Tensorfieldvisualiza0on–Myviewonthefield
January2016
GeilowinterSchool–Scien0ficVisualiza0on
IngridHotz–LinköpingUniversity
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Tensor field visualization – My view on the field
IngridHotz
Fromthegeneralperspec0ve…
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Tensor field visualization – My view on the field
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…todigginginthesand
Tensorfieldvisualiza0on–Myviewonthefield
January2016
GeilowinterSchool–Scien0ficVisualiza0on
IngridHotz–LinköpingUniversity
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Tensor field visualization – My view on the field
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Tensors–WhyDoWeCare?
§ Tensorsareeverywhere(Simula0ons,physicaltheories)§ Canhardlybeseenanywhere–nobodycares?
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Tensors–WhyDoWeCare?
Ifyou’resiWngatacocktailpartywithabunchofengineers,physicists,andmathema0cians,andyouwanttostartaheateddebate,
Justaskoutloud:“Whatisatensor?”
• Onepersonwillsaythat,forallprac0calpurposes,atensorisjustafancywordforamatrix.
• Thensomeoneelsewillpipeupindignantlyandinsistthatatensorisalineartransforma;onfromvectorstovectors.
• Yetanotherpersonwillsaythatatensorisanorderedsetofnumbers
thattransforminapar;cularwayuponachangeofbasis.
• Otherfolks(likeus)willstartbabblingabout“dyads”and“dyadics.”
[R. Brannon, Functional and Structured Tensor Analysis for Engineers, UNM Book Draft, 2003]
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Tensors–WhyDoWeCare?
• Vectorvisualiza0on• Intui0veno0onofavector
• PrevalentApplica0ons
• Somedrivingques0ons
• Longvisualiza;onhistory,wellaccepted
• Tensorvisualiza0on• Intui0veno0on?
• Manydifferentapplica0ons
• Onlyafewsharedques0ons
• Novisualiza;onhistory,hardlyknown
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Tensors–WhyDoWeCare?
Nocommonlanguageacrossapplica0onsButterminologymostlyrelatedtospecifictensorinvariants
àBasisofourvisualiza0onconcept
EigenvaluesEigenvectors
Oeenscien0stshavenogluehowtoexpectfromthetensordata
adaptedfromtalkbyMiriahMeyergivenatDagstuhl2011??
Data??
Head Computer
Fuzzy Crisp
Questions
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Tensor field visualization – My view on the field
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Tensors–WhyDoWeCare?
Tensorvisualiza0onisfieldthatiss0llinitsinfancy(maybeexceptforDiffusionTensorImaginghoweverthisdoesnotreallyhappeninthevisualiza0oncommunity)
Overview
I. TensorsAsmathema0calobjectsAsphysicaldescriptors
II. Somebasicvisualiza0onmethodsIII. TensorInvariantsforfeaturedefini0onIV. Thestoryofacollabora0on
I. Tensorsasmathema0calobjects
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Second-ordertensorTisabilinearfunc0on
Vn-dimvectorspace
v1,v2 ∈V2
T v1,v2( )∈!
T :V ⊗V → !
T (v,w) = wT ⋅T ⋅v, ∀v,w∈V , T∈!n×n
Second-ordertensorTaslin.operator
T :V →VT (v) = T ⋅v, ∀v∈V
ThisisnotatensorbutamatrixOeenatensorismixedupwithitsrepresenta0on
T =
t11 … t1n
! " !
tn1 # tnn
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
ITensorsasmathema0calobjects
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ITensorsasmathema0calobjects
Thetensorisuniquelydeterminedbyitsac0ononallunitvectorsRota0on,deforma0on,reflec0on
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Tensor field visualization – My view on the field
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ITensorsasmathema0calobjects
Symmetrictensor
Deforma0on,reflec0on–norota0on
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ITensorsasmathema0calobjects
Posi;vedefinite,symmetrictensor
Deforma0on–norota0on,noreflec0on
S
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ITensorsasmathema0calobjects
SymmetrictensorEigenvectorsandeigenvalues
§ Tensorscanbeanalyzedusinganyconvenientreferenceframe
§ Forspecificreferenceframes,thetensorrepresenta0onbecomesespeciallysimple
à Basisgivenbytheeigenvectorsofthematrix
Tei = λiei
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ITensorsasmathema0calobjects
TherearedifferenttensorDecomposi0on,whichareoeenthebasisforvisualiza0onmethods
§ Polardecomposi0on–stretch-rota0on
Concatena0onofmappings
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ITensorsasmathema0calobjects
Sumofmappings
sij =12tij + t ji( ).
S = 12T + TT( ),
Symmetricpart
aij =12tij − t ji( ),
A =12T − TT( ),
An0-symmetricpart
TherearedifferenttensorDecomposi0on,whichareoeenthebasisforvisualiza0onmethods
§ Symmetricandan0-symmetricpart
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ITensorsasmathema0calobjects
§ Manydifferentmeasuresforanisotropyavailable
TherearedifferenttensorDecomposi0on,whichareoeenthebasisforvisualiza0onmethods
§ Isotropicscaling,anisotropicdeforma0on(deviator)
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ITensorsasmathema0calobjects
Examples� Eigenvalues� Determinant� Trance� Allfunc0onsthatonlydependontheeigenvalues
Tensor Invariants
En00esthatdonotdependontherepresenta0onProper0esinherenttothetensor
ITensorasphysicaldescriptors
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ITensorasphysicaldescriptors
Firstorderdescriptorsofthedependenceoftwovectorfields
àfirsttermoftheTaylorexpansion
Theyprovideamoreorlessgoodapproxima0onofthereality
v(w) = T ⋅w + higher order terms
v,w
TensorsareeverywhereinPhysicsandEngineering
§ Notbecausetheworldislinear§ Butbecauselinearissimple
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Gradient tensor
Stress and strain tensor
x
z
y
σy σz
σx σx
τxy
σy σz τyx
Metric tensor
Curvature tensor
Diffusion tensor
ITensorasphysicaldescriptors
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StressTensor
Geo-andMaterialSciences
1. Solidblock,withtwoappliedforces2. Implantdesign,stresssimula0oninhumanbone3. Notchedblockwithexternalforces
Images: Kratz, ZIB
ITensorasphysicaldescriptors
3
2
1
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ITensorasphysicaldescriptors
Metric,Curvature,Stress
GeneralRela0vity• Simula0onofgravita0onalfieldofa
rota0ngblackhole,respec0velyneutronstar
Images: Benger, Kratz, ZIB
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ITensorasphysicaldescriptors
DiffusionTensor–Medicine
• Imagingmethod:basedonmagne0cresonancetomography(MRT)measuringthediffusionofwatermoleculesin0ssues
• Applica0on:Reconstruc0onofneuralfibersinhumanbrain(tractography)
Textured slice 3D Fiber tracking Glyph representations
Images: Kratz, Breßler, Hotz, ZIB
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ITensorasphysicaldescriptors
StructureTensor–ImageAnalysis
Image: Kratz, ZIB
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§ Indefinite§ E.g.stresstensor(forces/
area)§ TraceTàvolumechange
Describesadeforma;on
§ Posi0vedefinite§ E.g.deforma0ontensor§ DeterminantdetTàvolume
change
Describesageneratorofadeforma;on
v ' = T ⋅v v ' = etT ⋅v≈ 1+ tT( ) ⋅v
DifferentCharactersofTensors-Examples
ITensorasphysicaldescriptors
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ITensorasphysicaldescriptors
• Playafundamentalroleintheunderstandingofapplica0onspecifictensors
• Everyapplica0onhasitsowninvariantsthatareespeciallyimportant.Thyoeencomewithadomainspecificlanguage
àRelevantinvariantsshouldguidethechoiceofthevisualiza0on
Tensor Invariants
IISomebasicvisualiza0onmethodsWhathasvisualiza0oncurrentlytooffer
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IISomebasicvisualiza0onmethods
Commonprac;ce
• ColorRepresenta0onofderivedscalars,e.g.trace
• 2Dslicesandsurfaces
Stresssimula0on,construc0onpieceM.Stommel,UniSaarbrücken B.Hirschberger(2004),
UniKaiserslautern
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IISomebasicvisualiza0onmethods
Thetypicalglyph:Ellipsoid
LocalMethods-Glyphs
• Geometricobjectsrepresen0ngtensorcharacteris;cs
• Here:Ellipses• Mostfrequentlyusedvisualiza0on
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IISomebasicvisualiza0onmethods
Manydifferentglyphtypesinuse
Whentouse?
• Evalua0onofdataquality• Visualiza0onoverview• Probing
[Schultz2010,Kratz2014]
ExamplefromMaterialSciences
LocalMethods-Glyphs
Glyphplacement
� Howmanyglyphs?� Wheretoplacethem?
Glyphdesign
� Whatshouldberepresented?� Howshoulditbe
represented?
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IISomebasicvisualiza0onmethods
Howshoulditberepresented–generaldesignguidelines• Preserva0onofsymmetry
e.g.eigenvectorshavenoorienta0on,isotropictensorshavenotdis0nguisheddirec0on
• Con0nuitysimilartensorsshouldhavesimilarglyphrepresenta0ons
• Disambiguatetensorswithdifferentvaluesshouldbereflectedbydifferentglyphs
• Useglyphsthathavebeenusedinthecommunitybefor[SchultzandKindlmann2010]
Whatshouldberepresented–applica0onspecificguidelines• Useapplica0onspecificinvariantsforthedesign
LocalMethods-Glyphs
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IISomebasicvisualiza0onmethods
[Kindlmann2004]
LocalMethods-Glyphs
ResolvingPercep0onalIssuesSuperquadrics
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IISomebasicvisualiza0onmethods
[Schulz2010]
LocalMethods-Glyphs
ResolvingPercep0onalIssuesSuperquadrics
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IISomebasicvisualiza0onmethods
Images:Kratz,BresslerZIB,Amira
LocalMethods-Glyphs
Glyphplacement
� Howmanyglyphs?� Wheretoplacethem?
Anisotropicsampling
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IISomebasicvisualiza0onmethods
Par0clebasedandgeometricmethods onplanes,in3d,onsurfaces
[Feng2008] [Kindlmann2006] [Kratz2013]
LocalMethods-Glyphs
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IISomebasicvisualiza0onmethods
§ Integrallinessimilartostreamlines
§ Followoneeigenvectorfield
Tensorlines
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Tensor field visualization – My view on the field
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IISomebasicvisualiza0onmethods
Direc0onfieldisnotorientable.Itisnotavectorfield!!
Tensorlines
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Tensor field visualization – My view on the field
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IISomebasicvisualiza0onmethods
Dick2009
Frey2001
Tensorlines
Gravita0onalfieldoftwocollidingneutronstars
Images:Zobel,ZIB
Implantdesign
Forceappliedtosolidblock
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Tensor field visualization – My view on the field
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IISomebasicvisualiza0onmethods
TensorsfromdiffusionMRITrackingofneuralfibers• Majortensorlinescanbeusedto
approximatefibers• Linetracingonlyforregionsof
highanisotropy
Tensorlines–Fibertracking
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IISomebasicvisualiza0onmethodsTextures
FabricTexture
Samplingbasedtextures
HyperLIC-2dkernel
SplaWngglyphs[Benger2006]
FabricTexture
Topologybasedtextures
[Pang2003] [Hotz2004] [Eichelbaum2013]
Kratz2011 Auer2012
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IISomebasicvisualiza0onmethods
Textures–Principaldirec0onsandvalues• Beambendunderitsownload.• Simula0onrespec0ngevolvingdamage,30mesteps• Data:LouiseKellogUniversityofCalifornia.
Data:LouiseKellogg,DepartmentofGeology,UniversityofCalifornia,Images:LouisFengUCD.
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IISomebasicvisualiza0onmethods
Topologicalstructure§ Segmenta0onofdomaininareasofuniformdirec0onalbehavior§ Similartovectorfieldtopology–butdifferentbasicstructures
Topology
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Tensor field visualization – My view on the field
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IISomebasicvisualiza0onmethods
OnSurfaces[Auer2012]
§ TopologybasedSegmenta0on
§ TopologybasedTextures[Auer2013]
Tensorfielddesign[Zhang2007]
3Dtopology,cri0callines[Zheng2005]
Topology
Computa0on,Simplifica0on,Tracking[Tricoche2001-2003]
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Tensor field visualization – My view on the field
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IISomebasicvisualiza0onmethods
Sometechnicalstuff
• Glyphsfromallperspec0ves• Technicalgeneraliza;onofvectorfieldvisualiza;onmethods
• Textures
Whatiss;llmissing
• Applica0ons–linkformvisualiza0ontechniquestophysicalinterpreta0ons• No0onoffeatures• Ques0ons
Muchworkontensorprocessing
• Interpola0on/reconstruc0on• Morphology• Topology
Whathasbeendonesofar?
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Tensor field visualization – My view on the field
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IIITensorInvariantsforfeaturedefini0on
• Exploara;veframeworksuppor0ngmul0pleapplica0onswithdifferentques0ons
àDomainspecificfeaturespaces
• Structuringthedata
� Managecomplexityofthedata
� Highlighttrends
� Pointatcri0cal/atypicalbehavior
àDataatlasusingathumbnaillike
representa;on
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Tensor field visualization – My view on the field
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IIITensorInvariantsforfeaturedefini0on
• Explora;veframeworksuppor0ngmul0pleapplica0onswithdifferentques0ons
àDomainspecificfeaturespaces
• Structuringthedata
� Managecomplexityofthedata
� Highlighttrends
� Pointatcri0cal/atypicalbehavior
àDataatlasusingathumbnaillike
representa;on
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Tensor field visualization – My view on the field
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IIITensorInvariantsforfeaturedefini0on
Tensorinvariant
• Arethelanguageoftheapplica0ons
Tensorisuniquelydefinedby• 3eigenvalues à pointin‘shape-space’• 3eigenvectors à pointin‘direc;on-space’
Useapplica0onspecificinvariantstoparameterizetheshapespace
à shapedescriptorsIi(λ1,λ2,λ3)
à basisforthedefini0onsta0s0cviews,glyphdesign,similaritymeasure
Tensorinvariant
• Arethelanguageoftheapplica0onsTensorinvariant
• Arethelanguageoftheapplica0ons
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TheStressTensorandFailureModels
ExampleforfailureanalysisinmechanicalengineeringàMohrCircle
τmax
c
τ0
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Tensor field visualization – My view on the field
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IIITensorInvariantsforfeaturedefini0on
Example• Orderedshapespace
λ1 ≥ λ2 ≥ λ3
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Tensor field visualization – My view on the field
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IIITensorInvariantsforfeaturedefini0on
Example–Coulombfailure• Parametrizedby� Mohrcenter
� MaxShear
� Shapefactor
c = λ1 + λ32
τ = λ1 − λ32
R = λ1 − λ2λ1 − λ3
∈[0,1]
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Tensor field visualization – My view on the field
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Clusterinfeaturespace Clusterrepresenta0ves• Reynoldsglyphs
Spa0alview:Topviewofdata
IIITensorInvariantsforfeaturedefini0on
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Tensor field visualization – My view on the field
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Clusterinfeaturespace Clusterrepresenta0ves• Mohrcircles
Spa0alview:Topviewofdata
IIITensorInvariantsforfeaturedefini0on
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Tensor field visualization – My view on the field
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Top-viewofthetwo-pointload
IIITensorInvariantsforfeaturedefini0on
Clustering• Meanshie• Fullfeaturespace+spa0alcoordinates
Clustersta0s0cs• Clustersize
• Direc0onalhistogramλ1
xy
-z
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Tensor field visualization – My view on the field
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Two-pointload
Spa;al-view
IIITensorInvariantsforfeaturedefini0on
Clustering• Meanshie• Featurespace(shear,shape)+spa0alcoordinates
Clustersta;s;cs
• Clustersize
• Direc0onalhistogram
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Tensor field visualization – My view on the field
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IIITensorInvariantsforfeaturedefini0on
Feature Space Selection § Change of basis results in another set
of shape descriptors § Determines
§ Exploration space § Metric, similarity measure § Glyph design
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IIITensorInvariantsforfeaturedefini0on
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Tensor field visualization – My view on the field
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IIITensorInvariantsforfeaturedefini0on
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IIITensorInvariantsforfeaturedefini0on
TensorVisualiza0onDrivenMechanicalComponentDesignorThestoryofacollabora0on
3D CAD-modelVirtual evaluation:
Simulation +Tensor line visualization
Rapid Prototyping and experimental validation:
Test setupBoundary geometry:
Design space definitionPart design: Hand sketch
guided by tensor lines
Kratz,Schoeneich,Zobel,Burgeth,Scheuermann,Hotz,Stommel,PacificVis2014
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Tensor field visualization – My view on the field
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IVStoryofaCollabora0on
Star0ngpoint• Unspecificgoals• Different
language
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Tensor field visualization – My view on the field
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IVStoryofaCollabora0on
FirstExperiments• Everybodyshows
whattheyhave
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Tensor field visualization – My view on the field
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Visualiza0on–Framework
Images: Kratz, ZIB, Amira
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Tensor field visualization – My view on the field
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Engineering–ProductDevelopmentProcess
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Tensor field visualization – My view on the field
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TensorsinMaterialSciencesEngineering–ProductDevelopmentProcess
Adds a completely new perspective to the topic
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Tensor field visualization – My view on the field
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Engineering–ProductDevelopmentProcess
ConstraintsRequirementSpeci1icationsDesign4rulesExperience
Initial'structure'
Structure4optimization
Introduce4visualization4using4tensor4lines
to4support4this4process
Experimental'validation
CAD7Model
FE7'simulationvalidation
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Tensor field visualization – My view on the field
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IVStoryofaCollabora0on
FirstExperiments• Everybodyshows
whattheyhave
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Tensor field visualization – My view on the field
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!"#$%&'(#%$)*+,(&*-*#%./*0(1(0'%("#$
!"#$#%&'($)*+$*),'
-./,)#0,"$%&'1%%$#3"
4567832,&
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.%&,0%,&*5"/%(-(4'%("#8'$*95"#5*:/*&(*#0*5'#959*$(6#5&,3*$
IVStoryofaCollabora0on
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Tensor field visualization – My view on the field
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IVStoryofaCollabora0on
Hypothesisresul0ngfromfirstexperimentsanddiscussions
• Thetensorlinesforastresstensorfieldarerelatedtothemajorloadpathsfromtheopera;ngloadstothefixa;on
pointsofatechnicalpart
• Tensorlinescanbeusedtosupportthedesignofreinforcementstructures
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IVStoryofaCollabora0on
Design of a break lever • Buildreinforcementstructureonbasis
oftensorlinevisualiza0on• Combina;onofanautoma;canalysis
withtheexpert’sknowledge
• Manageablebutrealis0ccase
Evalua;on
• Comparisontoreferencestructure–samevolume
• Numericalvalida0on• Experimentalvalida0on
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IVStoryofaCollabora0on
Designoftheribstructure
• Newgeometriesbasedontensorvisualiza0on
• Noaddi0onalstepsintheworkflow
Stress Tensor Visualization
CAD - Model Vis supported manual design
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Tensor field visualization – My view on the field
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IVStoryofaCollabora0on
Numericalandexperimentaltests
• Newdesigns(red,green,yellow)• Referencegeometry(blue)
àAllnewgeometriesperformedbexerasthereferencegeometry
3D CAD-modelVirtual evaluation:
Simulation +Tensor line visualization
Rapid Prototyping and experimental validation:
Test setupBoundary geometry:
Design space definitionPart design: Hand sketch
guided by tensor lines
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Tensor field visualization – My view on the field
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IVStoryofaSuccessfulCollabora0on
AdvancesinbothfieldsCon0nuingStory• Manyexi0ngques0ons
andideas• Muchfun• Proposalsubmixed
Tensors as mathematical objectsproperties and theorems
Tensors as physical descriptors
Tensor invariantsas means for
communication
Visualizationgeneric <-> specific
Relevance and meaning of mathematical properties and theorems is application dependent
• AnisotropicSamplingofPlanarandTwo-ManifoldDomainsforTextureGeneraAonandGlyphDistribuAon,Kratz,Baum,Hotz,TVCG,2013
• Three-DimensionalSecond-OrderTensorFields:ExploratoryVisualizaAonandAnisotropicSampling(phdthesis),AndreaKratz,2013
• VisualizaAonandAnalysisofSecond-OrderTensors:MovingBeyondtheSymmetricPosiAve-DefiniteCase,Kratz,Auer,Stommel,Hotz,ComputerGraphicsForum-StateoftheArtReports,2013
• TensorInvariantsandGlyphDesign,Kratz,Auer,Hotz,Visualiza0onandProcessingofTensorsandHigherOrderDescriptorsforMul0-ValuedData(Dagstuhl'11),Springer,2014
• hVp://www.tensorvis.org/