Tetrahedral Mesh Generation with Good Dihedral Angles Using

99 Pages·2007·2.98 MB·English

Tetrahedral Mesh GenerationwithGood DihedralAngles
UsingPointLattices
by
Franc¸oisLabelle
B.Sc. (McGillUniversity)1997
M.Sc. (McGillUniversity)2000
A dissertationsubmittedinpartial satisfactionofthe
requirementsfor thedegreeof
DoctorofPhilosophy
in
ComputerScience
inthe
GraduateDivision
ofthe
UniversityofCalifornia, Berkeley
Committeeincharge:
ProfessorJonathanShewchuk,Chair
ProfessorJames F. O’Brien
ProfessorPanayiotisPapadopoulos
Fall2007
ThedissertationofFranc¸oisLabelleisapproved:
Chair Date
Date
Date
UniversityofCalifornia, Berkeley
Fall2007
Abstract
Tetrahedral Mesh GenerationwithGood DihedralAngles
UsingPointLattices
by
Franc¸oisLabelle
DoctorofPhilosophy
in
ComputerScience
UniversityofCalifornia, Berkeley
ProfessorJonathanShewchuk,Chair
Three-dimensionalmeshesarefrequentlyusedtoperformphysicalsimulationsinsci-
ence and engineering. This involves decomposing a domain into a mesh of small ele-
ments, usually tetrahedra or hexahedra. The elements must be of good quality; in partic-
ular there should be no plane or dihedral angle close to 0 or 180 degrees. Automatically
creatingsuchmeshesforcomplicateddomainsisachallengingproblem,especiallyguar-
anteeing good dihedral angles, agoal thathas eluded researchers for nearly two decades.
By using point lattices, notably the body centered cubic lattice, we develop two tetrahe-
dralmeshgenerationalgorithmsthat,forthefirsttime,comewithmeaningfulguarantees
onthequalityoftheelements.
Fordomainsboundedbyanisosurface,wegenerateatetrahedralmeshwhosedihedral
anglesareboundedbetween10.7and164.8degrees,or(withachangeinparameters)be-
tween8.9and 158.8degrees. Thealgorithmisnumericallyrobustandeasy toimplement
because it generates tetrahedra from a small set of precomputed stencils. The algorithm
isso fastthat itcan beinvokedat each timestepofasimulation,possiblyinreal timefor
smallmeshes. Thetetrahedraareuniformlysizedontheboundary,butintheinterioritis
possibletomakethemprogressivelylarger. Thiscombinationoffeaturesmakesthealgo-
rithm well suited for dynamic fluid simulation. If the isosurface is a smooth 2-manifold
with bounded curvature, and the tetrahedra are sufficiently small, then the boundary of
themeshisguaranteedtobeageometricallyandtopologicallyaccurateapproximationof
theisosurface.
For polyhedral domains, Delaunay refinement is a common mesh generation tech-
niquethatproducesguaranteed-qualitytetrahedrawithoneexception: sliver-shapedtetra-
hedra that can have dihedral angles arbitrarily close to 0 and 180 degrees. We show how
sliversawayfromtheboundarycanbeavoidedbyinsertinglatticepointswhilemaintain-
ingtheDelaunaypropertyofthemesh. Itispossibletocontrolthesizeoftetrahedra, this
timeboth in the interior and on the boundary, and a user may input a set of points which
1
must be vertices of the output mesh. The resulting dihedral angles are guaranteed to be
between 30and 135degrees, exceptnear theboundary.
Mostangleboundsareobtainedbyrecursivebisectionofthespaceofpossibletetrahe-
dron configurations together with interval arithmetic. They are guaranteed by computer-
assistedproofs.
Chair Date
2
Acknowledgments
I would like to thank my advisor Jonathan Shewchuk for his guidance and support,
and the Berkeley Graphics group for feedback. I also thank Nuttapong Chentanez for
providingisosurfacecodeandgeometricmodels,andCarloSe´quinfortheWhirledWhite
Web model.
This work was supported in part by the National Science Foundation under Awards
CCR-0204377, CCF-0430065 and CCF-0635381, and in part by an Alfred P. Sloan Re-
search Fellowship.
i
Contents
1 Introduction 1
1.1 TheMeshGeneration Problem . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 DomainRepresentation . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Mesh Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 QualityMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Size-Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Point Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Summary ofResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Previous Work 11
2.1 Background Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Surface Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 VolumeMeshes . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Quadtreeand Octree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Delaunay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 OtherMeshGeneration Techniques . . . . . . . . . . . . . . . . . . . . 18
3 Tetrahedral Meshing InsideanIsosurface 20
3.1 UniformTetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Creating theBackground Grid . . . . . . . . . . . . . . . . . . . 24
3.1.2 ComputingCut Points . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.3 Warping theBackground Grid . . . . . . . . . . . . . . . . . . . 25
3.1.4 TriangulatingtheBackgroundGrid . . . . . . . . . . . . . . . . 28
3.2 Mesh Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Computer-AssistedProofs ofAngleBounds . . . . . . . . . . . . . . . . 32
3.3.1 ExtremaofaFunctionofnVariables . . . . . . . . . . . . . . . 34
3.3.2 ExtremaofElementaryGeometricComputations . . . . . . . . . 35
3.3.3 Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.4 PlaneAngles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
ii
3.4 ApproximationGuarantees . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 Arbitrary Isosurfaces . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.2 Isosurfaces withBounded Curvature . . . . . . . . . . . . . . . . 42
3.5 Graded InteriorTetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 FourKindsofTetrahedra . . . . . . . . . . . . . . . . . . . . . . 48
3.5.2 Non-Standard Octree . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.3 ConditionsCloseto theIsosurface . . . . . . . . . . . . . . . . . 51
3.5.4 InteriorGrading . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5.5 MeshGeneration . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Delaunay Refinement WithoutSlivers 57
4.1 Delaunay Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.1 Delaunay TriangulationAlgorithms . . . . . . . . . . . . . . . . 59
4.1.2 DataStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Simplified MeshGeneration . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.1 Sizing Functionand LocalFeature Size . . . . . . . . . . . . . . 61
4.3 UniformTetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Graded Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 AddingVertex Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Bibliography 85
iii
Chapter 1
Introduction
Meshes of small elements are often used to simulate physical phenomena numerically
on a computer. In engineering, numerical simulations can help design and test planes,
bridges or components before they are built. This can decrease costs and increase de-
velopment speed where prototypes were once needed. In science, the same techniques
can be used for a wide range of problems like modeling tides or star interiors. Due to
thesuccessofthesemethods,moredifficultproblemsaretackled,forexampleinbiology
where geometry tends to be complicated and curved, hence harder to mesh. Figure 1.1
showsexampleapplications.
In computer graphics, the traditional thinking is that only the surface of an object is
visibleso onlysurface meshes are necessary. However,volumemeshesare used increas-
inglyasresearchersincorporatephysicalsimulationsingraphicstoautomaticallyanimate
liquids and gasses, or objects that bend, crack or break. As computing performance im-
proves, some of these simulations are starting to appear in computer games to make the
virtualenvironmentmorerealistic.
The finite element method (FEM) is the most common way of solving a partial dif-
ferential equation (PDE) overa meshed domain. The theory for ellipticproblems is well
understoodandenjoyssomeniceresults;forexampleapiecewiselinearapproximationis
sufficient to solve an elliptic PDE of order 2 (such as heat diffusion or elasticity), and in
generalapiecewiseapproximationofdegreemissufficienttosolveanellipticPDEofor-
der2m. SeethebookbyJohnson[48]foranintroductiontotheFEMandthemathematics
behindit.
The meshcan followthe material as it moves(Lagrangian formulation),or stay fixed
in space as the material flows through it (Eulerian formulation). Those two possibilities
arenormallyusedforsolidsandfluids,respectively,butthereverseistechnicallypossible.
There is a third possibility: the mesh is allowed to move but doesn’t necessarily have to
track the material. This extremely flexible formulation is called Arbitrary Lagrangian
1
Chapter 1. Introduction
Figure 1.1: Part of an exterior mesh used for computational fluid dynamics and
part of a mesh of a femur used for stress analysis.
Eulerian(ALE); seeforexamplethesurveyofDoneaet al. [30].
Inverydifficultapplications,somepeoplehavedevelopedmethodsthatdonotrequire
amesh,called“meshfreemethods”. Onecommonlycitedearlyexampleissmoothedpar-
ticle hydrodynamics, introduced in 1977 by Lucy [60] and Gingold and Monaghan [44].
Meshfreemethodshavesomedrawbacks: theyarecomputationallyexpensiveandbound-
ary conditions are harder to impose. Ironically, a mesh is often used at one step of the
method,for exampleto perform thenumerical integrationofthespecial basisfunctions.
1.1 The Mesh Generation Problem
Givenadescriptionofthedomaingeometry,thegoalistoproduceameshofelementsthat
fillthedomain. Sometimeswearealsogivenasizingfunctiondefinedoverthedomainto
controlelementsizes.
The mesher doesn’t usually depend on other details of the application (for example
which differential equation is going to be solved with the mesh). This means that the
samemesh generationcodecan beused formanydifferentapplications.
Formorebackgroundonmeshgeneration,seethesurveybyBernandPlassmann[10].
1.1.1 Domain Representation
Domaingeometrycanberepresentedinmanydifferentwayswhichweclassifyasexplicit
orimplicit.
2
Chapter 1. Introduction
Figure 1.2: A piecewise linear complex can be the input to a mesh generator
(J. Shewchuk).
ExplicitRepresentation
Inanexplicitrepresentation,wearegivenalistofvertices,segmentsandfacetsbounding
the domain. A common example is a piecewise linear complex [62] which can be used
torepresent polyhedraldomains(internalvertices,segmentsandfacetsarealsoallowed).
Self-intersectionisnotallowedunlesstheintersectionisexplicitlyresolvedwithvertices,
segmentsand facets. See Figure1.2.
When meshing such a domain, input vertices must be part of the output mesh. Input
segmentsand facets mustappearunchanged oras a unionofsubsegmentsorsubfacets.
Extension to curved segments and facets is possible [72, 15]. Sharp features are ex-
plicitlylistedandmustbeaccuratelyrepresentedbythemesh. Thisissimilartoboundary
representation(B-rep) in computer-aideddesign.
ImplicitRepresentation
In an implicit representation, the domain is defined as the volume bounded by a given
isosurface. Anisosurfaceisasurfaceofconstantvalue p : f(p) = c wheref : R3 R
{ } →
is a scalar field called a cut function that the mesher can interrogate. See Figure 1.3.
Minimally, the domain can be defined implicitly by a black box that tells if a point is
“inside”or“outside”withoutreturninganynumericalvalue.
This is a convenient representation in many cases. For example when simulating
a moving liquid, an inside/outside query at a point p for the current time step can be
answered by moving p backward in time using the estimated velocity field and querying
the previous time step—a method called semi-Lagrangian advection [7]. By contrast,
maintainingan explicitrepresentationofaliquidsurfacewouldbeconsiderablyharder.
Isosurfacesareproducedbyseveralalgorithmsforsurfacereconstruction[88,67,77].
Even for geometric models that do not use isosurfaces, it is usually possible to compute
3