The meccano method for simultaneous volume parametrization and mesh generation of complex solids

The meccano me hod o simul aneous olume pa ame iza ion and mesh gene a ion o complex solids R Mon eneg o1,JMCasc´on2,JMEscoba 1, E Rod ´ıguez1and G Mon e o1 1Ins i u e o In elligen Sys ems and Nume ical Applica ions in Enginee ing, Uni e si y o Las Palmas de G an Cana ia, Campus Uni e si a io de Tafi a, Las Palmas de G.C., Spain 2Depa men o Economics and Economic His o y, Facul y o Economics and Managemen , Uni e si y o Salamanca, Spain E-mail: { mon eneg o,jmescoba ,e od iguez,gmon e o}@siani.es, [email p o ec ed] Abs ac . The meccano me hod is a no el and p omising mesh gene a ion me hod o simul aneously c ea ing adap i e e ahed al meshes and olume pa ame iza ions o a complex solid. We highligh he ac ha he me hod equi es minimum use in e en ion and has a low compu a ional cos . The me hod builds a 3-D iangula ion o he solid as a de o ma ion o an app op ia e e ahed al mesh o he meccano. The new mesh gene a o combines an au oma ic pa ame iza ion o su ace iangula ions, a local efinemen algo i hm o 3-D nes ed iangula ions and a simul aneous un angling and smoo hing p ocedu e. A p esen , he p ocedu e is ully au oma ic o a genus-ze o solid. In his case, he meccano can be a single cube. The efficiency o he p oposed echnique is shown wi h se e al applica ions. 1. In oduc ion The au oma ic and adap i e mesh gene a ion is a c ucial aspec in fini e elemen applica ions [4, 16, 17, 27, 28, 29, 31]. Along he pas , he main objec i e has been o achie e high quali y adap i e meshes o complex solids wi h minimal use in e en ion and low compu a ional cos , bu he 3-D p oblem is s ill open [1]. I is well known ha mos mesh gene a o s a e based on Delaunay iangula ion and ad ancing on echnique, bu p oblems, ela ed o mesh quali y o mesh con o mi y wi h he solid bounda y, can s ill appea o complex geome ies. In addi ion, an app op ia e defini ion o elemen sizes is demanded o ob aining good quali y elemen s and mesh adap ion. Pa icula ly, local adap i e efinemen s a egies ha e been employed o mainly adap he mesh o singula i ies o nume ical solu ion. These adap i e me hods usually in ol e emeshing o nes ed efinemen . In his di ec ion, we ha e in oduced he new meccano echnique [2, 3, 24, 25] o cons uc ing adap i e e ahed al meshes o solids. We ha e gi en his name o he me hod because he p ocess s a s wi h he cons uc ion o a coa se app oxima ion o he solid, i.e. a meccano composed by connec ed polyhed al pieces (a pa icula case is when meccano is composed by connec ed cubes, i.e. a polycube.). The idea o he me hod is o build a olume mesh o he solid as a de o ma ion o an app op ia e e ahed al mesh o he meccano. WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 c 2010 Published unde licence by IOP Publishing L d 1 The new au oma ic mesh gene a ion s a egy uses no Delaunay iangula ion, no ad ancing on echnique, and i simplifies he geome ical disc e iza ion p oblem o 3-D complex domains, whose su aces can be mapped o he meccano aces. The meccano me hod combines a local efinemen /de efinemen algo i hm o 3-D nes ed iangula ions [21], a pa ame e iza ion o su ace iangula ions [9] and ou simul aneous un angling and smoo hing p ocedu e [6]. A p esen , he meccano echnique has been implemen ed by using he local efinemen /de efinemen o Kossaczky [21], bu he idea could be implemen ed wi h o he ypes o local efinemen algo i hms [18]. We no e ha he esul ing adap i e e ahed al meshes wi h he meccano me hod ha e good quali y o fini e elemen applica ions. Ou app oach is based on he combina ion o se e al o me p ocedu es ( efinemen , mapping, un angling and smoo hing) which a e no in hemsel es new, bu he o e all in eg a ion is an o iginal con ibu ion. Many au ho s ha e used hem in diffe en ways. T iangula ions o con ex domains can be cons uc ed om a coa se mesh by using efinemen /p ojec ion [26]. Adap i e nes ed meshes ha e been cons uc ed wi h efinemen and de efinemen algo i hms o e olu ion p oblems [8]. Mappings be ween physical and pa ame ic spaces ha e been analyzed by se e al au ho s. Significan ad ances in su ace pa ame iza ion ha e been done in [9, 11, 12, 23, 30, 32], bu he olume pa ame iza ion is s ill open. Floa e e al [13] gi e a simple coun e example o show ha con ex combina ion mappings o e e ahed al meshes a e no necessa ily one- o- one. La ge domain de o ma ions can lead o se e e mesh dis o ions, especially in 3-D. Mesh op imiza ion is hus key o keeping mesh shape egula i y and o a oiding a cos ly emeshing [19, 20]. In adi ional mesh op imiza ion, mesh mo ing is guided by he minimiza ion o ce ain o e all unc ions, bu i is usually done in a local ashion. In gene al, his p ocedu e in ol es wo s eps [15, 14]: he fi s is o mesh un angling and he second one o mesh smoo hing. Each s ep leads o a diffe en objec i e unc ion. The meccano me hod uses he imp o emen p oposed by [6, 7], whe e a simul aneous un angling and smoo hing guided by he same objec i e unc ion is in oduced. Some ad an ages o he meccano me hod a e ha : solid su ace iangula ion is au oma ically cons uc ed, he final 3-D iangula ion is con o ming wi h he objec bounda y, inne su aces a e au oma ically p ese ed ( o example, in e ace be ween se e al ma e ials), node dis ibu ion is adap ed in acco dance wi h he objec geome y, and pa allel compu a ions can easily be de eloped o meshing he meccano pieces. Howe e , ou p ocedu e demands an au oma ic cons uc ion o he meccano and an admissible mapping be ween he meccano bounda y and he objec su ace mus be defined. In his pape , we conside applica ions o he meccano me hod o complex genus-ze o solid, i.e. a solid whose bounda y is a su ace ha is homeomo phic o he su ace o a sphe e. In his case, we assume ha he solid geome y is defined by a iangula ion o i s su ace and ha he use only has o fix a meccano composed by a single cube and a ole ance ha fixes he desi ed app oxima ion o he solid su ace. In o de o define an admissible mapping be ween he cube aces and pa ches o he ini ial su ace iangula ion o he solid, we ha e in oduced an au oma ic me hod o decompose he solid su ace iangula ion in o six pa ches ha p ese es he same opological connec ions han he cube aces. Then, a disc e e mapping om each su ace pa ch o he co esponding cube ace is cons uc ed by using he pa ame e iza ion o su ace iangula ions p oposed by M. Floa e in [9, 10, 11, 12]. The shape-p ese ing pa ame iza ions, which a e plana iangula ions on he cube aces, a e he solu ions o linea sys ems based on con ex combina ions. In he nea u u e, mo e effo should be made in de eloping an au oma ic cons uc ion o he meccano when he genus o he solid su ace is g ea e han ze o. Cu en ly, se e al au ho s a e wo king on his aspec in he con ex o polycube-maps, see o example [23, 30, 32]. They a e analyzing how o cons uc a polycube o a gene ic solid and, simul aneously, how o define a con o mal mapping be ween he polycube bounda y and he solid su ace. Al hough ha monic WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 2 maps ha e been ex ensi ely s udied in he li e a u e o su ace pa ame e iza ion, only a ew wo ks a e ela ed o olume pa ame iza ion, o example a p ocedu e is p esen ed in [22]. In he ollowing Sec ion we p esen a b ie desc ip ion o he main s ages o he me hod o a gene ic meccano composed o polyhed al pieces. In Sec ion 3 we in oduce applica ions o he algo i hm in he case ha he meccano is o med by a simple cube. Finally, conclusions and u u e esea ch a e p esen ed in Sec ion 4. 2. The algo i hm o he meccano me hod The main s eps o he gene al meccano e ahed al mesh gene a ion algo i hm a e summa ized in his sec ion. A de ailed desc ip ion o he me hod can be analyzed in [2, 3, 24, 25]. The inpu da a a e he defini ion o he solid bounda y ( o example by a gi en su ace iangula ion) and a gi en ole ance (co esponding o he solid su ace app oxima ion). The ollowing algo i hm desc ibes he whole mesh gene a ion app oach. Meccano e ahed al mesh gene a ion algo i hm (i) Cons uc a meccano app oxima ion o he 3-D solid o med by polyhed al pieces. (ii) Define an admissible mapping be ween he meccano bounda y aces and he solid bounda y. (iii) Build a coa se e ahed al mesh o he meccano. (i ) Gene a e a local efined e ahed al mesh o he meccano, such ha he mapping o he meccano bounda y iangula ion app oxima es he solid bounda y o a gi en p ecision. ( ) Mo e he bounda y nodes o he meccano o he objec su ace wi h he mapping (ii). ( i) Reloca e he inne nodes o he meccano. ( ii) Op imize he e ahed al mesh wi h he simul aneous un angling and smoo hing p ocedu e. The fi s s ep o he p ocedu e is o cons uc a meccano app oxima ion by connec ing diffe en polyhed al pieces. Once he meccano app oxima ion is fixed, we ha e o define an admissible one- o-one mapping be ween he bounda y aces o he meccano and he bounda y o he objec . In hi d s ep, he meccano is decomposed in o a coa se and alid e ahed al mesh by an app op ia e subdi ision o i s ini ial polyhed al pieces. We con inue wi h a local efinemen s a egy o ob ain an adap ed mesh which can app oxima e he bounda ies o he domain wi hin a gi en p ecision. Then, we cons uc a mesh o he solid by mapping he bounda y nodes om he meccano aces o he ue solid su ace and by eloca ing he inne nodes a a easonable posi ion. A e hose wo s eps he esul ing mesh is angled, bu i has an admissible opology. Finally, a simul aneous un angling and smoo hing p ocedu e is applied and a alid adap i e e ahed al mesh o he objec is ob ained. We no e ha he gene al idea o he meccano echnique could be unde s ood as he connec ion o diffe en polyhed al pieces. So, he use o cuboid pieces, o a polycube meccano, a e pa icula cases. 3. Applica ions o he meccano me hod om a cube In his sec ion, we p esen he applica ion o he meccano algo i hm in he case o he solid su ace being genus-ze o and he meccano being o med by a single cube. We assume as da um a iangula ion o he solid su ace. We in oduce an au oma ic pa ame iza ion be ween he su ace iangula ion o he solid and he cube bounda y. To ha end, we au oma ically di ide he su ace iangula ion in o six pa ches, wi h he same opological connec ion ha cube aces, so ha each pa ch is mapped o a cube ace. These pa ame iza ions ha e been done wi h GoTools co e and pa ame iza ion modules om SINTEF ICT, a ailable in he websi e h p://www.sin e .no/ma h_so wa e. This code implemen s Floa e ’s pa ame iza ion in C++. In he ollowing applica ion we ha e used he mean alue me hod o he pa ame iza ion o he inne nodes o he pa ch iangula ion, and he bounda y nodes a e fixed wi h cho d leng h pa ame iza ion [9, 11]. WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 3 We ha e implemen ed he meccano me hod by using he local efinemen o AL- BERTA. This code is an adap i e mul ile el fini e elemen oolbox de eloped in C, see h p://www.albe a- em.de. This so wa e can be used o sol e se e al ypes o 1-D, 2-D o 3-D p oblems. ALBERTA uses he Kossaczky efinemen algo i hm [21] and equi es an ini ial mesh opology [26]. The ecu si e efinemen algo i hm could no e mina e o gen- e al meshes. The meccano echnique cons uc s meshes ha e i y he imposed es ic ions o ALBERTA in ela ion o opology and s uc u e. The minimum quali y o efined meshes is unc ion o he ini ial mesh quali y. The pe o mance o ou no el e ahed al mesh gene a o is shown in he ollowing applica ions. The fi s co esponds o an a madillo and he second o a bone. We ha e ob ained a su ace iangula ion o hese objec s om in e ne . 3.1. Applica ion o an a madillo The o iginal su ace iangula ion o he A madillo has been ob ained om he S an o d Compu e G aphics Labo a o y, h p://g aphics.s an o d.edu/da a/3Dscan ep/,andis shown in Figu e 1. I has 30000 iangles and 15002 nodes. The bounding box o he solid is defined by he poin s (x, y, x)min =(−60,−50,−26) and (x, y, z)max =(68,66,90). (a) (b) Figu e 1. O iginal su ace iangula ion o A madillo, (a) ini ial subdi ision in connec ed sub iangula ions using Vo onoi diag am and (b) compa ible pa i ion {T i S}5 i=0 ob ained a e applying ou educ ion algo i hm, emo ing di iding edges and smoo hing he in e aces be ween pa ches. We conside a uni cube as meccano. I s cen e is placed inside he solid a he poin (7.5,17.5,55.5). We ob ain an ini ial subdi ision o A madillo su ace in ele en maximal connec ed sub iangula ions using he Vo onoi diag am associa ed wi h he cen e s o he cube aces, see Figu e 1(a). In o de o ge a compa ible decomposi ion o he su ace iangula ion, WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 4 we use he i e a i e p ocedu e p oposed in [3] o educe he cu en ele en e ices o he g aph GS o six. Figu e 1(b) shows he esul ing compa ible pa i ion {T i S}5 i=0. (a) (b) Figu e 2. (a) Floa e ’s pa ame iza ion o {T i S}5 i=0 on co esponding cube aces, (b) cube e ahed al mesh ob ained by he meccano me hod a e efinemen . We map each su ace pa ch Σi S o he cube ace Σi Cby using he Floa e pa ame iza ion [9]. The defini ion o he one- o-one mapping be ween he cube and A madillo bounda ies is s aigh o wa d once he pa ame iza ion o he A madillo su ace iangula ion is buil , see Figu e 2(a). A de ail o he each pa ame iza ion o Ti Son he co esponding cube ace Σi Cis shown in Figu e 3. The cube is di ided in o six e ahed a. In o de o app oxima e he A madillo su ace wi h a ole ance ε2=0.1, see [3], we efine he cube mesh by applying 68 Kossaczky ecu si e bisec ions, see Figu es 2(b) and 5(a). This mesh con ains 144964 e ahed a and 33889 nodes, wi h 31316 iangles and 15660 nodes on i s bounda y. The mapping o he cube ex e nal nodes o he A madillo su ace p oduces a 3-D angled mesh wi h 10871 in e ed elemen s, see Figu e 4(a). The eloca ion o inne nodes by using olume pa ame iza ions educes he numbe o in e ed e ahed a o 186. We use he e ahed al mesh op imiza ion, p esen ed in [6], such ha he mesh is un angled in 6 i e a ions. The mesh quali y is imp o ed o a minimum alue o 0.08 and an a e age qκ=0.71 a e 6 smoo hing i e a ions. We no e ha he meccano echnique gene a es a high quali y e ahed al mesh, see Figu e 4(b): only 4 e ahed a ha e a quali y o less han 0.1, 150 less han 0.2 and 1046 less han 0.3. In o de o show he efficiency o he mesh op imiza ion echnique inside he A madillo, we display in Figu e 5 wo c oss sec ions be o e (b) and a e (c) i s applica ion. The loca ion o he cube can be obse ed in Figu e 5(b). We also no e ha , due o he high quali y su ace iangula ion ob ained wi h ou me hod, he mesh imp o emen is no significan i we p e iously apply he smoo hing su ace iangula ion algo i hm in oduced in [7]. The CPU ime o cons uc ing he final mesh o he A madillo is app oxima ely 112.78 seconds on a Dell p ecision 690, 2 Dual Co e Xeon p ocesso and 8 Gb RAM memo y. Mo e p ecisely, he CPU ime o each s ep o he meccano algo i hm is: 5.45 seconds o he subdi ision WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 5 (a) (b) (c) (d) (e) ( ) Figu e 3. De ail o Floa e ’s pa ame iza ion o A madillo su ace sub iangula ions {T i S}5 i=0. o he ini ial su ace iangula ion in o six pa ches, 1.01 seconds o he Floa e pa ame iza ion, 33.11 seconds o he Kossaczky ecu si e bisec ions, 2.25 seconds o he ex e nal node mapping and inne node eloca ion, and 80.96 seconds o he mesh op imiza ion. WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 6 (a) (b) Figu e 4. (a) Tangled mesh a e he mapping o he ex e nal nodes o he cube o he A madillo su ace and (b) esul ing alid mesh a e inne node eloca ion and mesh op imiza ion. (a) (b) (c) Figu e 5. C oss sec ions o he cube (a), and he A madillo e ahed al meshes be o e (b) and a e (c) he applica ion o he mesh op imiza ion p ocedu e. 3.2. Applica ion o a bone The o iginal su ace iangula ion o he Bone has been ob ained om h p://www-c.in ia. /gamma/download/a ichage.php?di =ANATOMY&name=ballJoin , and i can be ound in he CYBERWARE Ca alogue. This su ace mesh con ains 274120 iangles and 137062 nodes. WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 7 S eps o he meccano echnique a e shown in Figu e 6. The esul ing mesh has 47824 e ahed a and 11525 nodes. This mesh has 11530 iangles and 5767 nodes on i s bounda y and i has been eached a e 23 Kossaczky efinemen s om he ini ial subdi ision o he cube in o six e ahed a. A angled e ahed a mesh wi h 1307 in e ed elemen s appea s a e he mapping o he cube ex e nal nodes o he bone su ace. The node eloca ion p ocess educes he numbe o in e ed e ahed a o 16. Finally, ou mesh op imiza ion algo i hm p oduces a (a) (b) (c) (d) Figu e 6. The meccano (a) and a c oss sec ions o he Bone e ahed al mesh be o e (b) and a e (c) he applica ion o he mesh op imiza ion p ocedu e. (d) Resul ing e ahed al mesh o he Bone ob ained by he meccano me hod. WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 8 high quali y e ahed a mesh: he minimum mesh quali y is 0.15 and he a e age quali y is 0.64. We no e ha he meccano me hod uses o cons uc mo e han 1000 e ahed a pe second on a Dell p ecision 690, 2 Dual Co e Xeon p ocesso and 8 Gb RAM memo y. Finally, we summa ize a compa ison be ween ou me hod and s anda d e ahed al mesh gene a ion echniques [27, 28, 29]. On he one hand, one o he mos impo an con ibu ions o he meccano me hod is he esul ing olume pa ame iza ion o he solid. I can ha e in e es ing applica ions in solid modeling and nume ical simula ion. Fo example, he applica ion o isogeome ic analysis [1, 5] can be easie . On he o he hand, ou olume meshes can be u ilized in adap i e fini e elemen applica ions by using he Kossaczky’s algo i hm. The local efinemen s eps a e e y as because he sequence o solid meshes is defined om he coa se mesh o he meccano, i.e., he di iding edge o e ahed on bisec ion is known s aigh o wa d. In addi ion, we no e ha he minimum mesh quali y is bounded du ing all he mesh adap a ion p ocess, because simila elemen s appea a e h ee consecu i e bisec ions. Fo a gi en solid su ace iangula ion, we ha e checked ha a cons ained Delaunay e ahed aliza ion [29] can be as e han ou me hod, bu he esul ing meshes ha e lowe quali y. I he admissible minimum quali y is inc eased, many poin s can be added and he numbe o e ahed a inc eases o each he objec i e. I only a con o ming Delaunay e ahed aliza ion is desi ed, he numbe o e ahed a can be much highe han in ou me hod. A g ea ad an age o ou me hod is ha he adap i e node dis ibu ion on he bounda y and he inne o he solid is mo e s uc u ed, because he posi ions a e fixed wi h a sequence o nes ed meshes. In he case o ad ancing on [27], he esul ing mesh depends on he quali y o he gi en su ace iangula ion. We ha e seen ha he op imiza ion o his iangula ion, changing he mesh opology, can be oo cos ly. 4. Conclusions and u u e esea ch The meccano echnique is a e y efficien adap i e e ahed al mesh gene a o o solids whose bounda y is a su ace o genus ze o. We ema k ha he me hod equi es minimum use in e en ion and has a low compu a ional cos . The p ocedu e is ully au oma ic and i is only defined by a su ace iangula ion o he solid, a cube and a ole ance ha fixes he desi ed app oxima ion o he solid su ace. A c ucial consequence o he new mesh gene a ion echnique is he esul ing disc e e pa ame iza ion o a complex olume (solid) o a simple cube (meccano). We ha e in oduced an au oma ic pa i ion o he gi en solid su ace iangula ion o fixing an admissible mapping be ween he cube aces and he solid su ace pa ches, such ha each cube ace is he pa ame ic space o i s co esponding pa ch. The main ideas p esen ed in his pape can be applied o cons uc ing e ahed al o hexahed al meshes o complex solids. In u u e wo ks, he meccano echnique can be ex ended o meshing a complex solid whose bounda y is a su ace o genus g ea e han ze o. In his case, he meccano can be a polycube o cons uc ed by polyhed al pieces wi h compa ible connec ions. A p esen , he use has o define he meccano associa ed o he solid, bu we a e implemen ing a special CAD package o mo e gene al inpu solid. Acknowledgmen s This wo k was pa ially suppo ed by he “Sec e a ´ıa de Es ado de In es igaci´on” o Spanish Go e nmen , “Minis e io de Ciencia e Inno aci´on”, and FEDER, g an con ac s: CGL2008- 06003-C03 and UNLP08-3E-010. Re e ences [1] Bazile s Y, Calo V M, Co ell J A, E ans J, Hughes T J R, Lip on S, Sco M A and Sede be g T W 2008 Isogeome ic analysis: Towa d unifica ion o compu e aided design and fini e elemen analysis T ends in Enginee ing Compu a ional Technology (S i ling: Saxe-Cobu g Publica ions) chap e 1 pp 1–16 WCCM/APCOM 2010 IOP Publishing IOP Con . Se ies: Ma e ials Science and Enginee ing 10 (2010) 012018 doi:10.1088/1757-899X/10/1/012018 9