Scott A. Mitchell Patrick M. Knupp Unstructured Primal-Dual...
Transcript of Scott A. Mitchell Patrick M. Knupp Unstructured Primal-Dual...
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DOEASCRAppliedMathematicsPrincipalInvestigators'(PI)Meeting,Rockville,MD,September11-12,2017
A Delaunay triangulation (black) and its dual Voronoi polygon mesh (gray). The primal edge σ (red) is dual to the Voronoi edge *σ, which has green and blue half-edges contributed by the adjacent blue and green triangles.
The HOT energy depends on |σ| and |*σ|. HOT bounds the discretization error of the system of equations Ax=b given by the DEC formulation of the PDE. The error arises from the diagonalization of the dual Hodge-star operator. The error roughly corresponds to how well-
centered the mesh is, how close the dual vertices are to
the triangles’ barycenters.
Weinvestigatethemathandalgorithmsforprimal-dualmeshqualityimprovementandgeneration.Weadvancedsimultaneously-goodprimal-dualmeshquality,resamplingnodesformeshimprovement,anddualVoronoipolyhedralmeshgeneration.
Abstract Conclusions and Future Work
Motivation
Unstructured Primal-Dual Mesh Improvement and GenerationScott A. Mitchell1
Patrick M. Knupp2 1Sandia National Laboratories
2Dihedral
References1. MeshesOptimizedforDiscreteExteriorCalculus,techreport2. AConstrainedResamplingStrategyforMeshImprovement,CGF3. DiskDensityTuningofaMaximalRandomPacking,CGF4. Spoke-DartsforHigh-DimensionalBlueNoiseSampling,acceptedtoTOG5. ASeedPlacementStrategyforConformingVoronoiMeshing,CCCG6. VoroCrustSurfaceReconstructionConditions,manuscript7. VoroCrust:Clipping-FreeVoronoiMeshing,manuscript
Teammembers:A.Abdelkader(UMD),M.A.Awad(UCDavis),C.L.Bajaj(UTAustin),M.S.Ebeida(SNL),S.A.English(SNL),P.M.Knupp(Dihedral),A.H.Mahmoud(UCDavis),D.Manocha(UNC),S.A.Mitchell(SNL),S.C.Mousley(UIUC),C.Park(UNC),A.Patney(NVIDIA),J.D.Owens(UCDavis),A.A.Rushdi(SNL),D.Yan(CAS),andL.Wei(HKU).
ResultsPrimal-DualQualityMetricAnalysis,Design,andOptimization.WediscoveredthatwhiletheHOT-energymetricmaybeareasonablewaytoevaluatequality,thecommunitypracticeofdirectlyoptimizing itleadstoundesirablebehavior.Inparticular,HOThasnobarriertomeshinversionandbadlocalminimaexist.Becauseoftheselimitations,thecommunityreliesonpreconditioningbyLloyd’siterations,asinCentroidalVoronoiTesselation(CVT).
WedesignedametricHOTthatovercomestheseshortcomings,isunitlessandscaleinvariant,andhasbettervaluesovernon-Delaunaymeshes.Ournewmetricholdsthepromiseofbetterbehaviorunderoptimization,whilestillreducingdiscretizationerror[1].
Primal-dual meshes, otherwiseknownaswell-centeredmeshes,finduseinDiscreteExteriorCalculous(DEC)formulationsofPDE’sandplayaroleincompatiblediscretizations.Mirroringthegeometricorthogonalityofprimalanddualelements,thesearemostwellsuitedforsimulationsinvolvingorthogonalphenomena,suchaselectro-magnetism,andotherdivergence-curlphenomena.BesidesPDE’s,primal-dualmeshesareubiquitousingeometryprocessing,forapplicationsincludingthedesignofself-supportingstructures,andthedesignandmodelingoftightpackings.
Areas in which we need helpWeseekpartnershipswithsimulationgroupsbasedonDECorpolyhedralmeshes!Themajorgapweseektocloseistyingourresultstosimulationpractice.Inparticular,whilethetheoryconnectsmeshqualitytosimulationerror,wewishtodiscoverhowtightthisconnectionis,andwhethercoarsermeshesandworsequalitysufficeinpractice.
Dualpolyhedralmeshesareinterestingandusefulinandofthemselves,outsidetheirroleinprimal-dualmeshes.ThecellsofaVoronoitessellationboastmanypropertiesthatmakethemusefulforpolyhedralmeshgeneration,suchasconvexityandflatfacets.PolyhedralmeshgenerationfindsuseinfracturemechanicswithintheLabscomplex,andisnowwidespreadinindustryforfluidflowandsimilarphenomena,e.g.CD-adapco.Polyhedra’sefficiencyinfillingspace(i.e.alownode-to-cellratio),leadstocalculationefficiency.Polyhedraprovideflexibleelementformulations.
Areas in which we can helpWeseektodeliverageneral-purposeprimal-dualmeshimprovementlibrary,usefulacrossmanytypesofnext-generationsimulations.Whiletheliteraturehasdemonstratedsomeprimal-dualmeshoptimization,noopen-sourcecapabilitiesexist.
Wederivemathinsupportofgeneral-purposedualpolyhedralmeshgeneration.LANL’sCenterforNonlinearStudieshasexpressedinterestinVoroCrustmeshes.Sandia’sEngineeringSciencesCenterhasdemonstratedfracturemechanicsusingpriorversionsofourVoronoi-basedmeshing.
• Primal-dualmeshqualityanalysis,andoptimizationtoreducediscretizationerror• Sampling-basedframeworkforimprovingmultiplemeshqualitiessimultaneously• DualVoronoipolyhedralmeshgenerationalgorithms,and
derivationofmathematically-sufficientconditionsforprovablycorrectoutput
Wereachedadepthofmathematicalunderstandingofprimal-dualenergymetricsthatenabledthedesignofanewmetricthatshouldbehavebetterunderoptimization.Near-futureworkisactuallyoptimizingthisnewmetric,tuningforefficiency,andachievingthepropertiesthatarethemostbeneficialinpractice.Whileresamplingcanproduceawell-centeredprimal-dualmesh,weseektodemonstrateitoverenergymetricsmorecloselytiedtosimulationerror.Extendingtoweightedmeshesischallenging:weunderstandthemathematicsofthemetricanditsgradientwithrespecttonodeweights,butanalgorithmtosimultaneouslymodifybothweightsandpositionsischallengingduetotheirdifferentunitsandscales.
Besidesmeshes,wedemonstratedbuildinganaccuratemodelofafibermaterial.
ApproachThemajorstrengthswebuildonareourmathematicaldepthincomputationalgeometry,sampling,andmeshoptimization.
• Forthedesignandanalysisofmeshmetricsforqualityoptimization,weexploitmetricpropertiesthatmakethemwell-behavedunderiterativelocalmeshoptimization.
• Forsampling-basedmeshimprovement,weexploitthatsamplingisinsensitivetohowwellbehavedametricis,e.g.nogradientsarerequired,andtakeadvantageofhowconstraintsatisfactioneasilysupportsmultipleobjectives.
• FortheproofofVoroCrustcorrectnessweexploitthemathematicsoflocal-featuresize(lfs)samplingtoboundhowclosethemeshistothedomainboundary.
VoroCrustDual-PolyhedralMeshing.Wedevelopedthe“VoroCrust”theoryandalgorithmsforpolyhedralmeshgenerationandsurfacereconstruction[5,6,7].ItworksbygeneratingVoronoicellswhoseboundariesmatchaprescribeddomain.
Wediscoveredsufficientconditionsforprovablycorrect meshingandreconstruction,forbothsmoothmanifoldsandpiecewiselinearcomplexes.Sufficient Conditions for VoroCrust Polyhedral Meshing of Smooth Manifolds
VoroCrust produces a polyhedral mesh matching a prescribed set of triangles:
• The intersection of dual power spheres around three input samples (mesh nodes) produces two primal seeds.
• The seeds’ Voronoi cells have the input triangle as their common boundary.
Wasserstein distance
QualityImprovementbyResampling.Wedemonstratedeliminatingobtuseanglessotrianglesarewell-centered.Weproducedageneralframeworkforlocalresamplingofmeshnodes[2,3,4]• Supportmultiplequalityobjectives,ensurenonedegrade.• Nodesmoverandomlywithinthelocalfeasiblespace,areinsertedanddeleted.• Locally“satisfice”qualityratherthanoptimizeit.Constraintsactlikebarriers.• Gettingstuckinanundesirablelocalminimumispossible,butrare.
Our “Tuned MPS” material model, with correct fiber-to-epoxy ratio and random locations of fiber strands, gives a more accurate simulation of fracture response [3].
Transverse Strain [%]0 0.2 0.4 0.6 0.8 1
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SandiaNationalLaboratoriesisamultimissionlaboratorymanagedandoperatedbyNationalTechnologyandEngineeringSolutionsofSandia,LLC,awhollyownedsubsidiaryofHoneywellInternational,Inc.,fortheU.S.DepartmentofEnergy’sNationalNuclearSecurityAdministrationundercontractDE-NA0003525.
Two local minima, both collapse triangles!
HOT:
1. Replace formula-based bytheory-based extrapolation over non-Delaunay meshes.Energies stay positive, and partial barriers are created.
2. Scaling by 1/area2a. removes point “holes”
in barriers.b. improves landscape
“Horseshoe” Mesh PatchHOT behaves poorly
HOT behaves well
SAND2017-9545D
@
2
@x
2+
@
2
@y
2DEC
DiscreteExteriorCalculus
matrix AAx = b
�
⇤� = a+ ba
b
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HOT(�) = |�|| ⇤ �|W (µ�, µ⇤�)2 =a3|�|3
+a|�|3
12+
b3|�|3
+b|�|3
12
HOT(�) =1
area(N)2✓a3|�|3
+a|�|3
12
◆+
1
area(N)2✓b3|�|3
+b|�|3
12
◆
Features of HOT:
HOT (M(x , y)) ! 1 as (x , y) ! P(has inversion barrier)
HOT is dimensionless
HOT is scale invariant
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Optimizing HOT collapses triangles!
(0,-1)
(0,1)
(8,0)
Input
Landscape after 1, before 2
• Weighted ✏-sampling is ✏-sampling together with some larger ri = � lfs(pi)sample balls, for some constants ✏ and � with ✏ �.• Conflict Free. No ✏-radius sphere contains another sample with a larger lfs:8j 6= i, kpi � pjk � ✏min (lfs(pi), lfs(pj)) . [The smaller-disk condition.]• Sampling Conditions. Combining definitions, a conflict-free weighted✏-sampling with ✏ = 1/160 and � = 1/20 satisfies the sampling conditions.
• Theorem (Sampling Conditions are Su�cient.) VoroCrust produces ageometrically-close and topologically-correct reconstruction when the samplingconditions are satisfied. Moreover, with � = 8✏ 1/20, mesh M̂ ! M manifoldas ✏ ! 0 with Hausdor↵ distance h < 2 � sin(✓) lfs < 5.0 �2.