Application to complex solids of adaptive isogeometric analysis using T-splines

h p://www.dca.iusiani.ulpgc.es/p oyec o2012-2014 Applica ion o Complex Solids o Adap i e Isogeome ic Analysis using T-splines MINECO y FEDER P ojec : CGL2011-29396-C03-00 PEMEX & CONACYT-SENER P ojec , Fondo Sec o ial, con ac : 163723 R. Mon eneg o(1)* , J.M. Cascón(2) , E. Rod íguez(1) , J.M. Escoba (1) M. B o ka(1) , J.I. López(1) , J. Ramí ez(1) (1) Uni e si y Ins i u e SIANI, Uni e si y o Las Palmas de G an Cana ia, Spain (2) Depa men o Economics and His o y o Economics, Uni e si y o Salamanca, Spain 10 h WCCM , 8–13 July 2012, São Paulo, B azil Mo i a ion: Solid Modeling wi h T i a ia e T-splines ● INPUT: Su ace T iangula ion ● OUTPUT: T i a ia e T-spline ● 3-D T-Mesh o he Meccano The Meccano Me hod o Isogeome ic Solid Modeling Mo i a ion: Solid Modeling wi h T i a ia e T-splines ● INPUT: Solid Su ace ● OUTPUT: T i a ia e T-spline The Meccano Me hod o Isogeome ic Solid Modeling ● 3-D T-Mesh o he Meccano 16 h IMR (2007) Mo i a ion: Simul aneous Mesh Gene a ion and Volume Pa ame e iza ion 18 h IMR (2009) ● INPUT: Su ace T iangula ion ● OUTPUT: Te ahed al Mesh ● Meccano Te ahed al Mesh The Meccano Me hod o 3-D Mesh Gene a ion Algo i hm S eps: Su ace in o ma ion as inpu da a; explici in his case Isogeome ic Modeling o a Genus-one Solid Algo i hm S eps: The meccano app oxima ion Isogeome ic Modeling o a Genus-one Solid Algo i hm S eps: Coa se e ahed al mesh ( e -subdi ision o he polycube) Isogeome ic Modeling o a Genus-one Solid Algo i hm S eps: Local e ined e ahed al mesh Isogeome ic Modeling o a Genus-one Solid  Ini ial cube and i s subdi ision a e h ee consecu i e e ahed on bisec ion Local Re inemen : Kossaczky’s Algo i hm (JCAM 1994) Re inemen o a cube h p://www.albe a- em.de/, ALBERTA code F om a he i- h solid su ace iangula ion pa ch o he i- h meccano ace Su ace Pa ame e iza ion o M.S. Floa e (CAGD 1997) h p://www.sin e .no/ma h so wa e, GoTools om SINTEF ICT F ee node Local op imiza ion New posi ion o he ee node (x,y,z) (x,y,z) Objec i e: Imp o e he quali y o he local mesh N( ) by minimising an objec i e unc ion Local mesh N( ) Te ahed al Mesh Op imiza ion SUS Code: F eely-a ailable in h p://www.dca.iusiani.ulpgc.es/p oyec o2008-2011 Simul aneous Un angling and Smoo hing (CMAME 2003) Applica ion o he A madillo: A su ace iangula ion as inpu da um Meccano Me hod o a Complex Genus-Ze o Solid h p://g aphics.s an o d.edu/da a/3Dscan ep/, S an o d Compu e G aphics Labo a o y T T Physical Elemen T Ta ge Elemen T Op imiza ion ( o ge less dis o ion in he pa ame e iza ion) T-mesh and ancho α Bi a ia e Cubic T-spline Basis Func ion suppo o he T-spline 1 ξ 2 ξ × = ( ) ( ) ( ) 221121, ξξξξ ααα NNB = ( ) 11 ξ α N ( ) 22 ξ α N Kno s associa ed o ancho α : { } 1 6 1 5 1 4 1 2 1 1 1,,,, ξξξξξ α =Ξ { } 2 6 2 5 2 4 2 3 2 2 2 ,,,, ξξξξξ α =Ξ Isogeome ic Modeling and Analysis Example o T-mesh and T-splines in 2-D Cube e ahed al mesh Oc ee di ision o he cube Cube T-mesh Each cube o he oc ee does no con ain any node o he Kossaczky mesh in i s inne Ob ained by using he meccano me hod wi h Kossacsky e inemen Au oma ic Adap a ion o Inne and Bounda y Disc e iza ions Cons uc ion o he T-mesh o he Bunny In e pola ion poin s ( he ancho s) a e mapped o he solid by using he olume ic pa ame e iza ion ha was ob ained by he meccano me hod Mapping main aining ba icen ic coo dina es Mapping o he in e pola ion poin s Cons uc ion o he T-mesh o he Bunny ( ) ( ) ∑ ∈ = A R α αα ξξξξξξ 321321 ,,,, PS ( ) ( ) ( ) ∑ ∈ = A B B R β β α α ξξξ ξξξ ξξξ 321 321 321 ,, ,, ,, ( ) ( ) ( ) ( ) 332211321 ,, ξξξξξξ αααα NNNB = Wi h: Blending unc ions T i a ia e basis splines olume ic pa ame e iza ion ( ) ( ) AR A ∈∀= ∑ ∈ β α βααβ P S Con ol poin s a e calcula ed by sol ing he spa se linea sys em α P ( ) β S β The Spline In e pola ion Physical space loca ion Pa ame ic space loca ion Calcula ion o Con ol Poin s by Ful illing he In e pola ion Condi ions T-mesh o a bone T-mesh T-spline T-mesh and T-spline o he Bone Au oma ic Adap a ion o Inne and Bounda y Disc e iza ions C oss-sec ions o he Bone T-spline Au oma ic Adap a ion o Inne and Bounda y Disc e iza ions Adap i e Isogeome ic Re inemen Igea: T-spline o Nume ical Solu ion 2nd local e inemen 6021 cells, 9807 DOF Adap i e Isogeome ic Re inemen Igea: T-spline o Nume ical Solu ion 5 h local e inemen 6756 cells, 10838 DOF Adap i e Isogeome ic Re inemen Igea: Nume ical solu ion in a c oss sec ion o he pa ame ic space Ini ial T-mesh 5692 cells, 9304 DOF 5 h local e inemen 6756 cells, 10838 DOF 2nd local e inemen 6021 cells, 9807 DOF Adap i e Isogeome ic Re inemen uh: Ini ial T-mesh (g een) uh: 5 h local e inemen (blue) u: exac solu ion ( ed) Igea: Exac and nume ical solu ion in a c oss sec ion o he pa ame ic space Adap i e Isogeome ic Re inemen Igea: Ra e o con e gence E o es ima o (slope: -30.7) Exac e o in L2 no m (slope: -28.2) Exac e o in H1 semino m (slope: -17.2) Pa ame ic space Physical space Scaled Jacobian Final Commen s and Fu u e Wo ks Valid and In alid Con igu a ions in IGA Pa ame ic space Physical space Scaled Jacobian Valid and In alid Con igu a ions in IGA Rema ks: • In 2-D: P oblems could appea in he co ne s • In 3-D: P oblems could appea in he co ne s and edges Final Commen s and Fu u e Wo ks Pa ame ic space (Meccano T-mesh) Physical space (Tangled T-mesh) The Meccano Me hod o T-mesh: T-spline Op imiza ion Physical space (Op imized T-mesh) Final Commen s and Fu u e Wo ks Au oma ic Cons uc ion o he Meccano Final Commen s and Fu u e Wo ks Au oma ic Cons uc ion o he Meccano Final Commen s and Fu u e Wo ks