TOWARD QUALITY SURFACE MESHING

Transcription

1 TOWARD QUALITY SURFACE MESHING Jean Cabello EDS/PLM Soluton, 000 Eatman D, Mlfod, OH, USA ABSTRACT Th pape peent ecent poge and extenon to TQuaMeh (TQM [], tageted at povdng good qualty uface mehe: Inceaed obutne of the D meh geneato to handle hghly non lnea ze vaaton; nteo node geneaton dven by a ze vaaton ntepolaton doman; mpoved meh dtoton educton between the paamete pace and the phycal pace. The concept of Sze Contol, Sze Map and Tangle Map ae ntoduced to nceae the flexblty and the contol on the fnal meh. Thee concept ae geneal and apply to any mehng algothm, although they wll be llutated wth TQM. Keywod: uface meh geneaton, tangula, ze map, cuvatue adaptaton.. INTRODUCTION The mot common tetahedal mehng algothm, advancng font and Delaunay, eque the uface meh to be geneated ft, po to fllng n the nteo wth tetahedal element. Fo volume kn uface wth geodec dtance between two pont on the uface hgh compaed to the Eucldan dtance (.e. naow and hgh cuvatue paageway, adaptaton of the uface meh to the cuvatue mght be ctcal to the ucce of an automatc volume mehe by peventng geometcal uface meh nteecton. The ucce and the qualty of the volume mehe then dectly mpacted by the ucce and qualty of the uface mehe. Although temendou poge ha been made wth egad to mehng algothm n both two and thee dmenon, t tll eman a dffcult tak to uface meh any collecton of uface wth good qualty and ze contol. Many appoache ae avalable dependng on the uface defnton avalable (contnuou o dcete. Fo CAD paametc uface, a D paamete pace epeentaton of the uface avalable and uface mehng educed to a D mehng poblem. Howeve, the uface can be pooly paametezed leadng to hgh dtoton when mappng the meh back fom D to D pace. Some method have been peented to account fo the dtoton between the D and D pace ung the CAD Remannan uface evaluato []. Mot CAD ytem can expot an STL o facetted epeentaton of any paametc uface. Th a lowe level defnton of the uface that ha the advantage of a mple and common fomat ndependent of the CAD ytem. Fo dcete data epeentaton of the uface (STL data o legacy data, ome technque wok dectly on the D dcete data to obtan a good qualty meh [] whle othe ue a dvde and conque appoach to elect a egon and deve a paamete pace to educe the uface mehng poblem to D. The two mot common technque ued to deve a paamete pace ae: poecton technque, fo example Maxmum Aea Plane (MAP n I-DEAS, and flattenng technque baed on angle [4] o baed on length [5]. Adaptve mehng baed on eo etmaton anothe ntance whee contollng the meh ze vaaton to efne n aea of hgh eo and coaen n aea of low eo ctcal to obtan a good oluton wth educed node and element count. In th pape a mple method to account fo the dtoton between the D and D pace fo a uface epeented by STL data peented. Adaptve mehng baed on dcete uface cuvatue alo peented n ode to nceae the meh fdelty to the ognal uface at an economcal cot compaed to a contant ze meh.. TQM MESHING ALGORITHM IN A NUTSHELL. The TQM algothm a dvde and conque mehng algothm. Bounday loop ae dceted ung a D meh geneato. They ae then oned nto one ngle contou loop eultng n a loop of node. The contou loop ecuvely ubdvded nto two ub contou loop along a bet plt lne untl the ub contou loop ha been educed to a tval loop.e. a loop wth pont fo a tangle o a loop wth 4 pont fo quadangle. All the detal can be found n efeence [][6]. The emande of th ecton ecall the two man pont of nteet fo ou dcuon. Geneaton of contou pont: Let aume that we have a cuve Γ paameted ung the ac length wth total ac length L. Let aume that we have a contnuou gadng functon g( that epeent the gadng (o ze value along the paamete locaton. The D meh geneaton poblem can be tated a how many pont (npon to geneate? Compute the paamete locaton fo thee pont that wll atfy the gadng functon equement. In [] a oluton wa

2 g( g( g popoed aumng a Gadng functon known dcetely at (nsample ample o ba pont b the gadng functon aumed to be pecewe hamonc (fg. a. ( g 0 0 g + g g g ( co ( + [ ] 0, nsample ; nsample, + g g Fg. a gadng pecewe hamonc ntepolaton Gven nsample pont, the epectve paamete locaton and gadng value (, g g 4 g 4 (, the numbe of pont deved by equng to meet at bet the equdtbuton fo each nteval: ( g + g C te, npon ( π S npon C L d g( C nsample te te 0 In equaton ( the numbe of ample pont, the paamete locaton and the gadng value known. npon computed by oundng the eult of equaton ( to the neaet ntege. The paamete locaton of the D meh pont (fg.b ae obtaned by olvng the ytem: ( g + g ( g + g 0 0 ; npon + + L., npon deved fom the equdtbuton equaton (. Moe detal on the oluton of equaton (4 ae gven n ecton.. Once the bounday loop have been dceted, bounday node ae agned gadng value. The loop ae oned nto one ngle contou loop. A bet plt lne cteon ued to on the loop and the D meh geneaton technque appled to detemne how many pont and the locaton along the plt lne. The ample pont ae the two end pont of the plt lne whee the gadng value ae known. Th poce appled ecuvely. 8 7-Q α 4 α 6 ( g + g ( (4 α α 5 -P 4 g 0 0 Fg. b Nodal pont dtbuton Notce that n the equaton above and ae g g unknown a well a. Summng equaton ( ove all the nteval we obtan: ( npon Fg. Paamete fo the bet plt lne (P-Q and polygon damete Bet Splt lne: Gven a pont P, we fnd the et of admble pont Q that ae vble fom P. The bet plt lne fom P the lne [P,Q] that mnmze the obectve functon : F (Q W δ ( α + W δ ( l + W P angle Length ( 4, α, α, α npon δ (n The weght facto W ae contant. The ft tem, mnmal when the angle α (fg. tend to multple of 60 o 90 (epectvely fo tangle and quadlateal. The econd tem mnmal when the plt lne length tend to the mnmum damete of the bounday loop (.e. the damete of the mallet ncbed ccle pang though two bounday pont. The lat tem mnmal when the ound off n equaton ( to obtan the npon ntege mnmal. Othe choce fo the functon ae poble, ee [6], [7].

3 . TQM MESHING ALGORITHM SHORTCOMNGS In th ecton we ty to hghlght ome of the lmtaton of the TQM appoach a t cuently mplemented n I- DEAS.. D Bounday dcetaton Gven Npon, the locaton found by olvng the nonlnea ytem of equaton (4. One ha to fnd the oluton S (,, L, npon lnea equaton: f ( S 0 ;, npon that atfe the ytem of non ( (5 f S L D U (6 L D g( + ( g( + g( ; U g( + g( + g( + g( + Ung Newtown-Raphon method fo the non lnea ytem of equaton educe to olve the tdagonal ytem of equaton: T δs RHS T T T. (7,,, + f wth f f + + L U + g ( g ( ; RHS + D + g ( ( f ( S; δs S ( n The ndex n epeent the teaton numbe. The ytem olved ung LU decompoton wth fowad-backwad ubttuton. The ntal oluton taken a unfom dtbuton of the nteo pont. The convegence of the ytem depend on the matx T condton numbe, whch not known. Howeve, notce that f the ytem lneaed, ettng g ( 0, the matx become well condtoned and dagonally domnant. Baed on th obevaton, a tategy decbed n ecton. that mpove the obutne of the D bounday meh geneaton.. Bounday dven contol only The TQM algothm ft ceate node along bet plt lne and then ceate the element. The node dtbuton along the plt lne manly dven by the gadng value at S ( n the end pont. The effect that the bounday meh the man dve fo the nteo meh, and undeable bounday effect can popagate n the nteo. The ue ha a good contol on the element ze vaaton on the bounday, but the contol n the nteo and the meh tanton moe dffcult. Fo example, n I-DEAS, the ue can nput a local element length n the nteo, but he cannot contol t adu of nfluence. Alo, when the uface exhbt cuvatue n t nteo but t boundae ae flat thee no eay way to automatcally efne the meh wth epect to the cuvatue. In mot cae the ue wll have to manually add nteo local element length n thee aea to get the deed effect. To ovecome thee dawback and povde a mean to automatcally efne the meh n the nteo of a uface, wth no ue nteacton, TQM wa extended to wok wth a backgound meh, peented n ecton... Hgh dtoton TQM a D meh geneato that geneate a tangula o quadangula meh n a paamete doman. Th paamete doman can be developed dectly fom a CAD paamete pace o ndectly though poecton o flattenng technque. In many cae, the mappng between the paamete doman and the phycal doman not ometc and element ze and qualty need to be aduted n the D doman to eult n the deed meh ze and qualty n the D doman. Thee ae many appoache avalable to account fo the local meh dtoton dung the mappng. The mot common appoach [] ele on the CAD quey of contnuou opeato uch a Cuvatue. Thee can tun to be expenve quee. Cuently n I-DEAS, to mnmze the computatonal cot and account fo length dtoton between the D and D pace, each plt lne ampled wth nsample (0 nteo pont. Thee pont ae mapped back n D pace d υ / d whee dυ and we compute the vaaton the ac length vaaton of the cuve poly-lne n D pace and d the coepondng vaaton of the plt lne n D pace. Th local calng facto then ued to map the D meh ze to a coepondng D meh ze n the paamete doman. Th a vey mple and obut appoach, howeve one man dawback that the calng undectonal (along the plt lne. In ecton., a dffeent appoach to account fo the local dtoton dcued.. TQM EXTENSIONS AND ENHANCEMENTS The man uage of TQM n I-DEAS fo tuctual analy wth a contant meh ze. In th ange the oftwae pefom faly well. Fo bounday cuvatue adaptaton the poce automated but mght ometme become untable, eultng n poo node dtbuton tanton. The nteo uface cuvatue adaptaton not automated and ha to be done though ue nput of nteo local element ze.

4 Fom now on, a ttched teellaton epeentng accuately the D uface o uface to be mehed (fo example an STL epeentaton fom a CAD ytem aumed to ext. The coepondng D teellaton n the paamete doman, thu a dcete one to one mappng between D and D pace, alo aumed to ext. In hot, a tangle map, dcu n ecton 6, avalable (ee fgue 7a, 7b. They ae many poble anwe to the ctcal and common adaptve amplng queton: How many ample pont? Whee? What ze? One anwe could be to delegate the eponblty to the ue. To get the deed amplng adaptaton to the cuvatue both on the bounday and on the uface, t popoed to leveage the uface STL epeentaton. Fo the bounday cuve, the facet pont povde an adaptve amplng of the bounday cuvatue (ee fg. 4.. D bounday dcetaton Fo unfomly dtbuted and moothly vayng gadng value, the non-lnea tdagonal ytem (7 exhbt a unque oluton becaue the non-lnea tem cancel out and the matx tll well condtoned. Howeve, when ung an adaptve amplng pont tategy wth hgh gadng value gadent, the ytem become ll condtoned and mght neve convege to a oluton. The oluton algothm ha been enhanced by montong the convegence of the non lnea ytem and when the oluton ocllate and doe not convege, afte a fxed numbe of teaton, we etat the oluton ung the cuent oluton but th tme olvng the lnea ytem athe than the non lnea one. Th appoach ha poven to be vey obut and able to handle hghly non-lnea dtbuton and gadng vaaton, even exteme cae wth noy nput data. Th tategy ha the deed effect to mooth out the non-lneaty due to hgh fequency nput data.. Flexble backgound meh appoach In ode to povde bette contol ove the nteo element ze vaaton, a neceay n uface cuvatue adaptaton, the TQM algothm wa enhanced to wok n conuncton wth a backgound meh that povde an ntepolaton doman fo the ze vaaton. Thee ae two type of backgound mehe ued: a backgound meh eultng fom the flattened faceted epeentaton that ued a an ntepolaton doman fo the D to D mappng functon (ee fg. 7d and a backgound meh eultng fom an ntal ample meh (ee fg. 7e. The plt lne node geneaton wa modfed. A wa dcued n ecton., the plt lne tll ampled wth 0 unfomly dtbuted ample pont, howeve, the gadng at the ample pont detemned by ntepolaton. A the node ae equally paced along the taght lne, the ntepolaton qute fat nce the eult of the pevou node tangle locaton ued to tat a tangle walk to locate the next one. Alo, pecal attenton ha been gven to the obutne of the tangle walk algothm n ode to handle hghly tetched, even flat, tangle that often occu n STL data.. Dtoton coecton Ou appoach to account fo the local dtoton to ft ceate a ample meh n the D doman, map t back to D pace ung the facet tangle map and compute the length dtoton at the ample pont. Th gve a length cale D D facto λ l / l at each ample pont, S ( S ( that multpled to the D gadng value epeent the D gadng value. S( epeent the ng of ft neghbo node to node. The length cale facto povde a local and otopc etmate of the ze dtoton, computatonally nexpenve. One dawback that thee no attempt to ceate tetched element n the D pace, only ze vae. Gven the D ze vaaton, the calng facto appled. Fo example, a D contant ze n D pace wll eult n a vayng element ze n D pace. The D mehe then ntantated agan wth the D ample meh a the backgound meh wth computed D ample gadng value that dve the eultng fnal meh. The D fnal meh mapped back to D pace ung the facet backgound gd..4 Meh tanton Fo contant ze mehng, ung the ntepolaton doman to detemne the gadng value along the plt lne can lead to udden ump n meh ze. One could mooth out the feld of meh ze to get a moothe dtbuton. Intead, a paabolc dtbuton of the meh ze wa mulated by keepng the two end pont of the plt lne and addng a ample pont half way wth gadng value equal to the global ze. The gadng at the ample pont along the plt lne then obtaned a the mnmum value fom ntepolaton and the hamonc ntepolaton ung the gadng value at the end pont and the mdpont along the lne. Th tategy poved to be valuable n cae whee the plt lne connected two mall featue (.e. end pont have mall gadng value o that the mall featue ze dd not popagate along the plt lne. 4 SIZE CONTROL Thee ae vaou type of ze contol that a ue may want, each one wth dffeent computatonal cot. Thee type ae defned, angng fom the lowe cot to the hghe cot: None, contant, cuvatue. No ze contol: The ample meh a coae meh fomed by the bounday node and wth no addtonal nteo node. Th appoach fat and can be ued f the qualty/dtoton of the fnal meh not ctcal o f the pace tategy ued poduce vey lttle dtoton (fg. 8b, 8c, 8d. Contant ze contol: The ample meh the ntal meh obtaned wthout any account fo the dtoton. Dtoton at ample pont computed leadng to a D ze ntepolaton doman that dve the D mehe. Th the pefeed appoach f the fnal meh qualty/ze contol ctcal fo a gven contant ze (fg. 8e, 8f, 8g Cuvatue ze contol: The ample meh the ame a n the contant ze contol cae but th tme the D

5 meh ze computed a a functon of the cuvatue (fg. 9b, 9c. Although, we dcu a elf-contaned appoach wth the ample meh ntenally geneated, all the concept ae geneal and the ample meh could be povded a nput wth the feld of D ze deved fom an analy, a the cae of adaptve mehng to a oluton. Wth that data a nput, the meh geneato wll povde the deed meh. Smat ze contol [8] anothe mpotant vaaton on the ze contol that ha not yet been mplemented. 4. Cuve Sze Contol In ode to adapt the bounday to the cuvatue we ft need to compute the cuve cuvatue. Two opton ae poble: lne cuvatue o uface cuvatue. 4.. Lne cuvatue Fg 4. Poly-lne fomed by STL facet pont (uled uface : lne cuvatue uface cuvatue The lne cuvatue doe not take nto account the adacent uface cuvatue. The cuve ha a poly-lne epeentaton fomed by facet pont (fg. 4. At each nteo pont, to the cuve, the lne cuvatue computed a the nvee of the adu of the ccle pang though conecutve pont. When thee pont ae collnea, the adu et to nfnty. At the cuve end pont, the cuvatue computed by extapolaton. Futhemoe, the mnmum lne cuvatue at end pont taken fom all cuve that hae the vetex. 4.. Suface cuvatue In ecton 5 we wll peent dcete uface opeato to evaluate cuvatue. The mnmum adu of cuvatue ued at the bounday pont to etmate the local uface cuvatue. 4.. Lne cuvatue veu uface cuvatue. Ethe type of bounday cuvatue, lne o uface, an aveage o a mnmum can be choen dependng on the type of adaptaton the ue want. The lne cuvatue tend to hghly efne mall hole n flat aea. Thee can be vey mall geomety featue compaed to the meh ze that only need to be epeented wth a mnmum of to 4 pont (fg. 6b. In all example peented n th pape, only the uface cuvatue ha been ued Samplng efnement Exta ample pont g g + Fg. 5 Unde ampled bounday cuve Fo cuvatue ze contol, the facet pont epeentaton of boundae nea long flat egon cloe to fllet (fg. 5 need pecal attenton. In a ene, thee boundae ae unde ampled and ample pont need to be added to popely captue the flatne of the cuve. The algothm wok a follow : Gven a cuve, the ac length paamete locaton t of t ample pont P and a global meh ze S g C Loop ove egment [ t, t ] + L o Compute t length o Segment gadng aveage g ( g + g + λ g < S < λ L o If g g l Add exta ample pont at the md pont. Agn gadng value equal to S g at th exta ample pont. The paamete λg a contant epeentng the gadng ato between the local gadng and the global ze, whle the paamete λ epeent the nvee of the mnmum l numbe of nteval deed. The ft pat of the nequalty tate that the gadng at the egment end pont vey mall compaed to the global ze. The econd pat of the nequalty tate that the egment length lage compaed to the global ze. By addng a pont halfway, a paabolc node dtbuton wll eult. In the example, value of λ 4.0 and λ / have been choen. g l 5 SURFACE SIZE CONTROL Fo a contant ze contol the D meh ze a feld of contant value. Fo a cuvatue ze contol the meh ze become functon of the local uface cuvatue evaluated at the ample pont. 5. Contnuou uface opeato S K Gven a uface, the two pncpal cuvatue and K of the uface along the two othogonal pncpal decton vecto ( e, e ae the extema value of all the nomal cuvatue. The nomal cuvatue K N (α to the

6 S uface at a pont P wth unt nomal along a unt e tangent vecto α defned a the lne cuvatue of the ( P, N, e cuve fomed by nteectng the plane α wth the uface S. The mean cuvatue aveage of the nomal cuvatue: K H π π K N ( α dα 0 The Gauan cuvatue the two pncpal cuvatue: K G K.K K G K H N defned a the (8 defned a the poduct of D θ ( P α ( k k S ( wth θ P epeentng the total vetex angle. ( The Radu of cuvatue at a pont of the edge P P : P P < N, P P P along the decton ρ ( > and the mnmum adu of cuvatue at pont P : ρ mn ρ At each pont P one can compute the vetex angle exce π θ ( P that alo epeent the (total Gau (9 cuvatue at an nteo pont: and the mean cuvatue expeed a the aveage of the two pncpal cuvatue : K da π θ ( P (4 G The dcete gauan cuvatue at pont P can be appoxmated by: K H (K + K K G ( P (π θ ( P / Vo( P (5 5. Dcete uface opeato Gven a tangle map, the ft and econd ode attbute of the uface (nomal vecto, mean cuvatue, K Gauan cuvatue can be appoxmated [9], [0]. D pace P D pace G β d α d d P α 0 T l 0 Vo(P α d α t d l 0 d (0 K H Vo(P wth computed a the voono aea at a pont f all tangle ae acute and fo obtue tangle the contanment ccle ued ntead of the ccumcbed ccle ctea (.e. ntead of onng adacent edge mdpont to the cente of the ccumcbed ccle they ae oned to the mdpont of the (oppote longet edge (fg. 5. The dcete nomal cuvatue deved fom the fomula K H ( P. N( P da (cotα S ( d + cot β (6 d wth α and β the two angle oppote to the edge ( P, P n the two tangle hang the edge (ee fg. 5. The dcete mean cuvatue nomal gven by : K H ( P (cotα Vo( P S ( d + cot β d ( P ( P P P Fg. 5 Tangle map D to D pace (Voono aea wth modfcaton fo obtue tangle Ft we compute the dcete nomal a an angle weghted aveage of the nomal to the facet uoundng the pont. N α D k k S ( θ ( P N Tk ( (7 and the appoxmaton of the nomal cuvatue obtaned a: K P K H ( P (8 H ( All the above fomula ae the dcete countepat of the contnuou ft and econd ode attbute of the uface. In [] a meaue of the defomaton of a tangle between D and D pace popoed :

7 Def ( T cotα D D o l t D 0 D + cotα D l Aea( t D D + cotα D By ummng ove all the tangle, th fomula povde a meaue of the total dtoton nduced by the mappng ued n the tangle map between the D and D pace. The global meaue could be ued a ctea to elect the pace development tategy wth leat dtoton and/o to mpove an ntal paamete doman by mnmzaton of the global dtoton meaue. Cuently the cuvatue adaptaton tategy only conde the mnmum adu of cuvatue, but expementaton wth othe ctea undeway. The mappng between the dcete cuvatue and the D ze a follow [0]: Gven ε a pecent devaton to the ognal geomety. S Gven a global ze global Compute the contant γ ε( ε Look up n the tangle map fo the mnmum adu of cuvatue, ρ. D Compute local D ze: S ρ γ D S Bound, D S global < S < S dlenrato dlenrato 0 global A value of wa choen to contol the mnmum ze allowed dung cuvatue adaptaton. 6 TRIANGLE MAP The tangle map keep a map between the D meh and the D meh and alo povde a wealth of nfomaton about the uface. Thee ae two tangle map that we ue. The STL tangle map (fg. 7a, 7b and the ample meh tangle map. One tat wth a gven STL of a uface to meh (fg. 7a. The node coodnate n D pace and the meh connectvty ae toed n the tangle map. Ung a pace development tategy (n all the example peented a flattenng tategy ued, the D paamete doman (fg. 7e ceated and the map x(u,v, y(u,v, z(u,v toed fo the facet pont. The D paamete doman an ntepolaton doman fo the mappng between the D pace and the D pace. Next, all the dcete opeato ae computed a well a the dtoton between the D pace and the D pace. The STL tangle map alway ued to map the node back to D pace. Anothe ue of the STL tangle map dung the bounday node geneaton wth cuvatue ze contol. A mentoned n ecton 4., we ue the uface cuvatue that we obtan dectly n the look up table of the tangle map. The bounday node wee geneated n D pace and l D they need to be mapped to the coepondng D pace value. To do o one could pefom an exhautve D pont n tangle locaton. Intead, a the cuve epeented by a poly-lne of facet pont, we toe the paamete locaton of the facet pont along the cuve. Fo a meh pont geneated along the bounday cuve at the paamete locaton t, we fnd the facet pont nteval [t, t+] that contan t and ue a lnea ntepolaton to fnd the coepondng D paamete locaton (u,v. The D bounday node loop then mapped to the D plane. The gadng value at the bounday node computed a an aveage of the two adacent edge length at the pont. Th gve u the eal D ze that aleady account fo dtoton. The ample meh geneated n D pace ung the TQM mehng algothm wth the deed meh ze. At th tage, the ze map ha not been ceated yet. The gadng value along the plt lne ae computed ung the pecewe hamonc ntepolaton (equaton. The ample meh manly unfom (unle we ae ung no ze contol and povde a amplng feld of nteet fo the gven meh ze. The D meh tanfomed back to D pace ung the STL tangle map, and a ample meh tangle map ceated. Th latte tangle map povde a look up table to compute fo each pont n the ample meh, t dtoton, t uface cuvatue etc The data (dtoton, cuvatue etc computed at once fo the whole meh and toed n the tangle map. The bounday dcetaton of the ample meh and the fnal meh ae dentcal and need not be egeneated. The ze map ceated and wll be ued a a ze ntepolaton feld fo the fnal meh. 7 SIZE MAP The ze map a combnaton of the ze contol and the tangle map. The ze contol povde the D ze vaaton on the uface fo the ample meh whle the tangle map povde the ze dtoton. The ze map combne both data nto one ngle value. Fo example, fo a contant D ze meh, the ze contol ha a feld of contant value and only the dtoton facto vae at each pont of the ample meh. The D caled meh ze the poduct. 8 EXAMPLES Fg. 6a - Small hole

8 Fg. 6b - Small hole zoom Fgue 6 : mooth ze tanton fo mall featue. 8. Smooth tanton fom mall hole to lage contant ze Fgue 6a epeent the fnal meh fo a quae wth a mall hole. The damete of the hole whle the quae ze 00. A meh ze of.5 ha been ued. The hole ha been epeented by 5 element and the meh tanton moothly fom the mall to the lage ze (fg. 6b. poecton ued n I-DEAS. Fgue 7a epeent the STL of half a phee. Fgue 7b epeent the D paamete pace eultng fom the poecton technque. Notce the hgh dtoton along the bounday whee tangle have been quahed. Fgue 7c epeent on top the D fnal meh and on the bottom the coepondng D meh obtaned wth the opton of contant ze contol. Hghly dtoted element ae geneated along the bounday. Clealy, a poecton technque not atfactoy n local aea whee the nomal to the uface othogonal to the decton of poecton and a mall change dε n the D pace tend to nfnty n the D pace. Fgue 7d epeent the flattened STL meh. Th tme the length have been peeved along the bounday and the hghet dtoton eem to occu aound the pole. The fnal meh wth contant ze contol peented n fgue 7e. The flattenng pace tategy poduced a moe ometc mappng leadng to the good meh qualty. Poecton technque ae computatonally nexpenve but they ae etcted to doman that can be poected and theefoe wok well wth low cuvatue doman. On the othe hand, flattenng technque, dependng on the type of doman at hand, eult n moe ometc mappng (wok well fo developable doman ndependent of the cuvatue but ae n geneal moe computatonally expenve. They alo have the own lmtaton (cannot flatten a cloed uface wthout cuttng t, but they ae le tngent than poecton technque. Fg. 8a D STL of a dampe Fg. 7 a D STL of an hemphee 8. Developed pace dtoton compaon. Fgue 7 (b,c,d,e llutate the advantage of the flattenng technque [5] ove the Maxmum Aea Plane (M.A.P. 8. Sze Contol compaon. Th example llutate the eult obtaned wth vaou type of ze contol. The meh ze 5 n all cae. Fgue 8.a epeent the ntal STL. In fgue 8b and 8e the ample mehe fo no ze contol and contant ze contol opton ae peented. Fgue 8c and 8f povde a compaon of the eultng D fnal mehe fo epectvely no ze contol and contant ze contol. The meh n fg. 8f ha a moothe vaaton of the the ze than the one n fg. 8c, due to a che amplng of the cuvatue vaaton and theefoe a che ntepolaton doman fo the dtoton calng facto..

9 geneate a ample meh fo the nteo ha alo been peented. A natual way of gettng the ample meh, baed on the fnal meh global ze, poved to be a good amplng tategy. The pce to pay fo the addtonal qualty the cot of mehng the uface twce. It the ue choce whethe to ncu th exta cot. The cuvatue adaptaton peented obut and tanton moothly between hgh and low egon of cuvatue. Fnally, we have ted to olate ndependent concept uch a ze contol, tangle map and ze map that put togethe povde temendou flexblty. Fg. 9a D STL of a backet 8.4 Cuvatue adaptaton. Fgue 9a epeent a backet wth faly complex cuvatue patten. Fgue 9b demontate how the D mehe able to accuately adapt to the cuvatue patten and fgue 9c how that the efnement, when mapped back n D pace dd occu n the coect locaton. Notce alo, the D bounday cuvatue adaptaton and how the hole n flat egon wee not efned a the uface cuvatue, not the lne cuvatue, wa ued n thee example. Th wok tll at a pelmnay tage wth empha on flexblty and uface meh qualty. Futue wok hould nclude moothng technque that ae adaptaton peevng, a tudy of the vablty of ung the STL a ample meh, mat zng, and tudy and development of bet pactce/tatege to get a good qualty uface meh at lowet cot. 0 ACKNOWLEDGMENTS The autho expee h ncee thank to h colleague at EDS/PLM oluton fo the actve uppot. In patcula, Mchael Hancock, Nlanan Mukheee, Hu Xao, Radhka Vuputoo and Kk Beatty fo the actve nvolvement. REFERENCES. Fg. 0a- D STL of anothe backet Fgue 0a, 0b and 0c anothe example of cuvatue adaptaton. Notce n fgue 0b that the D mehe accuately captued the hgh cuvatue aea and tated to pck up the lowe cuvatue of the ea flap both on the bounday and the nteo. 9 CONCLUSION Recent poge and extenon to nceae TQM flexblty to handle lage vaaton n meh ze all aco a uface have been peented and demontated. An appoach that ue the uface STL data a ample pont fo the bounday dcetaton and automatcally [] A.J.G. Schoof, L.H.Th. M. Van Beukeng and M.L.C. Slute, A geneal pupoe two-dmenonal meh geneato, Advance n Engneeng Softwae, Vol ( pp.-6 (979 [] J.R. Ttano, S.J. Owen and S.A. Canann, "Advancng font uface meh geneaton n paametc pace ung a Remannan uface defnton",poc. 7 th IMR, pp (998. [] R. Lohne, "Regddng uface tangulaton", Jounal of Computatonal Phyc, Vol 6, pp.-0 (996 [4] A. Sheffe, E. de Stule, "Suface paametzaton fo mehng by tangulaton flattenng", 9 th IMR, pp.6-7 (000. [5] E.C.Shebooke, M.R. Laue and D.C. Goad, Membane ufacng : A tangulated G epeentaton fo featue-baed degn, NTI techncal epot 00-, (000 [6] J.A.Talbet and A.R. Paknon, "Development of an automatc two-dmenonal fnte element meh geneato ung quadlateal element and beze cuve bounday defnton", IJNME, Vol 9, pp (990. [7] J. Saate and A. Hueta, "Effcent untuctue quadlateal meh geneaton", IJNME, Vol 49, pp (000. [8] A. Cunha,.S.A Canann and S. Sagal, "Automatc bounday zng fo D and D mehe", Tend n

10 untuctued meh geneaton, ASME, AMD-Vol 0, pp.65-7 (997 [9] M. Meye, M. Debun, P. Schoede, A.H. Ba, "Dcete dffeental geomety opeato fo tangulated -manfold", VMath (00 [0] P.J. Fey and H. Boouchak, "Ctee geometque pou l evaluaton de tangulaton de uface", Rappot de echeche INRIA, N 0 95 (996 [] K. Homann, U. Labk and G. Gene, "Remehng tangulated uface wth optmal paametezaton", CAD, Vol, pp (00.

11 Fg. 7d D flattened STL. Fg. 7b D poected STL ung Maxmum Aea Plane (M.A.P. Fg. 7c M.A.P., contant ze contol. Top: D fnal meh. Bottom: D fnal meh Fgue 7e - Flattenng, contant ze contol. Top: D fnal meh. Bottom: D fnal meh Fgue 7 : Developed pace dtoton compaon between (left Maxmum Aea Plane and (ght flattenng.

12 Fg. 8b -D ample meh, ze contol none. Fgue 8e - D ample meh, ze contol contant Fg. 8c D fnal meh, ze contol none Fg. 8f - D fnal meh, ze contol contant Fg. 8d D fnal meh, ze contol none. Fg. 8g - D fnal meh, ze contol contant Fgue 8 Sze contol compaon between (left ze contol none and (ght ze contol contant.

13 Fg. 9b - D fnal meh, cuvatue ze contol Fgue 0b D fnal meh Fg. 9c - D fnal meh, cuvatue ze contol Fgue 0c D fnal meh Fgue 9 Cuvatue adaptaton of a backet wth complex cuvatue patten Fgue 0 Cuvatue adaptaton of a backet aound hgh and low aea of cuvatue.