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EUROGRAPHICS 2001 / A. Chalmers and T.-M. Rhyne (Guest Editors) Volume 20 (2001), Number 3 Tensor Topology Tracking: A Visualization Method for Time-Dependent 2D Symmetric Tensor Fields X. Tricoche, G. Scheuermann, H. Hagen Abstract Topological methods produce simple and meaningful depictions of symmetric, second order two-dimensional ten- sor fields. Extending previous work dealing with vector fields, we propose here a scheme for the visualization of time-dependent tensor fields. Basic notions of unsteady tensor topology are discussed. Topological changes - known as bifurcations - are precisely detected and identified by our method which permits an accurate tracking of degenerate points and related structures. 1. Introduction Tensor fields play a key role in many branches of physics and engineering sciences. Second-order tensor fields in par- ticular are involved in solid and fluid mechanics as well as in the theory of general relativity. Therefore, the visualiza- tion of second-order tensor fields is a challenging issue that has received an increasing interest in the last decade. A ma- jor contribution has been the PhD thesis of Delmarcelle 1 . For two-dimensional symmetric, second-order tensor fields, this author analyzed and visualized the topology of the re- lated orthogonal eigenvector fields. This work partially ex- tends the qualitative theory of dynamical systems dealing with (directional) vector fields to tensor and undirectional eigenvector fields. This has been motivated by the fact that topology-based approaches have already led to very success- ful methods for visualizing vector fields 3 . As in the vector case, drawing the topology of a symmetric, second-order tensor field yields a simple graph representation where the edges are formed by particular integral curves computed in the eigenvector field called separatrices and the vertices are singularities of the tensor field (where eigenvectors cannot be determined uniquely) called degenerate points. The ap- peal of this graph arises from decomposing the plane into sub-regions where all integral curves are structurally equiv- alent. A scaling method has been designed to simplify the topological graph in the case of turbulent tensor fields 4 and a topology-based technique has also been applied to the visu- alization of three-dimensional, symmetric, second-order ten- sor fields 5 . Now, when considering unsteady tensor fields, the method must be extended to depict structural changes affecting de- generate points and related separatrices during their mo- tion over time. Such changes are called bifurcations and have a fundamental mathematical meaning. To handle time- dependent tensor fields, Delmarcelle has suggested to suc- cessively analyze discrete time steps and to correlate the topologies found in a three-dimensional picture where time is visualized as third space dimension. Nevertheless, the sep- aratrices are not drawn which deprives the resulting pic- ture of an essential part of the topological information. Fur- thermore, this discrete approach fails to properly locate and identify the bifurcations that occur between two given time steps whereas these features are indispensable for under- standing the structures observed in discrete samples. In fact, what is required is to embed the discrete time steps in a three-dimensional space/time continuum to enable an ana- lytic tracking of tensor field topology. In previous work 7 , the authors presented a method based upon this basic principle to track the critical points of unsteady planar vector fields in order to visualize time-dependent topology. The precise detection and identification of local bifurcations was used as basis for the depiction of the topological graph. A simi- lar approach is used here for the purpose of tensor topology visualization. The paper is structured as follows: First, we introduce convenient notations to be used later on. Then we recall ba- sic notions of symmetric tensor field topology in the steady case. In the next section, we move to time-dependent ten- sor fields and focus on local bifurcations (i.e. bifurcations that only affect one or several degenerate points) that may be encountered in the context of our method. The following section presents the data structure used to model the con- tinuous space/time domain required for a precise topology c The Eurographics Association and Blackwell Publishers 2001. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.
Transcript of Tensor Topology Tracking: A Visualization Method for Time ...tricoche/pdfs/eg01.pdf · Tensor...
EUROGRAPHICS2001/ A. ChalmersandT.-M. Rhyne(GuestEditors)
Volume20 (2001), Number3
TensorTopologyTracking: A Visualization Methodfor Time-Dependent2D Symmetric TensorFields
X. Tricoche,G. Scheuermann,H. Hagen
AbstractTopological methodsproducesimpleandmeaningfuldepictionsof symmetric,secondorder two-dimensionalten-sor fields.Extendingprevious work dealingwith vectorfields,we proposehere a schemefor the visualizationof time-dependenttensorfields.Basicnotionsof unsteadytensortopology are discussed.Topological changes-knownasbifurcations- arepreciselydetectedandidentifiedbyour methodwhich permitsanaccuratetrackingofdegeneratepointsandrelatedstructures.
1. Intr oduction
Tensorfields play a key role in many branchesof physicsandengineeringsciences.Second-ordertensorfieldsin par-ticular areinvolved in solid andfluid mechanicsaswell asin the theoryof generalrelativity. Therefore,the visualiza-tion of second-ordertensorfields is a challengingissuethathasreceivedanincreasinginterestin thelastdecade.A ma-jor contribution hasbeenthe PhD thesisof Delmarcelle1.For two-dimensionalsymmetric,second-ordertensorfields,this authoranalyzedandvisualizedthe topologyof the re-latedorthogonaleigenvectorfields.This work partially ex-tendsthe qualitative theory of dynamicalsystemsdealingwith (directional)vector fields to tensorand undirectionaleigenvectorfields.This hasbeenmotivatedby the fact thattopology-basedapproacheshavealreadyledto verysuccess-ful methodsfor visualizingvectorfields3. As in the vectorcase,drawing the topology of a symmetric,second-ordertensorfield yields a simplegraphrepresentationwheretheedgesareformedby particularintegral curvescomputedintheeigenvectorfield calledseparatricesandtheverticesaresingularitiesof the tensorfield (whereeigenvectorscannotbe determineduniquely)calleddegenerate points. The ap-peal of this grapharisesfrom decomposingthe planeintosub-regionswhereall integral curvesarestructurallyequiv-alent.A scalingmethodhasbeendesignedto simplify thetopologicalgraphin thecaseof turbulent tensorfields4 andatopology-basedtechniquehasalsobeenappliedto thevisu-alizationof three-dimensional,symmetric,second-orderten-sorfields5.
Now, whenconsideringunsteadytensorfields,themethodmustbeextendedto depictstructuralchangesaffecting de-
generatepoints and relatedseparatricesduring their mo-tion over time. Such changesare called bifurcationsandhave a fundamentalmathematicalmeaning.To handletime-dependenttensorfields, Delmarcellehassuggestedto suc-cessively analyzediscretetime stepsand to correlatethetopologiesfound in a three-dimensionalpicturewheretimeis visualizedasthird spacedimension.Nevertheless,thesep-aratricesare not drawn which deprives the resulting pic-tureof anessentialpartof thetopologicalinformation.Fur-thermore,this discreteapproachfails to properlylocateandidentify thebifurcationsthatoccurbetweentwo given timestepswhereasthesefeaturesare indispensablefor under-standingthestructuresobservedin discretesamples.In fact,what is requiredis to embedthe discretetime stepsin athree-dimensionalspace/timecontinuumto enablean ana-lytic trackingof tensorfield topology. In previouswork7, theauthorspresenteda methodbaseduponthis basicprincipleto track the critical points of unsteadyplanarvector fieldsin order to visualizetime-dependenttopology. The precisedetectionand identificationof local bifurcationswas usedasbasisfor the depictionof the topologicalgraph.A simi-lar approachis usedherefor thepurposeof tensortopologyvisualization.
The paperis structuredas follows: First, we introduceconvenientnotationsto beusedlateron.Thenwe recallba-sic notionsof symmetrictensorfield topologyin thesteadycase.In the next section,we move to time-dependentten-sor fields and focus on local bifurcations(i.e. bifurcationsthat only affect oneor several degeneratepoints) that maybeencounteredin thecontext of our method.Thefollowingsectionpresentsthe datastructureusedto model the con-tinuousspace/timedomainrequiredfor a precisetopology
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tracking:Grid andcorrespondinginterpolationschemeareintroducedaswell as the relatedmethodto track degener-atepointsthrougha singlecell. After this, we describeouralgorithmfor trackingdegeneratepointsthroughthewholegrid and identifying bifurcationson the way. This permitstrackinganddepictionof thewhole topologicalstructureasexplainedin section7. Resultsareshown in section8.
2. Notations
In the following, we will focus on unsteadysecond-order,symmetrictensorfields definedover a boundedspace/timedomain.For convenience,we will usethe following nota-tions.Thetensorfield is denotedby T, whereT is a functionof thex andy coordinatesin theplaneandof time t.
T � �T11 T12T12 T22 �
The first order Taylor expansionof the tensorfield in thevicinity of an arbitrarypoint � x0 � y0 � can be written (for afixedvalueof time):
T11 � T222 � a � x � x0 � b � y � y0 � o ��� ��� x � x0 � y � y0 � ��� �T12 � c � x � x0 � d � y � y0 � o ��� ��� x � x0 � y � y0 � ��� �
3. TensorField Topology
We give a brief overview in this sectionof the fundamen-tal notions of second-oder, symmetric tensorfield topol-ogy as introducedby Delmarcelle2 1 andpresentthe newconceptsassociatedwith time-dependenttopology. Theba-sic idea is to studythe structureof a 2D symmetrictensorfield by consideringoneof its two (orthogonal)eigenvectorfields. Theseeigenvector fields have a sign indeterminacywhichdistinguishestheirtopologyfrom theoneof 2D vectorfields.Nevertheless,like in thevectorcase,tensortopologyis formed by singularities(called degenerate points) con-nectedby particularintegralcurves(or separatrices) bound-ing subdomainsin which all curvesarestructurallyequiva-lent. We focuson thesetopicsafter the introductionof thetensorindex.
3.1. TensorIndex
Theindex (alsoknown asPoincaréindex) playsakey roleinthequalitative analysisof vectorfield topology6. For tensorfields, thereexists a similar definition that will prove veryusefulin thefollowing.
Definition 3.1 The tensor index of a closed,nonself inter-sectingcurve lying in the domainof definition of a tensorfield T is thenumberof counterclockwiserevolutionsof theeigenvectorsof T while traveling oncealongthecurve in acounterclockwisedirection.
The major interestof the tensorindex is whenbeingcom-putedaroundadegeneratepoint (c.f. 3.2)asshown in Fig. 1.As a matterof fact, the index of a closedcurve enclosing
12
ee2
1
λ2λ1
θ
dθ =
Figure1: Index of closedcurvearounda degeneratepoint
no degeneratepoint is zerowhereasit is nonzerowhenen-closinga single simpledegeneratepoint. Furthermore,theindicescomputedon two curvesenclosingthesamedegen-eratepoint areequal.
3.2. DegeneratePoints
As saidpreviously, the visualizedtopologycorrespondstoa particulareigenvector field. Yet, this definition doesnotapply at positionswhere both eigenvalues are equal andthecorrespondingeigenspaceis two-dimensional.In suchacase,eigenvectorscannotbe(uniquely)determined:This isasingularity. Hence,we getthefollowing definition.
Definition 3.2 A degeneratepoint of a 2D symmetric,second-ordertensorfield definedon an opensubsetof IR2
is a pointwherebotheigenvaluesareequal.
It follows from thedefinitionabove, thata point is degener-ateif andonly if thecorrespondingtensorvalueis isotropic,that is of the form λI2, whereI2 is the identity matrix. Thisseconddefinitionis usedin practiceto detectandlocatede-generatepoints.With first orderprecision,the possibletypesof degeneratepointsarethetrisectorpoint andthewedgepointsasshownin Fig. 2. Computingthe tensor index of a closedcurvearoundan isolatedsingularity, one gets � 1
2 for a trisectorpointand 1
2 for awedgepoint.(Thesehalf integersaredueto thesignindeterminacy of eigenvectors.)
WEDGE POINTS
S1
S3S2TRISECTOR
S1 S2 S1 S2=
Figure2: FirstOrder DegeneratePoints
3.3. Separatrices
A trisectorpointhasthreehyperbolicsectors(whereintegralcurvesleadaway from thepoint in bothdirections)while a
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wedgepoint hasa hyperbolicsectoranda parabolicsector(where integral curves lead away in one direction and to-wardthepoint in theother),possiblyreducedto asingleline.Separatricesaredefinedasthecurvesthatboundhyperbolicsectors.Whenintegratingtheselinesaway from a singular-ity, onebuilds theedgesof thetopologicalgraph.Thelocationof theseparatricesin thevicinity of a degener-atepoint is baseduponthefollowing property(1, p.120).
Property 3.1With thenotationsof section2,consideringthefirst-orderTaylor expansionof T in thevicinity of a degen-eratepoint � x0 � y0 � , theanglesθk of possibleseparatricesskaregivenby therealrootsof thefollowing cubicpolynomial
dz3 � c 2b� z2 � 2a � d � z � c � 0
by invertingtherelationzk � tanθk.
Theanglesfoundin thatwaymustbecheckedto correspondto an actualboundarycurve of a hyperbolicsector. In thecaseof a degeneratepoint of index 1
2 , solving this poly-nomial enablesthe distinctionbetweena wedgepoint witha singleseparatrixanda wedgepoint with two separatricesandanactualparabolicsector.
4. Bifur cations
In the previous section,we have only consideredinstanta-neoustensorfields. When time is introducedasadditionalparameter, degeneratepoints move in the planeand inter-act,appearor vanish.This inducesstructuralchangesfor thetopology, calledbifurcations. Bifurcationsoccurfor particu-lar valuesof timeandentailtheinstantaneouspassageof thetopologyfrom a structurallystablestateto another.Unlikedirectionalvectorfieldswherebifurcationshavebeena major domainof interestto pureandappliedmathemati-ciansfor a long time, thebifurcationsof undirectionnalvec-tor fields have apparentlynot received as much attentionfrom mathematiciansand a generaltheoreticalframeworkseemsto be lessdeveloped.For this reason,we will onlydescribetwo kinds of local bifurcationsthat may occur inthe context of our methodand let apartfurther discussionaboutall possiblecases.In general,bifurcationsthatdo notfall into oneof thefollowing categorieswill bedetectedbyourmethodaswell but their identificationwill beimpossible(seesection7).
4.1. Pairwise Creation/Annihilation
Ourfirst kind of bifurcationis theso-calledpairwiseannihi-lationof degeneracies.It occurswhentwo degeneratepointsof oppositeindex, that coexist closeto another, merge. Atthis moment,a singledegeneratepoint canbeobservedthatis of nosimpletypeandthatdisappearsright afterthefusion:No singularityexistsfrom now on in theneighborhood.Re-versingthe directionof time, onegetssimilarly a pairwisecreationof degeneracies:Startingwith no singularity, theresuddenlyappearsadegeneratepointwith complex structure.
This instantaneousstateis thenreplacedby two simplede-generatepointsthat move away from another. Several con-figurationsexist for the intermediate,unstablestate.Fig. 3shows two of them.
Figure3: Pairwiseannihilationof degeneraciesof oppositeindex
4.2. WedgeBifur cation
The other bifurcation that mostly occursin our context isthe transition from one type of wedgepoint to the other.Thepassagefrom a wedgepoint with two separatricesto awedgepoint with onesingleseparatrixmeansfrom a struc-turalpoint of view thedisappearanceof thewholeparabolicsectorpreviously locatedbetweenthetwo separatrices:Thissectorwas2-dimensionalandbecomes1-dimensional,i.e. itis reducedto a singleline. Inverting time direction,oneob-servesthecreationof a parabolicsectorin thevicinity of awedgepoint from anoriginal singleseparatrix(seeFig. 4).
Figure 4: Creationof a parabolic sectorin thevicinity of awedge point
5. Data Structur e
As mentionedpreviously, our methodhandles2D, symmet-ric, second-ordertensorfields. Thesetensorfields dependon time andaregiven in a discretemanner:For eachtimestep,we get the tensorvaluesat theverticesof a planartri-angulationthatremainsconstantover time(thepositionsareunchangedfrom one time step to the next). At eachtimestep,the discreteinformationis piecewise linearly interpo-latedover thetriangulationto resultin aninstantaneouspla-narC0 tensorfield. Now, the time stepsmustbe embeddedin a space/timecontinuumto enabletheprecisedetectionof
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structuraltransitionsin thetopology. This is describedin thepresentsection.
5.1. SpaceTime Grid
As in ourworkontime-dependentvectorfields7, theintegra-tion of thediscretedatain a continuousmathematicalspaceis achievedby consideringtime asthird dimensionandcon-structinga 3D structuredgrid, madeof prismcellsthatcon-necttogethercorrespondingtrianglecells lying in consecu-tive time stepsasshown in Fig. 5. The prismcell hasbeenpreferredto a tetrahedrisationof thegrid to reproducebetterthe topologicalcontinuity insideeachtrianglecell, trackedthroughtime.
T i+1T i
prism cell
Figure 5: Space/timegrid connectingidentical triangula-tionsover time
5.2. Inter polation
An interpolationschemesuitedto this type of cell hasal-readybeenusedearlier7. It is designedto beconsistentwiththetensorfieldgivenateachtimestep.Basically, weproceedtheinterpolationof symmetric,second-order2D tensorval-ues(3 independentscalarcomponents)overa3D grid.Moreprecisely, for eachfixedtime value,the interpolantmustre-turna piecewiselineartensorfield.For a givenprismcell lying betweent � ti andt � ti � 1, letf j � x � y� � α j β jx γ jy, j ��� i � i 1� be the affin linearinterpolantscorrespondingto a particularscalarcomponentof the tensorfield in the triangularfacesof theprism lyingin theplanes� t � ti � and � t � ti � 1 � , respectively. Thenwedefinetheinterpolantover thewholeprismcell by linearin-terpolationover time:
f � x � y� t � � α � t � β � t � x γ � t � ywhere
α � t � � ti � 1 � tti � 1 � ti
αi t � titi � 1 � ti
αi � 1
(idemfor β andγ). This formulaensuresobviously, for eachfixedvaluet, that ft is affin linearin x andy.
5.3. Singularity Location
For a particularprism cell, finding the locationsof a sin-gularity consistsof determiningthe equationof the curve(or path)thatadegeneratepointdescribesin its motionovertime.It alsorequirestheprecisedetection,locationandiden-tificationof thechanges(or bifurcations)thatmayaffect thenatureof thesingularity.As saidin section3.2,locatinga degeneratepoint is equiva-lent to finding a positionin theplanewherethe tensorfieldis isotropic,thatis of theform λI2, whichcanbewrittenas�
T11 � T22 � 0T12 ��� T21 � � 0 �
As we deal,for each(fixed)valueof t, with a piecewiselin-eartensorfield, thisequationsystemis equivalentto thefol-lowing linearsystemin x andy.�� � � α11 � t � � α22 � t ��� � β11 � t � � β22 � t ��� x � ������ � � β11 � t � � β22 � t ��� y � 0
α12 � t � β12 � t � x γ12 � t � y � 0
Solving this systemfor x andy yields biquadraticrationalfunctionsin t. (Note that the linearity of the interpolantforany fixedt ensuresthepresenceof auniquedegeneratepointfor eachvalueof t.) But thisequationis only valid insidethecell, sowe areinterestedin thesectionsof curve that inter-secttheinterior domainof thecell. A practicalandefficientway to achieve it, is to considerthe singularitieslying onthe facesof the prism cell: 2 triangles(in the planest � tiandt � ti 1) and3 quadrilaterals(joining both planesto-gether).Inducedby the interpolantdefinedover the prismcell, thesefacesare linearly (resp.bilinearly) interpolated,which permitsa fastsingularitysearch.Thepositionsfoundonthecell boundaryarethenassociatedpairwiseto describesuccessive entry andexit positionsof the paththroughtheprism(seeFig. 6).
yexit
exit
entry
entry
entry
exit
xT
T
exit
entry
exit
entry
entry
exit
y
x
Figure6: entryandexit pointsof a paththrougha prism
5.4. Bifur cation Detection
Wemustnow considerthecorrespondingsuccessive naturesof this singularityto geta completepictureof its evolutionthroughthecell. Fromsection3.2,we know thatthedegen-eratepoints are of two major types(trisectorand wedge).
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Furthermore,we know that trisectorsandwedgeshave op-positeindices.Now, thetensorindex of a closedcurve is aninvariantin continuoustensorfields,asis thePoincaréindexfor vectorfields (this is intuitively clearbut we have foundno formal proof of this propertyso far). That meansthat ifonetakesthe(triangle)cell boundaryasclosedcurveto com-putethe tensorindex, this index will beconstantaslong asno degeneratepoint reachestheboundary, i.e. aslong asnodegeneratepointentersor leavestheprismcell.For ourcase,it entailsthata wedgepoint remainsa wedgepoint andthata trisectorpoint remainsa trisectorpoint.That is thereasonwhy, no type swap canoccur in the context of our methodinsidea cell. Thesameresultholdsif thesingularityleavesthecell to enteroneof its neighborsby takingtheboundarycurve of bothcellsto computethe(constant)index.Furthermore,as a linear tensorfield canhave at mostonedegeneratepoint, several singularitiescannot meetor splitin the interior of a cell but only at thecommonboundaryoftwo cells. This restrictsthe possiblelocationof a pairwisecreation/annihilationto the boundaryof a prism cell. Thelastrestrictionconcernsthemerging (or splitting) of severaldegeneratepointsof the sametype at the commonedgeoftwo cells.To show it, we first needthefollowing property.
Property 5.1If two simpledegeneratepointslie in two cellssharinga commonedge,then they must have oppositein-dices.
δBCθ
δCAθ
δABθ
δACθ
δDAθ
δCDθ
A
B
C
D
Figure7: Degeneratepointsin neighborcells
Proof: Considerfigure7: Let thepointsdepictedin bothcells correspondto degeneratepoints.By definition of thetensorindex (see3.1),computingtheindex alongtheedgesof ABC, onegets
IABC � δθAB δθ
BC δθCA
whereθ is theanglebetweentheeigenvectorsandthex-axisandδθ
AB is the changeof θ along �AB� . Likewise, in ACD,onegets
IACD � δθAC δθ
CD δθDA
But of course, δθCA ��� δθ
AC. Supposefor instancethatδθ
CA � 0, then δθAC � 0. Now, we know that the angle
changesalong a closed curve encompassinga singleisolatedsimpledegeneratepoint occuralways in the samedirection(1, Corollary4, p. 119).That is, δθ
AB andδθBC must
be both positive and δθCD and δθ
DA must be both negative.Consequently, we get in ABC a degeneratepoint withpositive index andin ACD, we find a degeneratepoint withnegative index which provestheproperty.
Consequently, theonly bifurcationthatcanbeobservedintheinterior of a prismcell is whatwe calleda wedge bifur-cation (see4.2).Theexact locationof sucha bifurcationisa difficult algebraicproblem(onemustfind a valuet0 suchthat the cubic polynomial equationin 3.3 hasexactly onereal root for t � t0 and3 real roots for t � t0, which is inour casea polynomialequationof degree6!) thatwe solvenumerically:Wecomputesuccessive positionsof thedegen-eratepointaccordingto theformulaof 5.3andcheck,in thecaseof a wedgepoint, if thenumberof its separatriceshaschanged.At this point, we starta binary searchto approxi-matetheexactbifurcationposition.
6. Degeneracies’Tracking
Theprevioussectiondescribedthepreprocessingstepof ourmethod:Eachprismcell is checkedfor apathof adegeneratepoint throughit andtherelatedentry andexit positionsaresavedaswell asthewedgebifurcationson theway. Thenext taskis to reconnectthesepiecesto obtainthewholepathof eachsingularity togetherwith its possibletype changes.Furthermore,we still have to identify the bifurcationsthatoccuron thesidefacesof theprismcellsandthathave notbeendetectedsofar.Thedegeneracies’trackingalgorithmprocessestwo consec-utive time stepsat once(which avoidshaving all time stepsloadedin memory).It startsby investigatingall trianglesly-ing in the first time stepand tracksevery found degener-atepoint throughthecorrespondingprismcell until it eitherreachesthenext timestepor returnsto thepreviousone.Wegivenow asketchof this forwardtrackingalgorithmstartingwith time stepsTS(i) andTS(i+1). (ThebooleanarrayToCheck indicatesif a cell lying in TS(i) muststill bechecked.ThebooleanarrayReached tells if a cell lying inTS(i+1) hasbeencheckedwhile forwardtracking.At thebeginning, one setsfor all j, ToCheck[j]=TRUEandReached[j]=FALSE.)
foreach PrismCell P(j)TrackingDirection = FORWARDif (ToCheck[j] &&
TriangleFace in TS(i) containsSingularity)
track singularity until it leaves P(j)get ExitPointCurrentTime = time(ExitPoint)CurrentPoint = ExitPointif (ExitPoint lies in TS(i+1))Reached[j] = TRUE
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elseget next PrismCell P(l) :1track singularity until it leaves P(l)get ExitPointif (time(ExitPoint) < CurrentTime &&trackingDirection == FORWARD)add PairwiseAnnihilation
at CurrentPointTrackingDirection = BACKWARDelse if (timeExitPoint > CurrentTime &&trackingDirection == BACKWARD)add PairwiseCreation at CurrentPointTrackingDirection = FORWARDCurrentTime = time(ExitPoint)CurrentPoint = ExitPointif (ExitPoint lies in TS(i+1))Reached[l] = TRUE
else if (ExitPoint lies in TS(i))ToCheck[l] = FALSE
elsegoto :1
endifendif
endifToCheck[j] = FALSE
endforeach
Oncethis hasbeendone,we investigatethe trianglesly-ing in thenext time stepTS(i+1) with a backward track-ing. Only cells j with Reached[j]=FALSE are consid-ered.Otherwise,the triangleshave beenprocessedalready.The codeof this backward tracking is very similar to theoneof the forward tracking.The transitionto the next pairof time stepsis doneby substitutingToCheck=ReachedandsettingReached=FALSE. At the endof this scheme,all piecesof degeneracies’pathslying betweenconsecutivetime stepsarereconnectedtogether(throughtime) with thecorrespondingbifurcationsthat have beenencounteredontheway.
7. TopologyDepiction
Now, as degeneraciesare only part of the topology in the2D steadycase,thepathswehave trackedover timemustbeassociatedwith thecorrespondingseparatricesto depictthewholetopologicalstructureof thetensorfield in theunsteadycase.Practically, theseparatrixcurveschangeover timeandthusspanseparatrixsurfacesthatboundvolumesof thetime-dependenttopology.
7.1. Structur esTracking
Drawing thesesurfacesassumesthepreviousdeterminationof eachsingularpathandfor eachdiscretepositionalongthepaththecomputationof thestartdirectionsof theseparatri-ces(see3.3). In our method,thesedirectionsarecomputedduringthepathtrackinginsideeachprism(see5.3).Thede-pictionof theseparatrixsurfacesis thenbasedupontheinte-grationof separatrixcurvesalongthisdirectionononehand,
andon thecorrelationof eachseparatrixcurve with anothercurve at the previous position along the path on the otherhand.This correlationis neededif onedealswith a trisectorpathor a wedgepathwith two separatricesto avoid bindingcurvesthatdonotfit in a topologicalsense.This is achievedby looking for the bestapproximationof currentseparatrixangleamongthepreviousones.The integration itself is done numerically by a 4th orderRunge-Kutta integration with adaptive stepsize. Comput-ing integral curvesin tensorfieldsis a challengingissuefornumericssinceeigenvectorshave neithernormnor orienta-tion. A stop criterion must thereforebe basedupon topo-logical informationto ensurepropertreatmentof thedegen-eratepoints.The computationcanendin two ways:Eitherthe integral lines reachthe domainboundaryor they con-vergetowardsa wedgepoint insideits parabolicsector(thatis, integral curvescanonly convergetowardsa wedgepointwith two distinctseparatrices).We thusconsiderfor a givencurve,thattopologicalconsistency hasbeenmaintainedwithits predecessorif they both leave thegrid at closepositionsor both converge towards the samewedgepoint. In othercases,consistency hasbeenlostandweinfer thepresenceofaglobalbifurcation(thatwe cannot identify morepreciselycurrently)if no local bifurcationwasfound at this position(see4).
7.2. SurfaceDrawing
Onceseparatriceshave beenassociatedover time, onegetsa setof curvesthatspana surface.Theconstructionof thissurfaceis donein two steps.Originally, the curvesthat re-sult from numericalintegrationarepiecewise linear. There-fore, they arefirst replacedby interpolatingNURBScurvesof degree2 to increasesmoothness.In thesecondstep,theseNURBS curves are embeddedin a NURBS surface,afteruniformizationof their parametrisations.
8. Results
We presentherethe resultsof our methodappliedto twodatasets:Thefirst oneis anartificial symmetrictensorfieldthat we useto demonstratethe different aspectsof the al-gorithmandto illustratethetopologicalconceptsintroducedpreviously. The secondone is provided by a CFD simula-tion of a turbulent tensorfield andgivesus theopportunityto commenttheapplicationof this techniquefor thevisual-izationof real,complicatedCFD data.
8.1. Artificial TensorField
Our first datasetis geneneratedon a small rectilinearpointsetof 25 verticesthat we triangulateto get a grid with 32cells.We startwith a topologycontaining3 trisectorpointsand4 wedgepoints(3 with asingleseparatrixand1 with twoseparatrices)asshown in Fig. 8. We process11 consecutivetimesteps.
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Figure8: StartTopology
Tracking the degeneraciesover time, we get the pictureshown in Fig. 9. Color codingis as follows: Trisectorsaredepictedin red,wedgepointswith two separatricesin darkblueandwedgeswith a singleseparatrixareshown in lightblue. Bifurcations are marked as small balls: greenonesindicatepairwisecreation,red onesillustrate pairwisean-nihilation andyellow balls designatewedgeswap bifurca-tions. Five Bifurcationshave beendetected:two pairwisecreations,two pairwiseannihilationsandonewedgeswap.
Figure9: Degeneracies’paths
Adding theseparatrixsurfacesto thedegeneracies’paths,wecompletethetopologydepictionandobtainthestructurespresentedin Fig. 10 (separatrixsurfacesemanatingfrom atrisectorpoint arecoloredin redwhile thosecomingfrom awedgepointaredisplayedin blue).
If we cut instantaneoustime planesout of this three-
Figure 10: Picture of the completetopology evolution (seealsoFig.17)
dimensionalpicture,wecanexhibit discretesamplesaroundabifurcationpoint.As anexample,we depictin Fig. 11partof thetopologybefore,duringandaftera pairwisecreation.A wedgebifurcationis presentedin Fig. 16.
8.2. Rateof Deformation TensorField
Ourseconddatasetstemsfrom aCFDsimulation.This is therateof deformationtensorfield of a swirling jet simulationprovidedby WolfgangKollmannfrom UC Davis. Thecom-pletestructuredgrid has250 x 244 cells. We considerthesymmetricpart of theoriginal data.To avoid visual clutter,weprocessacell groupcontaining45x 60cellsasshown onFig. 12.
Processingthis setof cellsover 25 timesteps(seeFig. 13to getan impressionof the3D structure),we obtainthede-generacies’pathsshown in Fig.14.Wehavereducedthesizeof theballsindicatingbifurcationsto avoidconfusingthede-piction. As a matterof fact,asonecanseeon this picture,trackingthetopologyof this turbulent tensorfield resultsina large numberof encounteredbifurcations.This effect isprobablyrelatedin someextent to the linear schemeusedfor time interpolation.If onezoomsinto thedatato look atsmall topologicalfeatures,oneobservesstructureslike theoneshown in Fig. 15. Thepresenceof suchstructuressug-geststhatapost-processingstepwouldbehelpfulto simplifytheresultingtopologyby suppressingfeaturesof smalltimeor spacescale.Nevertheless,structuresof larger scalemaybe easily recognizedin the global picture becauseof theirtemporalpersistence.
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Figure11: Topology in theneighborhoodof a pairwisecre-ation
9. Conclusions
We have presenteda topology-basedmethodfor the visu-alization of time-dependent,two dimensional,symmetric,second-ordertensorfields.Degeneratepoints- depictedbyconventionaltopologymethodsfor steadyfields- aretrackedover time aswell asthe relatedstructures.An essentialas-pectof this taskis to find andidentify thebifurcationsthatoccuron the way andaffect the position,natureandnum-ber of degeneratepoints.Their detectionis madepossibleby theuseof a temporalinterpolationthatembedsthegivendiscretetime stepsin a space/timecontinuum.The method
Figure12: Starttopology andconsideredpart
Figure13: Degeneracies’pathsand3D grid
resultsin three-dimensionalpictureswheretopology is vi-sualizedasdegeneracies’pathsandseparatrixsurfaces.Wethink thatthis techniqueprovidesanefficient way to under-standandinterpretthetemporalevolutionof time-dependenttensorfields that could be completedby a post-processingstepto suppressinterpolationartifacts.
Acknowledgements
The authorswish to thankWolfgangKollmann,MAE De-partmentof the University of California at Davis, for pro-viding therateof deformationtensordataset.
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Figure14: Degeneracies’paths
Figure15: Local topological structure
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Figure16: A wedge bifurcation
Figure17: Completetopology in anotherperspectivethanFig. 10
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