Mesh generation forcomputational analysisPart I — Electromagnetic and technicalconsiderations for mesh generation

by T. I. Boubez, W. R. J. Funnell, Prof. D. A. Lowther, A. R. Pinchukand Profl P. P. SilvesterMcCill University

Geometric meshes used in the finite-element method and relatedtechniques must meet three requirements: they must modelgeometric shapes adequately; they must respect physical phenomenaof importance in the problem (for example crack singularities); andthey must provide high local accuracy in regions of particularimportance. Only simplex elements can be guaranteed to modelgeometry correctly if used with fully automatic mesh generators.Only adaptive mesh refinement can model in accordance with thecharacteristics of the physical solution. Technical specificationsrequiring high local accuracy can be accommodated in automaticmesh generation by permitting the user to specify a locally varyingaccuracy index.

Introduction

Numerical analysis of continuum prob-lems in engineering design usuallyinvolves discretisation of geometricstructures into numerous topologicallysimilar sub-sections. This requirementis clearly fundamental to the finite-element method, but it arises in otheranalytic techniques as well.

Mesh generation, the sub-division of

a given shape into elementary seg-ments, can be carried out either man-ually, using the extensive editing aidsnow available, or automatically. Two-dimensional problems are commonlyhandled by a combination of automaticmeshing and manual editing. In threedimensions, editing is difficult althoughoften necessary; automatic mesh gen-eration appears more attractive forrapid working but still requires manual

Fig. 1 Complex geometric structure of a rotating electric machineOne quarter section is shown, including the rotor and stator with winding slots

control. This paper proposes a compro-mise: automatic adaptive mesh genera-tion controlled by hand editing ofaccuracy control parameters.

Triangular meshes and simplicialcomplexes

The finite-element, as well as othermethods which rely upon decomposi-tion of complicated geometric shapesinto simpler standardised parts,employs many different kinds of ele-ments. There are few inherent limita-tions on element shape or type. Indeed,the literature contains articles recom-mending anything from infinite iso-parametric brick elements to lineartriangles (Refs. 1 and 2). Each elementtype has distinct advantages under cer-tain circumstances. However, if meshgeneration is to be automated fully, it isnecessary to employ a family of ele-ments which is guaranteed to be suit-able for the discretisation of arbitrarygeometric shapes. The only elementshapes for which proofs of convergenceand error bounds have been obtainedto date are simplexes (Ref. 3), and thediscussion to follow will therefore berestricted to simplicial complexes.

Formally, an m-dimensional simplexis a region of dimension m defined bythe smallest closed convex set whichcontains m+1 vertices in N-dimensionalspace, where m is at most equal to N.For example, a three-dimensional sim-plex in 3-space is a tetrahedron; a2-sim-plex is a triangle. A simplicial complex isa geometric object composed of sim-plexes. It is easy to show that every poly-gonal or polyhedral structure can besub-divided exactly into a finite set ofsimplexes. In this paper, any represen-tation of an N-dimensional object as asimplicial complex will be referred to asa triangulation, provided the intersec-tion of any two simplexes contained in itis a simplex of lower or the same dimen-sion. For example, in two dimensions

190 Computer-Aided Engineering Journal October 1986

triangles may not meet except along fulledges (1-simplexes) or at vertices(O-simplexes).

Triangulations are the only elementaldecompositions for which there existsthe necessary background to permitfully automatic mesh generators to bewritten. They are also reasonably effi-cient in their demand for computationalresources.

When triangulations are stored, thememory required varies linearly withthe number of simplex nodes. Astoragetechnique which closely approachesthe theoretical minimum is first to storethe n co-ordinate sets of the n nodeswhich appear in the triangulation assimplex vertices, and then to store adescription of each simplex as a set ofpointers to the node set. The wellknown Euler-Poincare formula statesthatthe numberof simplexes present ina given topology is proportional to thenumber of nodes. Consequently, thedatabase must contain at least O{kNn)= O(n) entities, where k is a propor-tionality constant dependent upon thetype of triangulation employed.

Triangulation is not a costly process incomputer time, taking only O(nlogn)operations in two dimensions (Ref. 4).The theoretical bound for computa-tional complexity can in fact beapproached fairly closely by practicalprograms.

Meshing criteria

A major strength of finite elements andrelated solution methods is that a non-uniform discretisation is easilyarranged, thus allowing certain regionsto be more or less refined. This feature,known as selective refinement, reducesboth the mesh storage requirementsand the computational effortthroughout the analysis. What con-stitutes a good triangulation is prin-cipally governed by three classes ofcriteria:

• The geometric structure may vary inlocal complexity, thus requiring a dis-cretisation of varying fineness to repre-sent the device adequately.• The physical laws underlying theproblem may require that certainregions be discretised more finely thanothers in order to model the solutionwithin the prescribed accuracy.• The technical specifications to bemet may indicate that a specific regionof the model is of outstanding import-ance and particularly error sensitive,requiring a discretisation with low localerror.

Mesh generators now exist which areable to satisfy the f i rst or geometric classof criteria. The region to be meshed is

'. ^4 -I *"V*"U j i „*»„!>-* till".

Fig. 2 Triangulation to model a conducting plate with an infinitely thin crackOnly one half of the plate is discretised. The bold broken line is a symmetry plane while theremainder of the boundary is an equipotential. Fine discretisation is used in the regionsurrounding the abrupt change in boundary conditions

approximated by a polygon, and the for analysing a rotating electricpolygon is triangulated. The mesh thus machine. The complex geometry of theproduced is minimal, and selective slots requires a large number of ele-refinement is usually required to satisfy ments to model the geometric structurethe other two classes of criteria.

Geometric representationGeometric constraints are necessarily

imposed on the shape of elements andtheir overall distribution. For instance,the numerical algorithms used forfinite-element calculations maybecome unstable if triangular elementswith very small angles are included inthe mesh. Anisotropy may be intro-duced into finite-element solutionswhere none actually exists, by skewingof the mesh; it is usually avoided bymaking all element edges roughlyequally long, i.e. keepingtriangularele-ments more or less equilateral.

This requirement on the aspect ratiois imposed because a series of shortedges can model a random fluctuation,while longer edges are constrained tovary as the elemental interpolationfunction. For example, with linear ele-ments, many short edges couldaccurately represent a non-linear vari-ation, while a long edge can only modela linear variation. Therefore havingmany long edges along a particulardirection and short ones in anotherdirection will influence the solution,thus reducing the generality of themethod.

Cases do occur in practice, often inthree-dimensional problems but some-times also in two dimensions, where thegeometric complexity of the device tobe modelled is so high that the minimalmesh represents almost the finest meshthat can be dealt with within thecomputer memory and time available.For example, Fig. 1 shows a mesh used

Computer-Aided Engineering Journal October 1986

accurately.

Physical laws and principlesThe physical laws to be satisfied in any

particular problem impose require-ments on the mesh to be used. Forexample, in many problems involvingelliptic partial differential equations,field singularities may occur atre-entrant corners. The use of finely dis-cretised meshes or special corner ele-ments, or both, is usually indicated.

As an example drawn from elec-tromagnetics, Fig. 2 depicts the meshused for analysing the field in a con-ducting plate with a non-conductivecrack. The infinitely thin crack is model-led by an abrupt change in boundaryconditions. Although the geometry ofthe model is simple, the physical natureof the problem requires that in theregion adjacent to the crack the ele-ments must be finely discretised inorder to solve for the complex field pat-tern caused by the crack.

Although a re-entrant corner or crackis a geometric feature of the problemregion and has nothing to do with theequations being solved, the need to dis-cretise finely arises from the physics,not the geometry, of the problem. InFig. 2, fine discretisation would not berequired if the electrical boundary con-ditions were such as to make currentflow parallel to the crack, yet it is essen-tial if the boundary conditions attemptto force currents to cross the crack.

There are two ways of dealing withthis problem: foreknowledge and adap-tation. The human analyst uses fore-knowledge when building meshes by

191

hand, placing fine meshes where pastexperience indicates that it will be nec-essary. Mesh generators have now beenbuilt to make meshes adaptively, solv-ing the problem on a coarse mesh ini-tially and refining stepwise as the needfor refinement becomes evident fromthe solution itself. It is conceivable thatrule-based 'expert' field solvers mightbe built to mimic the human analyst; forthe present, however, exploitation offoreknowledge and experience remainsthe exclusive province of mankind.

Technical requirementsTriangulations of given geometric

shapes can be produced automaticallywith guarantees of correctness, andadaptive meshing can be employed totake care at least partially of the require-ments imposed by physics. The thirdclass of mesh goodness criteria, how-ever, is harder to satisfy by automaticmeans. This third class, termed techni-cal requirements in this paper, typicallyflows from the performance require-ments of the device being analysed.

To proceed again by example, Fig. 3shows one-quarter of a particle acceler-ator magnet cross-section. To computeparticle trajectories accurately, veryhigh solution precision is needed in aparticular region of interest: the mag-net bore. Ideally, the magnet should bedesigned so that the field in the bore isuniform, see Fig. 3a. Were this not anarea of prime interest a very coarse dis-cretisation using simple linear elementswould suffice in this region, preciselybecause the field is nearly uniformthere and the error fairly low even if acoarse geometric model were used.However, because the main interest isto verify the uniformity of field within ahigh precision, the bore region shouldbe modelled using a fine discretisation,

as in Fig. 3b.To complicate matters, several quite

disjoint design criteria may attach to thesame device. For example, if therequired field uniformity is found toobtain in the magnet bore, trje designerwould probably next wish to determinethe local field values near the supercon-ductive coils of the magnet, so as to findthe current levels at which quenchingmight occur. For this purpose, intenseinterest attaches to the geometric areacontaining the coils themselves. Model-ling the magnet bore region crudely,and the coils in great detail, is nowdesirable.

Many other examples can be broughtto illustrate circumstances in which dif-ferent models of one and the samephysical situation are required toanswer the various different questionsto which the analyst may wish to findanswers. There is obviously no likeli-hood of including any but a small frac-tion of the possible choices in anautomatic program, even if the programcan be strongly rule based; after all,there is no way a program can deducefrom rules what question the analyst isthinking of! It appears, therefore, thatmanual intervention in the modellingprocess is essential for effective designand analysis. However, it is to be hopedthat this intervention can be carried outat a high level, not by detailed manip-ulation of mesh nodes and elements.

Mesh generation and editing

The meshing requirements discussedabove carry clear implications for thestructure of mesh generator software.The feasibility of meeting require-ments, and some methods for doing soexplored by the authors (as well as byothers) will be examined next.

Interactive editingMany of the requirements for the

mesh, specifically the selective refine-ment, are most easily satisfied by allow-ing an intelligent user to control themesh construction directly, by prescrib-ing the locations of mesh nodes andelements. The final mesh then reflectsuser wishes exactly. In this manner ofworking, the computer and graphicsscreen are used as a mesh editingdevice which allows the user to createand alter a triangulation interactively(Ref. 5).

The screen display provides at alltimes a visually clear representation ofthe current state of the model, and sup-porting software can be designed torelieve the user of repetitive or mechan-ical tasks. Furthermore, the mesh data iscalculated directly from an unam-biguous representation of the geome-try. At all times the user retains thecapability of altering any aspects of themesh. Interactive mesh constructionwith graphic echo has been very suc-cessful for two-dimensional mesh gen-eration (Refs. 5-7). Several interactivethree-dimensional mesh generatorsalso exist (Refs. 8 and 9).

Whereas interactive mesh generationwith graphic echo is now well under-stood in two dimensions, difficulties areoften encountered when attempting todo so in three dimensions. A majorproblem is the inability to display, in acomprehensible fashion, large num-bers of intersecting three-dimensionalmesh lines on a two-dimensionalgraphic screen or plotter. A typical dis-play from a three-dimensional meshpackage is shown in Fig. 4. In contrast todisplays that accompany geometricmodellers, the use of hidden-line (orsurface) removal is not a feasible sol-ution to thedisplay problem. These hid-

Fig. 3 Quarter section of a particle accelerator magneta Flux lines and graph illustrate the uniform magnetic field in the magnet bore, which is surrounded by the superconducting coils andthe armatureb Triangulation used for analysis when technical specifications concern the central region

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den lines are precisely what the userrequires in order to generate a finite-element mesh! This implies thatcrowded, line-filled screens are to beexpected, a situation not conducive toerror-free working.

Present methods of manual pre-processing require a knowledge of boththe problem and the solution tech-nique. Whereas in two-dimensionalanalysis flux and contour plots are wellunderstood and provide considerableinsight to the solution, in a three-dimensional analysis most users do notfeel equally at home. This implies thatthe user of an interactive three-dimen-sional mesh generator, even if capableof specifying the regions for refine-ment, would not always have the physi-cal intuitution to know where theproblem requires fine discretisation.

One attractive solution is to providethe user with a geometric modeller forentering the model description andwhich is also capable of superimposingan independent model for the technicalspecifications. Once the geometricstructure, the physical problem and thetechnical specifications have beenentered, the mesh is generatedautomatically.

Automatic mesh generationMost present-day interactive mesh-

ing systems employ some form of auto-matic mesh generation, thus relievingthe user of the drudgery of enteringlarge numbers of elements. However,for fully automatic mesh generation,both the point set and triangulationmust be generated without any userintervention. There are two major ques-tions in automatingthis step: the choiceof node location and the choice of ele-ment placement. Choosing a set ofnodal points does not uniquely estab-lish a triangulation, so the second ques-tion is largely independent of the first.

Assumingforthe moment that a set ofnodal points has been chosen — ini-tially at least, the point set can be usedwhich minimally describes the geo-metric object to be meshed — the taskof triangulation becomes that of placingelements on the point set in a way whichwill meet the computational require-ments discussed above. Since the meshgeneration is to be automatic, thealgori-thm used must be guaranteed toproduce valid meshes over the pointset. In addition, it must meet the geo-metric criteria to the best capabilities ofthe underlying set of points.

A triangulation algorithm whichmeets requirements is the Delaunay tri-angulation (Ref. 10) or the dual graph ofthe Voronoi diagram (Ref. 11). Severalcomputationally optimal algorithms forcomputing the Delaunay/Voronoi dia-

Fig. 4 Three-dimensional finite-element mesh as displayed on a graphics screen

grams have been reported in the litera-ture (Refs. 4 and 12-15). Thecharacterising property of the Delaunaytriangulation is that any hyperspherecircumscribed over the N+1 nodes ofany N-simplex must not contain anyother nodes of the data set. Delaunayrefers to this as the 'empty spheres'property (Ref. 10).

The Delaunay triangulation at firstglance appears to be no different fromany other triangulation. However, the'empty spheres' property causes smallangles to be avoided and usually pro-duces triangles with good aspect ratios.Both of these features are desirable infinite-element analysis and elsewhere.The same property also ensures that themesh is locally equiangular, a termintroduced by Lawson (Ref. 16), mean-ing that, for any two adjacent triangleswhich unite to form a convex quad-rilateral, the minimum internal angle ismaximised through correct choice ofthe diagonal.

A computationally optimal algorithmfor generating the Delaunay triangula-tion is to derive it from the Voronoi dia-gram (Ref. 4), which is a polygonaltessellation based upon the point set.Delaunay proves (Ref. 10) that given aDelaunay triangulation a uniqueVoronoi diagram may be constructed.Conversely, given the Voronoi diagramthe unique Delaunay triangulation maybe obtained.

Having the Delaunay triangulationand its dual graph also offers manyother possible computations at very lit-tle expense (Refs. 4 and 17). The addi-tional information provided directly bythe Delaunay and Voronoi diagramsmust not be overlooked. For example,once the Voronoi diagram has been cal-culated for a set of points, computa-tionally important information such asthe nearest neighbour graph and theEuclidean minimal spanning tree maybe calculated in O(n) complexity (Ref.4). These topological entities and

Computer-Aided Engineering Journal October 1986

closely related ones are neededrepetitively during computationalanalysis, especially at the solving andpostprocessing stages. Since this infor-mation is available from the triangula-tion, it can be used to good advantage atessentially no cost.

The Delaunay triangulation method isused in many automatic mesh genera-tors at the present (Refs. 5, 18 and 19).However, for the mesh construction tobe fully automatic the nodes them-selves must also be generated with con-sideration of the geometric restrictionsspecified for the triangulation. Theserestrictions must direct the point gener-ation so as to avoid badly distorted ele-ments (Ref. 20). They could take theform of a set of rules which each tri-angulation must obey, and subject towhich points may be moved or added to,satisfy both the geometric andDelaunay restrictions.

Automatic adaptive refinementThree fundamental groups of criteria

affect triangulation: geometric, physi-cal and technical. Part II of this paper(Ref. 21) discusses automatic triangula-tion based solely on geometric struc-tural considerations. The two otherclasses are considered further in thissection.

Successive refinement by adaptivemeshing is normally guided by localerror estimates. Such estimates, ofcourse, can only be obtained once themesh nodes are generated! For-tunately, it is knowledge of the error inan approximate solution, rather thanthe solution itself, that is essential forguiding the mesh refinement. There-fore a minimal triangulation whichunambiguously represents the physicalproblem may be used to calculate anestimated error distribution. Regionswhich are not adequately refined arethen identified and improved iter-atively. Obviously the estimated errorplays an important role in this process.

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Fig. 5 Shaded regions identify the areas of concern as per the technical specificationsa Bore region is specified for particle trajectory analysisb Regions surrounding the superconducting windings are important for quench analysis

Hence much attention in the area ofautomatic adaptive mesh generationturns to finding error bounds. Oncethese bounds are obtained an adaptivealgorithm then improves upon an exist-ing mesh until a convergence criterionis met.

Adaptive mesh generation requiressome sort of refinement process duringeach iteration. Adaptive methods useestimates of the errors in an approxi-mate solution to indicate where refine-ment of the existing coarse mesh isnecessary. Two types of error estimatesare required: global and local. Theglobal error estimate bounds the dif-ference between the exact and theapproximate solutions in someLipschitz norm and therefore signifieswhen the mesh may be considered tohave converged. The local error esti-mate represents the error in each ele-ment of the mesh and indicates whereadditional refinement is required.Together, these estimates indicatewhen and where to refine the existingmesh.

The topic of error estimation hasbeen approached from many differentdirections. Some authors derive theerror estimates based on knownapproximations which are imposed bythe analysis method (Refs. 22-24).Another interesting approach to errorestimation is based on calculating twocomplementary solutions: one whichbounds the exact solution from above,and the other which is known to boundit from below. These upper and lowerbounds indicate the maximum possibledeviation of a calculated solution fromthe exact answer. Error estimates basedupon these techniques have also beenproduced by several workers (Refs.25-29).

When the estimated global error dis-tribution throughout the model is used

as the refinement criterion, it may actu-ally mislead the mesh refinement. If aprecise error bound is available theremaining question is which elementsshould be refined to improve upon thiserror distribution. It may initially appearthat those elements should be sub-divided which exhibit the highest error.However, every useful error estimatedeals with the difference in values ofsome quantity such as stored energy. Alarge estimated error may in fact merelyreflect a large energy in a particularregion or element, yet the relative errormay actually be small compared withthe relative error in other regions. Totalerror is therefore not invariably a goodcriterion to guide mesh refinement.

Using the local value of relative error,on the other hand, can be equally mis-leading. Elements or regions containinglow stored energies are usually of theleast concern in an analysis, yet they willoften show the largest estimated rela-tive errors (because the datum value ofstored energy is small). No fully accept-able solution to this problem appears inthe literature to date, perhaps account-ing for the curious meshes occasionallyproduced by current commercialsoftware.

Technical specifications may requirerefinement of certain problem regionseven though the estimated local errormay be small. Any such requirementsmust be included in the overall schemeof the mesh generator and should notbe repeated during each iterative stepin the adaptive process. A simple andyet flexible method of incorporatingthis information, in addition to the geo-metric and physical description of themodel, is to define a scalar field over theproblem space to indicate the import-ance of solution accuracy. This accuracyindex is then used in mesh refinementas a scalar weighting coefficient.

This approach is attractive both forreasons of flexibility and for its easyincorporation in software. Users notinterested in specifying regions of par-ticular importance may ignore the mat-ter altogether, provided the accuracyindex is initially assigned a default valueof 1 everywhere. The user can then editits local values (up or down the defaultvalue) in exactly the same way as thesoftware system permits assigningmaterial properties to specified regions— or leave them at the default value.The regions over which the accuracyindex is assigned specific values are ofcourse independent of the materialboundaries in the model.

The particle accelerator magnetalready discussed above may be used toillustrate the principle involved in localaccuracy specification. In this example,the analyst first demands high accuracyin the bore region of the magnet, so asto ensure precision in particle trajectorycalculations. The region of concern inthis case may be specified by a circle, asin Fig. 5a. When the fields surroundingthe superconducting windings are ofconcern to a design engineer, the inter-est region surrounds the winding as inFig. 56.

It may be noted that neither regionneed coincide with any naturally occur-ring material boundary in the problem.In two-dimensional mesh editing theinterest regions can be specified tocoincide with element boundaries, butfor greater flexibility it is preferable tospecify them with the geometric model-ler used to define material shapes. Thischoice is preferable in three-dimen-sional problems where mesh-basedediting is quite difficult. However, it isalso important in two dimensionsbecause the technical specificationsshould not change as the mesh isrefined during the adaptive process;

194 Computer-Aided Engineering Journal October 1986

hence fully automated adaptive meshgeneration is not compromised.

Conclusion

Mesh construction for finite-elementanalysis can be carried out by manualediting, or meshes can be generatedautomatically from geometric descrip-tions of the objects to be discretised. Intwo dimensions, interactive editing is avaluable tool which can be usefully sup-ported by automatic meshing pro-grams. In three-dimensional problems,the difficulties of display and interactivemanipulation appearto make automaticmesh generation the fundamentalworking tool, to which manual editingprocedures can lend support andguidance.

The criteria which guide mesh gener-ation include:

• adequacy of representation of geo-metric detail• error levels in the computedsolution• accuracy distribution as dictated bythe user's needs.

The first can be satisfied by an automaticsimplicial complex generation pro-gram, given the problem geometry asfurnished by a geometric modellingprogram. The second requires, in addi-tion to problem geometry, a knowledgeof the physical laws to be satisfied; itcan be met by an adaptive mesh genera-tor working iteratively in alternationwith a physical problem solver.

The third can be satisfied only by per-mitting the user to define accuracyrequirements on a local basis and there-fore requires an interactive adaptivemesh generator. By including useraccuracy requirements in the problemdatabase in a manner similar to materialparameters or source densities — quan-tities locally defined and subject to for-mal editing processes — it appearsfeasible to build such mesh generationprograms. Experimental prototypesappear to be workable, and programs ofindustrial quality should be feasiblewithin the foreseeable future.

Although expert systems might atsome future date assess and prescribelocal accuracy distribution, such a pos-sibility must, for the moment at anyrate, be considered a long-term goalrather than a realistic requirement forsoftware systems to satisfy. For theimmediate future, the target mustremain computer-aided design andengineering, not design by computer.

Acknowledgment

A portion of this work was carried out atthe University of Cambridge and at the

Imperial College of Science and Tech-nology, London, whose generous helpthe authors wish to acknowledge with

thanks. This work was supported finan-cially by the Natural Sciences and Engin-eering-Research Council of Canada.

References

1 ZIENKIEWICZ, O. C : 'The finite element method' (McGraw-Hill, 1977)2 SILVESTER, P. P., and FERRARI, R.: 'Finite elements for electrical engineers' (Cambridge

University Press, 1983)3 ALEXANDROFF, P.: 'Elementary concepts of topology' (Dover, 1961)4 SHAMOS, M. I.: 'Computational geometry'. Ph.D. Thesis, Yale University, New Haven,

CT, USA, 19785 LOWTHER, D. A., and SILVESTER, P. P.:'Computer-aided design in magnetics' (Springer-

Verlag, 1985)6 SIMKIN, J., and TROWBRIDGE, C. W.: 'Electromagnetics CAD using a single user

machine', IEEE Transactions on Magnetics, 1983, MAG-19, (6), pp. 2655-26587 SABONNADIERE, J. C , MEUNIER, G., and MOREL, B.: 'FLUX: a general interactive finite

elements package for 2D electromagnetic fields', ibid., 1982, MAG-18, (2), pp. 624-626

8 BRYANT, C. F., and FREEMAN, E. M.: 'A highly interactive three-dimensional meshgenerator', ibid., 1983, MAG-19, (6), pp. 2531-2534

9 REQUICHA, A. A. G., and VOELCKER, H. B.: 'Solid modelling: a historical summary andcontemporary assessment', IEEE Computer Graphics & Applications, 1982, 2, Mar.,pp.9-24

10 DELAUNAY, B.: 'Sur la sphere vide', Bulletin del'Academiedes Sciences del'URSS, Classedes Sciences Mathematiques et Naturelles, 1934, 52, (6), p. 793

11 VORONOI, G. M.: 'Nouvelles applications des parametres continus a la theorie desformes quadratiques', Journal fur die Reine und Angewandte Mathematik, 1908,134,(3)

12 GREEN, P. J.,andSIBSON, R.: 'Computing Dirichlet tessellations in the plane', ComputerJournal, 1977, 20, (2), pp. 168-173

13 BOWYER, A.: 'Computing Dirichlet tessellations', ibid., 1981, 24, (2), pp. 162-16614 NELSON, J. M.: 'A triangulation algorithm for arbitrary planar domains', Applied Mathe-

matical Modelling, 1978, 2, pp. 151-15915 WATSON, D. F.: 'Computing the n-dimensional Delaunay tessellation with application to

Voronoi polytopes', Computer Journal, 1981, 24, (2), pp. 167-17216 LAWSON, C. L.: 'Software for C surface interpolation'. Third Symposium on Mathemati-

cal Software, University of Wisconsin, Milwaukee, USA, Publication 39, March 197717 TOUSSAINT, G. T.: 'Pattern recognition and geometrical complexity'. Proceedings of

Fifth International Conference on Pattern Recognition, Miami Beach, FL, USA, Dec. 1980,pp. 1324-1347

18 HERMELINE, E.: 'Triangulation automatique d'un polyedre en dimension N', RAIROAnalyse Numerique/Numerical Analysis, 1982, 16, (3), pp. 211-242

19 CSENDES, Z. J., SHENTON, D., and SHAHNASSER, H.: 'Adaptive finite element meshgeneration usingthe Delaunay algorithm', IEEE Transactions on Magnetics, 1983, MAG-19,(6), pp. 2551-2554

20 CAVENDISH, J. C : 'Automatic triangulation of arbitrary planar domains for the finiteelement method', International Journal for Numerical Methods in Engineering, 1974, 8,pp. 679-696

21 BOUBEZ, T. I., FUNNELL, W. R. J., LOWTHER, D. A., PINCHUK, A. R., and SILVESTER, P.P.: 'Mesh generation for computational analysis. Part II — Geometric and topologicalconsiderations for three-dimensional mesh generation', Computer-Aided EngineeringJournal, 1986, 3, (5), pp. 196-201

22 KELLY, D. W., DE, S. R., GAGA, J. P., ZIENKIEWICZ, O. C , and BABUSKA, \.:'A posteriorierror analysis and adaptive processes in the finite element method. Part I — Error analysis.Part II — Adaptive mesh refinement', International Journal for Numerical Methods inEngineering, 1983, 19, pp. 1621-1656

23 VIVIANI, A.: 'Grid optimization in finite difference and finite element methods in mag-netic problems', IEEE Transactions on Magnetics, 1978, MAG-14, (5), pp. 461-463

24 FREDRICK, C. O., WONG, Y. C , and EDGE, F. W.: 'Two-dimensional automatic meshgeneration for structural analysis', International Journal for Numerical Methods in Engin-eering, 1970, 2, pp. 133-144

25 PINCHUK, A. R.: 'Automatic adaptive finite element mesh generation and error estima-tion'. M.Eng. Thesis, McGill University, Montreal, Canada, 1985

26 PINCHUK, A. R., and SILVESTER, P. P.: 'Error estimation for automatic adaptive finiteelement mesh generation', IEEE Transactions on Magnetics, 1985, MAG-21, (6), pp. 2251-2254

27 THATCHER, R. W.: 'Assessing the error in a finite element solution', IEEE Transactions onMicrowave Theory & Techniques, 1982, MTT-30, (6), pp. 911-915

28 PENMAN, J., and GRIEVE, M. D.: 'An approach to self adaptive mesh generation', IEEETransactions on Magnetics, 1985, MAG-21, (6), pp. 2567-2570

29 SYNGE, J. L.: 'The hypercircle in mathematical physics' (Cambridge University Press, 1957)

T. I. Boubez, Prof. D. A. Lowther, A. R. Pinchuk and Prof. P. P. Silvester are with theComputational Analysis & Design Laboratory of the Department of Electrical Engineering,McGill University, 3480 University Street, Montreal, Quebec, Canada H3A 2A7, and Dr. R.Funnell is with the Biomedical Engineering Unit, Mclntyre Medical Sciences Building, McGillUniversity, 3655 Drummond, Montreal, Quebec, Canada H3G 1Y6.

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