Introduction to Geometric

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1 Manel Ventra Shp Desgn I MSc n Marne Engneerng and Naal Archtectre Smmary. Parametrc Cres 2. Parametrc Srfaces 2

2 Parametrc Cres. Mathematcal Formlatons Cbc Splnes Bézer B-Splne Beta-Splne NURBS 2. Interpolaton and approxmaton of cres 3. Analyss of cres 3 Cbc Splne Consderng the wooden splne rote a thn elastc beam and for small deflectons the Eler law relates the deflecton of the beam axs yx wth the bendng moment Mx by the expresson: x y M x EI where: E Modls of Yong I Moment of nerta of the beam secton 4 2

3 Cbc Splne 2 Assmng that the beam s smply spported on the weghts then the bendng moment ares lnearly between them.e. Mx Ax B. Replacng n the expresson and ntegratng reslts yx M x dx 3 2 Ax B dx Ax Bx Cx D EI EI In each segment the cre can be defned as a fncton of the parameter t normalzed for the nteral [] P t At 3 Bt 2 Ct D The constants can be obtaned from the followng bondary condtons: P P P P p p T T 5 Cbc Splne 3 Fnally the cre can be represented n the matrx form as where 3 2 t [ t t t ][ H ][ G] P [ H ] [ G] p p T T 6 3

4 Bézer Cres The cres generally nown as Bézer reslted from separate research from Castela Ctroen and Perre Bézer Renalt n the begnnng of the 96s. A Bézer cre s defned by: n P t C Bn t for t where B n are the Bernesten base fnctons of degree n B n n! t! n! n n n t t n for... t 7 Bézer Cres 2 The Bézer cre s tangent to the frst and last segments of the control polygon The cre order s eqal to the nmber of ertces of the control polygon. The cre s entrely contaned n the conex hll of the control ponts. 8 4

5 B-Splne Cres They were stded by N. Lobatchesy n the XIX centry Ther se for cre fttng to expermental data began n 946 wth Schoenberg They were frst ntrodced n CAD systems by J. Fergson Boeng n 963. N N t para C t t t N t t t n t t < t t t t P N t t t Where C are the ponts of the control polygon and N are the B-Splne base fnctons of order that can be compted by the recrse expresson from Cox/de Boor: N t Defned oer a not ector {... } X t t 2 t 3 t m 9 B-Splne Cres 2 The not ector s a non-decreasng seqence of nmbers The not ector can be classfed as: Unform the ncrement between nots s constant { } Perodc the ncrement s constant and eqal to { } Non-Perodc the ncrement of the nteror nots constant and eqal to and the nots of the extremtes wth mltplcty eqal to the order { } Non-Unform - the ncrement of the nteror nots not necessarly constant and the nots of the extremtes wth mltplcty eqal to the order { } 5

6 B-Splne Cres 3 The B-Splne cres hae the followng propertes: Lnear precson Conex hll n consecte control ponts Varaton dmnshng Are narant when sbmtted to affne transformatons When the order of the B-Splne s eqal to the nmber of control ponts the not ector conssts only n the ales of the extremtes wth the mltplcty eqal to the order { } and the B-Splne base fnctons are eqalent to Bernesten fnctons and the cre degenerates nto a Bézer cre. Beta-Splnes Os cbc Beta-splnes were ntrodced on 98 by Barsy They are a generalzaton of the B-Splnes based n notons of geometrc contnty and n the mathematcal modelng of tenson The reqrements of parametrc contnty of the 2ª order C 2 between the B-Splnes segments s replaced by the reqrements of geometrc contnty of 2ª order G 2 of the nt tangent ector and of the cratre ector Ths orgnates dscontntes of the st and 2nd parametrc derate that are expressed as fnctons of the parameters β and β2 desgnated by bas and tenson respectely. 2 6

7 Beta-Splnes 2 A Beta-splne cre s defned by: b C br r2 r 2 g 3 β β ; P p / < 2 r where b γ are the base fnctons g β β2; cgr β β p / < e r 2 The parametrc contnty reflects the far araton of the parameterzaton and not necessarly of the cre The geometrc contnty s a measre of the contnty that s ndependent from the parameterzaton 3 NURBS Cres C n n P. w N w. N p p N para < N p N p p p N p p { } U 2 n n... n 4 7

8 Representaton of Conc Shapes A NURBS cre of the 2nd degree wth 3 ponts represents a conc shape f the conc form factor c defned by: 4 c 4 c 4 c w. w 2 4. w c 3 Has one of the followng ales 2 <. elpse. parabola >. hperbole 5 Representaton of Conc Shapes 2 To represent a crclar arc the 3 control ponts [P P2 P3] mst be oer the ertces of a trangle sosceles The arc rads obtaned s compted by: R 2 4b 4b Complete crcmferences can be represented onng arcs Wth 9 ponts 4 arcs of 9 can be oned X W {. } P P P P P P P P P P { } { } where: 2 b 2 6 8

9 Representaton of Conc Shapes 3 The preos representaton can be smplfed remong the repeated nots.25 and.75 The reslt s a crcmference represented by only 7 control ponts X W P { } { } { P P P P P P P} Representaton of Conc Shapes 4 A crcmference can also be obtaned onng 3 arcs of 2 defned by 7 control ponts. X W { } { } P P P P P P P

10 Representaton of Conc Shapes 5 A complete ellpse can be represented applyng an affne transformaton to a crcmference for example one represented by 7 control ponts eepng the dstrbton of the weghts and the not ector. X W P { } { } { P P P P P P P} Smmary - Parametrc Cres Adantages Dsadantages Obs. Cbc Splne Interpolates data ponts Can present nexpected nflectons Bézer The control polygon les otsde the data ponts Global behaor Degree ncreases drectly wth the ncreasng nmber of control ponts 2

11 Smmary - Parametrc Cres 2 Adantages Dsantages Obs. B-Splne Beta-Splne NURBS Local behaor Degree ndependent of the nmber of control ponts Two addtonal parameters to control bas and tenson Accrate representaton of concs Can NOT represent conc shapes accrately It s dffclt to tae adantage of the addtonal coordnate weght Used n farng methods State of the art. Used n most exstng CAD systems 2 Cre Generaton Interpolaton cre contans all the data ponts Approxmaton cre tres to mnmze the dstance to all the data ponts 22

Analyss of Cre Cratre The cratre of a space cre s defned by: κ t x t x t x t 3 The dstrbton of ths cratre along the cre can be represented sng the method of the porcpne ectors

12 Analyss of Cre Cratre The cratre of a space cre s defned by: κ t x t x t x t 3 The dstrbton of ths cratre along the cre can be represented sng the method of the porcpne ectors wth modles proportonal to the ales of the cratre at each pont normal to the cre at that pont orented to the opposte sde of the centre of cratre 23 Analyss of Cre Cratre

13 3 25 Parametrc Srfaces. NURBS Srfaces 2. Srfaces Generaton Extrson Loftng Sweepng Reolton Grd Interpolaton Prmtes 3. Srface Analyss Shadng Contors Cratres Isophotes Reflecton Lnes 26 NURBS Srfaces A NURBS srface of degree l n the drectons s defned by the expresson: n n M N w M N w P S l m l m... < / N N N p N < / M M M p M l l l l l l

14 Extrson Traectory drectrz Profle geratrz 27 Loftng Sectons 28 4

the loft room sala do rsco The prmte bldng process was smlar to the modelng process sng

15 Loftng n Shpbldng The desgnaton loftng has orgn n shpbldng Desgnates the deelopment of the shp hll srface nterpolatng the shape of a set of cross sectons that was carred ot n the loft room sala do rsco The prmte bldng process was smlar to the modelng process sng the frames to shape the hll srface form. 29 Sweepng Profle geratrz Traectory drectrz 3 5

16 Srfaces of Reolton Profle geratrz Axs Exo de rotação 3 Edge Cres Srfaces defned by 2 3 or 4 edge cres 32 6

17 Grd Interpolaton Srface generated from a reglar grd of cres Prode a better control oer the nner shape of the srface 33 Elementary Prmte Shapes Box Cone Cylnder Sphere 34 7

18 Srface Analyss - Shadng Shadng 35 Srface Analyss - Contors Contors 36 8

Srface Analyss - Isophotes Isophotes: Analyze/Srface/Zebra Lnes of constant lght ntensty created by a set of parallel lght sorces wth a gen drecton L n L cosα 37 Srface Analyss Cratre Mean Cratre 2FM

19 Srface Analyss - Isophotes Isophotes: Analyze/Srface/Zebra Lnes of constant lght ntensty created by a set of parallel lght sorces wth a gen drecton L n L cosα 37 Srface Analyss Cratre Mean Cratre 2FM EN GL H 2 2 EG F Gass Cratre LN M K EG F 2 2 Cratres expressed as a fncton of the max. and mn. cratres H κ mn κ max 2 K κ κ mn max st Fndamental Form Coeffcents Srface normal nt E r. r F r. r G r. r ector 2nd Fndamental Form Coeffcents r r n p/ r r r r L n. r M n. r N n. r 38 9

20 Mean Cratre Mean Cratre dstrbton 39 Gass Cratre Gass Cratre dstrbton K < K K > Srface wth doble cratre saddle shape Deelopable srface Srface wth sngle cratre concae or conex 4 2

An Improved Isogeometric Analysis Using the Lagrange Multiplier Method

An Improved Isogeometric Analysis Using the Lagrange Multiplier Method An Improved Isogeometrc Analyss Usng the Lagrange Mltpler Method N. Valzadeh 1, S. Sh. Ghorash 2, S. Mohammad 3, S. Shojaee 1, H. Ghasemzadeh 2 1 Department of Cvl Engneerng, Unversty of Kerman, Kerman,