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

Tran

sver

se S

tress

[MPa

]

0

20

40

60

80 HexagonalSquareTuned MPSExperimental

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

September 6, 2017 1 / 1

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

September 6, 2017 1 / 1

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.