Generalizations of Non-Uniform Rational B-Splines: Theory, Software and Applications

Transcription

1 Generalatons of Non-Unform Ratonal B-Splnes: Theory, Software and Applcatons Alrea H. Taher 1, Saeed Abolghasem 2, Krshnan Suresh 1* 1 Department of Mechancal Engneerng, U-Madson, Madson, sconsn 5376, USA. 2 School of Mechancal Engneerng, Shahrood Unversty of Technology, Shahrood, Iran. * Correspondng author, emal: ksuresh@wsc.edu Abstract: e ntroduce a new class of curves and surfaces by explorng multple varatons of Non-Unform Ratonal B-Splnes. These varatons whch are referred to as Generaled Non-Unform Ratonal B-Splnes (GNURBS) serve as an alternatve nteractve shape desgn tool, and provde mproved approxmaton abltes n certan applcatons. GNURBS are obtaned by decouplng the weghts assocated wth control ponts along dfferent physcal coordnates. Ths unexplored dea brngs the possblty of treatng the weghts as addtonal degrees of freedoms. It wll be seen that ths proposed concept effectvely mproves the capablty of NURBS, and crcumvents ts defcences n specal applcatons. Further, t s proven that these new representatons are merely dsgused forms of classc NURBS, guaranteeng a strong theoretcal foundaton, and facltatng ther easeof-use. An nteractve MATLAB toolbox has been developed demonstratng these generalatons. Keywords: NURBS, soparametrc, weghts, decouplng, generalaton. 1

2 1. Introducton Non-Unform Ratonal B-Splnes (NURBS) are perhaps the most popular curve and surface representaton method n Computer-Aded Desgn/Computer-Aded Manufacturng (CAD/CAM). They were frst ntroduced n 1975 by Versprlle [1] as ratonal extenson of B-splnes. NURBS form the backbone of CAD, and are consdered the domnant technology for engneerng desgn [2]; further, they have also been extensvely used n several applcatons ncludng sogeometrc analyss (IGA) [3], shape optmaton [4], topology optmaton [5,6], materal modelng [7,8], reverse engneerng [9], G-code generaton [1] etc. Recent generalatons of NURBS-based technology nclude T-splnes [11,12] whch consttute a superset of NURBS, and provde the local refnement propertes by allowng for some unstructured-ness. An alternatve generalaton of NURBS, referred to as Generaled Herarchcal NURBS (H-NURBS), were ntroduced n 28 by Chen et al. [13] by extendng the dea of herarchcal B-splnes to NURBS. Smlar to T-splnes, H-NURBS prmarly brng the possblty of local refnement wth tensor-product surfaces. A novel shape-adjustable generaled Béer curve wth multple shape parameters has been recently proposed by Hu et al. [14], and ts applcatons to surface modellng n engneerng has been studed. Most recent class of splnes whch removes the lmtatons of T-splnes are Unstructured-splnes (U-splnes) that have been developed by Scott [15]. Other generalatons of NURBS have also been suggested n the lterature, even though these representatons have not ganed popularty. For nstance, ang et al. [16] propose a generaled NURBS curve and surface representaton wth the prmary advantage of representng smooth surfaces wth genus ero usng only one surface patch. Ths also provdes a new method to exactly generate conc curves and revoluton surfaces. Further, t smplfes modellng local features such as creases and ruled patches. Hstorcally, NURBS were prmarly ntroduced to represent concal shapes precsely. Ths s the crtcal advantage of NURBS over other polynomal-based classes of splnes, and the man reason for ts prevalence. Ths s acheved by the ntroducton of weghts nto the bass functons n a ratonal manner. The applcatons of ths ratonal form, however, s not lmted to precse constructon of concs. Accordng to the lterature, there are other applcatons where the weghts have been employed as addtonal degrees of freedom for mproved flexblty. 2

3 For nstance, the weghts can be employed as addtonal desgn varables for nteractve shape desgn so that one can utle both control pont movement, and weght modfcaton to attan local shape control [17]. Many studes suggest employng the weghts as addtonal desgn varables n data-fttng for better accuracy [9,18]. Carlson [18] develops a non-lnear least square fttng algorthm based on NURBS, and dscusses multple methods for solvng ths problem. Hs numercal results demonstrate sgnfcant mprovement n the accuracy of approxmaton compared to B-splnes, especally n the case of rapdly varyng data. Ths s n fact one of the other man advantages of NURBS over B-splnes. hle smooth pecewse polynomals such as B-splnes are poor n the approxmaton of rapdly varyng data and dscontnutes, employng ratonal functons s an effectve tool for addressng ths class of problems [18]. In order to avod solvng a non-lnear optmaton problem, Ma [9,19] develops a two-step lnear algorthm for data approxmaton usng NURBS. Despte beng an effectve technque for mprovng the performance of NURBS, there s a wde range of applcatons where treatng the weghts as extra desgn varables s ether mpossble or can be problematc. For nstance, Dmas and Brassouls [2] pont out that a bad choce of weghts n approxmaton can lead to poor curve/surface parameteraton. Pegl [21] mentons that mproper applcaton of the weghts can result n a very bad parameteraton, whch can destroy subsequent surface constructons. Further, there are numerous applcatons where employng the weghts as addtonal desgn varables s essentally mpossble. e wll dscuss some of these applcatons n Secton 5. The focus of ths paper s to develop new generalatons of NURBS to prmarly address ths shortcomng. These proposed generalatons mprove the performance of NURBS, and provde an alternatve concept for removng these defcences of NURBS. It wll be shown that, unlke T-splnes, these generalatons are only varatons of classc NURBS, and do not consttute a new superset of NURBS, makng t easy to ntegrate and deploy them n modern CAD/CAM systems. The remander of ths paper s organed as follows: n Sectons 2 and 3, we ntroduce two dfferent generalatons of NURBS, and develop ther theoretcal propertes. e explore some of the applcatons of GNURBS n Secton 4, and compare ther performance aganst classc NURBS. Further potental areas of applcatons and extensons of GNURBS are also dscussed n ths 3

4 secton. An nteractve MATLAB toolbox for GNURBS s dscussed n Secton 5, and fnally conclusons are drawn n Secton Generaled NURBS Curves: a non-soparametrc approach e recall that the equaton of a NURBS curve s parametrcally defned as where n ( ) ( ) C ξ = R ξ P, a ξ b (1) = p, P are a set of n + 1 control ponts and R p, are the correspondng ratonal bass functons assocated wth th control pont defned as R, p N w p, = (2) n j= N j, p w j where w are the weghts assocated wth the control pont, and N, are the B-splne bass p functons of degree p, defned on a set of non-decreasng real numbers Ξ = { ξ, ξ1,..., ξ n + p} called knot vector. N, s recursvely defned as: p 1 f ξ ξ < ξ+ 1 N, = otherwse ξ ξ ξ ξ N N N + p+ 1 = ( ξ) + ( ξ) p, p, 1 + 1, p 1 ξ+ p ξ ξ+ p+ 1 ξ+ 1 (3) The NURBS curve n (1) s a vector equaton whch, assumng P = [ x y ] T, could be wrtten n the followng expanded form n 3D space ( ξ) x x n p, ( ξ) (4) y = R y = Observe that NURBS curves are soparametrc representatons where all the physcal coordnates are constructed by lnear combnaton of the same set of scalar bass functons n parametrc space. Ths s the case for all the other popular CAGD representatons, e.g. all dfferent types of splnes; ths ensures sgnfcant propertes such as affne nvarance and convex hull whch are of nterest n geometrc modellng. 4

5 e ntroduce here the concept of Generaled Non-Unform Ratonal B-Splnes (GNURBS) by the extenson of the above equatons as follows n ( ) ( ) C ξ = R ξ P, ξ 1 (5) = p, where denotes component-wse product of two vector varables. In ths equaton, ( ξ) = ( ξ) ( ξ) ( ξ) x y T R p, [ R R R ] s now a vector set of bass functons whch s defned as, p, p, p p, ( ξ) R ( ξ) x T, p x y Np, ( ξ) w Np, ( ξ) w Np, ( ξ) w y = R =, p n x n y n N, ( ), ( ), ( ) j j p ξ wj N j j p ξ wj N j j p ξ w = = = j R, p R (6) The expanded equaton of a GNURBS curve n 3D space could be accordngly wrtten as ( ξ) x R x, p n y ( ξ), p (7) = x y = R y R, p Comparson of the above equaton wth that of classc NURBS shows that the man dfference of the proposed generaled form s assgnng ndependent weghts to dfferent physcal coordnates of control ponts. As can be seen, the above leads to a non-soparametrc representaton. Ths modfcaton suggests n loss of propertes such as strong convex hull and affne nvarance. However, t wll be establshed that GNURBS are only dsgused forms of hgher-order classc NURBS,.e., all propertes of NURBS can be recovered through a sutable transformaton, thus ensurng a strong theoretcal foundaton. In the followng secton, we develop the theory of GNURBS, and dscuss the propertes of ths non-soparametrc representaton. 2.1 Theory and propertes It can be easly shown that many propertes of NURBS curves elaborated n [17] such as endponts nterpolaton, contnuty, etc. are smlarly satsfed n GNURBS. However, when treated n the drect form, some of the NURBS propertes wll be modfed or even volated. e frst dscuss these, and later show how a smple transformaton can be appled to recover all NURBS propertes. 5

6 1. Affne nvarance: Due to coordnate-dependence of the bass functon n GNURBS, applyng an affne transformaton drectly to the control ponts wll not result n the same curve as applyng the same transformaton to the curve; hence, ths property s lost. 2. Strong convex hull: A GNURBS curve need not le n the convex hull of ts control ponts. e demonstrate ths graphcally n Fg. 1 for a cubc curve ( p = 3 ) constructed on the knot vector Ξ = { ξ, ξ,..., ξ } = 1 2 { 3 3 } 1 9,,,,,,1,1,1,1. (a) (b) 6

7 Fg. 1. (a) A NURBS knot-span les nsde the convex hull of ts control ponts. (b) A GNURBS knot-span need not le nsde the convex hull of ts control ponts. Fg. 1(a) shows a B-splne curve and a NURBS curve wth { w,..., w } = { 1, 5,1,1,1,1} 5 constructed usng the same control polygon. As observed, by ncreasng w1 the mddle knot span ξ [ ξ4, ξ5) always les wthn the convex hull of control ponts {P1, P2, P3, P4}. Fg. 1(b) llustrates an example where the same knot span of a cubc GNURBS curve constructed wth the same control polygon but a decoupled set of weghts { x x w,..., w 5} = { 1, 5,1,1,1,1} and { y y,..., 5} { 1,1,1,1,1,1} w w = exts the convex hull of ts control ponts. However, we prove that t satsfes a weaker condton referred to as strong boundng box property descrbed below. The functon spaces correspondng to Fg. 1 are depcted n Fg. 2. Observe that the functon space assocated wth the NURBS curve n Fg. 1(a) s dentcal for both x and y physcal components,.e. R( ξ ). Nevertheless, n the case of GNURBS curve shown n Fg. 1(b), the x-coordnate s constructed usng the ratonal set of bass functons R( ξ ), whle the y- coordnate s constructed usng the set of B-splne bass functons N( ξ ). Fg. 2. Cubc functon spaces correspondng to Fg. 1: B-splne functon space N( ξ ), and NURBS functon space ( R ξ ) wth {,..., } { 1, 5,1,1,1,1} w w =. 5 7

8 3. Strong boundng box: Every GNURBS knot span les wthn the coordnate-orented boundng box of ts correspondng control ponts. That s, f [, ) wthn the boundng box of the control ponts { p,..., } Proof: Eq. (7) can be easly wrtten n the followng form: ( ) P P. ξ ξ ξ + 1 x x n n n y ξ = R ξ + R ξ y + R ξ, then C (ξ ) les x y ( ) ( ) ( ), p, p (8), p = = = Accordngly, Eq. (5) could be wrtten as where C, C and x ( ) ( ) ( ) ( ), x y C ξ = C ξ + C ξ + C ξ ξ 1 (9) y C are smply classc NURBS curves. From a geometrc standpont, each of these curves s the projecton of the orgnal non-soparametrc curve onto the correspondng physcal axs. The followng fgure shows a graphcal representaton of above equatons for a 2D curve. 8

9 Fg. 3. Graphcal representaton of the boundng box property of a 2D cubc GNURBS curve wth { x x w,..., w 5} = { 1, 5,1,1,1,1} and { y y,..., 5} { 1,1,1,1,1,1} w w =. Snce each of these curves s a classc NURBS curve, they satsfy the convex hull property. Therefore, the mddle knot span of the curve whch s marked n Fg. 3, must le wthn the convex hulls of ts correspondng control ponts on both projected curves. That s, f ) ξ 1, 2, then ( 3 3 x ξ ) C les wthn the convex hull of the control ponts { x,..., x } 1 4 whch s the regon between the two vertcal lnes n Fg. 3. Smlarly, C y les wthn the convex hull of the control ponts {,..., } y y whch s the area between the two 1 4 horontal lnes n ths fgure. Consequently, C (ξ ) s contaned n the ntersecton of these two convex hulls, whch s the rectangular area shown n Fg. 3, referred to as the boundng P P. box of {,..., } Local Modfcaton: Smlar to NURBS, one can show that, n GNURBS, f a control pont d P s moved, or f any of the weghts w( d= xy,, ) s changed, t affects only the curve segment over the nterval [ ξ, ξ + p 1). However, unlke NURBS, changng the weghts wll only affect the parameteraton of the curve along the correspondng physcal coordnate d, whle the curve parameteraton n the other drectons wll be preserved. Ths s, n fact, the key dfference between GNURBS and NURBS whch ncreases control. Assumng ξ [ ξ, ξ + ), f p 1 d w s ncreased (decreased), the curve wll move closer to (farther from) P. Further, for a fxed ξ, a pont on C moves along a horontal (vertcal) straght x y lne as a weght w ( w ) s modfed; see Fg. 1(b). Ths can be easly concluded from the proposed decomposton n (8) and the propertes of classc NURBS curves. 5. Varaton Dmnshng Property: Due to loss of convex-hull property, ths property s also not preserved n the drect form of GNURBS; that s, snce the curve does not need to le wthn the convex hull of ts control ponts, there can be a plane (lne n 2D) whch ntersects the curve multple tmes wthout havng any ntersectons wth the control polygon. 9

10 6. NURBS Incluson: If the weghts n all drectons are equal for each control pont, then the GNURBS curve reduces to a NURBS curve. Havng dscussed the propertes of GNURBS n the drect form, we now consder a transformaton of GNURBS nto an equvalent NURBS of a hgher order, where all propertes are preserved. Towards ths end, we frst revew two lemmas on the multplcaton of Be er, as well as B-splne functons. The proofs of these lemmas can be found n [22]. Lemma 1: Let fb and gb be two Béer functons of degree p and q, respectvely. Ther product functon h s a Be er functon of degree p+q whch can be computed as [23] where B b k, p q p+ q k = k, p+ q ( ) b h ( ξ) = f ( ξ) g ( ξ) = B ξ h (1) b b b + denotes k th Be er bass functon of degree p+q, and k h b k mn(, ) = pk j= max(, k q) p q j k j p+ q k fg j k j (11) End of Lemma 1 Lemma 2: Let f and g be two unvarate B-splne functons of degree p and q, respectvely. Ther product functon h s a B-splne functon of degree p+q,.e. 1 n h k = k, p+ q where h are the ordnates of the product B-splne functon. k End of Lemma 2 ( ) h( ξ) = f( ξ) g( ξ) = N ξ h (12) Specfc to Lemma 2, numerous algorthms have been proposed n the lterature for evaluatng the ordnates; see [24 27], for nstance. In ths paper, we wll use a straghtforward algorthm proposed by Pegl and Tller [23] ncludng three steps of - Performng Be er extracton - Computaton of the product of Be er functons k

11 - Recomposton of the Be er product functons nto B-splne form usng knot removal. The product of Be er functons n the second step can be computed analytcally employng Lemma 1. Further, one can construct the knot vector of h as descrbed n [23]. A more advanced algorthm referred to as Sldng ndows Algorthm (SA) recently proposed by Chen et al. could be found n [25]. The above two Lemmas lead us to followng the crtcal theorem. Theorem: Every GNURBS curve of degree p and dmenson m can be transformed exactly nto a NURBS curve of degree m p. Proof. e provde the proof here for a 2D curve, however, t can easly be extended to any hgher dmenson. The proof reles on the lemma that the summaton of two NURBS curves s a hgher order NURBS curve [23]. e rewrte Eq. (8) for a 2D curve n the followng form: n ( ) ( ) x ( ) x n ( ) y n x ξ Np, ξ w Np, ξ w = x + y y ξ = w ( ξ) = w ( ξ) y (13) x x y where w ( ξ)= N ( ξ) w and ( ξ ( ξ) = p, n w )= N w. = Extractng the common denomnator leads to: p, y x = y = n = n = x Np, ( ξ) w x w x y w ( ξ) w ( ξ) y Np, ( ξ) w y w y x w ( ξ) w ( ξ) y x ( ξ) ( ξ) (14) As can be observed, evaluaton of (14) nvolves performng the multplcaton of unvarate B- splne functons. Accordng to Lemma 2, the product functons n (14) are B-splne functons of degree 2p. Therefore, we can obtan the equvalent hgher order NURBS representaton of (13) n the followng form where ( ) nˆ x X = R,2 p (15) y ξ = Y 11

12 N,2 R,2 p = nˆ = p N,2 p n whch ( X, Y, ) are the coordnates and weghts of the equvalent hgher order NURBS curve, whch can be obtaned usng the algorthm descrbed n Lemma 2, and n ˆ + 1 s the number of control ponts. End of proof In the specal case of Ratonal Be er (R-Be er) curves, one can obtan straghtforward analytcal expressons for the coeffcents of the equvalent hgher order R-Be er curve n (15). For ths case, Eqs. (15) and (16) can be wrtten as (16) where 2 p x X = y = Y R, 2 p (17) B,2 R,2 p = p nˆ = B,2 p Usng relatons (1) and (11) n Lemma 1, the weghts and control ponts n these equatons are obtaned as (18) X = mn( n, ) j= max(, n) 1 = 1 Y = mn( n, ) j= max(, n) mn( n, ) j= max(, n) λ ww x y j j j λ xww x y j j j j λ y ww y x j j j j (19) n n j j where λj =. 2n Fgure 4 shows a quadratc GNURBS curve, and ts equvalent quartc NURBS curve obtaned usng the above theorem. 12

13 x x Fg. 4. Equvalence of a 2D quadratc GNRUBS curve wth {,..., } { 1,2.5,1.5,3} y y {,..., 3} { 1,1,2.5, 2} w w = and 3 w w =, wth a quartc NURBS curve wth {,..., } { 1.,1.75, 2.3,3.19,3.81, 4.4,5.25, 6.} w w =. 7 It needs to be ponted out that, despte the apparent volaton of crtcal propertes, property 7 establshes that GNURBS are merely dsgused form of hgher order classc NURBS, thereby nhertng all the propertes of NURBS ndrectly. 2.2 Partal decouplng for 3D curves One can easly extend the above theorem and formulaton to 3D curves wth ndependent weghts along all three physcal drectons. However, a more practcal case, whch wll be the emphass for the rest of ths paper, s to perform partal decouplng of the weghts. In partcular, n 3D, one can use the same set of weghts n x and y drectons, denoted by drecton w. Accordngly, Eq. (7) could be wrtten as ( ξ) 13 xy w, and a dfferent set of weghts n xy R x, p n xy ( ξ), p (2) = x y = R y R, p

14 where R xy, p N w xy p, = (21) n xy N j j, p w = j Observe that owng to ths decouplng of the n-plane and out-of-plane weghts, unlke n classc NURBS, one can now freely manpulate the weghts along drecton, for nstance, wthout perturbng the geometry or parameteraton of the underlyng curve n x-y plane. For better nsght, we provde a graphcal vsualaton of desgnng a 3D curve wth an n-plane crcular shape n Fg. 5. Fg. 5. A 3D GNURBS curve wth an underlyng precse crcular arc: { xy xy w,..., w 3 } = { 1,.8536,.8536,1} and {,..., 3} { 1,1,1,1} w w =. As can be clearly seen n Fg. 5, treatng the ndependent set of out of plane weghts can provde better flexblty and control. As a smple example, one can use ths representaton as an ntermedate nteractve shape desgn tool, and fnally convert t to a hgher order classc NURBS, f desred, to recover affne nvarance and other propertes. In ths paper we wll focus on 14

15 demonstratng superor approxmaton abltes of ths representaton n certan applcatons where a heght functon, feld or set of data ponts need to be approxmated over an underlyng 2D curve. To derve the equvalent hgher order NURBS representaton of (2), we rewrte ths equaton n the followng form x xy x n n Np, ( ξ) w Np, ( ξ) w y = y xy + = w ( ξ) = w ( ξ) (22) Followng a very smlar procedure as for 2D curves, we can easly derve the expressons for the equvalent hgher order NURBS curve to the generaled form n (2) as where x X nˆ y = R,2 p Y (23) = Z N,2 R,2 p = nˆ = p N,2 p ( X, Y, Z, ) n these equatons can be obtaned usng a smlar algorthm as for 2D curves n the followng form (24) mn( n, ) = λ w w (25) j= max(, n) xy j j j and X 1 = 1 Y = Z 1 = mn( n, ) j= max(, n) mn( n, ) j= max(, n) mn( n, ) j= max(, n) λ xw w xy j j j j λ yw w xy j j j j λ ww xy j j j j (26) 15

16 n n j j where λj =. 2n It should be noted here that the propertes of classc NURBS whch are lost n ths proposed generalaton are not crtcal or even of nterest n many applcatons of NURBS. Nevertheless, n some applcatons, these propertes can be crucal. In order to make GNURBS applcable to such applcatons, we develop an alternatve varaton of NURBS whch can be drectly derved from the generalaton proposed above. 3. Generaled NURBS curves: an soparametrc approach va order-elevaton Note that the equvalent hgher order NURBS representaton n (15) or (23) tself provdes another varaton of NURBS whch can be drectly employed as another alternatve to NURBS wth better flexblty n some applcatons. In order to clarfy how these equatons provde addtonal flexblty than classc NURBS, we frst derve a more generc form of these equatons va an alternatve approach usng an extenson of order elevaton technque. Assume a 2D R-Be er curve of degree p s gven as follows ( ) x B w x p xy p, = xy (27) y ξ = w y In order to elevate the degree of ths curve by q, we can smply multply both numerator and denomnator of ths equaton by any arbtrary expresson n the followng form q w ( ξ)= B w (28) = Recallng Lemma 1, we can obtan the hgher order R-Be er curve wth q degree elevatons as q, ( ξ) where nˆ x X = Rp, + q (29) y = Y 16

17 n whch ˆn p q = + and (,, ) Bp, + q R p, + q = nˆ = B p, + q X Y can be obtaned usng (31) and (32) (3) mn( p, ) = λ w w (31) j= max(, q) xy j j j X Y 1 = 1 mn( p, ) j= max(, q) λ xw w xy j j j j mn( p, ) xy = λj j j j= max(, q) yw w j (32) p q j j where λj =. p+ q Observe that ths procedure can be seen as a trval extenson of the classc order elevaton technques n the lterature [17,28]. In fact, one can smply recover the common order elevaton algorthm by assgnng w = 1, n (28). e wll refer to ths procedure as generaled order elevaton hereafter. Now suppose we ntend to add another dmenson to the representaton n (29) n an soparametrc manner. Agan, ths extra dmenson can be vewed as the heght functon of a parametrc curve n 2D, or may represent a feld or set of data ponts whch needs to be approxmated over a 2D curve. For ths purpose, we extend (29) as x X nˆ y = Rp, + q Y (33) = Z It s nterestng to notce that, although Eq. (33) apparently seems to be a classc R-Be er curve, t provdes addtonal flexblty. Observe that n the above procedure, w are arbtrary varables whch can be freely chosen wthout perturbng the geometry or parameteraton of the underlyng curve n x-y plane. 17

18 In order to better demonstrate the effect of these weghts on the behavor of GNURBS curves, we generate a 3D quartc GR-Be er curve by performng the above process wth q = 2 on a quadratc R-Be er crcular arc and assgnng the heghts of control ponts as shown n Fg. 6. (a) Fg. 6. A 3D soparametrc GNURBS curve wth (a) { w 1,w 2,w 3} = { 1,1,1} { w 1,w 2,w 3} = { 1, 2.5,1}. (b) 18, and (b)

19 w,w,w = The obtaned results wth { w 1,w 2,w 3} = { 1,1,1} (classc order elevaton) and { 1 2 3} { 1, 2.5,1 } are represented n Fgs. 6(a) and (b), respectvely. As observed, the heghts of control ponts n both cases are dentcal. For more clarty, the se of control ponts s plotted proportonal to ther weghts. Further, the correspondng sets of bass functons are plotted n Fg. 7. Comparng Fgs. 6(a) and (b), t can be notced that by ncreasng w 2, the weghts of the three nteror control ponts are ncreased whch results n out of plane deformaton of the curve as depcted n Fg. 6(b). However, as ths fgure shows, ths leads to automatc n-plane rearrangement of control ponts n such a manner that the n-plane geometry of the curve (as well as ts parameteraton) remans unchanged. Fg. 7. The functon spaces correspondng to GNURBS curves n Fg. 7. The above algorthm can be extended to NURBS n a straghtforward manner usng a smlar three step algorthm explaned n Lemma 2. That s, Eq. (33) also holds true for NURBS wth the ratonal bass functons defned as Np, + q R p, + q = nˆ = N e here note that whle the varables w n (33) or (34) can be drectly treated as desgn varables for mproved flexblty, the physcal meanng and local support of the weghts n ths varaton are lost. Hence, t mght not be sutable for beng used as an nteractve shape desgn tool. However, 19 p, + q (34)

20 as wll be shown n the next secton, t can stll be effectvely employed as an enhanced tool for approxmaton purposes where the decson on the optmal values of the weghts s made by a numercal algorthm. 4. Applcatons The proposed generalatons of NURBS n (2) and (33) provde alternatve tools to NURBS whch can be useful n certan applcatons such as IGA. Explorng these advanced applcatons, however, s beyond the scope of ths paper. In ths secton, we however nvestgate functon approxmatons as an applcaton. Hereafter, we wll persstently refer to (2) as the frst generalaton of NURBS or non-soparametrc GNURBS, whle we wll refer to (33) as the second generalaton of NURBS or soparametrc GNURBS. Both these varatons prmarly provde the common and sgnfcant possblty of treatng the outof-plane weghts as addtonal desgn varables, wthout perturbng the underlyng geometry or ts parameteraton. However, the dfference between them should be clear snce the frst form s obtaned va explct decouplng of the weghts along dfferent physcal coordnates resultng n a non-soparametrc representaton wth the propertes elaborated n Secton 2, whle the second varaton s obtaned by mplct decouplng of the weghts wthn the soparametrc set of bass functons; thereby preservng the propertes of NURBS. As dscussed above, the generaton of these mplctly decoupled set of weghts n the second varaton requres order elevaton a pror. Fnally, we emphase that although these new representatons fnally le n the NURBS space, obtanng ther results n certan class of applcatons by drectly makng use of NURBS does not seem possble. 4.1 Approxmaton over curved domans There are varous applcatons where the data or a functon needs to be approxmated over a parametrc curved doman. For nstance, there are numerous studes n the lterature for the approxmaton of scattered data or functons on curved surfaces; see [29,3] for a rgorous revew. A smlar problem arses n other applcatons such as modellng helcal curves and surfaces [31 33], treatng the non-homogenous essental boundary condtons n IGA [34 37] etc. In all these applcatons the lmtaton of preservng the underlyng parameteraton apples. Therefore, employng the weghts as addtonal desgn varables s dsallowed. In ths secton, we nvestgate 2

21 the performance of GNURBS versus NURBS n ths class of problems for two cases of approxmatng a smooth functon as well as a rapdly varyng one Least-square mnmaton usng NURBS and GNURBS Suppose an n-plane crcular arc s gven n the followng parametrc form π π C ( ξ) = r cos( ξ),sn( ξ) ξ 1 (35) 2 2 where r s the radus of the crcular arc. Eq. (35) can be precsely constructed usng NURBS. Now, assume a heght functon h needs to be approxmated over ths arc wth mnmum error. Ths can be easly posed as a least-square approxmaton problem leadng to optmal accuracy n L2-norm. Assumng { ξ (, ): } s s s f to be mnmed s defned as x y s s the set of ns collocaton ponts, the error functon ξs ξs 2 L ξs L ξs 2 s 2 s s L f = ˆ( ) ( ) = R ( ) ( ) where ˆ s the approxmated NURBS functon, ξ are the correspondng collocaton ponts n s the parametrc space, s s the set of ndces of non-ero bass functons at ξ. s In the case of NURBS, the only unknowns to consder are control varables and the problem L leads to a lnear least square problem whch can be solved for the 1 by proper choce of collocaton ponts. 2 2 (36) n + unknowns λ = { },..., n To mprove the accuracy of approxmaton, nvokng the proposed varatons of NURBS, we can treat the out-of-plane weghts w as extra desgn varables wthout perturbng the geometry or parameteraton of the underlyng precse crcular arc. e may refer to these varables as control weghts hereafter. th the frst generalaton n (2), the vector of desgn varables becomes {,..., n, w,..., wn} λ =, where the postvty constrants on control weghts ( w >, ) are often desred to be satsfed for numercal stablty. To avod solvng a non-lnear problem, we employ a two-step algorthm developed by Ma [9,19], whch leads to two separate lnear systems of equatons; a homogenous system whch yelds the optmal control weghts and a non-homogenous 21

22 one that yelds the correspondng optmal control varables. The extenson of ths algorthm for GNURBS s straghtforward, and hence s not presented here. th the second generalaton n (33), however, the development of a lnear algorthm does not seem easly possble. Therefore, a non-lnear least square algorthm needs to be used to fnd the optmal set of desgn varables. Further, snce the dervaton of analytcal Jacoban matrx becomes complcated n case of havng nternal knots, we lmt our study to GR-Be er. The vector of desgn varables for ths smplfed case becomes = { Z,..., Zn, w,..., wq} λ where n= p+ q. The mposton of the least square problem s qute straghtforward; hence, we do not present t here. The dervaton of Jacoban matrx components wth the respect to control weghts, however, s non-trval and requres evaluatng the senstvty usng the followng expressons p q xy k k k, ( ) w f p k = p q + wk Otherwse The ntal condtons for solvng the least square problem are specfed as follows (37) As prevously dscussed, by changng λ =,,...,,1,1,...,1 (38) n+ 1 q+ 1 w durng the optmaton process, the n-plane coordnates of control ponts also vary at each teraton. However, snce the n-plane geometry and parameteraton are always fxed, one may only re-evaluate and update these coordnates after the termnaton of the optmaton process accordng to the obtaned optmal set of soparametrc bass functons. It s mportant to note that ths algorthm yelds the combnaton of optmal weghts and the correspondng arrangement of control ponts whch results n the best approxmaton over a gven parameteraton. To our knowledge, no such nvestgaton has been reported n the lterature thus far. In the next secton, we approxmate varous heght functons over the crcular arc n (35) modelled precsely wth NURBS. In all cases, the nterpolatng end control ponts are prescrbed to le on the heght functon. Further, we employ 1 unformly dstrbuted sample ponts n the parametrc 22

23 space for settng up the least square problem. The numercal mplementatons are performed n MATLAB. Fnally, the relatve L2-norms of the error are calculated usng the followng relaton error = ( ( ˆ( ) ) 1 ξ) ( ξ) dγ dγ 2 2 where the numercal ntegratons are calculated usng Gaussan quadrature A smooth functon: helx modellng As the frst numercal example, we consder approxmatng a smooth heght functon as h( ξ bϕ b π )= = ξ 2 over the parametrc curve n (35). In the above equaton, ϕ s the center angle of the crcular arc n x-y plane and b s a constant. Eq. (39) together wth (35) represent a segment of a helcal curve, shown n Fg. 8 for b = 1, and s a classc problem n geometrc modellng. e here demonstrate how the proposed varatons of NURBS can be useful for mproved modellng of such type of problems. (39) (4) Fg. 8. A smooth helcal curve. 23

24 Helcal curves and surfaces do not have an exact representaton n terms of polynomals or ratonal polynomals [38]. A hgh accuracy of approxmaton by NURBS usng the mnmal number of control ponts s of nterest, and wll make the helx more convenent to use n current CAD/CAM systems [32]. There s a large number of studes n the lterature addressng ths problem usng R- Be er or NURBS; see e.g. [31 33] for a revew of these studes. Havng examned these studes, t can be found that there are several consderatons for a sutable approxmaton of helx such as the accuracy of normal angle, curvature, torson and heght, besdes meetng certan geometrc condtons at the end ponts of each segment [32]. However, we only focus here on approxmatng the heght functon wth maxmum accuracy, for smplcty. Further, t s desrable that the fttng curve precsely les on the cylnder surface of the helx [33]. Snce ths s a geometrc modellng problem, the propertes of NURBS are mportant to be preserved for ths partcular applcaton. Therefore, t s an deal canddate for employng the second varaton,.e. soparametrc GR-Be er, as the obtaned optmal desgn s drectly n the NURBS space. The obtaned results usng the above-dscussed algorthm for dfferent degrees of bass functons are presented n Table 1 for comparson. Table 1. Error of approxmatng the helx heght functon usng R-Be er versus GR-Be er n relatve L 2-norm. Curve type Degree n= p+ q ( ) No. of control varables No. of control weghts Error R-Be er E-2 2 nd GR-Be er 2.41E-2 R-Be er E-4 2 nd GR-Be er 2 1.5E-4 R-Be er E-4 2 nd GR-Be er E-6 R-Be er E-6 2 nd GR-Be er 4 1.1E-8 Error rato As the table shows, the accuracy of approxmaton by GR-Be er over R-Be er ncreasngly mproves by elevatng the degree, as a larger number of control weghts are added to the desgn 24

25 space. In case of p = 3, however, no mprovement n the accuracy s ganed. Ths mples that the optmal values of the control weghts for ths case are equal to 1; that s, cubc R-Be er obtaned va order elevaton s concdentally optmal for the approxmaton of ths heght functon. The ntal and optmal sets of bass functons for approxmaton wth dfferent degrees are represented n Fg. 9. As can be observed n ths fgure, n both cases, the optmal sets of bass functons are only slghtly dfferent than the ntal ones, however, ths small devaton results n dramatc mprovement of the accuracy of approxmaton as reported n Table 1. (a) Fg. 9. Intal and optmal bass functons for approxmatng the helx heght functon usng 2 nd GR- Be er wth degree (a) n = 4 and (b) n = 5. e remnd that n the case of soparametrc generalaton (2 nd GR-Be er), the bass functons are dentcal along all physcal coordnates. As prevously explaned, ths leads to automatc rearrangement of the n-plane coordnates of control ponts, depcted n Fg. 1, n such a manner that the n-plane geometry and ts parameteraton reman unchanged. (b) 25

26 (a) (b) (c) Fg. 1. Intal and optmal control nets for approxmatng the helx heght functon wth (a) R-Be er of degree n = 2, and 2 nd GR-Be er of degree (b) n = 3 (c) n = 4 and (d) n = 5. e also nvestgate the performance of GNURBS compared to NURBS wth respect to refnng the knot sequence. For ths experment, we use the frst varaton (non-soparametrc), for smplcty and as t provdes better flexblty. The obtaned results for p = 2 are represented n Fg. 11. (d) 26

27 Fg. 11. Convergence rate of 1 st GNURBS versus NURBS for approxmatng the helx heght functon. As the fgure shows, by ncludng the control weghts to the desgn space, the convergence rate s mproved by one order resultng n dramatc mprovement n the accuracy especally when larger numbers of control ponts are employed. However, as prevously mentoned, n the case of GNURBS there s an extra computatonal cost for obtanng the optmal weghts va solvng an addtonal homogenous system of equatons A rapdly varyng functon As the second example, we nvestgate the performance of the proposed varatons of NURBS n capturng rapdly varyng functons. e consder the problem of approxmatng a rapdly varyng functon as n (41) over the same crcular arc whch s plotted n Fg. 12 for α = ( ) (.5) (.8) h( ξ)= ϕ 1 + e + e, ϕ = ( ) ξ (41) 2 α ϕ α ϕ π 27

28 Fg. 12. A rapdly varyng functon over a crcular arc. Employng the frst proposed varaton of NURBS, we approxmate the heght functon usng dfferent degrees of bass functons. The obtaned results are presented n Table 2. All these models are obtaned by performng unform knot nserton over an ntal R-Be er arc and therefore possess maxmal contnuty. Table 2. Error of approxmatng the rapdly varyng functon n (41) usng NURBS versus 1 st GNURBS n relatve L 2-norm. Curve type Degree (p) No. of control varables No. of control weghts Error NURBS 6.86E st GNURBS E-3 NURBS 5.35E st GNURBS E-3 NURBS 6.27E st GNURBS E-3 NURBS 5.48E st GNURBS E-3 Error rato

29 Accordng to the table, the accuracy of approxmaton usng NURBS does not change notceably by elevatng the degree. On the other hand, the obtaned results wth GNURBS persstently mprove by elevatng the degree, whch reveals the superorty of approxmaton of GNURBS over NURBS n capturng rapdly varyng felds. The approxmaton results for p = 5 are plotted n Fg. 13. The fgure clearly shows the mprovement of approxmaton n the case of GNURBS especally n the vcnty of exstng sharp transtons n the feld. (a) (b) Fg. 13. Approxmaton of the rapdly varyng functon wth quntc (a) NURBS and (b) 1 st GNURBS. Further, the correspondng bass functons are represented n Fg. 14. It s nterestng to note that, unlke the prevous case of approxmatng a smooth functon, there s a sgnfcant change between the ntal and optmal bass functons. As can be seen, ths dfference s more substantal for the bass functons effectng the behavor of the curve around the exstng sharp local gradents, mplyng that the correspondng weghts tend to take the extreme values n these regons. 29

30 (a) Fg. 14. (a) Intal, and (b) optmal sets of quntc bass functons assocated wth Fg. 13. (b) 4.2 Extensons and further applcatons hle, n ths paper, we lmted our study to applyng the proposed generalatons to NURBS curves, they can be smlarly appled to surfaces and volumes whch s the subject of our future research. Moreover, due to fundamental smlartes between dfferent varatons of splnes, these generalatons seem plausble to other ratonal forms of splnes such as T-splnes, Tr-angular Be ers, etc. In addton to the dscussed applcatons n CAD, there are other areas of applcatons of NURBS where employng the weghts as addtonal desgn varables for better flexblty can be problematc or sometmes mpossble. For nstance, we lmted our numercal experments to approxmaton over curved domans, GNURBS may also help crcumventng the dffcultes of consderng the weghts as degrees of freedom n general curve/surface fttng problems. As prevously studed n [2,21], employng the weghts as addtonal degrees of freedom n data approxmaton can deterorate the surface parameteraton, and lead to undesrable results. In ths regard, exstng studes suggest mposng boundng constrants on the varaton of the weghts explctly or va regularaton [9,18,19], to avod ths ssue. However, ths lmts the obtaned mprovement n the accuracy of approxmaton, especally n the case of problems contanng rapd varaton n data or feld where the weghts tend to take extreme values. 3

31 On the other hand, employng the suggested varatons of NURBS, one can create a good parameteraton and preserve t whle ncludng the control weghts as desgn varables for fttng the curve/surface to 3D data ponts, wthout mposng any lmtatons on the values of the weghts. Owng to the nherent propertes of NURBS, they have been extensvely used n computatonal mechancs for the optmaton of dfferent felds of nterest over a computatonal doman. For nstance, Qan [39] employs B-splne bass for the representaton of densty feld n FEM-based topology optmaton as an ntrnsc flterng technque. thn the framework of IGA, numerous studes have been performed where the same NURBS based parameteraton of computatonal doman has also been used for the representaton of dfferent felds whch need to be optmed over the doman n varous applcatons such as se optmaton of curved beams [4 42], topology optmaton [6,43 45], optmaton of materal dstrbuton n functonally graded materals (FGMs) [46,47] etc. Havng examned these studes, t can be notced that n ths class of applcatons, the parameteraton of the desgn doman must reman fxed throughout the optmaton process. Moreover, many of them requre lnear parameteraton of the desgn doman and acheve ths by placng the control ponts at ther Grevlle abscssae, see e.g. [39,46]. Hence, they are only able to treat the out-of-plane coordnates of control ponts as desgn varables, as the varaton of weghts alters the underlyng parameteraton whch s dsallowed. Owng to the proposed GNURBS representatons wth decoupled weghts, one can now treat the out of plane weghts as addtonal desgn varables whle settng up the optmaton problem and stll preserve the underlyng geometry as well as ts parameteraton unchanged. As the presented numercal results suggest, ths dea can lead to sgnfcant mprovement n the flexblty n both cases of smooth as well as rapdly varyng felds. Explorng these applcatons s the subject of future studes. 5. MATLAB Toolbox: GNURBS Lab In order to facltate understandng the behavor of GNURBS and further abltes they provde, a comprehensve nteractve MATLAB toolbox, GNURBS Lab, has been developed. Ths toolbox s developed va the extenson of an exstng NURBS toolbox n MATLAB, Bsplne Lab, avalable as an opensource package under GNU lcense at gthub.com. 31

32 A snapshot of the GNURBS Lab envronment s depcted n Fg. 6, whch demonstrates some of the avalable features n ths software. The fgure shows an example of desgnng a quadratc GNURBS curve wth 5 control ponts constructed over a unform knot-vector. Employng the provded tools, one can easly manpulate any defnng parameter of the curve, ncludng the locatons of control ponts, knots or weght components, and observe the changes nteractvely n both the orgnal GNURBS and ts equvalent hgher order counterpart, smultaneously. Fg. 15. A snapshot of GNURBS lab. The opensource toolbox s avalable for download at Detaled nstructons for usng ths toolbox s also avalable as an addtonal document Manual.pdf va the same lnk. 6. Concluson e presented two generalatons of NURBS, referred to as GNURBS, by decouplng of the weghts assocated wth the control ponts along dfferent physcal coordnates. These generalatons, whch can be obtaned usng ether a non-soparametrc or an soparametrc concept, mprove the flexblty of NURBS and crcumvent ts defcences by provdng the 32

33 possblty of treatng the weghts as addtonal desgn varables n specal applcatons. It was proved that these representatons are only varatons of classc NURBS and do not consttute a new superset of NURBS. The superor approxmaton abltes of these varatons for both smooth and rapdly varyng functons were shown va smple examples. However, as ponted out n Secton 4.2, there are many other areas of applcatons whch can potentally beneft from GNURBS. A comprehensve MATLAB toolbox, GNURBS Lab, was developed to demonstrate the behavor of GNURBS n a fully nteractve manner. Further, although we lmted our study to NURBS curves, smlar extensons are applcable to surfaces and volumes, as well as perhaps any other ratonal form of splnes. Overall, GNURBS provdes a new powerful technology wth superor flexblty whle ncludng NURBS as a specal case. Acknowledgements The authors would lke to thank the support of Natonal Scence Foundaton through grant CMMI

34 References [1] Versprlle KJ. Computer-aded Desgn Applcatons of the Ratonal B-splne Approxmaton Form. Syracuse Unversty, [2] J. Austn Cottrell, Thomas J. R. Hughes YB. Isogeometrc Analyss: Toward Integraton of CAD and FEA. John ley & Sons; 29. do:1.12/ [3] Hughes TJR, Cottrell JA, Balevs Y. Isogeometrc analyss: CAD, fnte elements, NURBS, exact geometry and mesh refnement. Comput Methods Appl Mech Eng 25;194: do:1.116/j.cma [4] Qan X. Full analytcal senstvtes n NURBS based sogeometrc shape optmaton. Comput Methods Appl Mech Eng 21;199: do:1.116/j.cma [5] Leu QX, Lee J. A mult-resoluton approach for mult-materal topology optmaton based on sogeometrc analyss. Comput Methods Appl Mech Eng 217;323: do:1.116/j.cma [6] Taher AH, Suresh K. An sogeometrc approach to topology optmaton of mult-materal and functonally graded structures. Int J Numer Methods Eng 217;19: do:1.12/nme.533. [7] Coelho M, Roehl D, Bletnger K-U. Materal Model Based on Response Surfaces of NURBS Appled to Isotropc and Orthotropc Materals. In: Muño-Rojas PA, edtor. Comput. Model. Optm. Manuf. Smul. Adv. Eng. Mater., Cham: Sprnger Internatonal Publshng; 216, p do:1.17/ _13. [8] Coelho M, Roehl D, Bletnger KU. Materal model based on NURBS response surfaces. Appl Math Model 217;51: do:1.116/j.apm [9] Ma, Kruth J-P. NURBS curve and surface fttng for reverse engneerng. Int J Adv Manuf Technol 1998;14: do:1.17/bf [1] Kanna SA, Tovar A, ou JS, El-Mounayr H. Optmed NURBS Based G-Code Part Program for Hgh-Speed CNC Machnng. ASME 214 Int. Des. Eng. Tech. Conf. Comput. Inf. Eng. Conf., Amercan Socety of Mechancal Engneers; 214. [11] Sederberg, T.., Cardon, D.L., Fnngan, G.T., North, N.S., Zheng, J., Lyche T. T-splne smplfcaton and local refnement. ACM Trans Graph 24;23: [12] Sederberg, T.N., Zhengs, J.M., Bakenov, A., Nasr A. T-splnes and T-NURCCSs. ACM Trans 34

35 Graph 23;22: [13] Chen. Generaled Herarchcal NURBS for Interactve Shape Modfcaton. VRCAI 8 Proc. 7th ACM SIGGRAPH Int. Conf. Vrtual-Realty Contn. Its Appl. Ind., vol. 1, 28, p do:1.1145/ [14] Hu G, u J, Qn X. A novel extenson of the Béer model and ts applcatons to surface modelng. Adv Eng Softw 218;125: do:1.116/j.advengsoft [15] Scott M. U-splnes for Unstructured IGA Meshes n LS-DYNA 218:1 5. [16] ang Q, Hua, L G, Bao H. Generaled NURBS curves and surfaces. Proc - Geom Model Process 24 24: do:1.123/b:jmsc ee. [17] Pegl L, Tller. The NURBS Book. 1st ed. Sprnger-Verlag Berln Hedelberg; do:1.17/ [18] Carlson N. NURBS Surface Fttng wth Gauss-Newton. Lulea Unversty of Technology, 29. [19] Ma. NURBS-based computer aded desgn modellng from measured ponts of physcal models. Catholc Unversty of Leuven, [2] Dmas E, Brassouls D. 3D geometrc modellng based on NURBS: a revew. Adv Eng Softw 1999;3: [21] Pegl L. On NURBS: A Survey. IEEE Comput Graph Appl 1991: [22] Mehaute A Le, Schumaker LL, Rabut C, edtors. Curves and Surfaces wth Applcatons n CAGD. 1st ed. Vanderblt Unversty Press; [23] Pegl L, Tller. Symbolc operators for NURBS. Comput Aded Des 1997;29: do:1.116/s1-4485(96)74-7. [24] Che X, Farn G, Gao Z, Hansford D. The product of two B-splne functons. Adv Mater Res 211;186: do:1.428/ [25] Chen X, Resenfeld RF, Cohen E. An algorthm for drect multplcaton of B-splnes. IEEE Trans Autom Sc Eng 29;6: do:1.119/tase [26] Lee ETY. Computng a chan of blossoms, wth applcaton to products of splnes. Comput Aded Geom Des 1994;11: [27] Mørken K. Some denttes for products and degree rasng of splnes. Constr Approx 1991;7:195 35

36 28. do:1.17/bf [28] Farn G. Curves and Surfaces for CAGD A Practcal Gude. 5th ed. Morgan Kaufmann; 21. [29] Alfeld P, Neamtu M, Schumaker LL. Fttng scattered data on sphere-lke surfaces usng sphercal splnes. J Comput Appl Math 1996;73:5 43. do:1.116/ (96)34-9. [3] Fasshauer GE, Schumaker LL. Scattered Data Fttng on the sphere. Nashvlle, TN: Vanderblt Unversty Press; do:1.116/j.powlec [31] Pu X, Lu. A subdvson scheme for approxmatng crcular helx wth NURBS curve. Proceedng 29 IEEE 1th Int Conf Comput Ind Des Concept Des E-Busness, Creat Des Manuf - CAID CD 29 29:62 4. do:1.119/caidcd [32] Yang X. Hgh accuracy approxmaton of helces by quntc curves. Comput Aded Geom Des 23;2: do:1.116/s (3)74-8. [33] Juhas I. Approxmatng the helx wth ratonal cubc Beer curves. Comput Des 1995;27: [34] Shojaee S, Iadpenah E, Haer A. Imposton of Essental Boundary Condtons n Isogeometrc Analyss Usng the Lagrange Multpler Method. Int J Optm Cv Eng 212;2: [35] ang D, Xuan J. An mproved NURBS-based sogeometrc analyss wth enhanced treatment of essental boundary condtons. Comput Methods Appl Mech Eng 21;199: do:1.116/j.cma [36] Anand Embar, John Dolbow IH. Imposng Drchlet boundary condtons wth Ntsche s method and splne-based fnte elements. Int J Numer Methods Eng 21;83: do:1.12/nme [37] Chen T, Mo R, Gong Z. Imposng Essental Boundary Condtons n Isogeometrc Analyss wth Ntsche s Method. Appl Mech Mater 211; : do:1.428/ [38] Pottmann H, Leopoldseder S, Hofer M. Approxmaton wth Actve B-splne Curves and Surfaces. 1th Pacfc Conf Comput Graph Appl 22 Proceedngs 22:8 25. do:1.119/pccga [39] Qan X. Topology optmaton n B-splne space. Comput Methods Appl Mech Eng 213;265: do:1.116/j.cma [4] Nagy AP, Abdalla MM, Gürdal Z. Isogeometrc sng and shape optmsaton of beam structures. 36

37 Comput Methods Appl Mech Eng 21;199: do:1.116/j.cma [41] Nagy AP, Abdalla MM, Gürdal Z. Isogeometrc sng and shape optmsaton of beam structures. Comput Methods Appl Mech Eng 21;199: do:1.116/j.cma [42] Lu H, Yang D, ang X, ang Y, Lu C, ang ZP. Smooth se desgn for the natural frequences of curved Tmoshenko beams usng sogeometrc analyss. Struct Multdscp Optm 218. do:1.17/s [43] Hassan B, Khanad M, Tavakkol SM. An sogeometrcal approach to structural topology optmaton by optmalty crtera. Struct Multdscp Optm 212;45: do:1.17/s [44] ang Y, Benson DJ. Isogeometrc analyss for parametered LSM-based structural topology optmaton. Comput Mech 216;57: do:1.17/s [45] Dedè L, Borden MMJ, Hughes TJRT. Isogeometrc analyss for topology optmaton wth a phase feld model. Arch Comput Methods Eng 212;19: do:1.17/s [46] Leu QX, Lee J. Modelng and optmaton of functonally graded plates under thermo-mechancal load usng sogeometrc analyss and adaptve hybrd evolutonary frefly algorthm. Compos Struct 217;179: do:1.116/j.compstruct [47] Taher AH, Hassan B, Moghaddam NZ. Thermo-elastc optmaton of materal dstrbuton of functonally graded structures by an sogeometrcal approach. Int J Solds Struct 214;51: do:1.116/j.jsolstr