Lin ear-Com plexity Hexahedral Mesh Gen eration David Eppstein - - PDF document

Lin ear-Com plexity Hexahedral Mesh Gen eration

David Eppstein

• Dept. In form ation an d Com puter Scien ce

Un iv. of Californ ia, Irvin e http:/ / www.ics.uci.edu/ ∼eppstein /

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Mesh gen eration in theory

• Fin d trian gulation

– add diagon als to quadtree – in crem en tal Delaun ay refin em en t – circle packin g

• Prove som ethin g

– good elem en t quality – approxim ation to n um ber of elem en ts

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Mesh gen eration in practice

• Fin d prelim in ary m esh

– Often quadrilaterals or hexahedra in stead of trian gles an d tetrahedra

• Mesh im provem en t

– Move Stein er poin ts (sm oothing) – Split/ m erge elem en ts (refinem ent) – Other topological chan ges (flipping)

• Use the m esh

– Com putation al fluid dyn am ics – Other fin ite elem en t problem s – Fun ction in terpolation

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How to fin d a hexahedral m esh?

Why n ot just partition tetrahedra? Sharp an gles on boun daries can ’t be sm oothed Prefer to get good boun dary m esh, then fill

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Problem statem en t

Given polyhedron with quadrilateral sides Fin d hexahedral m esh respectin g boun dary

Does this octahedron have a hexahedral m esh?

Because problem is hard, relax it: fin d topological m esh (w/ curved cells) then worry about geom etric em beddin g

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Mitchell-Thurston solution

Theorem. A polyhedron (form inga topological ball) has a topological hexahedral m esh iff it has evenly m any quadrilateral sides. Proof: If: by duality from existen ce of span n in g surfaces. On ly if: every hexahedron has six sides. Every in tern al boun dary pairs up two sides. So extern al faces m ust be even . ✷

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Quadrilateral duality

Con n ect opposite quad edge cen terpoin ts Form s arran gem en t of curves

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Hexahedron duality

Fin d curve arran gem en ts on hexahedron faces Con n ect by squares m eetin g in hexahedron cen ter

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Mitchell-Thurston algorithm

• Fin d dual curves on boun dary
• Pair curves w/ odd self-in tersection s
• Span by surfaces
• Fix up so it has a valid dual
• Dualize to form m esh

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What’s wron g with Mitchell-Thurston ?

Produces too m any hexahedra Ω(n3/2) : Ω(n2) : Doesn ’t result in geom etric em beddin g.

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New Algorithm

• I. Cover Boun dary w/ Hexahedral Tiles

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New Algorithm

• II. Tetrahedralize in terior an d

partition tetrahedra in to hexahedra

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New Algorithm

• III. Fix up boun dary tiles
• Subdivide sides, leave outside faces un chan ged
• Use m atchin g in dual graph

to m ake all tiles have even # quads

• Apply Mitchell-Thurston to tiles

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New results

• Com plexity boun d for topological hex m esh:

If polyhedron has 2n quadrilateral sides, it has a m esh with O(n) hexahedra.

• Som e exten sion s to polyhedra

that don ’t form topological balls (if boun dary form s bipartite graph)

• Som e progress in geom etric em beddin g

(reduction to fin ite case an alysis)

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Open Problem s

• Geom etric m esh (con vex polyhedral hex’s)?

– More com plicated boun dary layer – May be Ω(n2) in terior hexahedra – Can ’t apply Mitchell-Thurston

• Non -sim ply-con n ected dom ain s?

– We have som e sufficien t con dition s – Not both n ecessary an d sufficien t

• Quality of elem en ts?

– How good is result of algorithm ? – How easy is it to sm ooth?

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