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