Fun sheet matching: towards automatic block decomposition for hexahedral meshes

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1 DOI /s ORIGINAL ARTICLE Fun st mtn: towrs utomt lok omposton or xrl mss Nols Kowlsk Frnk Loux Mttw L. Sttn Stv J. Own Rv: 19 Frury 2010 / Apt: 22 Dmr 2010 Ó Sprnr-Vrl Lonon Lmt 2011 Astrt Dpnn upon t numrl pproxmton mto tt my mplmnt, xrl mss r rquntly prrr to ttrrl mss. Bus o t lyr strutur o xrl mss, t utomt nrton o xrl mss or rtrry omtrs s stll n opn prolm. Ts lyr strutur usully rqurs topolol motons to propt lolly, tus prvntn t nrl vlopmnt o msn lortms su s Dluny s lortm or ttrrl mss or t vnn-ront lortm s on lol sons. To utomtlly prou n ptl xrl ms, w lm tt ot lol omtr n lol topolol normton must tkn nto ount n t ms nrton pross. In ts work, w propos tortl lsston o t lyrs or sts prtptn n t omtry ptur prour. Ts sts r ll unmntl, or un-sts or sort, n mk t onnton twn t lol lyr strutur o xrl mss n t omtr surs tt r ptur urn t msn pross. Morovr, w propos rst nrton lortm s on un-sts to l wt 3D omtrs vn 3- n 4-vlnt vrts. N. Kowlsk F. Loux (&) CEA, DAM, DIF, Arpjon, Frn -ml: rnk.loux@.r N. Kowlsk -ml: nols.kowlsk@.r M. L. Sttn S. J. Own Sn Ntonl Lortors, Aluqurqu, USA -ml: mlstt@sn.ov S. J. Own -ml: sjown@sn.ov Kywors Hxrl ms Blok-strutur ms Dul ms Funmntl ms Ms nrton 1 Introuton Computtonl [1] smultons pn upon t numrl pproxmton mto us, su s nt lmnt, nt rn, n nt volum mtos (to nm w), tt nl moln o pnomn o ntrst to t sn n nnrn ommunts. Crtl to o ts numrl pproxmton mtos s t srtzton o t pysl omn nto ms. Dpnn on t numrl pproxmtons mtos, xrl mss r prrr to ttrrl mss. T utomt nrton o xrl mss or ny omtr 3D ojt s, owvr, stll n opn prolm. In rnt s, rnt ppros v n ollow to nrt ull xrl ms or ny omtr 3D ojt. Strtn rom prms ounry sur som utors v propos pur omtr ppros lk plstrn [2] or H-morp [3] lortms, wl otr utors v ollow pur topolol ppro proposn rnt lortms [4 9] wt lmt suss. Som unll vts n rmn or nvrt lls, n ntv Jon lls n nrt ns t ms. By rlxn t onstrnt o workn wt pr-ms ounry, ns-out lortms [10 12] or unonstrn plstrn [13] v prov nourn xmpls o ll-x mss or rtrry omtrs. In spt o ts, ts mtos v tr wknsss. For nstn, ns-out lortms [10 12] o not onsr t lol topolol strutur n nrlly pl t worst qulty lmnts nr t ounrs wl t qulty o lmnts n ts rs n vry mportnt or numrl smulton.

2 T ulty o utomt xrl ms nrton s mnly u to t lyr strutur o xrl mss [14] tt n rqur topolol motons to propt lolly. Bus n utomt nrl ll-x soluton tt rlly prous -qulty lmnts or rtrry omtrs s not n vlop, t s otn nssry to ollow n-m ppro wt sotwr su s CUBIT [15] (mnly s on t swpn lortm [16 18]) or ICEM-CFD [19] (lok-strutur omposton) wr t usr rvs t xrl ms nrton y lvr omtry ompostons. Wt su tools, t usr must mnully trmn t rltonsp twn t omtr turs o t 3D ojts n t xrl lyrs to orm t nl ms. To utomtlly prou n ptl xrl ms, w lm tt w must onsr ot lol omtr n lol topolol normton n t ms nrton pross. Rnt works v monstrt t n or ts ppro [13, 20]. For xmpl, Sttn t l. [13] strt rom t ounry n ul omtr lyrs o xr prlll to t surs. Fnl stps o t prour nvolv ronzn n ttn strutur xrl tmplts su s swpn, mppn n mpont suvson to ll t rmnn vty. Altou promsn, ts omtr lortm msss t tortl ountons to nsur tt ny rmnn vty n 3D omtry n ll wt xr o sunt qulty. Ro n Srrt [20] strt rom ttrrl ms n ul ul surs onsrn lol ul ontrutons. T ul surs tt r ntrou orrspon to lyrs o xr lon t omtr surs. On ts lyrs r ult, xrl lok strutur n rv n strtorwr mnnr. Bot o ts ppros ttmpt to prou lyrs o xr prlll to t omtr surs y usn omtr n topolol lol normton. Exprn s sown, owvr, tt vn lol knowl o topoloy s not nou. T lol topolol strutur must onsr to rlly prou qulty xrl ms. Flur to mn lol topoloy my rsult n unsolv onurtons, wr t s mpossl to ll vty wt xr [13]or rsolv topoloy ontons n t 3D mol [20]. In ts work, w propos tortl lsston o t lyrs or sts prtptn n t omtry ptur prour. Ts lsston llows us to lnk omtr n lol topolol onstrnts n unqu onpt known s unmntl sts, or un-sts or sort. Funsts wll us to rv rst nrton lortm or t utomt ll-x prolm n srv s t ounton or ortomn works. T ppr s ornz s ollows. In St. 2, w rvw n nn som s ms strutur ntons rqur or t unrstnn o t lortm w prsnt n St. 3. In St. 3, w prsnt our nw lortm or utomt nrton o lok ompostons. In St. 4, w prov onlusons n sussons on utur work. 2 Funmntl st tory Unlk ttrrl mss, xrl mss v n nrnt lyr strutur, w mks ot lol motons [21 25] n utomt nrton o ll-xrl mss ult. T lyr strutur n xrl mss s t prmry rson or roustnss ssus wt prvous ttmpts t ll-xrl msn w rly on pr-ms ounry. 1 Ts lyr strutur n xrl mss s ormlly n s t ul, or t Sptl Twst Contnuum. W lm tt vlopmnt o rll ll-xrl msn lortm or rtrry omtry nnot on wtout tkn ts strutur nto ount. Morovr, ts strutur must onnt to t omtr turs o t pysl omn tt s to ms. In orr to ormlz ts onnton, Spr [27] n t noton o unmntl mss. 2.1 Bkroun T trtonl rprsntton o nt lmnt ms, ompos o xr, qurltrls, s n nos, s known s t prml ms. In most ss, t s sunt to sr xrl ms s n st o onnt xr tt srtzs 3D ojt (orrsponn to pysl omn usul or numrl smulton). Mor prsly, w n [28]: Dnton 1 (Hxrl ms) A xrl ms s 4-tupl (H, F, E, N) wr H s non-mpty st o xr, F s t non-mpty st o ll qurltrls nnt to on or mor xr n H, E s t non-mpty st o ll s nnt to on or mor xr n H, n N s t non-mpty st o ll nos nnt to on or mor xr n H. Ts s wt w nrlly ll t prml ms. Hxrl mss r us to srtz pysl omn X: T noton o srtzton rqurs tt ny omtr pont n X lons to only on ms ll n t ms wolly lls X: Ts noton s ptur y t ollown nton. 1 Unr t ptl ontons o vn volum somorp to ll n n vn numr o qurltrls on t ounry [26].

3 Dnton 2 (Hxrl ms srtzn 3D oun omn) Lt X 3D oun omn, xrl ms M = (H, F, E, N) srtzs X n only 8ð 1 ; 2 Þ2H 2 ; 1 \ 2 ¼;; 8x 2 X; 8 [ 0; 9 2 H; B x; \ 6¼;wr B x, s t -rus ll ntr n x. Trouout ts ppr, t s ssum tt ny mnton o xrl ms wll stsy ot o ts ntons. Intrstnly, ntr o ts prml ntons rrs to t lyr strutur nrnt n xrl mss. In ontrst, t Sptl Twst Contnuum [26], or ul ms, s s on ts lyr strutur. T ul s n rrnmnt o surs. Evry sur s ll st n s t ul o lyr o xr. T ntrston o two sts s or, w s t ul o olumn o xr. A or, or olumn o xr, s orm t t ntrston o two sts, or t t sl-ntrston o snl st. Howvr, or t purposs o ts ppr, w prr to work n t prml ms. A xrl ll H s 12 s ornz s tr sts o our topololly prlll s E 1, E 2, n E 3 (s F. 1). Tus, two s o H r s to topololly prlll on to t otr n only ty ot lon to t sm sust E 1, E 2, or E 3. Consrn xrl ms M = (H, F, E, N) n n [ E, w v H ¼ 1 ;...; n t xr nnt to, n w not E == ¼ ¼1;...;p t sust o E su tt V [ [1, p], A k [ [1, n], n r topololly prlll n k. T sust E // ns topolol sur tt n xtn to t jnt xr n so on, nn prml st. Exmpls o prml sts r vn n F. 2. F. 1 A xrl ll.,, T tr sts o our topololly prlll s o ts ll. As s n 0 lon to t sm sust pt n, ty r s to topololly prlll Dnton 3 (Prml st) Lt M = (H, F, E, N) xrl ms. Consrn n [ E, lt us n E s n t smllst sust o E su tt [ E n 8 2 E ; E == E : T st o xr nnt to ny n E orms t prml st rltv to. Smlrly, xrl ll s sx s ornz s tr prs o topolol oppost s F 1 ; F 2 ; n F 3. Two s r s to topololly prlll n only ot s lon to t sm xr n orm tr F 1 ; F 2 ; or F 3. E o ts sts ns lol olumn tt n xtn to t jnt xrl lls n so on, nn prml or. Exmpls o prml ors r vn n F. 3. F // nots t st o s topololly prlll to. 2 F. 2 On t lt, xrl ms nrt usn swpn lortm [16, 17]; on t rt, tr prml sts o ts ms r lt Dnton 4 (Prml or) Lt M = (H, F, E, N) xrl ms. Consrn [ F, lt us n F F. 3 On t lt, two prml ors r sown wl t 2 In xrl ms srtzn 3D oun omn, ts st s ru to on or two s. orrsponn prml sts tt ntrst otr r sown on t rt

4 sust o F su tt [ F n 8 2 F ) F == F : T st o xr nnt to ny n F orms t prml or rltv to. Som otr usul ntons or our purpos r ms surs n ms lns. Dnton 5 (Ms surs) Lt M = (H, F, E, N) ms srtzn oun omn X; ms sur o M s st o prws jnt s o F ormn 2-mnol. Dnton 6 (Ms lns) Lt M = (H, F, E, N) ms srtzn oun omn X; ms ln o M s st o prws jnt s o E ormn 1-mnol. Lt us now n 3D omtr ojt n BRp rprsntton. In, n n nustrl n prmt ontxt, t 3D oun-omns w wnt to srtz r nrlly rprsnt y su ounry rprsnttons. Dnton 7 (BRp rprsntton o 3D omtr ojt) A 3D omtr ojt s 3-tupl (S,C,V) wr 1. S s non-mpty st o omtr surs nlosn 3D sp n su tt 8ðs 1 ; s 2 Þ2S 2 ; s 1 \ s 2 ¼;; 2. C s t non-mpty st o urvs nnt to on or mor surs n S; 3. V s t non-mpty st o vrts nnt to on or mor surs n S. Consrn ts nton, w n now ssot omtr surs n omtr urvs, rsptvly, to ms surs n ms lns. Ts ssoton s n xtnson o t lsston 3 noton ntrou y Rml n Spr [29] or mss. Dnton 8 (Gomtr ssoton) Lt M = (H, F, E, N) xrl ms srtzn t BRp omtr ojt G = (S, C, V). A ms sur s M F s ssot wt omtr sur s [ S n only ll t s n s M, ll t s n nos jnt to o s M r omtrlly on sur s wtn tolrn, n s M srtzs sur s (.., vry pont x [ s s ontn n xtly on, [ s M, n s M wolly lls s). A ms ln l M E s ssot wt omtr urv [ C n only ll t s n l M n ll t nos jnt to n o l M r omtrlly on urv wtn tolrn, n l M srtzs urv (.., vry pont x [ s ontn n xtly on, [ l M, n l M wolly lls ). 3 In topoloy-s moln ts noton s ll mn. Impltly, ts nton nts tt two omtr surs s 1 n s 2 o BRp omtr ojt sr urv, tn t s o t ms ln ssot wt urv r ot ssot wt surs s 1 n s 2 too. 2.2 Funmntl xrl mss Typlly, nlysts rqur oo sp n topoloy o xr nr t ounry o 3D ojts. Tt s t rson wy ns-out lortms [10 12] r not sutl or mny numrl smultons. In t, ns-outs r normlly post-pross wt t nsrton o sts lon t ounry to t ttr sp xr [12]. T nton o unmntl mss s n ttmpt to rtrz wt t mns or ms to v wll-sp xr nr t ounry. T rst nton o unmntl mss ws vn y Spr [27] n tl y Loux n Spr [30]. Intutvly, n ts ntons, w onsr tt xrl ms s unmntl n only vry omtr sur s oun or ptur y snl prml st n vry omtr urv s oun or ptur y on or mor prlll prml ors. In ts ppr, w propos tortl lsston o t prml sts prtptn n t omtry ptur prour. In ts wy, w otn mor omplt nton o unmntl ms tn t ons vn y Loux n Spr [25] Cptur o omtr surs In orr to n t noton o unmntl prml sts, w ntrou t nton o pturn o omtr surs. Dnton 9 (Prml st pturn omtr sur) Lt M = (H, F, E, N) xrl ms o 3D omtr ojt G = (S, C, V), lt s G [ S omtr sur o G n s M F t qurltrl sur ms ssot wt s G, lt H s t st o ll t xr nnt to o s M, prml st P s st pturn s G n only H s P n P s smply onnt lolly to sur s G. T rst onton nsurs tt t omtr sur s ptur y snl prml st. For nstn you onsr F. 4, ll t xr trvrs y t r ul st r lon snl urv sur. In otr wors, ts omtr sur s ptur y t orrsponn prml st. In F. 4, ts sur s prtlly ptur y sx prml sts. Son onton urnts tt t orrsponn ul st s lolly 2-mnol. Ts son proprty s ssntl to vo prtulr ss smlr to t 2D s sown n F. 5.

5 F. 4 A snl st pturs snl omtr sur n, wl svrl sts ptur t n. Courtsy o Json Spr, Sn Ntonl Ls [27] F. 6 A lvl 1 un st pturs on omtr sur; t pturs tr omtr surs jnt to urv G lon to snl prml or. Ts nton sps notn out t xr vn n ssot wt urv G ut no s on t omtr ounry. In, ny numr o ors n unmntl to t sm omtr urv Funmntl sts F. 5 2D Exmpl o sl-ntrstn st tt pturs t top sur wtout stsyn t son onton o Dnton Funmntl ors In orr to n wt unmntl ms s, Loux n Spr [30] ntrou t notons o unmntl prml ors n unmntl prml sts. In tr ntons, unmntl prml sts orrspon to t nton o prml sts pturn omtr sur (Dnton 9), wl unmntl prml ors r prml ors prtptn to ptur omtr urv n vn som o tr s ssot wt omtr sur jnt to t omtr urv to ptur. Dnton 10 (Funmntl prml or) Lt M = (H, F, E, N) xrl ms o 3D omtr ojt G = (S, C, V), lt G [ C omtr urv o G n ðs G1 ; s G2 Þ2S 2 t jnt surs o G. Lt ðf G1 ; F G2 ÞF 2 t sts o s jnt to G n lonn to s G1 n s G2 ; rsptvly, lt H H t st o ll t xr srn o F G1 (rsptvly F G2 ), prml or H s unmntl prml or o G ssot wt s G1 (rsptvly s G2 ) n only H s nlu nto n s smply onnt lolly to urv G. Ts nton vs us rlton twn omtr urvs n prml ors tt s quvlnt to t rlton twn omtr surs n prml sts w n n Dnton 9. Intutvly, ts nton nsurs tt you onsr omtr urv G lmtn two omtr surs, tn, on ot surs, qurltrls W now nn t nton o unmntl sts, ornlly propos y Loux n Spr [30] y sutorzn tm nto tr lvls. T sts o t unmntl rst lvl v t rtrst o n wolly jnt to t omtr ounry. To smply, w nm tm lvl 1 un sts. 4 Dnton 11 (lvl 1 un st) Lt M = (H, F, E, N) xrl ms o 3D omtrl ojt G = (S, C, V), lt s G [ S omtr sur o G n s M F t qurltrl sur ms orrsponn to s G, lt H s t st o ll t xr srn o s M. A prml st P s lvl 1 un st or s G n only P pturs s G n vry xron [ P - H s prtpts to ptur notr omtr sur o G. T rst onton nsurs tt st P pturs omtr sur s G, wl son onton nsurs tr P only pturs s G ( P = H s )orp pturs on or mor omtr surs o G. As onsqun, tr xsts on-to-mny mppn rom t lvl 1 unmntl prml sts o H to t omtr surs o G. Fur 6 sows xmpls o lvl 1 un sts: In F. 6, t rprsnt prml st pturs on omtr sur, wl n F. 6 t pturs tr omtr surs. Alon, lvl 1 un sts r not sunt to n ll possl unmntl prml sts. W lvl 2 un sts tt r prml sts pturn t lst on omtr sur n n not lvl 1 unmntl sts. It mns tt som xr ormn lvl 2 un st o 4 W opt t sm nmn or son n tr lvls o unmntl prml sts.

6 F. 8 Two lvl 3 un sts prtptn to ptur t sm omtr urv. E o ts st s rltv to omtr sur jnt to t ptur omtr F. 7 On t lt, t lvl 2 un st pturs on omtr sur; on t rt, two omtr surs r ptur y t sm lvl 2 un st A lvl 3 un st lps to ptur on or mor omtr urvs. not prtpt n t ptur o omtr sur. For xmpl, F. 7 llustrts two sts, ot o w ontn xr ntror to t volum w r not ssot wt ny omtr sur. Dnton 12 (lvl 2 un st) Lt M = (H, F, E, N) xrl ms o 3D omtrt to ny omtr sur, ojt G = (S, C, V), lt s G [ S omtr sur o G. A prml st P s lvl 2 un st or s G n only P pturs s G n P s not lvl 1 un st. Lvl 1 n 2 un sts r qut to ptur ll omtr surs. Howvr, to ptur ll omtr urvs, w lvl 3 un sts. Dnton 13 (lvl 3 un st) Lt M = (H, F, E, N) xrl ms o 3D omtr ojt G = (S, C, V), n lt G [ G omtr urv o G. A prml st P s lvl 3 unmntl prml st or G n only tr xsts unmntl or n lvl 1 or lvl 2 un st P 2 su tt s t ntrston o P n P 2. Comprn to lvl 1 n lvl 2 un sts, lvl 3 un sts just lp to ptur omtr urvs n not omtr surs. T two sts llustrt n F. 8 r only jnt to ounry s lon t omtr urv ty lp to ptur. Not tt prml st n ot lvl 2 n lvl 3 un st. In, t nton o su prml sts s mnly lol to omtr sur (lvls 1 n 2) or omtr urv (lvl 3). As onsqun, prml st n ptur ot on or mor omtr surs n on or mor non-nnt omtr urvs. To onlu ts ston out unmntl sts, w summrz t mportnt turs tt stnus our tr lvls o un sts: A lvl 1 un st pturs on or mny omtr surs, n ll ts xr r on t ounry; A lvl 2 un st pturs on or mny omtr surs, ut som o ts xr o not prtpt n t ptur o omtr sur; 2.3 Funmntl ms Consrn t tr lvl o unmntl sts n prvously, w n now n wt unmntl ms s Dnton 14 (Funmntl ms) Lt M xrl ms o 3D omtr ojt G = (S, C, V), M s unmntl ms o G n only 1. All omtr surs n S r ptur y lvl 1 or lvl 2 un sts n M; 2. All omtr urvs n S r ptur y lvl 3 un sts n M. Ts nton ormlzs wt w xpt rom xrl ms srtzn omtr ojt: tr xsts t lst on prml st lon omtr surs, n omtr urvs r lso wll-ptur. 3 Gnrton o xrl lok-strutur mss usn unmntl sts Consrn Dnton 14 o unmntl mss, w propos n lortm or nrtn unmntl mss or 3D omtr ojts vn 3- n 4-vlnt omtr vrts. Dnton 14 o unmntl ms s only s on t noton o unmntl sts. As onsqun, w n wrt n lortm nrtn unmntl mss y only onsrn sts n not ors. Wt only t prvous lmt nton o unmntl sts prov y Loux n Spr [30], t ws ult to vlop onstrutv lortm w woul nrt -qulty lmnts vn on smpl onv ojts. Howvr, t nn nton o unmntl sts prov n St. 3, provs t rounwork nssry or su onstrutv lortm. Strtn rom 3D omtr ojt G = (S, C, V), t prnpl o our onstrutv lortm s s ollows (s F. 9):

7 1. Trnsorm n ntl ttrrl ms o G (s F. 9) nto n ll x ms, M = (H, F, E, N) usn t THx tmplt [31]. T qulty o ts ntrmt ms wll nsunt or most numrl smultons. 2. Dn tr sts ðs 1 ; S 2 ; S 3 Þ o unmntl sts, rsptvly, orrsponn to lvl 1, 2, n 3 un sts y solvn lol topoloy onstrnt stston systm. 3. Dn st, S s ontnn ll non-unmntl (.., sonry) sts n M. 4. On-y-on, nsrt unmntl st, s 2ðS 1 [ S 2 [ S 3 Þ; nto M usn t pllow oprton [32]. Ts oprton onssts n ntroun prml st lon sust o F ormn 2-mnol sur. At ts stp, w t unmntl ms o G (s F. 9, wr Lpln smootn lortm s n ppl). 5. On-y-on, xtrt non-unmntl st, s j 2 S s usn t st xtrton oprton [33] to rt strutur lok xrl ms (s F. 9 ). T ulty s to v roust mplmntton o t xtrton oprton. In, wr n trky sts n our urn st xtrton. Morovr, n som spl ss w o not rmov ll t sonry sts. For nstn, you onsr spr, you just v on un st tt os not sl-ntrst. A x s t ntrston o tr sts. Tus, you rmov ll t sts ut t on unmntl st, you o not kp ny xron. 6. Rn lok to t sr nsty. In ts ston, w ous on Itm 2, w s t rt o t lortm. T two mn ssus tt n to onsr r: 1. T mnmnt o vrts wt vln rtr tn 3. In ts ppr, w rstrt our ous to 4-vlnt vrts; 2. T trmnton o t nsrton pt o ntrnl prml sts (lvl 2 n 3) ns t omtr volum. F. 9 In ts ur, r vn rnt stps o our lortm. W nrt ttrrl ms o our omtr ojt; ts ttrrl ms s onvrt nto THx ms y pplyn t trtnl pttrn trnormn ttrron nto our xr; unmntl sts r nsrt to t unmntl ms; Lpln smootn lortm s ppl; rsults o sussv xtrtons o prml sts to t lok-strutur xrl ms n. All t xtrtons r not sown. In ts xmpl, 16 xtrtons r nssry ntrston o two lvl 1 un sts,.., on snl prml or, n vrtx s ptur y tr lvl 1 un sts,.., on snl xron. Fur 10 sows n xmpl o su s. T prolm s wn you v to l wt non-onvx omtrs n/or r vlnt omtr 1 un sts (rprsnt y t rn, yllow n r ul sts) w prws ntrst lon omtr urvs to orm prml ors vrts. In, t lst n - 2 xr 3.1 On t ounry: mnmnt o 4-vlnt vrts T nsrton o unmntl sts n ors prov xplt ptur o omtr urvs n surs. But wt out omtr vrts? In t, tr s no sp rqurmnt or omtr vrts, w r ptur y on or mny xr. For onvx omtrs wt only 3-vlnt vrts, w n put on lvl 1 un st pr omtr sur. Tn urv s ptur y t F. 10 T omtr vrtx s ptur y snl xron, w s t ntrston o tr lvl 1 un sts (rprsnt y t rn, yllow, n r ul sts) w prws ntrst lon omtr urvs to orm prml ors (olor ur onln)

8 r nssry to ptur n-vlnt onvx omtr vrtx [27, 34]. In ts ppr, w lmt our ous to t 4-vlnt vrts onurtons vn n F. 11. In, two xr ptur on omtr vrtx, wl our xr ptur t n n. T onsqun s tt lvl 1 un sts r not sunt to ptur 4-vlnt omtr vrts. In t two prvous ss, w v: 1. In, two xr r nrt y two lvl 1 un sts (rn n yllow ul sts) n two lvl 3 un sts (lu n r ons); 2. In n, our xr r nrt y on lvl 1 un st (r ul st) n our lvl 3 un sts (ry, lu, yllow, n rn ons). Tr s no xplt rrn to omtr vrts n t unmntl ms nton, ut ty v mportnt rstrtons n t mnnr to n t unmntl sts to nsrt. Morovr, t lol topolol strutur o xrl mss nors t onton wr t lol onurton t vrts must propt lon t jnt urvs n surs. For nstn, onsr F. 12 wr 4-s s pyrm s rprsnt. W ous on t ln smnt jonn t 4-vlnt top vrtx n t 3-vlnt ront vrtx o t s. Altou t s 3-vlnt, ts vrtx nnot ptur y snl xron u to t propton o onstrnts mntn rom t top vrtx, w s ptur y two xr. I w onsr un sts, ts two xr r nrt y two lvl 1 un st n two lvl 3 un sts s llustrt n F. 12. T onstrnts tt r propt lon omtr urvs n ntrprt s onstrnts on t omtr urvs tmslvs. Mor prsly, w n tr typs o urvs: Typ 1 T urv s lolly ptur y two lvl 1 un sts (s F. 13); Typ 2 T urv s lolly ptur y on lvl 1 un st n two lvl 3 un sts (s F. 13); Typ 3 T urv s lolly ptur y two lvl 2 un sts n two lvl 3 un sts (s F. 13). T lolty trm n ts tr ss s u to t t tt, lolly to urv, t s mpossl to stnus F. 12 Propton o omtr vrts onstrnts lon omtr urvs. A omplt pyrm s sown n, wl ous on t ln smnt jonn t top vrtx n t ront vrtx o t s s vn n lvl 1 rom lvl 2 un st n lvl 3 rom lvl 2 un st. Morovr, propton o onstrnts lon omtr urvs n surs n moy ts lsston. W v lrtly slt only tr typs o urvs to t qut smpl lortm. Extnsons o ts lortm wll proly l to ntrou som otr typs o urvs. Vrts lso v prsr st o vl onurtons. In ts ppr, w lmt our ous to t ollown: Typ V3A A 3-vlnt vrtx wt ll jnt urvs o Typ 1 (s F. 10); Typ V3B A 3-vlnt vrtx wt two jnt Typ 1 urvs n on jnt Typ 2 urv; Typ V3C A 3-vlnt vrtx wt ll jnt urvs o Typ 3; Typ V3D A 3-vlnt vrtx wt two jnt Typ 3 urvs n on jnt Typ 2 urv; Typ V4A A 4-vlnt vrtx wt two oppost jnt urvs o Typ 1 n t otr two oppost jnt urvs o Typ 2 (s F. 11); Typ V4B A 4-vlnt vrtx wt two oppost jnt urvs o Typ 1 n t otr two oppost jnt urvs o Typ 3 (s F. 11); Typ V4C A 4-vlnt vrtx wt two oppost jnt urvs o Typ 2 n t otr two oppost jnt urvs o Typ 3 (s F. 11); Typ V4D A 4-vlnt vrtx wt ll jnt urvs o Typ 2 (s F. 11); Typ V4E A 4-vlnt vrtx wt ll jnt urvs o Typ 3 (s F. 11). F. 11 A 4-vlnt omtr vrtx s r ptur y two xr () or our xr (, ). In, t our xr r sown; wrs n, t orrsponn ul sts r rprsnt

9 F. 13 T tr typs o urvs w nl. A Typ 1 urv s ptur y two lvl 1 un sts n orrspons to snl unmntl or. A Typ 2 urv s ptur y on lvl 1 un st n two lvl 2 un sts n orrspons to two unmntl ors. A Typ 3 urv s ptur y two lvl 2 un sts n two lvl 3 un sts n orrspons to tr unmntl ors As n t s o urvs, t s mpossl to stnus, lolly to vrtx, lvl 1 rom lvl 2 un st n lvl 3 rom lvl 2 un st. Tt s t rson wy w v t sm lol onurtons o xr roun vrts o typs V4A, V4B, n V4C on on n, n V4D n V4E on t otr n. As sr ltr n ts ppr, t lsston Typ ssn to omtr urv s mo untl ll omtr vrts n t mol mt on o ts vrtx typs. 3.2 Ins t volum: propton o ntrnl prml sts Lvl 2 n 3 un sts propt ns t ms. It must trmn wr ts sts r pl n ow ty n onnt totr ns t ms. For xmpl, onsr t 2D xmpl n F. 14. T pts o lvl 1 un sts (lu) v n ompltly trmn. Hnn wskrs rprsnt our prtlly n lvl 2 un sts (r) n our prtlly n lvl 3 un sts (rn). T ol o our lortm s to trmn ow ts nn wskrs r pr n t pt trvrs to pr tm. Drnt solutons n slt lk solutons A, B, n C prsnt n F. 14. Altou not t prmry ous o ts ppr, w must trmn t sptl pt o t un sts s ty propt nto t volum. In our urrnt lortm, ts propton s s on omtr rtr n ollows plnr projtons o ounry normls. In spt o t lmttons o ts smpl ppro, t llows us to nrt ms y onsrn lol proprts n not just lol proprts. Ts pont s rul; on o t mn ults n xrl msn s to norport qut onsrton o t lol topolol strutur o xrl mss. 3.3 Our lortm: rom ounry to ns T propos lortm s s on lsston o omtr vrts n urvs. As xpln n St. 3.1, F. 14 In 2D, un st onnton onssts n onntn un ors. For ts xmpl, tr solutons o t mny possl solutons r llustrt. Soluton B sms to t xpt on or most usrs onstrnts on vrts r mnly propt lon urvs. Ts propton ls to lsston o urvs nto tr tors wl w lssy omtr vrts y onsrn tr vln. T skt o t lortm trmnn t sts to nsrt s s ollows: 1. W ntlz vry urv to Typ 1 t s onvx or Typ 3 otrws; 2. Tn w trmn w vrts r nvl (.., o not mt on o t vrtx lsston Typs: V3A, V3B, t.); 3. W trtvly n urv lsston Typs, on t tm, untl ll nvl vrts mt on o t vrtx lsston Typs. Ts n tmporrly nvlt som ntlly vl vrts; 4. Fnlly, w omtrlly los som yls o Typ 2 urvs tt wr rt urn t prvous stp. To los ts yls, w us omtr ntrston (urrntly st o plns), n w n som vrtul urvs. W now llustrt n tl ts lortm on t omtr ojt vn n F. 15. T rst stp o t lortm onssts o ssnn n ntl strtn lsston

10 to vry omtr urv. In F. 15, t omtr ojt s onvx; tus, ll t s r ntlly o Typ 1. For onvnn, Typ 1 urvs r rwn n lk, n Typ 2 urvs r rwn n r n F. 15. Nxt, vrtx s nlyz to s w vrtx Typ ty r (V3A, V3B, t.). In F. 15, non o t 4-vlnt vrts mt ny o t vrtx lsstons n r tus to lst, V,o nvl vrts: V ¼; ; ; ; : T prnpl ol o t lortm s to moy t Typ lsstons o urvs, on-y-on untl vrtx mts on o t vl vrtx typs (V3A, V3B, t.). In our xmpl, to mk vrtx vl, w v to n t Typ lsston o t lst two oppost urvs jnt to t. T rsultn onurton s pt n F. 15 wr Typ 2 urvs r rwn n r. In ts nw onurton, vrtx s ptur y two xr n s o Typ V4A. T lst o nvl vrts oms tn: V ¼; ; ; : T nxt vrtx to l wt s vrtx. Cnn t Typ lsston o [, ] to Typ 2 wll mk vrtx Typ V4A n tror vl (s F. 15). Now, vrts n r, rsptvly, ptur y two xr. Lst V s tn {,, }. T sm moton s prorm roun vrtx to t t onurton o F. 15 n V = {, }. Nxt, solvn t nvl onurton t vrtx n on y tr rlssyn urvs [, ] n [, ] to Typ 2 urvs or y rlssyn urvs [, ] n [, ] to Typ 2 urvs. Our lortm slts t rst opton to t t onurton n F. 15. V = {, } sn vrtx s no lonr vl. Vrtx s m vl y urv [, ] to om Typ 2 urv (s F. 15). Vrtx s tn ptur y our xr n s now Typ V4D. Lst V s ru to {}. Fnlly, vrtx oms vl y rlssyn urv [, ] to Typ 2 urv (s F. 15). W otn two opn yls o Typ 2 urvs n F. 15: t rst on strts n vrtx to n n vrtx, t lttr on strts n vrtx to n n vrtx. Ty v to los us yl orrspons to splt t omtr ojt n two volums y nsrtn two lvl 3 sts lon t yl. W los ot yls y onsrn t t tt tr n vrts r on t sm omtr sur (tus w smply vrtul urv to los yl), n w t rsult o F. 15. I t n vrts r not on t sm omtr sur, w rt svrl plns n y t vrts o t opn yl, n w us ts st o plns to rv t lvl 3 un sts nsrtons. Ts nsrtons, s ll t st nsrtons w prorm, r m rtly n t THx ms o t ntl omtr ojt. W t rsult o F. 16. Not tt our urrnt lortm s ry lortm tt mks os twn rnt possl solutons t mny stps n t lortm. Ts sons n l to rnt vl unmntl mss. Tus, n most ss, tr s not unqu soluton n pns on t mto us or rsolvn non-vl onurtons. For nstn, n t prvous xmpl, w my not t t sm rsult w strt t lortm y xn t nvl onurton n vrtx nst o vrtx. F. 15 Smpl xmpl sown ow w trmn t typ o t omtr urvs n orr to x nvl onurtons roun omtr vrts

11 F. 16 T ms tr vn nsrt t un sts n prvously; two lvl 3 un sts nsrt lon t sm Typ 2 urv r xt; t rsultn lokstrutur xrl ms s sown F. 17 T omtry o ts xmpl s t sm s t prvous xmpl, ut ornr ws rmov to t nononvx omtry. T rsultn strutur-lok xrl onurton s prsnt wt two rnt ponts o vw n two srnkn prmtrs F. 18 T non-swpl omtry to ms;, t nrt strutur-lok xrl ms n two ponts o vw l wt non-onvx omtrs vn 4-vlnt omtr vrts. 4 Conluson n utur works F. 19 T omtry to ms s sown n ; n, w t t rsultn lok-strutur xrl ms; nl xrl ms wr vry lok s n rms s sown (Not tt t Lpln smootn us n ts xmpl os not prov t st omtr rsults) 3.4 Exmpls Furs 17, 18, n 19 prov som rsults otn wt t propos lortm. Ty sow tt t lortm n T mn ulty to ovrom n xrl msn s tt xrl mss v lol topolol strutur tt ny msn lortm must tk nto ount. Ts rtrst s vry rstrtv n xplns wy w nnot just onsr lol onurtons. In ts work, w v propos mor omplt nton o unmntl mss. Ts nton s s on lsston o unmntl sts n tr lvls n llows us to lnk omtr onsrtons n topolol strutur o xrl mss. W lv tt t provs stron ounton to ul upon n orr to nrt xrl ms or ny omtr ojt wt wll-sp xr lon t omtr ounry. Strtn rom t nton o unmntl mss, w v propos n lortm to ms ny omtr ojt

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