Efficient Adaptive Meshing of Parametric Models

Alla Sheffer

ComputationalScienceandEngineeringUniversityof Illinois atUrbana-Champaign

Urbana,IL, 61801e-mail: sheffa@uiuc.edu

Alper Ungor

Departmentof ComputerScienceUniversityof Illinois atUrbana-Champaign

Urbana,IL, 61801e-mail: ungor@cs.uiuc.edu

Abstract

Parametricmodelingis becomingthe representationof choiceformostmodernsolidmodelers.However, whengeneratingthefinite-elementmeshof themodelfor simulationandanalysis,mostmesh-ing toolsignoretheparametricinformationanduseonly thebound-ary representationof the model for meshing. This resultsin re-meshingthe modelbasicallyfrom scratcheachtime a parametricchangeis instantiated,which happensnumeroustimesthroughoutthedesignprocess.

In this paper we look at ways to use the parametricinfor-mation during the meshingprocedureto prevent unnecessaryre-meshing.Thepaperexaminesexisting meshingtechniquesdevel-opedfor otherpurposes,which canbe appliedto this problem. Italso suggestsseveral new meshmodification techniquesspecifi-cally designedfor efficientmeshadjustmentafterparametricmodelchanges.

Keywords: Meshgeneration,Parametricmodels,Adaptivity

1 Intr oduction

Engineeringdesignis an iterative processencompassingrequire-ment definition, conceptdesign,designvalidation and analysis.Parametricand variationalmodels[23, 30] provide the tools foreasymodificationof the modelparameters,basedon analysisre-sultsor designdecisions.Recentextensive researchmakes thosemodelssignificantlymorerobustandreliable[28, 8]. A testamentto the popularityof this representationis the growing numberofcommercialsolid modelingandCAD systemsusingthis represen-tation.TheseincludePro-E[27], IDEAS [35], UG/Solid-Modeling[43], to namejusta few.

With further integrationof theproductdatamanagementappli-cations,solid modelsgeneratedby the designareuseddirectly inthefinite elementanalysis(FEA) andsimulationapplications.Theuseof CAD geometryfor FEA had increasedfrom 11% in 1991to nearly80% in 1996[15]. This encouragesparametricstudies,which addressthe impact of designvariableson productbehav-ior. The integration of CAD and FEA dependson the ability toefficiently generatehigh quality finite-elementmeshesof theCADmodels.Therehave beensignificantadvancesin thedevelopmentof meshgenerationalgorithms[26], however, theprocessis still farfrom full automation[15, 36, 4].

Thetwo mainareasrequiringmanualinterventionaregeometryadjustmentprior to meshingandhexahedralmeshgeneration[1].Adjustmentof modelgeometryprior to meshingis often requiredto improve the resultingmeshquality. Additionally, geometryad-justmentis a prerequisitefor many of theavailablemeshingalgo-rithms. This makesthemeshgenerationa majorbottleneckin theengineeringdesignprocess.

Whengeneratingthefinite-elementmeshof themodelfor anal-ysisandsimulationpurposes,mostmeshgenerationalgorithmsuseonly the boundaryrepresentationof the model. Hence,paramet-ric changesarenot directly considered.As a result,mostparame-terchangesrequiresre-meshingthemodelpracticallyfrom scratch.Thishappensevenif mostof theboundarymodelremainsthesame,sincechanginga boundaryentity causesmostsystemsto removethemeshon thatentity andall higherdimensionalentitiesthatrelyon it. This leadsto a significantslow down in the designprocessasmultiple repetitive parametricchangesareoftenperformeduntilthedesiredmodelis generated.

In additionto efficiency concerns,preservingthe links betweenoriginalandnew mesheswhenparametricchangesoccuris requiredwhenthechangesarepartof anautomaticshapeoptimizationpro-cedure.Shapeoptimizationapplicationsrepeatedlymodify modelparametersbasedonsimulationresults.Sincethesimulationresultsareoftenaffectedby mesh-basednoise,minimizing thechangesofthemeshin betweentheoptimizationiterationsimprovestheopti-mizer’sestimates.

In this paperwe examinethe tools that can be usedto avoidfull re-meshingdue to parameterchanges. The suggestedtoolsusetheinformationprovidedby theparametricrepresentationbothwhengeneratingthe modelmeshandwhenadjustingit followingthe changein parameters.Both hexahedralandtetrahedralfinite-elementmeshesareconsidered.

This work examinesthe possibility of using existing meshingtoolsdevelopedfor otherpurposesto beusedto facilitateefficientmeshadjustment. The researchintroducesseveral new tools formeshadjustment,which are particularly well suitedfor efficientmeshmodificationin thecontext of parametricmodelchanges.

Theapproachpresentedhereis basedon usinga dual represen-tationof themodel,containingbothaboundaryrepresentationanda parametricone,suchas the onesuggestedby RaghothamaandShapiro,[28]. This way the meshgenerationandmodificational-gorithmscan continueto usethe boundaryrepresentation,whiletheparametricmodelwill provide theinformationrequiredduringthemeshadjustmentprocedure.Theparametricrepresentationwillprovidethemappingbetweentheoriginalandnew modelentitiesaswell asthesize/shapechangesfor eachof theentities.Notethatinthecontext of thispaperparametricchangesareviewedaschangeswhich do not affect themodelconnectivity. Changesin connectiv-ity tendto requiremajorchangesin themodelmeshfor theanalysispurposes,andoften requirehumaninteractionto decideon thosechanges.

Therestof thepaperis organizedasfollows. Section2 describesthe v� arious issueswhich needto be addressedwhen adjustingamodelmeshafter parameterchanges.Section3 addressesmodelgeometryadjustmentfor meshing.In Section4, themeshcopyingfrom theoriginal to themodifiedmodelis described.In Section5,we describetools for meshimprovementfor both tetrahedralandhexahedralmeshes.Section6 providestwo examplesof generatinga meshfor a complex model after a parameterchange. Finally,Section7 summarizesthepaperanddiscussestheeffectivenessofusingtheparametricinformation.

2 Parameter Modification - Meshing Is-sues

In thissectionweconsidertheissuesinvolvedin generatingameshfor a new model,basedon anexisting meshof anoriginal model,wherethenew modelwascreatedby changingsomeof theparam-etersizesin the original one. The problemsandthe solutionsareaddressedin detailsin thefollowing sections.Theprocessof gen-eratinga meshfor a new model after parametricchangesto theoriginalone,involvesthefollowing steps.

Thefirst stepis adjustingthenew CAD modelfor meshing.Asmentionedabove to makeamodelsuitablefor meshingit oftenhasto bemodified.To distinguishbetweenthemodelgeneratedby theCAD systemandthemodifiedmodelusedfor meshingwe refertotheformerasthedesignmodelandto thelaterasthemeshmodel.The meshmodelmaydiffer significantlyfrom the original designone. Therequiredmodificationsincludesuppressionof minor de-tails, redefinitionof ill conditionedCAD geometry, suchassurfaceslivers,andgeometrydecomposition.If the modificationsarenotpreserved whenthe modelparameterschange,the editingprocesswill needto berepeated,requiringa lengthyandonly partly auto-matedprocedure.Thepreservationof themeshmodelis addressedin Section3.

After a meshmodelis definedthenext stepprior to meshingitis theassignmentof meshingparameters.Most meshingsoftwaresrequiresettingof suchparametersasmeshsizingandtypeof meshgenerationalgorithmto beusedfor eachmodelboundaryentity, in-cludingthemodelasawhole.Settingthoseparametersis necessaryto generatethemeshof thenew model. If thenew meshmodelisidenticalto the original one,thoseparameterscanbecopiedfromeachoriginalmodelentity to its counterpartin thenew model.

Whenthe meshmodels’connectivity is identicalanda one-to-onemappingof themodelentitiesexists,thenew meshcanbegen-erated,by finding the locationsfor thenew nodesandcopying theoriginal connectivity. This stepin thegenerationof a meshfor thenew modelis explainedin Section4.

The final stepin the meshingprocessis meshconnectivity ad-justment. After the meshis copiedto the new modelit might nolongerhave anacceptablequality or observe thedesiredmeshsiz-ing. In this casethemeshconnectivity needsto bemodified. Therequiredamountof work dependson the type andsizeof the pa-rameterchanges.Theprocessis describedin Section5. This stepmightnotbenecessaryif themodelchangesareminorandthemeshquality if sufficiently high after the meshreassignmentstep. Ontheotherside,if theparameterchangesaretoo big, a completere-meshingmightbenecessary.

Figure2 shows the stepsfor generatingthe meshgiven a newmodel,usingthemeshof theoriginalmodel(Figure1). Thetools,whichcanbeusedin eachstep,arediscussedbelow.

3 Preser ving the Mesh Model

As explainedabove,whenpreparingCAD modelsfor meshing,themodelsoftenneedto bemodifiedto suit themeshingneeds.Such

(a) (b) (c)

Figure1: Meshingan“L” shapedmodel.(a)Originaldesignmodel.(b) Meshmodel(aftersimplification).(c) Hexahedralmesh.

(a) (b)

(c) (d)

Figure2: Adjustingthemeshof “L” shapein Figure1 to differentparametersizes.(a)Designmodel.(b)MeshModel(sametopologyas original meshmodel). (c) ReassignedMesh. (d) Mesh afterconnectivity adjustment(whisker-sheetsplitting,Section5.2.2).

adjustmentusuallyincludes[15, 37]:

� Suppressionof design details that are too small to affecttheanalysisresultsbut that imposesevereconstraintson themeshingprocedure. Sincemeshgenerationalgorithmsareguidedby theB-rep,themeshconnectivity hasto follow thefaceandedgeboundariesin themodel.Henceif themodello-calfeaturesizeis smallcomparedto thedesiredmeshelementsize,it adverselyeffectsthemesh,resultingin reducedmeshquality andlarger numberof meshelementsthannecessary.An exampleof detailsuppressionis shown in Figure1, whereafillet narrower thanthedesiredmeshsizeis removed.

� Redefinition of ill conditionedCAD geometry like sliverfaces,narrow cornersand minute edges. Suchgeometrieshave andadverseeffect on meshessimilar to the minor de-tailsabove,andhaveno designor analysissignificance.

� Decompositionof complex modelsinto meshableparts. De-compositionis often requiredto enablethe useof availablemeshingalgorithmsandto improve themeshquality. It is es-peciallycommonwhenhexahedralmeshesaregenerated,ashexahedralmeshingalgorithmarelessgeneric.

Numeroustools[3, 37, 41] weredevelopedto performthosead-justments.Thetoolsoffer partialautomationof themodeleditingprocessbut someuserinteractionis still required.Examplesof typ-ical necessaryeditingareshown in Figure3.

Sincethosemodificationsareperformedfor meshingpurposesonly, thedesignmodelis unchanged.In thecurrentsolid-modeler

(a)

(b)

(c)

Figure3: A pumphousingmodel. (a) Designmodel. (b) Meshmodel(aftersimplification).(c) A tetrahedralmeshof themodel.

systemsthe meshand designmodelsare not connectedand theparametricchangesareappliedonly to the designmodel. Hence,the meshmodelneedsto be explicitly rebuilt after the parameterschange.Thereare treealternatives to avoid editing yet againthedesignmodelto fit themeshingneeds.

The first is to maintainthe connectionbetweenthe meshanddesignmodelswithin the system,in orderfor parametricchangesto beappliedto both. This is probablytheoptimalsolution,but totheauthorsknowledge,no existing commercialsoftwareembracesit.

Anotherapproachis to maintaina historylog of themodeledit-ing process,andwhenthe designmodel is modified,to rerunthesequenceof editingcommandsbasedonthelog. Thisapproachcanwork for many models,but it is likely to encounterentity namingproblems[6], sincea log re-runrelieson the entity namesto per-form the desiredoperationsequence.Namesmight often change,sinceparameterchangescancausedifferentorderingof operations,especiallyfor sizebasedprocedures,likedetailsuppression.

An alternative approachwe suggestis to performtheeditingus-ing avirtual topologymodel[37]. Thevirtual topologymodelrep-resentationseparatesthe model geometryandconnectivity infor-mationandallows topological(connectivity) entitiesto correspond

to a partor a setof geometricentities(e.g. a facecancorrespondto a part of a modelsurfaceor to a setof adjacentsurfaces).Thevirtual topologyeditingoperatorsconstructnew topologicalentitieson top of theexistingones,thuskeepingtheoriginal modelunder-neathintact. Thevirtual topologyoperatorswereusedfor editingthemodelsin all theexamplesshown in this paper. Two examplesof editing complex modelsareshown in Figure3 andFigure17.Note,that in all cases,themodelgeometryis unchangedandonlytheconnectivity is modified.

Using the virtual topologyediting tools for generatinga meshmodel from a designmodel [36, 41], constructsthe meshmodelon-topof thegeometric,designmodelwithout changingit. Hence,given a new designmodel,differing from the original only in pa-rametersizes,thevirtual topologymodelcanbesimply copiedontop of thenew designmodel.This would only requirecopying thetopologicalentityhierarchyfromtheoriginalmodelto thenew. Thecopy operationtogetherwith thevirtual topologymodelrepresenta-tion andothervirtual topologyoperatorswereimplementedaspartof theGambitanalysispre-processor[12]. An exampleof topologycopy is shown in Figure2(b), wherethe topologicalhierarchyasbuilt for the original designmodel(Figure1(b)) wascopiedontothenew model(Figure2(a)).

(a) (b) (c) (d)

Figure4: Featuresize impacton meshgeneration.A notchnar-rowerthanprescribedelementsize(a),haslittle impactonanalysis,andhencecanbesuppressedin themeshmodel(b). Whenthenotchwidth increasesabove themeshsize(c), it startsto have animpactonanalysisresults.Hence,themodelneedsameshreflectingit (d).

Prior to generatingthemeshmodelfor thenew designmodel,aparametercheckis requiredto seeif thecriteriausedfor generatingthemeshmodelstill apply. For exampleaminorfeaturesuppressedin theoriginalmodel,might in thenew modelgrow to beof impactfor theanalysis(Figure4). Thischeckcanbedoneusingthemodelparametersizesand the meshsizing. If someof the criteria nolongerapply, meshadjustment,asproposedhere,will not generatea meshsuitablefor analysis,asit will not follow importantmodelfeatures.In suchacaseto generateagoodqualitymesh,themodelneedsto bere-editedandre-meshed.

Oncethenew meshmodelis constructed,sincetheoriginal andnew meshmodelshave identical connectivity, the meshcan becopiedto thenew model.

4 Mesh Reassignment and Smoothing

4.1 Mesh Reassignment

Given two modelswith the sameconnectivity, the next stepis toassigna copy of theoriginalmeshto thenew model.This requiresto position the new nodesand thengeneratea meshconnectivitybetweenthemidenticalto the original one. To positionthe nodesthemodelboundaryrepresentationis used.Eachnodelocationisgeneratedbasedon themodelentity theoriginalnodewason. The

positioningprocessis donebasedon theentity hierarchy, first ver-tex nodes� areplaced,thanedgeonesandsoon.

� Thenew nodescorrespondingto originalmeshnodeslocatedat the model vertices,are placedat the correspondingnewmodelvertices.

� New edgenodesareplacedon the correspondingnew edge,at thesamelengthwiseparameterasweretheoriginal nodes.(Lengthwiseparameterizationis independentof theactualge-ometriccurve definition.)

� Facenodesareplacedon the appropriatenew face. Whenasurfaceparameterizationfor the faceexists, the ����� coordi-natesof the original nodecanbe usedto placethe nodeonthenew face.However, in somecasessuchparameterizationis not available,or is highly non-uniform. In this case,theplacementstrategy in Section4.1.1canbeused.

� Nodesontheinteriorof thesolidareplacedlast.Sincethereisrarelya volumeparameterization( ������� ) present,theplace-mentstrategy describedbelow hasto beused.

4.1.1 Placing interior mesh nodes for faces and solids

Below wedescribetwo approachesto placinginteriornodesonfaceor insidea solid, basedon theboundarynodes.Thefirst approachis to usea backgroundmesh[40] (Figure5). First, a backgroundmeshof theoriginal boundaryis generated,by constructinga De-launaytriangulationof theboundarynodes.Than,a similar back-groundmeshis assignedto the new boundary, by connectingthenew boundarynodesusingthesameconnectivity asgeneratedfortheoriginal ones.To placeaninterior node,its containingtrianglein the backgroundmeshof the original faceis found. Than, thebarycentriccoordinatesof thenodein thetriangleareusedto placeit on thecorrespondingtrianglein thenew backgroundmesh.

(a) (b) (c) (d)

Figure5: Copy of interior facenodesusinga backgroundmesh.(a) Meshof theoriginal face. (b) New, modifiedfacewith copiedboundarynodes.(c)Backgroundmeshof (a). (d)Backgroundmeshandcopiedinteriornodeon(b).

For a solid thesameprocedureis usedonly a tetrahedralback-groundmeshis generatedinstead. The generationof the back-groundmeshwhile relatively expensive canbe doneoncefor theoriginal modelandthen,it canbeusedrepeatedlyfor differentpa-rameterchanges.Theuseof thebackgroundmesh,providesagoodestimatefor thenew nodelocations,but doesnotguaranteeoptimalplacement.Often, meshsmoothingasdescribedbelow shouldbeperformedto improve thenodelocations.

Thesecondalternative is to keepthenodelocationsasthey werein theoriginalmodelandlet themeshsmoothingprocedure,whichfollows, to find theoptimalnodelocationsbasedon thelocationoftheboundarynodesandtheconnectivity [19]. Thisapproachavoidsthe complexity of generatingthe backgroundmesh,but might re-quiresignificantlymoreeffort at thesmoothingstage.It is likely tobemoreefficientwhenonly minorparameterchangesaredone,butwill requirestrongsmoothingtoolswhenchangesarelarge.

4.2 Smoothing

The meshcopy doesnot ensureoptimal nodelocations,with re-spectto the meshquality andfurther processingis often requiredto improve it. Node locationsdefinethe shapeof the meshele-ments,which in turn determinethemeshquality, i.e. its suitabilityfor analysis.Meshqualitymeasuresevaluatetheeffectof theshapeof the elementsin the meshon the correctnessof the analysisre-sults. Therehave beena large numberof measuressuggestedformeasuringthequalityof hexahedralelements[12, 5]. Somerecentresultsby Knupp [20], suggesta systematicapproachfor measur-ing qualityby useof algebraicmeshqualitymetrics.Thework pro-posesseveral metricslike shape,skew, volume,andcombinationsof those. Usually, to answerthe questionif the meshis suitablefor analysisseveralmeasureshave to bechecked,aseachmeasurecapturessomeaspectsof theelementshape,but nonecapturesall.The suitability for analysisdependson both the average(median,etc.) andthe worst valuesof the quality measures.For example,if a singleelementwith a negative or nearlyzeroJacobianexists,theentiremeshis invalid. Additionally meshsizinghasto becon-sidered,sothat themeshconformsto a prescribedsizingfunction,definedby theaccuracy requiredby theanalysis.

If after themeshcopy is performed,themeshquality is not ac-ceptable,oneoptionto improve it is to performasmoothingproce-dure. Smoothingis a generalterm for meshimprovementprocessin whichmeshnodesarere-locatedbut themeshconnectivity is un-changed.Thereis a largevolumeof work on differentsmoothingalgorithms,asreviewed in [26]. The methodscanbe roughly di-videdinto twogroups,globalandlocal. Globalalgorithmsarethoseusedto move every nodein themesh[18]. Suchmethodsusuallyperformrelatively simplecomputationsfor eachnodeandperformseveraliterationsof those.They areusuallyrelatively cheapto per-form but do not handlewell extremelydistortedmeshesandoftenachieve sub-optimalresults.Localmethods,like in [20, 5, 13] per-form morecomplex computationsper node,includingapplicationof optimizationtechniques.They achieve muchbetterresultsandcanhandlevery distortedmeshes,eventhosethatincludeelementswith negativeJacobian.Howeverthey aresignificantlymoreexpen-sive andhence,areusuallyusedonly in smallregionsof themesh,wherethequality is low.

Whentheparameterchangesaresmall(e.g. Figure6) themeshcopy togetherwith smoothingis usually sufficient to generateahigh quality meshof the new model. However, becausesmooth-ing doesn’t changethemeshtopology, its ability to improve meshqualityis limited (eg. Figure2). Moreover, whenmodelparameterschange,theelementsizingmight no longersatisfytheanalysisre-quirements.In suchcasesa modificationof themeshconnectivityis required,asdescribedin thenext section.

(a) (b)

Figure6: Adjusting the meshfor a simplebracket. The changedparametersinclude: holeradii, blendsradius,andbasewidth. Thechangesarerelatively small, hencemeshcopy with smoothingissufficient to geta goodquality mesh.(a) Original model. (b) Newmodel.

5 Mesh Connectivity Adjustment

Changingthemeshconnectivity is the lastoptionavailable,whenaftermeshcopy andsmoothingthemeshqualityremainsunaccept-able.Whenchangingmeshconnectivity therearesignificantdiffer-encesbetweentheoptionsavailablewhendealingwith hexahedralor tetrahedralmeshes. The possiblemodificationtechniquesforeachmeshtypearedescribedbelow.

5.1 Simplicial Meshes

This sectionsurveys existing triangular/tetrahedralmeshingmeth-odsto provide an efficient solutionfor adaptive meshingof para-metricmodels.Thesetechniqueswereintroducedto handlemeshadaptationwhenthe meshsizing functionchanges,herewe applythem to the problemof parametricchangesin the model. Morespecificallywe addressthe issueof handlingthe copiedmeshonthenew model,whichasresultof theparameterchangesno longersatisfiesthemeshquality requirementsor themeshsizing.

Sincemostparametricchangesapplyto only apartof themodel,a large part of the modelmeshmaintainsthe requiredsizing andquality. The simplicial adaptive techniques,describedbelow, canbeappliedsolely in theareaswherethemeshchanged,sinceall ofthemhaveonly localeffect.

5.1.1 Refinement Techniques

Recentunstructuredmeshgenerationliteraturewitnessedmany re-finementtechniques.Most of themaredesignedto generatequal-ity guaranteedtriangular/tetrahedralmeshes.Among them,Rup-pert’s Delaunayrefinementmethod[31] wasthe first to provide aquality guaranteefor optimalsizemeshesin two dimensions.Thismethodstartsby generatingtheconstrainedDelaunaytriangulationof theinput domain.Then,badquality trianglesareeliminatedbyinsertingtheir circumcenterswhile maintainingthe Delaunaytri-angulation. The work proves a

�� �� ���minimum angleguarantee.

Shewchukextendedthis approachto 3D, but with no guaranteeonavoiding slivers [38]. Several new algorithmswererecentlysug-gestedto remove slivers[7, 11].

The sink insertionmethod[10] alsousespoint insertionto im-prove the triangulationquality, however it behaves selectively indecidingwhichcircumcentersto insert.Alternativeto circumcenterinsertion,Rivara[29] suggestedto split thelongestedgesto gener-atequality triangulations.

We suggestusingthe refinementtechniquesfor adaptive mesh-ing of theparametricmodels.After theoriginal meshis copiedtothenew model,thequality of thenew meshcanbeunsatisfactory,i.e. if the original meshis a Delaunaytriangulationthe resultingmeshis no longer guaranteedto be Delaunay. The meshrefine-mentmethodsdiscussedcanbe usedto improve it. This processavoids completere-meshing.Henceit tendsto requirelessworkwhentheparameterchangesaresmall. If theparametricscalingofthemodelpartsis morethanone,this immediatelyimpliesacoarsetransformedmesh. Hence,thereis no unnecessaryrefinementinthefinal mesh.If thescalingis smallerthanonehowever, thefinalmeshcouldhave a largerthannecessarynumberof elements.Thisissueis discussedin Section5.1.3anda solutionis suggested.Wesuggestthefollowing four stepalgorithmasa formalizationof ourrefinementbasedadaptive parametricmodelmeshingmethod.

1. Copy themeshof theoldmodelto thenew modelasexplainedin Section4.

2. Performfaceswapsto improve thequality, e.g. to geta De-launaytriangulation.

3. Recomputethequalityof eachsimplex element

(a)

(b)

(c)

Figure7: Adaptive re-meshingof a 2D gradedmeshusingthe re-finementtechnique. (a) Original meshwith 169 nodes. (b) Themodel is stretchedby a factor of 2 along X direction. This cre-ates54 badelementsin themesh.(c) Final meshis a resultof 75circumcenterpointsinsertions.

4. While thereexistsa simplex T thathasa badradiusto short-estedgeratio, insertthecircumcenterof T andperformfaceswapsto maintaintheoptimality.

Thefaceswapoperationsuggestedin thesecondstepof theal-gorithm is discussedin computationalgeometryandmeshgener-ation literature[9]. It is usedboth for computingthe Delaunaytriangulationandfor improving meshquality. The last two stepsof this methodsuggestthe useof a classicalDelaunayrefinementtechnique. It could be replacedwith other refinementtechniquesdiscussedabove,e.g.longestedgerefinementmethod.Thelasttwostepscould be omittedif the Thereareseveral criteria to measurethe quality of a simplex, including aspectratio (inradiusover cir-cumradius),andminimumangle.Theabovealgorithmemploys thecircumradiusovershortestedgeandtheillustrationsin Figure7 and8 aregeneratedusingthis criteria. Thesethreequality criteriaareknown to beequivalentin 2D.However, circumradiusshortestedgecriteriadoesnotpreventtheexistenceof sliversin 3D.Thisrequiresthepost-processingof themeshto remove sliversusingoneof thetechniquessuggestedin [7, 11] .

5.1.2 Maintaining a Sphere Packing

Spherepackingbecameapopulartool in meshgenerationresearch[21,22, 24, 39]. It enablesgeneratingwell-spacedpointssetswhich

(a) (b) (c)

Figure8: Surfacemeshof a dodecahedron.(a) Original meshwith 290nodes.(b) Themodelis stretchedby a factorof 2 alongX direction(for illustrationpurposesthetop view is chosen).Thestretchingcreates98 badsimplices.(c) Inserting43 circumcenterpointsremovesallthebadsimplices.

in turn guaranteesquality meshes[24]. Thespherepackingmeth-odspackthedomainwith sphereswherethesizeof thespherecen-teredatapoint � is relatedto thedesiredmeshsizeat � . Thedesiredmeshsizeis determinedasa combinationof thegeometricproper-tiesof the domainandthenumericalpropertiesof the simulation.Geometriccomponentof thespacingis usuallyrepresentedby thelocal feature sizefunction introducedby Ruppert[31]. The localfeature sizeat a point � , ����������� is the radiusof the smallestdiskcenteredat � that intersectstwo non-incidentverticesor segmentsof thedomain.TheLipschitzpropertyof this function,which im-pliesaslow changein thefunction,liesbehindthetechnicalproofsof Ruppert’s qualityguaranteedtriangulationresults[31]. Later, Liet al. [22] definedananalogousfunctionwhich alsohasLipschitzpropertyto denotethenumericalspacing:

� ���������! #"%$ ��&�� ���(')�����*�+"%$ �-,�&.� ����'/��01��24353 �7680-353 9/9 (1)

where � ����'/�:� is the numericalspacingdriven by the error analy-sis.Globalspacingfunctioncanbedefinedsimply theminimumofthesetwo spacingfunctions.

;<� �������= >"?$ �@& �����)�����*� � ����AB����9 (2)

This sectionreviews the spherepackingbasedquality guaran-teedmeshgenerationmethodsanddescribestheir usefor adaptivemeshingof parametricmodels.

Suppose;<� ���:� is thedesiredsizefunctionof awell-shapedmeshfor a domain C . Let DE������F/� denotethespherecenteredat point �with radiusF . Let G beapositive realconstant.A set H of spheres,DE�JI�� ;<� �)�JI���K � � , IML%C is a G -packing[22] if

� No two spheresin H overlap;and

� Thereare no big gaps,i.e. NO4LPC , Q a spherein H thatoverlapswith DR��O<��G ;<� �)��O/��K � � .

Thestructuretheoremof Miller et al. [24] statesanequivalencerelationshipbetweenG -spherepackingandwell-shaped(boundedaspectratio) meshes. Basedon this resultseveral algorithmsaresuggestedto constructwell-shapedmeshesin 2D andboundedcir-cumradiusshortestedgeratio meshesin 3D [21, 22, 24]. Thefourstepalgorithmwe suggestbelow is similar to simultaneousrefine-mentandcoarseningwork in [21].

1. Copy themeshof theoriginalmodelto thenew modelasex-plainedin Section4.

2. Recomputethe ;<� �)�:� function to reflect the new geometryandthenumericalspacingneeds.

3. Modify the packingto becomea goodpacking. This canbeachievedin variouswaysanddiscussedbelow in detail.

4. ConstructtheDelaunaytriangulationof thecentersof thenewpackingspheres.

Therearevariouswaysto maintaininga goodpacking.In [21],the oversamplingidea is shown to generatea good packing foradaptivemeshes.First,themethodinsertsnew points(andtheircor-respondingspheres)to thedomainto ensurethereareno big gaps.Next, overlappingspheresareeliminatedresultingin a goodpack-ing. Alternatively, onecanuseparticlesimulationapproachof Shi-madaandGossardtogetherwith insertionandremoval of spheres(particles)during the simulation[39] . A modifiedversionof thebiting methodin [22] canalsobeusedfor maintainingapacking.

Figure9 illustratesthe packingbasedapproach.The model isstretchedin theY directionwith a factorof 2, implying largergapsbut no overlapsin thedistortedpacking.Also notethat,sinceonlya few point (sphere)insertionsare sufficient to get a goodpack-ing, thetriangulationdoesnot needto berecomputed.In this casethenew triangulationcanbecomputedby point insertion/removaloperationson thedistortedtriangulation,eliminatingcompleteDe-launayre-meshing.

5.1.3 Coarsening via edge contraction

Whenneitherthe parametricgeometrychangesnor the numericalspacingimposesmallerelementsin the new meshmodel, refine-ment basedmethods,discussedin Section5.1.1, might generateover-refinedmeshes.An exampleis givenin Figure10. Thespherepackingbasedmethodsdiscussedin Section5.1.2avoid this prob-lem. This sectionsuggestscoarseningstrategiesto avoid theover-refinement.

Coarseningstrategies are very popular in computergraphicscommunity. Thesemethodsaimatmeshsimplificationfor efficientvisualization.Both theprogressivemeshesof Hoppe[17] andthequadratic error metric basedsimplificationof GarlandandHeck-bert [14] useheavily theedgecontractionoperationto coarsenthemeshes.This operationremoves an edge,merging the two end-pointsof it andmodifying theconnectivity accordingly. Themeth-odscanbeapplieddirectly to ourproblem.

(a)

(b)

(c) (d)

Figure9: Maintainingaspherepacking(a)Originalmeshgeneratedby a spherepacking. (b) Packingis no longergoodasbig gapsinthe domainare introduced. (c) Packingis modifiedto be a goodpacking. (d) DelaunayTriangulationof the points in the packinggivesagoodqualitymesh.

5.2 Hexahedral Meshes

Whendealingwith adjustmentof hexahedralmeshesthe modifi-cationoptionsavailablearemuchmorelimited. Automatichexa-hedralmeshgenerationcontinuesto beanopenproblem[1]. Gridbasedmeshgeneration[33] is theonlyautomatedhexahedralmesh-ing algorithmfor generalsolid models,matureenoughto beusedin practice.However, this algorithmtendsto generatelow qualityelementsnearmodelboundariesanddoesnot preserve conformitybetweenmesheson adjacentsolids. Theseareseveredrawbacks,whichpreventits wideuse.Toolsprovidedby existingmeshgener-ationsoftware[1, 26], arelimited in scope,i.e. they only applytoa subsetof geometries.Modelsthatcanbe meshedautomaticallytypically includeuni-axial combinationsof sweptsolids,andsub-mappablesolids,i.e. solidswhichcanbedecomposedinto topolog-ical cubes.Due to the difficulty of generatinghexahedralmeshesin thefirst place,theavailableresearchon modifying suchmeshesis limited to templatebasedmethods[26]. Hence,for the moregeneralcases,new techniqueshave to bedeveloped.

Below, we proposeseveral new ideason handlingpartial re-meshingandmodificationof hexahedralmeshes,giventhe limitedoptionsavailable.

(a)

(2)

Figure10: Refinementalgorithmmaycreateover-refinedmeshes.(a)Originalmesh.(b) Refinementbasedmethodcomputesmeshofthesamemodelshrunkin Y directionby a factorof 1/2.

5.2.1 Partial re-meshing

Partial re-meshingconsistsof removing a partof themodelmesh,followed by generationof a new meshfor the region conformingto the existing mesh. It canbe performedwhena meshin a spe-cific region of themodelis distorted.This approachwasusedforadaptationof tetrahedralmeshesby Hassanet. al[16].

An algorithmwhich canmeshthe emptyregion conformingtotheexistingboundarymeshis necessaryto performthere-meshing.Therequestto maintainconformitywith therestof themesh,pre-ventstheuseof grid basedmethods[33]. This,sincein grid basedmeshingthemeshisgeneratedfromthesolidinterioroutwards,pre-scribingtheboundarymesh.Therefore,only theregionsthatcanbemeshedby mapping/sub-mappingor sweepcanbere-meshedusingmostexistingsoftware[26]. Hence,asopposedto tetrahedralmesh-ing,aregionof poorqualityelementscannotbere-mesheddirectly.A largerregion,containingit, whichfits theconstraintsimposedbysub-mappingor sweephasto beused.Clearlywe would like thisregion to remainrelatively small. Givena region of themeshit ispossibleto checkif it canbe meshed,asmany hexahedralmesh-generationcodes,suchasGambit [12] by Fluent,Inc. or CUBIT[32] by SandiaNationalLaboratories,areable,givena solid (withor without someof themodelfacesmeshedahead),to checkif thesolidcanbemeshedby availablemeshingalgorithms.Thequestionto beansweredis how to find suchmeshableregions.An extensivesearchstartingwith the badly shapedmeshelementsand addingelementsoneor moreata time,canclearlybeexponential.

A possibleapproachis, giventhemeshof theoriginalmodel,tosubdivide it into partsmeshableby basicalgorithms.Then,givenaregion of poorshapedelements,find which part (or parts)containit. Suchsubdivision is usuallygeneratedwhenmeshingtheorigi-nal model. Themodelsareoftendecomposedby theuserin orderto useoneof thosebasicalgorithmsfor the meshing. The moreadvancedmeshingtechniques,suchasmethodsfor meshingacom-binationof sweptvolumes[2], aregeneratingsuchsubdivision aspartof themeshingprocess.By takingadvantageof this subdivi-

sion the re-meshingcanbe performedautomatically. An exampleof thisS approachis shown in Figure11, wheretheoriginal meshissubdividedinto two sweepableregions.While, this approachoftendoesnotproducethesmallestpossibleregions,for complex modelsit resultsin significantsavings comparedto re-meshingthe entiremodel. Methodsto detectsmallermeshableregionsshouldbe in-vestigatedto make thepartialre-meshingmoreefficient.

(a)

(b) (c)

Figure11: Adjustingthemeshfor amodelcomposedof two sweep-able parts. The parameterthat changesis the tube height (from2 to 5). Applying only copy andsmoothingwill generatehighlystretchedelementson thepipe. (a) Original model. (b) Meshaftercopy. (c) Meshafterre-meshingthepipepart.

5.2.2 Adaptive Hexahedral Meshing

As mentionedearlier, publishedresearchon meshadaptivity ismostlyrestrictedto tetrahedralmeshes.Dueto thestructureof hex-ahedralmeshes,localmodificationstypically propagatethemselvesto more than the immediatevicinity. As a result most works onmodificationof hexahedralmeshes,asreviewedby Owen[26], arelimited to local, templatebasedchanges[34, 25]. Thosemethodsusuallyhave very strict requirementson themeshstructure,andasa resulta very limited scope.In this work, we suggesttaking ad-vantageof themeshstructureto enableglobalmodificationsof thehexahedralmesh,while ensuringthemeshvalidity.

TheSpatialTwist Continuum(STC)wasintroducedby Tautgeset. al[42] asa representationof the connectivity of a hexahedralmesh.TheSTCis aninterpretationof thegeometricdualof a hex-ahedralmeshasanarrangementof intersectingsurfaces,whichbi-secthexahedralelementsin eachdirection.Tautgeset. al[42] calledthosesurfaceswhisker sheets(Figure12). Themeshdualis thecellcomplex inducedby the intersectionof the whisker sheets.Eachwhisker sheetis dual to a layer of meshelements.Eachhexahe-dral meshelementbelongsto threewhisker sheetscorrespondingto its threeaxes. A curve formedby theintersectionof two sheetsis calledawhisker chord. A chordrepresentsacolumnof elementsthatpropagatesthroughthemesh.TheSTCfully definesthecon-nectivity of a hexahedralmesh.Themodelmeshcanbe modifiedby modifyingtheSTCstructure,subjectto validity constraints[42].Theconstraintsfor topologicalvalidity of ahexahedralmeshbasedon theSTCare:

1. eachintersectionof a sheetwith the geometricmodel is aclosedcurve;

2. theintersectionof twosheetsis acurvewhichiseithercircularor hasbothendson theoutsidesurfaceof themodel;

3. no morethanthreesheetsintersecttogetherat a point andnomorethantwo alonganintersectioncurve;

4. eachsheethasat leastoneintersectionpoint with two othersheets.

ab

cd

(a) (b)

Figure12: (a) Four whisker sheets(a-d) in a two elementhexahe-dralmesh.(b) Two whiskersheetsin acylindermesh(highlighted).

Here,we considertwo suchmodificationoperations,basedonmanipulatinga whisker sheet(Figure14). To defineoperationsona sheetwe usethetermwhisker sheetaxisasthedirectionperpen-dicularto thethemeshlayerrepresentedby it (Figure13).

X

Z

Y

E

S

Figure13: A whisker sheetH with T pointing in the directionofthesheetaxis.

Thetwo operationsare:

� Sheetremoval - removal of an entiremeshlayer associatedwith agivenwhiskersheet(Figure14 (a) to (b));

� Sheetsplit - separationof a layer into two layers,by split-ting eachhexahedralelementinto two in thedirectionof thewhiskersheetaxis(Figure14(a) to (c)).

To ensurethattheoperationspreserveavalid meshconnectivity,we needto verify that thefour conditionsabove hold for themeshafter the operation.For a sheetremoval, the first threeconditionswill hold for any removing any sheet. The only conditionwhichneedsto be ensuredis that eachmeshsheethasto have at leastoneintersectionpoint with two othersheet.Hencea sheetcanberemovedonly if all othersheetshave intersectionpointswhich donot include it. Exampleof a sheetwhich can’t be removed is asheetcontaininganelementwith two oppositefaceson themodelboundary.

For thesheetsplit operation,thenew sheetsarecopiesof theoldoneandU henceall propertiesof the old sheetstill hold, i.e. theyintersectthe samesheetsthe old one did at identical curves andpoints,henceall theconditionshold. Sheetsplit canbeperformedfor any whisker sheetin themesh.

To decidewhena whisker sheetshouldbe split or removed weintroduceameasureof sheetaspectratio. It isdefinedastheaverageof theaspectratiosof thesheetelementsin thedirectionof thesheetaxis.For theexamplein Figure13, thesheetratioof elementV is

WYX Z\[�] 2 Z^[ ,

� Z\[�_ (3)

where[a` �:b?L & ����0���c<9 arethelengthsof theelementedgesin theb direction.

Theremoval andsplit operationsareappliedto sheetswith ex-tremelyhigh or low aspectratiosrespectively. The whisker sheetremoval operationleadsto the deletionof pressedelementsin re-gionswheretheparameterchangeleadsto contractionof themodel.Thesheetsplitting leadsto theadditionof new elementsin regionswherethe model is expanded. A more advancedapplicationofthoseoperationscanrely directly on theparameterschange,usingtheratioof thenew andoriginalsizesto decidewhichoperationstoperformandhow many times.

(a) (b) (c)

Figure14: Whisker sheetremoval andsplit. (a) Initial model. (b)Modelafterasheetremoval. (c) Modelafterasheetsplit.

After a sheetremoval or split operation,the meshneedsto besmoothedagainto diffusethechangeto thesurroundingarea.Thewhisker sheetbasedmeshmodificationis usedfor the modelsinFigure2, Figure15,andFigure17. In Figure2 thesheetsplit oper-ation is usedto split stretchedelementsinto two. In Figure15 thesheetremoval is usedto remove layersof pressedelements.Theeffectof thesmoothingcanbeseenin Figure15(c)to (d).

Theseoperationsarehighly successfulwhenthe modelparam-eterschangedareperpendicularto someof the meshlayersin themodifiedregion. They will be lesseffective whenno suchlayersexist.

6 Examples

Figure17 shows the re-meshingprocessfor a crank-shaftmodel.The meshingof the original modelrequiredsignificantmodelad-justment. The blendshadto be suppressedandthe modelhadtobedecomposedinto five partsto enablehexahedralmeshingusingtheCooperTool algorithm[2] (Figure17 (b)). Thedecompositionincludedseparatingthe two “tips” of the modelby usingsplittingplanesandseparatingthecentralboardfrom thetwo side“wheels”.Oncetheeditingwascompletedthemeshwasgeneratedautomat-ically. The editing processandmeshgenerationtook aboutthirtyminutes.TheeditingandmeshingwereperformedusingtheGam-bit analysispre-processor[12].

Thenew model(Figure17 (d)) differsfrom theoriginal in threeparameters.First, the width of the centralboardwasreducedby

(a) (b)

(c) (d)

(e)

Figure15: Adaptingahexahedralmeshusingthewhiskersheetre-moval operation.(a)Meshof theoriginal model.(b) Copiedmeshof thenew model(cylinder radiusof 5, replacedby two radii of 7and3). (c) Meshaftersinglesheetremoval. (d) Meshaftersmooth-ing. (e) Final meshafteronemoresequenceof sheetremoval andsmoothing.

third. Second,thecylindrical tip on theright wasmodified,reduc-ing its radiusby third andincreasingits heightby a factorof two.And third, theangleof theblendon theleft “wheel” wasincreasedfrom d � to d.e � . Thecopiedmeshis shown in Figure17(e)). Meshadjustmentwasrequiredto restorethe meshquality in the centralboardandthe right tip cylinder. The adjustmentusedthewhiskersheetediting operations.The entireprocedurecanbe fully auto-matedasdescribedabove,asopposedto theinitial meshgenerationthatrequiredmanualmodelsimplificationanddecomposition.

Figure Figure 16 shows anotherexample of remeshingafterparametricmodification.In this examplea tetrahedralmeshof themodelis constructed.Thefigureshows a meshedpartof a boosterrocket. It includesthesolid propellantandthecavity insideit. Thecavity is star-shapedto optimizetheburningrateof thepropellant.Thecavity is alsomeshedto simulatethebehavior of thegasesgen-eratedduring propellantburning. Theoriginal meshof themodelin Figure16 (a) has6829nodesand36726tetrahedra.Figure16(b) shows themodelstretchedby a factorof 1.5alongX andZ di-rections.This parametricchangecauses1085of the tetrahedratobecomebad,i.e. their circumradiusto shortest-edgeratio is largerthan2.5. Delaunayrefinementis performedon thedistortedmeshby inserting2170circumcenterpointsandall of thebadsimplicesareremoved(seeFigure16 (c)).

(a)

(b) (c)

Figure16: Volumemeshof partof aboosterrocket containingthesolidpropellantandthestarshapedcavity insideit. Thefinsof thestarareabouttwo elementsthick. (a)Originalmeshwith 36726tetrahedra.(b) StretchingthemodelalongX andZ directionsmakesabout3f of thetetrahedrabad.(c) Inserting2170circumcentersremovesall of thebadsimplices.

7 Summar y

Thepaperaddressedtheissuesinvolvedin re-meshingCAD modelsafterparametermodification.It demonstratedthatusingparametricinformationcaneliminateuserinteractionandsignificantlyshortenthemeshgenerationprocess.

The paperpresentsa comprehensive schemefor re-meshingamodelafterparameterchanges,startingwith thegeometricadjust-mentsrequiredandconcludingwith themeshadaptationnecessaryto maintainthemeshquality. It examinestheapplicationof severalexisting meshingtechniquesto this new problemand introducesseveral new methodsfor topologicalmeshadjustmentfor hexahe-dral andtetrahedralmeshes.Many of the methodsreferredin thepaperwerenot implementedby theauthorsanda largeamountofwork is still requiredto fully implementthe automatedmeshad-justmentprocedure.Implementedandtestedpartsof the methodincludegeometryadjustmentusingthevirtual topologyapproach,meshcopy andsmoothing,Delaunayrefinement,spherepacking,and the whisker sheetbasedediting of hexahedralmeshes. Themainaresfor futureresearcharequantitative analysisandcompar-isonof thesuggestedapproaches.

Thiswork furtherencouragestheuseof theparametricanddualmodelrepresentationsthroughouttheproductdatamanagementcy-cle,by showing theadvantagesof usingtheparametricinformationfor applicationsnotconsideredearlier.

8 Ackno wledgments

The work reportedherewassupported,in part, by the CenterforProcessSimulationandDesign(NSFDMS 98-73945)andtheCen-ter for Simulationof AdvancedRockets (DOE LLNL B341494).We would like to thank DamrongGuoy for providing his refine-mentsoftwareto testouralgorithms.

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