Metric control of spatial mappingsV.A.GaranzhaComputingCenter, RussianAcademyof Sciences40,Vavilov str., 119991Moscow, Russia,garan@ccas.ru

1. General requirementsfor variational principle

Variationalgrid generationmethods[1]-[5] have becomeimportanttools inmany real-life applications. Despiteconsiderableprogressin this area,lots ofunsolvedproblemsstill exist. Someof themarethesubjectof this paper.

Let us denoteby���

domainin coordinates���������� � � �������� . The spatialmappingof interest����������� � � � � is constructedasa minimizer of the func-tionaldependingon thegradientof themapping:!"$#&% �('�)*���,+-�.� % �/� ��01� � ��� (1)

where ' ) � denotesmatrix with entries 2�3542 )76 .Comprehensivereview of mathematicalformulationof minimizationproblemsfor suchclassof functionals,including analysisof conditionsof wellposedness,regularity of solutions,restrictionson domainsandboundaryconditionscanbefoundin [15].

The variationalprinciple usedfor grid generationand geometricmodellingobviously differs from thosein mechanicsandotherareasandshouldsatisfythefollowing basicrequirements:

1.1 Variationalproblemshouldbe well posed,its solution shouldexist andshouldbestablewith respectto input data.Sotheminimal requirementis theel-lipticity condition.However it is well known thatfor highly nonlinearfunctionalsof interesthere,ellipticity conditionsdo not guaranteewell posednessof varia-tionalproblem.

1.2 Variationalprinciple shouldnot admit singularmappingsasminimizers,hencetheclassof admissiblemappingsconsistsof locally invertiblequasi-isometricmappings[7]. Thequasi-isometrymeansthatfor any two sufficiently closepointsin � coordinates,say 8 and 9 , thefollowing inequalityholds::;=< 9?>@8 <AB< ���C9D�E>@�.�F8G� <1A : ; < 9H>@8 < � (2)

where;JILK

is theconstant,<*M/< is theEuclideanlengthand:

is thelengthscale.1.2Solutionsof variationalproblemshouldbeassmoothaspossible.

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1.4Ability to constructquasi-uniformmappingsis a key propertyin grid gen-eration.Invariantdefinitionof a quasi-uniformmappingreadsasfollows:NOQPSRUT1VXWCY[Z�\^]GR N O P`_ (3)

where a$b aresingularvaluesof ' ) � and; IcK is a constant. In fact this is

equivalent definition of quasi-isometricmapping,provided that inequalitiesarevalid almosteverywhere.From(2) it follows that �.�C��� is locally Lipschitz,so itsgradientbelongsto dDe andinequality(3) holds.On theotherhand,any functionwith gradientbelongingto dDe is locally Lipschitz. In what follows we will use(3) asthedefinitionof quasi-isometricmapping.

1.5 Deviation of optimalsolutionsfrom the targetonesshouldbeboundedinmaximumnorm,which alsonaturallyleadsto quasi-isometryconcept.

1.6Solutionsof variationalproblemshouldbelocally invertibleandwith properboundaryconditionsshouldbe globally one-to-one.From the practicalpoint ofview thismeansthatrelatively easilycheckedand/orguaranteedalgebraicproper-ties of solutionsshouldleadto requiredglobal topologicalproperties.Importantexamplesof suchrelationsaredueto Reshetnyak [8] andBall [11]. In [8] it wasshown that a mapping ������� from a certainSobolev classsatisfyingboundeddis-tortion inequalityf1g�h '�)�� ILikjmln

h�o �('�)*� T '�)*���,prq st� KvuLi A l (4)

almosteverywhereis openand discrete. Namely imageof any openset underthis mappingis openset,andany point canhaveonly finite numberof preimages.In [11] it wasprovedthata mapping�.�C��� from a certainSobolev classsuchthat

f1g�h '�)*� ILKalmosteverywhereandcertainfunctionalsimilar to (1) is bounded,then ���C�1� isglobally one-to-one.Soif it is possibleto prove thatsolutionof variationalprob-lem doessatisfysuitablealgebraicconstraints,theglobalinvertibility conditionisguaranteed.Quasi-isometricmappingssatisfyboth Reshetnyak andBall condi-tions.

1.7Thereareseveralwell establishedapproachesto constructionof mappings.Harmonicmappingsapproachis relatively simpleandbasedon convex function-als [16]. In many practically important2-D casesit guaranteesconstructionofone-to-onemappings.However in the presenceof nonsmoothboundariesthedi-latationof harmonicmapping(i.e. theratioof maximalto minimalsingularvaluesof Jacobimatrix) nearboundariescanbe unbounded.This phenomenonis very

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well known in mechanicsasstressconcentration.In the caseof harmonicmapwith nonsmoothmetricstheunboundeddilatationcanbepresentfar from bound-aries.Whensingularitiesarepresenttheharmonicmapsarenotstablewith respectto input data.

Theconformalmappingstechniqueprovidescompletesolutionto theproblemin thesimplecases.In [7] it wasshown how to constructmappingof curvilinearquadrangleonto squarewhich is both conformalandquasi-isometric.Howeverin this approachtotal metricdistortion,i.e. constant

; is far from minimal, thisapproachcannotbe appliedfor domainswith nonsmoothboundariesandcannotbegeneralizedto 3-D case.

2. How to obtain well posedvariational problem

The basicprinciplesof well-posedvariationalproblemsfor constructionoflocally invertiblemappingsareformulatedin the context of mathematicaltheoryof elasticitywith finite deformations.In facttheideaof usingvariationalprinciplesfrom mechanicsin grid generationwasexploitedquiteextensively [3]. Howevervariationalprinciplesin mechanicsgenerallyviolate the above requirementsandshouldnot beappliedasit is. But mathematicaltoolsdevelopedin mechanicsarequite universalandshouldbe applied. Let us briefly review basicmathematicalprinciplesfor well posedvariationproblem(1) formulatedby Ball [9]:

1. Function% �F' ) ��� shouldbepolyconvex, i.e. it canbewritten asa convex

functionof minorsof ' ) � . In 2-D caseit meansthatthereexistsconvex functionw � M � M � , suchthat% �F' ) ���m� w � f1g�h ' ) ���5' ) �1� . In 3-D casethe existenceof con-

vex function w � M � M � M � is assumed,suchthat% �(' ) �1�m� w � f1g�h ' ) �.�7' ) �.��x f�y ' ) � ,

where x f�y�z � z�{ T

f1g�h.zdenotestheadjugatematrix.

2.% �F'�)*��� shouldbeboundedfrom below andshouldtendto |t} whenfea-

sible ���C�1� tendsto theboundaryof thefeasibleset,for examplewhen

f1g�h '�)*� �| K .3.% �F'�)���� shouldsatisfycertaingrowth conditions[9].

4. Thesetof addmissiblemappingsshouldbedefinedby polyconvex inequal-ity [13]. An exampleof suchinequalityis (4), which definesa convex setin then�~ | l -dimensionalspaceof matrix ' ) � entriesplus its determinant.Anotherexampleis inequality

f1g�h ' ) � I�K whichdefinesin thesamen�~ | l -dimensionalspacethehalf-spacewhich is convex aswell.

Theaboveconditionsallow to proveexistencetheoremfor functionalof inter-est. The mostfundamentalpropertyis thepolyconvexity. Any polyconvex func-tional is alsorankoneconvex, namely.% �(���&�| � l >������ ~ � A � % �(�&���|�� l >���� % �(� ~ ��� where

o x����.�F�&�>=� ~ �-� l When

% � M � is smoothenoughthe rank oneconvexity is equivalentto Legendre-82

Hadamard(ellipticity) condition. Functionalsviolating the polyconvexity condi-tion shouldberejected.

3 Variational principle for quasi-isometricmappings

In [6] it wassuggestedaquasi-isometricmappingdefinitionwhich is polycon-vex in theabovesense

f�g�h ' ) � I����.� � f1g�h ' ) ���5' ) �1��� Kvu�� A l � (5)�.� �,��������� �$� lnh�o ��� T ����� q s | � l >@�/�� �C�m| � ~� �`� Kvu � u l �

and � I�K is aconstantwhichdefinesthetargetaveragevalueof

f1g�h ' ) � . As wasshown in [6], (5) allowsdirectevaluationof constant

; from (3), namelyOQP-� W��`�`�m� � ��E��� ]1�q W�� P �m� � � P ��� ] q�� �q _ � PG  �-� W �-�H¡ ]F¢¡ ¢ _ �`�   ���H¡ ¢W ���H¡ ]F¢ _ N  �£ �q�¤We seethatvalue l*¥ � hasthemeaningof totalmetricdistortion.

Now the variationalprinciple for constructionof quasi-isometricmappingslooksasfollows! " #&% �('�)*���,+-�.� % �¦� l > � � � � � f1g�h '§)��.�5'§)����f�g�h ' ) �§> ���.� � f1g�h ' ) �.�7' ) ��� (6)

Thisvariationalprinciplesatisfiesall requirementsformulatedin previoussection.It is well posed[13]. If feasibleset(5) is not emptythenminimizerexistsandis aquasi-isometricmapping[13]. Dueto similarity with elasticityproblemsit is nat-ural to expectthatglobaluniquenessof solutionandconvergenceof finite elementapproximationsis provablefor problemswith very smoothboundaryconditionsandsolutioncloseto identity mapping.However numericalexperimentssuggestthat convergenceof finite elementapproximationsis observed in quite generalcasesso further theoreticalanalysisis necessary. Unlike elasticityproblems,noLavrentiev phenomenonis possiblefor (6) sinceminimally regular solutionsarestill locally Lipschitz[13]. TheLavrientev phenomenongenerallymeansdifferentminimizersin differentfunctionalspaces.The unsolvedtheoreticalproblemsin-cludethestability conditionsfor minimizers,theproof (if any) thattheminimizerlies strictly insidethe feasibleset,andsomewhat relatedproblemof existenceofweakvariationalformulationof Euler-Lagrangeequationsof thefunctional.Theseproblemsarenot fully solvedin elasticityaswell [10].

4. Control of the mappingspropertiesvia compositionof mappings

Up to this point we have consideredthevariationalprinciple for constructionof quasi-uniformmappings.However in practiceit is necessaryto constructmap-pingsandgrids with controlledvariationsof local shape,sizeandorientationof

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elements.We do not considerhereorientation/alignmentcontrol which is verycomplicatedandessentiallyunsolvedproblem.More generalfunctionalallowingfor constructionof quasi-isometricmappingswith orientationcontrol was sug-gestedin [18]. Hereonly shapeandsizecontrol is considered.Generalideais tousecompositionof mappingsin orderto modify thepropertiesof solutions.It isillustratedon Fig.1.

¨ ©� �

© � ¨ �¨ �C��� { �ª��� ¨ � ��� © ����C�1�Q�L��� © � ¨ ���������

� ¨ ��«¬ � ¨ ��� � © �7­®� © ���«¬ � ¨ ���¯'±°�� T '§°�� ­®� © �Q��'�²�� T '�²��Fig. 1. Compositionof mappings. ��� © � and ��� ¨ � areprescribedmappings,

{ meansinversemapping.

Themapping�.�C��� is representedasacompositionof mappings̈ �C�1� , © � ¨ � and��� © � . It is assumedthat¨

and © arecoordinatesin n -dimensionalspace,whilemoregeneralcaseof � coordinatesin ³ -dimensionalspace,³µ´ n is consideredaswell. In particular��� © � maydefineaparameterizationof surface.Themappings��� © � and��� ¨ � arespecifiedwhile thefunctionis © � ¨ � is thenew unknownsolution.We assumeagainthat ��� © � and ��� ¨ � arequasi-isometricmappings,but possiblywith largeconstants

; . Usingthenotations¬ �¯'±°������=��'§° © � z �¯' ² �.�&� ��'�)��.�Q�?� f�g�h �¶�we get � � z � ¬ { ���?�

f1g�h.z�f1g�h �f1g�h ¬ �E+-�®� f�g�h ¬ + ¨ �andfunctional(6) is simply rewrittenas!" % � z ' ° © ¬ { � f�g�h ¬ + ¨ � (7)

where ��� � ��� ��� . Notethatfunction%

dependsonly on orthogonalinvariantsofmatrix � T � , andusingequalities� T � � ¬ { T � T ­S� ¬ { �E­®� © �Q� z T

z ��«¬ � ¨ �Q� ¬ T¬ �

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andthe fact that

h�o �C·¶¸��±� h7o �F¸±·[� we get that%

canbe written via orthogonalinvariantsof thematrix [6] � T ­S��«¬ { �andfrom the formal point of view we have a familiar formulationof a mappingbetweentwo Riemannianmanifolds � ¨ ��«¬ � ¨ ��� , � © �7­®� © ��� [16], whichis illustratedonFig.1(right).This formulationis quitegeneralsincematrices

zand¬

now canbenon-squareonesandmetrictensors «¬ � ¨ ����­®� © � arenot assumedto besmooth.Theonly restrictionis thattheir eigenvaluesarepositive andshouldhave uniformlower andupperbounds. We will show later that even when input control dataarejust two metric tensors,discreteapproximationto functionalon eachsimplexelementnaturallyemploys factorizedrepresentation(7).

5. Fundamental discretization problems

Unlikeother“standard”problems,sayrelatedto finite elementsolutionof lin-earelliptic problems,thediscretizationproblemsin themappingstheoryarefun-damentalandmostlyunsolved.

5.1 Surprisinglythe problemof approximationof locally invertible mappingby a converging sequenceof locally invertible piecewise affine mappingsis notsolvedevenfor quasi-isometricmappings,notsayingaboutgeneralSobolev map-pings[10].

5.2 Compositionof mappingsis not closedwith respectto Sobolev norms.Hencewe do not know generalsolutionfor very simpleproblem:how to approx-imateby locally invertible piecewise affine mappinga compositionof a generalquasi-isometricmappinganda locally invertiblepiecewiseaffinemapping(whichis quasi-isometricby default).

5.3More difficult problem:givenamappingwith metricdistortionbelow cer-tain thresholdl*¥ � (5). Is it possibleto constructa sequenceof converging piece-wiseaffine mappings,eachbeingbelow thesamethresholdl�¥ � ? If it is possible,how to do it in practice?

5.4 Variationalprinciple for surfacegrid generationshouldbe invariantwithrespectto surfaceparameterization,including the caseof nonsmoothparameter-izations. How this propertycanbe implementedon the discretelevel? What isthecounterpartof thefinite elementpatchtestcondition[14] in thecaseof spatialmappings?

Ourhypothesisis thatthesolutionsto abovediscretizationproblemsshouldbebasedon Alexandrov theoryof metricswith boundedcurvature(seecomprehen-sivereview of Alexandrov theoryin [19]. Roughlyspeaking,everymetricallycon-nectedspacewith boundedcurvatureis thelimit of convergingsequenceof spaceswith polyhedralmetrics,eachhaving boundedcurvature. The geodesictriangles

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in Alexandrov spacesarewell definedsowe canusethis conceptfor constructionof discreteapproximationto functional(7).

6. Geodesictriangles and patch test for grid functionals

Let usconsidernow thecaseof surfacesin 3-D, so n � � �&³c�ª¹ . Werequirethatdiscreteapproximationto functionalshouldsatisfysimplecompatibilitycon-dition. Let us considera mappingbetweentwo developablesurface. Thenwithproperboundaryconditionsfunctional(7) shouldattainabsoluteminimumon theisometricmapping.We requirethatdiscretefunctionalattainsabsoluteminimumon thesameisometricmapping.

Sincewe considerheretheintrinsic geometryproblem,theparticularshapeofsurfaceis not relevant.

A

B

C

A

B

C

A

B

C

Fig. 2. Undistinguishablegeodesictrianglesondevelopablesurfaces.In particulardifferentdevelopablesurfacesareundistinguishablewhich is il-

lustratedon theFig. 2.In orderto build discreteapproximationto (7) fully employing all introduced

controlsweassumethatcertaintriangulationis availablein coordinates̈ andcon-structa compositionof mappings,similar to thosein (7).

¨ ©� �

© � ¨ �¨ �C�1� { �L��� ¨ � ��� © ����C�1�

�

z � ¬ { z¬Fig. 3. Compositionof mappingsandtheir approximationby compositionofaffinemappings.Matrix

z � ¬ { is supposedto approximategradient' ) � . ºsymbolsmarktheunknownsof thevariationalproblem.

Our unknowns are imagesof the nodesof this triangulationin coordinates86

© . For eachtriangle mapping ������� is alreadyassumedto be affine. Howevermappings��� © � and ��� ¨ � arenotaffineonesandshouldbesomehow approximatedon eachtriangle. The ideaof approximationis very simpleand is attributed toAlexandrov aswell:

A

B

y(A)

y(B)

y(C)

C

flatteningpreserving edge length

Q

geodesic triangle

Fig. 4. Constructionof affineapproximationto ' ² ��� © �� Let ustakethreepoints ·±��¸»� ; in © -coordinatesandcomputetheir imagesun-

dermapping��� © � . Connectingthesethreepointsvia geodesicsweobtaingeodesictriangleon thesurface ��� © � . And finally replacingthis triangleby a flat onewithedgelengthscoincidingwith thoseof geodesictriangle,we obtainlocalaffine ap-proximationto ��� © � andnaturalformulafor matrix

zwhichcanbeeasilyderived

and is omittedhere. This operationis alwayswell definedsincedistancealonggeodesicssatisfiestrianglerule. Note that thespatialorientationof lastflat trian-gle is irrelevantsincefunctionaldependsonly on matrix

zT

z.

Supposethat we have a certaindevelopablesurfacein 3-D space� and twodifferentflatteningof this surface.Thefirst oneis isometric(seeFig. 5 (left)) andthesecondoneis distortedandis quasi-isometric(seeFig. 5 (right)).

1

2

3

4

1

2

3

4

12

3

4

Fig. 5. Patch test: different flatteningsand parameterizationsof the devel-opablesurfaceandthesameoptimalmeshon surface.

Sincein thefirst case­¼�¾½ , discretefunctionalwill attainits absolutemin-

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imum providedthat targetshapesfor eachplanetrianglearecorrectlychosen.Itcanbeeasilyshown that in thesecondcasethesolutionvia compositionof map-pingscanbereducedto thefirst oneandfunctionalalsoattainsabsoluteminimum,soindependentlyof surfaceparameterization,optimalgeodesicgrid onthesurfaceis thesame(of courseprovidedthatall geodesicsareunique,which is not alwaysthecase).

7. Global parameterization of surfacesvia unfolding and flattening

In exampleshown on Fig. 5 thesurfaceparameterizationwasconstructedviaflatteningoperation.This is very powerful tool in geometricmodelling[12]. Inparticularit canprovide global parameterizationfor surfacesdefinedby mutiplepatches(in reallife upto 4-5 thousands).Any triangulatedsurfacehomeomorphicto diskwith holescanbeflattened[12]. Moregeneralstatementcanbeformulatedasa hypothesis:every manifold with boundedcurvature homeomorphicto diskwith holesadmitsquasi-isometricflattening.

In orderto build surfacemeshsingleglobalparameterizationis necessary. Twobasicapproachesarepossiblehere:

A. Surfaceunfoldingbasedon constructionof geodesictriangulation,flatten-ing of eachtrianglevia edgestraightening,connectedunfoldingof trianglesontheplane,meshingof polygonwith consistentmeshesalongcutsandbackwardfold-ing andmappingonto thesurface.This is in fact famouscut andpasteoperationdueto Alexandrov.

B. Surface flattening basedon constructionof geodesictriangulation,flat-teningof eachtrianglevia edgestraightening,global quasi-isometricflattening,anisotropicmeshingof planedomains[17], meshimprovementusingvariationalmethoddescribedabove, mappingof planemeshback to the surface. Robustmethodfor quasi-isometricflatteningwassuggestedin [18].

flattening unfolding

gluing rule

Fig. 6. Flatteningvs. unfoldingfor surfaceparameterization.

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A: Advantages:minimalmetricdistortionin unfoldingoperation.Drawbacks:lotsof internalcuts,complicatedcouplingconditionsalongcuts,notpracticalfor largenumberof geodesictriangles.B: Advantages:nocuts,singleglobalplanemesh.Drawbacks:globalflatteningisnot simple,in somecaseslargemetricdistortionsareinevitable.

It seemsthat the optimal strategy is to usea finite numberof cuts andper-form globalquasi-isometricflatteningcombinedwith unfolding,to constructplanemeshandto do backwardfolding andmapping.Findingoptimalcutsis very hardandunsolvedproblem.

Both flatteningoperationandvariationalimprovementof planegrids canbedoneusingthecompositionof mappingframework. In flatteningoperationthetar-getshapefor eachplanetriangleis givenvia matrix

¬which in turn is computed

via flatteningof geodesictriangleof thesurface.In planemeshimprovementop-erationflatteninghavealreadydefinedglobalparameterizationof surface��� © � , soit is necessaryto recomputematrix

zfor eachplanetriangleduringoptimization

procedure.If additionaladaptationis necessary, sayadaptationto curvatureof sur-face,thenwe just havedifferentdefinitionof metrics ­®� © � , but thesamerulesforconstructionof matrix

z, sincelengthof geodesicsis definedonly by ­®� © � .

8. Numerical experiments

The practicalminimizationmethodfor suggestedfunctional is basedon theideaof frozenmetrics ­ . Similar ideawassuccessfullyusedin [?]. Obviouslymatrix

¬for eachtriangleis computedonly onceand“accompany” eachtriangle

during the minimizationprocess.We can interpretit asshape/sizedefinition in“Lagrangian”coordinates.On theotherhandmatrix

zchangeseachtimetriangle

verticesareupdated,it behavesmorelikeshape/sizedefinitionin “Eulerian” coor-dinates.So the ideaof iterative solutionis to solve partialminimizationproblemfor functionalwherefor eachtriangle

% � % � zS¿À{ � ¿ ¬ { � , Á beingtheiterationnumber. Generallythis partial minimizationprobleminvolvesjust 1-2 iterationsof preconditionedgradientmethod[6]. This approachwasfoundvery stableandconvergedquitefast.

In practiceonecanhavesurfacedescriptiondifferentfrom geodesictriangula-tion, saytesselationwhich approximatesgeometrywith prescribedchordalerror.Thenthebasicproblemis to computequasi-isometricflatteningof extremelyill-conditionedtriangulation[18].

An exampleof meshingprocedurebasedon global flattening is shown onFig. 7. Note thedifferencebetweenmesheson subfiguresA andB. In bothcasesmeshquality is quitegood,but in thecaseB discretefunctionalviolatedthepatchtestandtheproblemswith meshconvergenceareexpected.

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¸

·

Fig. 7. Surfacemeshingvia quasi-isometricflattening,planegrid generationandvariationalimprovement.A - meshconstructedusingdiscretefunctionalsati-fying patchtest;B - patchtestis violated.

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Fig. 8. Patchstitchingwith repairof topologyandgeometryof thesurface.

Fig. 9. Fragmentsof repairedsurface tesselations. Left - separatetesselatedpatches,right - repairedtriangulations.

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Fig. 10. Optimizedsurfacemesh.

Fig. 11. Surfacemeshfragments:left - no patchtest,right - patchtestis satisfied.

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Numericalexperiencesuggeststhat not badquality facets,but surfacewrin-kles, especiallyneedle-like arefatal for flatteningquality. In this caseGaussiancurvatureof surfaceis very largeandflatteningwith low metricdistortionsimplydoesnot exist. Soa key operationin repairof geometryis smoothingof parasiticwrinkles which by itself is quite nontrivial. The wrinkles canbe createdduringassemblyof surfacefrom CAD patches.An exampleof fault tolerantmeshingprocedureis shown on Figures8, 9, 10,11. In thiscasetheflatteningprocedureisverygood,but still mapping�.� © � is nonsmoothdueto slightly differentflatteningof very large facets. In the optimizedmeshshown on Fig.10 the quasi-isometryconstant

; is about l  which seemsto bequitecloseto unimprovableresult.

9. Conclusions

Unsolvedproblemsin grid generationwereconsideredandsomesolutionsandhypothesesweresuggested.Themainconclusionis thatgrid generationfield alongwith othermethodsshoulduseideasandmethodsfrom theoryof mappingsandtheoryof manifoldsof boundedcurvature.

References

[1] Godunov S.K., Prokopov G.P. Calculationof conformalmappingsandgridgeneration.Zh.Vych. Mat. Mat. Fyz.1967,7(6):902–914.

[2] Brackbill J.U.,SaltzmanJ.S.,Adaptivezoningfor singularproblemsin twodimensions.J. Comput.Phys.1982,46(3):342–368.

[3] JacquotteO.P. A mechanicalmodel for a new grid generationmethodincomputationalfluid dynamics.Comp.Meth.in Appl. Mech. andEng. 1988,66:323–338.

[4] Liseikin V.D. Constructionof regulargridson n -dimensionalmanifolds.Zh.Vych. Mat. Mat. Fyz.1991,31(11):1670–1683.

[5] K. Hormannand G. Greiner. MIPS: an efficient global parameterizationmethod.In: Laurent,Sablonniere,Schumacher(Eds).Curve andsurfacede-sign.VanderbiltUniv. Press,Nashville,2000,pp.163-172.

[6] V.A. Garanzha. Barrier variational generationof quasi-isometricgrids.//Comp.Math.andMath.Phys.V.40,N.11,2000,PP.1617-1637.

[7] S.K. Godunov, V.M. Gordienko and G.A. Chumakov. Quasi-isometricparametrizationof a curvilinearquadrangleanda metricof constantcurva-ture.//SiberianAdvancesin Mathematics.1995,V.5, N.2, P.1-20.

93

[8] Reshetnyak Y.G. Mappings with boundeddeformationas extremalsofDirichlet typeintegrals- SiberianMath.J.,9, 1968,487-498.

[9] Ball J.M.,Convexity conditionsandexistencetheoremsin nonlinearelastic-ity, Arch. RationalMech.Anal. 63,1977,337-403.

[10] J.M. Ball. Singularitiesandcomputationof minimizersfor variationalprob-lems,1999.Preprintof Math. Instituteof Univ. of Oxford.

[11] J.M. Ball. Global invertibility of Sobolev functionsandthe interpenetrationof matter, Proc.Royal Soc.of Edinburgh,88A, 315-328,1981.

[12] M.S. Floater. Parametrizationandsmoothapproximationof surfacetriangu-lations.Comp.AidedGeom.Design1997,14:231–250.

[13] V.A. Garanzha,N.L. Zamarashkin.Quasi-isometricmappingsasminimizersof polyconvex functionals.In: “Grid generation:theoryandapplications”,eds.S.A. Ivanenko, V.A. Garanzha,2002,pp.150-168(in Russian).

[14] G.Strang,J.Fix.Analysisof finiteelementmethod.Prentice-Hall,EnglewoodCliffs, NJ,1973.

[15] P.G. Ciarlet, Mathematicalelasticity, V.1: Three DimensionalElasticity,Stud.Math.Appl. 20,Elsevier, New York.

[16] EellsJ.E.andL.LemairL. Anotherreportonharmonicmaps.Bulletinof theLondonMathematicalSociety. 1988,20(86):387-524.

[17] GeorgeP.L. andBorouchakiH. Delaunaytriangulationandmeshing.Appli-cationto finite elements.Paris.Hermes.1998.413p.

[18] V.A. Garanzha.Max-normoptimizationof spatialmappingswith applicationto grid generation,constructionof surfacesandshapedesign. “Grid gener-ation: new trendsandapplicationsin real-world simulations”,eds.S.A. Iva-nenko, V.A. Garanzha,Comm.on Appl. Math., ComputerCentre,RussianAcademyof Sciences,Moscow, 2001

[19] Reshetnyak, Yu: Two-DimensionalManifolds of BoundedCurvature.In:ReshetnyakYu.(ed.),GeometryIV (Non-regularRiemannianGeometry),pp.3-165.Springer-Verlag,Berlin (1991).

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