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 / 1

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 2

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 3

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 4

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 5

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 . ✷ 6

Quadrilateral duality Con n ect opposite quad edge cen terpoin ts Form s arran gem en t of curves 7

Hexahedron duality Fin d curve arran gem en ts on hexahedron faces Con n ect by squares m eetin g in hexahedron cen ter 8

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 9

What’s wron g with Mitchell-Thurston ? Produces too m any hexahedra Ω ( n 3 / 2 ) : Ω ( n 2 ) : Doesn ’t result in geom etric em beddin g. 10

New Algorithm I. Cover Boun dary w/ Hexahedral Tiles 11

New Algorithm II. Tetrahedralize in terior an d partition tetrahedra in to hexahedra 12

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 13

New results • Com plexity boun d for topological hex m esh: If polyhedron has 2 n 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) 14

Open Problem s • Geom etric m esh (con vex polyhedral hex’s)? – More com plicated boun dary layer – May be Ω ( n 2 ) 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? 15