to use Delaunay edges for the electric field

Hn+0.5j − Hn−0.5

j

∆tAD

j = −µ

MDj∑

k=1

Enij,k

lDij,k, j = 1, . . . , NV

s

(4)where En

i is the projection of the electric field onto the ithDelaunay side at the instant ∆t n; Hn+0.5

j is the projec-tion of the electric field onto the jth Voronoı side at the in-stant ∆t (n + 0.5); lDi and AV

i are, respectively, the lengthof the ith Delaunay side and the area of the correspondingVoronoı face; lVj and AD

j are, respectively, the length of thejth Voronoı side and the area of the corresponding Delau-nay face; ji,k, k = 1, . . . , MV

i are sides of a Voronoı facecorresponding to the ith Delaunay edge and MV

i is theirnumber (Figure 2a); ij,k, k = 1, . . . , MD

j are sides of a De-launay triangle face correspondent to the jth Voronoı edgeand MD

j = 3 is their number (Figure 2b); NDs and Ne

s arethe numbers of Delaunay and Voronoı sides in the mesh.

p p

i ji,4

1 2

ji,2 ji,3

ji,1

ji,6 ji,5

e e1 2

j

ij,1 ij,2

ij,3

a b

e e1 2

j

ij,1ij,2

ij,3

c

Figure 2. The ith Delaunay side connectingnodes p1-p2 and corresponding Voronoı faceformed by Voronoı sides ji,1, . . . , ji,6 (a). Thejth Voronoı side connecting circumcentra ofelements e1-e2 and corresponding Delaunayface formed by Delaunay sides ij,1, ij,2, ij,3 (b-c). In (c) the Voronoı side does not intersectthe corresponding Delaunay face.

Solving (4) for Hn+0.5j and (3) for En

i , we obtain anexplicit scheme staggered in space and time.

This co-volume method diverges if the time step ∆t istoo large. For a grid of cubis, the stability criterion is de-rived in [13] and c∆t < l/

√3 where c = 1/

√εµ is the

wave speed and l is the edge length. For an unstructuredmesh, there is no such simple criterion. In the 2D case, the


stability property could be estimated from the relation

c∆t < Sf mini,j

{lVi , lDj } (5)

where Sf is a safety factor that depends on the mesh andneeds to be found experimentally. For different meshes itcan vary from 0.8 to 2.0, where Sf = 0.8 for an ideal meshand the larger value of Sf , as a rule, can be taken for worsequality meshes. For a 3D, mesh we will use the same sta-bility criterion. There is some suggestion that the criterioncould depend on the Voronoı and the Delaunay face areaalso.

If the mesh is Delaunay degenerate, i.e. more than fourpoints are on the same circumsphere, then, formally, someVoronoı edges have zero length and some Voronoı faceshave zero area. Nevertheless, the co-volume scheme canremain stable, after a Voronoı zero length edge and the cor-responding orthogonal Delaunay face are excluded from theformulae above. The same is true for Voronoı face of zerotharea and the corresponding Delaunay edge. Tetrahedron el-ements are considered as been merged into polyhedra withvertices lying on the same sphere. A cubic grid used for theclassical Yee’s scheme is an example of such a degenerateDelaunay grid (see Figure 1, right) where 6 tetrahedra aremerged to form a cube. MD

j = 4 in this case.In the co-volume scheme, the values of the electric and

magnetic field are taken at points of intersection of the edgeprojected onto the corresponding face. If the circumcentreof a Delaunay element is outside the element, then one ofVoronoı edges connected to its circumcentre does not inter-sect the corresponding Delaunay face. In this case, the con-tour integral

∫lH dl in (1) is approximated by a value of

the magnetic field outside the corresponding Voronoı edge,but at the point where the continuation of the edge intersectsthe corresponding Delaunay face (Figure 2c). This approx-imation of the integral cannot guarantee even first order ac-curacy with respect to edge length. Thus, when generatinga mesh for a co-volume scheme, it is necessary to minimizethe number of ‘bad’ elements, and, if possible, to removethem completely.

From the above it follows, that there are two main char-acteristics of mesh quality for use with co-volume integra-tion scheme. The first is

Q = βDmin{lD, lV }

δ(6)

Here, βD is a normalization coefficient, which depends onthe dimension of the mesh and gives Q = 1 for an idealmesh (see below); δ is the the prescribed element size,which provides the necessary accuracy for a given electro-magnetic wavelength λ (say, δ = λ/15). Traditional meshgeneration methods (see below) just provide that the meanDelaunay edge length 〈lD〉 is close to δ. Therefore, wecan substitute δ by 〈lD〉 in (6). The second characteristic

is that the number of elements with circumcentre outsidecompared to the total number of elements in the Delaunaymesh, i.e.

rbad =Nbad

e

Ne. (7)

Thus, in the generation the mesh we must obtain

Q → max, rbad → min . (8)

3 Traditional Meshing Methods

Traditional unstructured mesh generation methods, suchas the advancing front technique (AFT) [9] and the Delau-nay triangulation [5] as well as their combinations [7], gen-erate meshes in which the Delaunay edge lengths are ac-ceptable, but the corresponding Voronoı diagram is oftenhighly irregular, having some very short Voronoı edges and,in 3D, also some Voronoı faces of small area. Mesh im-provement methods, based on swapping, reconnection [3]and smoothing, improve essentially the quality of 2Dmeshes. Nevertheless, a significant number of very shortVoronoı edges and bad elements, located mainly near thedomain boundary, remain in the final mesh [11]. In 3D, tra-ditional methods produce meshes with about 50% of badelements and these are unsuitable for co-volume integrationschemes.

A promising approach is the construction of the cen-troidal Voronoı tessellation (CVT) and its dual Delaunaymesh. The CVT relocates the generated nodes to be at themass centroids of the corresponding Voronoı cells with re-spect to a given density function [2]. A new Voronoı tes-sellation of the relocated nodes is produced. This process,which is called Lloyd’s algorithm [6], can be repeated untilall nodes are close enough to the corresponding centroids.Lloyd’s algorithm needs an initial mesh, which can be pre-pared by any method. However, in addition to relocating thenodes, the CVT scheme changes the mesh topology. Al-though the quality of final mesh is much higher than thequality of the initial mesh, it is normally not suitable forthe successful application of co-volume solution schemes:in 3D the share of bad elements is usually around 10% ofthe total elements, although this can be reduced to 3–5% forspecially prepared initial meshes.

An alternative approach is the stitching method [11]. Inthis approach, the problem of triangulation is split into a setof relatively simple problems of local triangulation. Firstly,in the vicinity of boundaries, body fitted local meshes arebuilt with properties close to those regarded as being ideal.An ideal mesh is employed, away from boundaries, to fillthe remaining part of the domain. The mesh fragments arethen combined, to form a consistent mesh, with the outerlayer of the near boundary elements stitched to a regionof ideal mesh by a special procedure, in which the high


compliance of mesh fragments is used. This will result inhigh quality meshes compared to those built by other meth-ods [11].

4 Ideal Mesh

In 2D, a mesh of equilateral triangles is an ideal mesh.The Voronoı edge length lV = lD/

√3. Thus β2D =

√3 in

(6).To obtain the 3D analogue of the ideal mesh, we consider

first a mesh of cubes, every element of which is split intosix tetrahedra. Then some Voronoı edges are of zero length(see above). Now we skew every cube into a parallelepipedmaximizing the minimal Voronoı edge. As a result, we ob-tain a mesh consisting of equal non-perfect tetrahedra, eachface of which is an isosceles triangle with one long sideand two shorter sides with the ratio 1 :

√3/2 :

√3/2. All

Voronoı edges have the same length lV ≈ 0.38 〈lD〉. Ev-ery Voronoı cell is a polyhedron with 6 square faces and 8perfect hexagon faces (see Figure 3a). Computation givesβ3D = 2

√2(3 + 2

√3)/7 ≈ 2.6 in (6).

The 2D and 3D ideal meshes satisfy the requirementslisted above, but do not necessarily fit the boundaries. Nev-ertheless, they allow us to demonstrate the advantages of theco-volume scheme in some problems.

The ideal mesh can form a right angle parallelepipedwith two ragged faces. It can be used to simulate propa-gation of an electromagnetic (EM) pulse in a rectangularwaveguide. A fragment of the such waveguide is shown inFigure 3a. The waveguide is located along the x-axis andis of width Wy = λ/

√2 and of height Wz = Wy/2 where

λ = 2πc/ω is the wavelength in free space at the excitingfrequency ω. The electric field was set at the tetrahedronedges of the the initial section using

Ex = Ey = 0, Ez ={

0 t < 0sin(ωt − kxx0) t > 0

(9)where x0(y, z) is the local x co-ordinate of the initialsection, kx is the x component of the wavevector k ={kx, ky, kz}; in this case, only the TE+x

1,0 mode is excited.

In the simulations c = 1, ω = 2π, then λ = 1, kx = π√

2.The mesh spacing is λ/16. The length of the waveguidewas taken to be 170λ, which is large enough to avoid anyreflection from the end of the waveguide. Comparison be-tween analytical and numerical solutions are presented inFigure 3b. Here the Ez component at the axis after 200 cy-cles is plotted, in the region of transition from the forerunnerto the main signal with carrier frequency.

The second example is EM wave scattering on a 2λ do-decahedron fitted into the 3D ideal mesh. The shape of thisdodecahedron can be obtained if we consider a body formedby all Delaunay elements in the 3D ideal mesh containing

a

115 120 125 130 135 140 145

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x/λ

Ez

b

Figure 3. Fragment of the rectangular waveg-uide designed by the 3D ideal mesh (a):surface Delaunay faces and internal Voronoıcells are shown. Example of integrationbased on this mesh (b): blue dots indicatethe numerical and red curve the analytical re-sults.

the same node. The mesh is taken big enough to neglectthe reflection from the outer boundaries during 3 cycles. InFigure 4a, the Ez component of the computed EM field isshown in the vertical plane and on the surface of the dodec-ahedron. In Figure 4b, a comparison is made of the RCScomputed by FE (red) and the co-volume (blue) methods.One can see that the local discrepancy in RCS obtained bythe co-volume scheme between 15 (solid) and 8 (dashed)elements per wavelength is much less than that obtainedby the time domain finite element method [8] between 30(dashed) and 15 (solid) elements per wavelength. Only20 seconds is needed to compute RCS by the co-volumescheme. The TDFE method is a hundred times more expan-sive and requires a finer mesh.


a

0 60 120 180 240−15

−10

−5

0

5

10

15

20

FE15FE30CV15CV 8

b

Figure 4. Mesh and z component of electricfield the scattered by the dodecahedron (a).Radar Cross Section (RCS) in the verticalplane (b).

5 Near-boundary triangulation

If the domain boundary is smooth enough (i.e. if its cur-vature radius is much greater than δ) then building the firstfew layers is an elementary task in the 2D case: the near-boundary mesh has the same topology as the 2D ideal mesh.A well tuned 2D advancing front method produces thisnear-boundary high-quality mesh. Some modifications ofthis method which improves the mesh quality if the bound-ary curvature is not small are described in [11].

The analogous 3D problem is not elementary. Considerthe simplest, plane boundary with a 2D ideal boundary tri-angulation and consider how to build a single layer of highquality tetrahedra.

Following the 3D advancing front technique (which isregarded to be a good method for placing points) we build aperfect tetrahedron on every boundary triangle. We see that

the first layer contains double the amount of points (tetrahe-dra apexes) compared to the number of the boundary points.These new points form a hexagonal structure (like Voronoıvertices of the 2D ideal mesh). No matter how we connectthe new and boundary points, we cannot produce a propermesh for the first layer, as it necessarily contains a largeshare of bad tetrahedra.

Analyzing the structure of the 3D ideal mesh we can con-clude that the best placing of a new point is above the edgeshared by two conjugate surface triangles. To reproducethe topology of the 3D ideal mesh we split all boundarytriangles by non-intersecting pairs sharing the same edge,and locate a new point above this edge. A few single iso-lated triangles (having no paired ones) can remain. Thena new point is located above its centroid as in the stan-dard Advancing front method. Investigation shows that af-ter some optimization, a high quality near-boundary layer,which does not contain bad elements, can be produced if theboundary has small curvature and the quality of the bound-ary triangulation is high enough.

6 Example for the spherical layer

Some simple domains, like a spherical layer, can betriangulated if the near boundary triangulation describedabove is repeated until it fills the total domain [10]. Asa result, an unstructured dual mesh is generated where allcircumcentra are inside the corresponding Delaunay tetra-hedron: rbad = 0 (see Figure 5a). The mesh quality basedon the stability criterion is Q ≈ 0.1.

The mesh has been used to compute scattering of EMwave on a 2λ PEC sphere. Results for the RCS computedby the use of this mesh can be seen in Figure 5b. The errorin the RCS computation is within 0.2 dB.

7 Conclusion

A new method of placing points in a 3D triangulationis proposed. The method produces a high quality near-boundary triangulation and can be used to triangulate cer-tain types of domains. It has been demonstrated that thequality of the obtained mesh is enough to employ a co-volume integration scheme. The example presented in thepaper demonstrates that the problem of building a mesh fora 3D domain for the use of co-volume scheme is solvable.Work on generalization of this method for more compli-cated domains is under development.

References

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a

0 60 120 180 240 300 360−20

−15

−10

−5

0

5

10

15

20

Co−VolumeExact

b

Figure 5. Unstructured 3D mesh for thespherical layer: boundary Delaunay facesand internal Voronoı cells are shown (a).Radar Cross Section (RCS) in the verticalplane (b).

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[4] S. Gedney and F. Lansing. Full wave analysis of printedmicrostrip devices using a generalized Yee algorithm. InProceedings of the IEEE Antenas and Propagation SocietyInternational Symposium, pages 1179–1182. PennsylvaniaState University, Ann Arbor, 1993.

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[7] D. Marcum and N. Weatherill. Aerospace applications ofsolution adaptive finite element analysis. Computer AidedGeometric Design, Elsevier, 12:709–731, 1995.

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[9] J. Peraire, M. Vahdati, K. Morgan, , and O. Zienkiewicz.Adaptive remeshing for compressible flow computations.Journal of Computational Physics, 72:449–466, 1987.

[10] I. Sazonov, O. Hassan, K. Morgan, and N. Weatherill.Smooth Delaunay-Voronoı dual meshes for co-volume in-tegration schemes. In P. Pebay, editor, Proceedings of 15 In-ternational Meshing Roundtable, pages 529–541. Springer,2006.

[11] I. Sazonov, D. Wang, O. Hassan, K. Morgan, and N. Weath-erill. A stitching method for the generation of unstruc-tured meshes for use with co-volume solution techniques.Computer Methods in Applied Mechanics and Engineering,195(13-16):1826–1845, 2006.

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