Imposition of the essential boundary conditions in transient heat conduction problem based on Isogeometric analysis

Transcription

1 Imposton of the essental boundary condtons n transent heat conducton problem based on Isogeometrc analyss S. Shojaee, E. Izadpanah, S. Nazar Department of Cvl Engneerng, Shahd Bahonar Unversty, Kerman, Iran Abstract The purpose of ths paper s usng the proposed two step method to mpose essental boundary condtons for mprovng the accuracy of soluton feld. In the proposed approach, mposng essental boundary condtons n transent heat flow wthn two-dmensonal regon s extended n two steps. The essental boundary condtons are defned on Drchlet boundary as determned temperatures and ndependent of tme. In the frst step, Drchlet boundary condtons are weakly bult nto the varatonal formulaton, choosng weght functon, approprately. In the second step, wth fxed condton, the system of equatons s approprately adjusted. For nvestgaton of the effcency of the proposed approach, several 2D numercal examples have been performed. The results demonstrate sgnfcant mprovement n accuracy and rate of convergence n comparson wth drect mposton of essental boundary condton. Keywords: Isogeometrc analyss, essental boundary condton, transent heat flow, NURBS. 1. Introducton Isogeometrc analyss (IGA) s a recently developed computatonal approach that offers the possblty of ntegratng ntegratng NURBS-based Computer Aded Desgn (CAD) tools nto the conventonal fnte element analyss. The concept of the IGA n mechancal problems s poneered by Hughes and hs co-workers as a novel technque for dscretzaton of partal dfferental equatons [1]. The basc dea and the core of IGA are to utlze the bass functons that are able to model geometry exactly, from the CAD ponts of vew, for numercal smulatons E-mal address of the correspondng author: saeed.shojaee@uk.ac.r (S. Shojaee) Fax Number: , Telephone:

2 of physcal phenomena. Ths can be acheved by usng the B-splnes or Non Unform Ratonal B- splnes (NURBS) for the geometrcal descrpton and nvoke the soparametrc concepts to defne the unknown feld varables. The IGA-based approaches have been constantly developed and shown many great advantages for solvng varety of dfferent problems n a wde range of research areas, such as flud structure nteracton, shells, structural analyss and so on [2-9]. In spte of these advantages, the IGA method suffers from some defcences. One of the most sgnfcant drawbacks arses from mposton of essental boundary condtons. Due to the non-nterpolatng nature of NURBS bass functons, the Kronecker Delta propertes are not satsfed, and as a consequence, the mposton of essental boundary condtons needs specal treatment. In consderng ths, several methods have been proposed for mposng essental boundary condtons n IGA. Ths ssue for NURBS-based sogeometrc analyss was frst dscussed by Hughes et al. [1]. In ther research, the essental boundary condtons were mposed to the control varables by evaluatng the functon of boundary condton at the spatal locatons of the control ponts. In current study, ths approach s referred to as Drect Method (DM), mentoned by Wang and Xuan [10]. Ths method s effcent for homogenous boundary condtons, but t s not relable for non-homogenous boundary condtons. In addton, when the poston of boundary control ponts s not located on the desred boundary, t s not even reasonable to enforce the gven boundary values to the correspondng boundary control varables [10]. Therefore, the enhancement of essental boundary condtons n IGA needs to be researched more thoroughly [1]. Wang and Xuan [10] have proposed an mproved method for mposton of essental boundary condtons n IGA, whch s based on concepts of the mxed transformaton method that was orgnated by Chen and Wang [11]. In ther work, nstead of evaluatng functon of boundary on specal locaton of control ponts, boundary value of control ponts were calculated by nterpolaton on Drchlet boundary. Ths method produces more accurate results and convergence rates n comparson wth DM [10]. Although Wang and Xuan [11] mentoned that a set of boundary nterpolaton ponts can be selected to construct the approprate transformaton matrx t should be consdered that selected boundary ponts can result n sngular transformaton matrx[12]. Imposng essental boundary condtons n tme dependent problems s appled n IGA wth Hughes et. al. [5]. They appled drect method n structural vbratons and wave propagaton 2

3 problems. As mentoned above, DM s able to mpose homogenous essental boundary condtons accurately. Bazlevs and Hughes [13] weakly enforced Drchlet boundary condtons and compared wth strongly enforced condtons for boundary layer solutons of the advecton dffuson equaton and ncompressble Naver Stokes equatons. In ther proposed method, they developed stablzed formulatons, ncorporatng weak enforcement of Drchlet boundary condtons. They also appled ths method n computaton of flows about rotatng components [2]. Cottrell et. al. [7] appled drect method n structural vbratons and learned that nhomogeneous boundary condton and the boundary values must be approxmated by functons lyng wthn the NURBS space. Ths resulted n strong, but approxmated satsfacton of the boundary condtons. In ths paper, mposng essental boundary condtons n transent heat flow wthn twodmensonal regon s extended n two steps. The essental boundary condtons are defned on Drchlet boundary as determned temperatures and ndependent of tme. In the frst step, Drchlet boundary condtons are weakly bult nto the varatonal formulaton, choosng weght functon, approprately. In the second step, wth fxed condton, the system of equatons s approprately adjusted. Ths paper s organzed as follows: Frst, the NURBS-based IGA s brefly revewed. Then, the framework of sogeometrc analyss, dealng wth transent heat conducton s dscussed. Two steps of proposed approach used n transent heat conducton are descrbed and the formulaton of such process s presented. Subsequently, several numercal smulatons are llustrated to demonstrate the robustness and effcacy of the present method. 2. Isogeometrc analyss based on the NURBS bass functons The tradtonal fnte element formulatons are based on nterpolaton schemes wth Lagrange or Hermt polynomals to approxmate the geometry, the physcal feld and ts dervatves. Ths approach often requres a substantal smplfcaton of the geometry, partcularly n the case of curved boundares of the analyss doman. Generally, adaptve refnement of the dscretzed doman s appled to better approxmate the boundary and to acheve suffcent convergence. The concept of IGA s based on applyng the NURBS bass functons n accurate modelng of 3

4 geometry and approxmaton of soluton space. The NURBS bass functons are weghted functons whch orgnate from B-splne nterpolaton. The B-splne functons are generated from a knot vector, whch s a non-decreasng sequence of coordnates n the parameter space wrtten as, 1, 2,..., n p 1, (1) where s the th knot value, n and p are the number and the order of bass functons defned on knot vector, respectvely. The half open nterval, [, 1), s called knot nterval. If 1, then the length of knot nterval s equal to zero. If 1 and n p 1 are repeated p+1 tmes n a knot vector, the resultng knot vector s called open knot vector. B-splne bass functons are startng from pecewse constants N,0 1 ( ) 0 f otherwse 1 (2) and arbtrary polynomal degree j can be generated recursvely wth N ( ) N ( ) N ( ) j1, j, j1 1, j1 j j11 j 1,2..., p 1,2,..., n p 1 j (3) n whch N, js the th bass functon wth a j order. The frst order dervatve of B-splne s d j j N ( ) N ( ) N ( ) d, j, j1 1, j1 j j1 1. (4) The B-splne bass functons whch are constructed from the open knot vectors have the nterpolaton feature at the ends of the parametrc space. A cubc B-splne bass functons wth the nterpolaton feature at the ends of the parametrc space are shown n Fg. 1. 4

5 The NURBS bass functons are made from B-splne functons by followng equaton: Rp, ( ) N w, p, (5) W ( ) n whch w s the weght correspondng to th control pont and W ( ) s the weght functon as defned by: n W( ) N w. (6) 1, p The bvarate NURBS functons on knot surface are defned by: R pq,, j (, ) 1,2..., n j1,2..., m N ( ) M ( ) w, p j, q, j, (7) W (, ) n whch M jq, and N, p( ) are respectvely the th p-order and j th q-order functons on and knot vectors. w, j s the weght correspondng to j control pont and W (, ) s the bvarate weght functon, whch s gven by: n m W (, ) N ( ) M ( ) w. 1 j knot nserton, p j, q, j (8) Smple and straghtforward refnement s one of the great advantages of IGA n comparson to classcal fnte element method. It s very straghtforward for ncreasng number of elements and elevatng degree of NURBS bass functons. In ths paper, knot-nserton or h-refnement s employed for convergence study. In each refnement step, knots are added to the knot spans. Knot nserton s a procedure that arbtrary new knots are added to a knot vector wthout any change n the shape of the B-splne curve. If there are m n p 1 knots n the knot vector of the B-splne curve, where n s the number of control ponts and p s the order of B-splne 5

6 curve, by addng a new knot, a new control pont must be added. Also, some current control ponts must be redefned. Consder a knot vector of p. Let ˆ, steps [14]:,,..., m n p wth n control ponts P1, P 2,..., P n and the order k k 1 be a desred new knot. The knot nserton procedure has the followng 3 1. Fnd k such thatˆ belongs to, 1 k k. 2. Fnd p 1control ponts P, P 1,..., P k p k p k. 3. Compute p new control ponts Q from the above p 1 control ponts by usng Eq. (9). Q (1 ) P P, (9) 1 where s obtaned from: ˆ for k p 1 k. (10) p By performng the above procedure, the new knot vector and control ponts are obtaned by: 1, ˆ 2,..., k,, k 1,..., m, (11) P1, P2,..., Pk p, Qk p1, Qk p2,..., Qk, Pk, Pk 1,..., Pn. (12) Now, ths knot nserton algorthm s extended to a NURBS curve. For ths purpose, a gven NURBS curve n d-dmensonal space s converted nto a B-splne curve n (d+1)-dmensonal space, then by applyng the knot nserton algorthm n ths B-splne curve, the new control ponts are obtaned. These new control ponts should then be projected to d-dmensonal space to obtan the new control ponts of the NURBS curve. Consder control ponts P ( x, y ) wth correspondng weghts of w, by convertng these control ponts to 3-dmensonal space, P ( w x, w y, w ), the new control ponts are then computed from Eq. (13). w Q (1 ) P P. (13) w w w 1 6

7 The locaton of control ponts n 2D are obtaned by the followng projecton technque: Q (1 ) P (1 ) w P, (14) w w w 1 1 and the weghts are: w (1 ) w w. (15) Q 1 3. Governng equatons and dscretzaton In ths secton, the governng and dscretzed equatons for the transent heat conducton are brefly presented. 3.1 Transent Heat Conducton Consder the tme dependent equaton governng the transent heat transfer n a homogeneous two dmensonal regon wth total boundary ; u c u, c t k, (16) wth boundary condtons: u g on D u h on n N (17) Here k denotes heat transfer coeffcent, s the specfc heat of the materal and s ts densty (mass per unt volume). As shown n Eq. (17) essental and natural boundary condtons are defned as specfed tme ndependent temperature and heat flux, respectvely. It s assumed that the boundary of an admssble regon can satsfy the followng condtons: D D N N (18) where D s the admssble Drchlet boundary and that the materals are homogenous and tme ndependent. N s the Neumann boundary. It s assumed 7

8 3.2 Varatonal approxmaton Several well-establshed weghted resdual methods such as Galerkn method, the method of least square, collocaton and subdoman methods can be used to approxmate the soluton. Here, we consder the Galerkn method to seek an approxmate soluton to transent heat transfer equaton. Assume S and V to be the subspaces of functon space wth a contnuous second dervatve; 1 ( ), D 1 ( ), 0 D S f f H f g V r r H r where H 1 ( ) s Sobolev space, whch can be defned as: 1 2 H ( ) u D u L ( ), 1. (20) The sem dscrete weak varatonal formulaton of Eq. (16) over s gven by u. (21) 2 w d c wh d w. u d t N The man dfference between the proposed method and the other methods s, how to choose W (.e. weght functon). In the conventonal method, weght functon s consdered from V, but n the proposed two-step method, t s consdered from S space. In ths subsecton, the conventonal method s dscussed, ntally. Consderng u v g and u u, and substtute n Eq. (21) t t (19), (22) 2 w v d w g d c t wh d w. v d w. g d N where g s prescrbed temperature on D and v belong to V. The NURBS approxmaton of v, g, v, f b and w are gven by v R v (23a) g R g (23b) b v R d (23c) b g R d (23d) b w R w (23e) 8

9 where R and Rb are nteror and boundary bass functon matrces, respectvely (see [10, 12] for more detals); v and g are vector of nteror and boundary control ponts temperature respectvely. d and d b are vector of nteror and boundary control ponts temperature gradent. w s the weght vector of nteror control ponts. to be constant by the tme on obtaned by substtutng Eqs. (23) to Eq. (22): t s tme nterval. The temperature s assumed D that means g d b 0. The matrx form of equatons can be Kd F F F, (24) n1 N D n and, T K R. R d (25a) 2 T FN c t R h d (25b) N 2 T FD c t R. Rb d g (25c) n 2 T n F c t R. R d v (25d) where n shows the nth tme nterval. We can use two approaches to estmate g. In the frst approach, the temperature of boundary control ponts s mposed by evaluatng the functon of boundary condton at the spatal locatons of the control ponts. Ths method suffers from two essental drawbacks. When the poston of boundary control ponts s not located on the desred boundary and t s not reasonable to enforce the gven boundary temperatures to the correspondng boundary control varables. In addton, the non-nterpolatng nature of NURBS bass functons does not allow for the satsfacton of nhomogeneous boundares n a straghtforward approach, and offers a lower rate of convergence. In the second approach, the vector g s obtaned by nterpolaton of functon on boundary. It offers a far hgher rate of convergence n comparson wth the frst approach. However, t should be consdered that selected boundary ponts can result n a sngular transformaton matrx. Ths drawback s more sgnfcant when there are many actve control ponts on the desred boundary, whch requres a more precse selecton procedure. As mentoned by Wang and Xuan [10], a set of boundary nterpolaton ponts can be selected to construct the approprate transformaton matrx. 9

10 3.3 Proposed Method Step1 As dscussed earler, for the conventonal methods, sncew V, the ntegraton on Drchlet boundary becomes zero and the usual weak formulaton (15) can be obtaned. In the proposed method, the weght functon, W belongs to S, and a change n the weght functon causes a correspondng change n weak formulaton as: u t u n. (26) 2 w d c wh d w d w. u d N D The functons w and u are approxmated by total NURBS functons, u w Ru (27a) Rw (27b) where R s the total bass functon matrx. Comparng Eq. (27b) and Eq. (23e) shows the man dfference between conventonal and proposed methods that s the space of w functon. In other words f you choose w V the other hand selectng w t leads to strongly mposton of essental boundary condton and on S leads to weakly mposton of essental boundary condton. The second term on the rght sde of Eq. (25) s the ntegraton on Drchlet boundary; where n s the outward unt normal to Drchlet boundary. u n w d wnx n y u d D. (28) D One of the problems n the proposed method s the characterstc determnaton of vector n. The vector n s formed by cross product of t and s, where these vectors are perpendcular to the plane of problem and tangent to the surface, respectvely. The correct drecton of vector n depends on exact choce of the drecton of t. As shown n Fg. (2), f the gven drecton of t shall be changed, the drecton of n s changed. The exact selecton of t s defned as cross product of vectors s and m. m s a vector from boundary to regon and defned thorough space near to the p b. pb and p ponts. pb s a pont on Drchlet boundary and p s a pont n 10

11 x l t s m ( p pb), (29) y l where l s the drecton of Drchlet boundary n parametrc space (see Fg. (3)). It s worthwhle to note that, f p s selected napproprately ncorrect drecton of t shall be obtaned. Two stuatons may occur; 1. p may be placed along the s vector that results to t 0 ( see Fg. (4a)). 2. Incorrect placement of p leads to the ncorrect drecton of t ( see Fg. (4b)). It s clear that the approprate selecton of p n the physcal space s not smple, and therefore, t s selected n the parametrc space. Assume that pˆ b( b, b) and pˆ (, ) are mages of pb ( xb, yb ) and p ( x, y ) n the parametrc space, respectvely. The approprate placement of pˆ (, ) s selected as follow: p, f l, L b b f l, L b b (30) where L and L are the lengths of parametrc space n and drectons. Parameter s a constant that belongs to (0,1]. Fnally, the vector n can be obtaned by: n y ( x y y x ) x ( x y y x ) j, (31) l l b l b l l b l b () Where () l denotes l and () denotes ( ) ( ) b b. Accordngly, the varatonal statement of Eq. (25) leads to the matrx form as; Kd F F F, (32) n1 n n N D where matrx K, vectors F N and n F are defned n Eq. (24). The evaluaton of these parameters can be performed usng the complete NURBS bass functons. The vector n FD s defned as: F c t R R d u. (33) n 2 T n D D 11

12 Here, R s the matrx of the NURBS basc functons and vector n n th tme step, and R s defned as: n u s the control ponts temperature yl R ( xl yb yl xb ) R x. (34) l t s the tme step sze and should be suffcently small to acheve the numercal stablty, due to the Courant Fredrchs Lewy (CFL) condton Step2 As dscussed above n ths paper the essental boundary condtons are mposed n 2 step. In prevous subsecton frst step s descrbed, and n ths subsecton the second step s explaned completely. In ths paper the essental boundary condtons are tme ndependent. So, we must modfy the system of equatons to enforce ths condton. Accordng to Eq. (24) t can be done f the temperature varaton of boundary control ponts s equal to zero. 3.4 Intal Condton Solvng the transent heat transfer needs to see how the temperature change from the ntal state (.e. t = 0) to the fnal steady state as a functon of tme. In fnte element method, the ntal temperature at each node s equal to temperature dstrbuton functon at node. But n IGA, the non-nterpolatng nature of NURBS does not allow for determnng the ntal temperature of control ponts at the spatal locatons. Here, usng nterpolaton nsde the physcal regon, the ntal temperature of control ponts s evaluated. Assume H( ) to be the ntal temperature functon and m be the number of NURBS functons on doman. Approprate choce of m nterpolaton ponts leads to the ntal temperature of control ponts, whch s gven by: T u T H 0 H R (, ) j j H( x, y ) (35) 12

13 where (, ) s th nterpolaton pont n the parametrc space and ( x, y ) s the projecton of ths pont n the physcal space. A set of nterpolaton ponts can be selected to construct the soluton. Approprate selecton procedure s requred to prevent sngularty n transformaton matrx. For approprate placement, a set of nterpolaton ponts n the maxmum ponts of NURBS functons can be selected. 4. Numercal examples In ths secton, the accuracy and the convergence of the proposed method through several numercal examples s verfed. The results obtaned by the proposed method are also compared wth drect and fnte element methods. Thrd order NURBS s used n all examples. For numercal ntegraton 33gauss quadrature rule s appled n the doman and 5 gauss quadrature rule on the boundares. The materal s alumnum, k (heat transfer coeffcent), (the specfc heat of the materal) and (densty or mass per unt volume) are consdered, ( g ) 3 cm 2.7, J ( ) J ( ) gk. cm.. s K 0.9, k heat conducton n annulus dsk In ths example, the heat conducton n annulus dsk s modeled. Geometry and boundary condtons are shown n Fgure 5. As shown n Fgure 5, heat transfer s prevented between dsk boundares and other domans, and D as: s assumed. The ntal temperature n entre doman as shown n Fgure 6 s consdered H( ) 50 2 x. (36) H( ) s used to approxmate the functon H ( ) and ths s obtaned by calculatng ntal temperature n the control ponts. Fgure 7 llustrates the percentages of the error dstrbuton for H ( ). For the convergence study, the h-refnement strategy s employed and meshes wth 60, 200, 240 and 360 (Fgure 8) elements n IGA and 52, 172, 503 and 1008 n FEM are nvestgated (see Table 1). 13

14 As shown n Fgure 9(a), the two-step and drect methods have smlar convergence rate, where no essental boundary condtons are defned. In other words, when there are no essental boundary condtons, the drect and proposed methods have the same accuracy. Further, t s shown that, the convergence rate of the proposed and drect methods s better than the classc fnte element method. The dfference between the results of each step and the result of fnal step T T4 s shown n Fgure 9(b) The parameter δ s defned as: 100, for =1 to 4. It s clear T that the proposed and drect methods converge faster than FEM: Temperature dstrbuton at t=3sec and temperature varaton wthn tme at p (1.5,0) are shown n Fgures 10 and 11, respectvely. 4.2 heat conducton on square plate wth crcular hole The next example refers to a stuaton that the ntal temperature s defned by mult functons. Here, the mult patch technque s used to dvde a square plate nto four patches, havng a hole n the center as shown n Fgure 12. A partcular ntal temperature s consdered for each patch are lsted n Table 2. The outer boundary of the plate s nsulated. The essental boundary condtons are defned on boundary of hole as a specfed and tme-ndependent temperature. Fgure 13(a) shows the ntal temperature dstrbuton on doman. Fgure 13(b) confrms that approxmated ntal temperature s very close to exact one. Smlar to the pervous example, the convergence study s carred out ncreasng the number of elements and cells lsted n Table 3 n four steps. The fnal meshng ncludes 720 elements wth 880 degree of freedom n IGA (see Fgure 14), and 4913 elements wth 5131 degree of freedom n FEM. As shown n Fgure 15(a), the two-step and fnte element methods converge to the same temperature. However, the result of the drect method n the fnal meshng s a lttle dfferent. The reason s essental boundary condtons. Snce the functon on Drchlet boundary s not constant or lnear, mposng essental boundary condtons through drect method s not accurate. 14 4

15 It also can be found that, n a coarse meshng DM s more accurate than FEM but when the number of elements s ncreased the result of FEM s more accurate. Fgure 15 (b) confrms that the proposed method converge very faster than two other methods. It s clear that the result of two-step method n frst step s very close to fnal step result. The temperature dstrbuton at t=3sec and temperature varaton wthn tme at p= (1.5, 0) are plotted n Fgures (16) and (17), respectvely: 5. Concluson In ths paper, a new method s proposed for mposton of the essental boundary condtons n transent heat conducton problem based on sogeometrc analyss. Ths method s named as twostep method, because the essental boundary condtons are mposed n two steps. The frst step ncludes calculatng the force vector correspondng to the Drchlet boundares, and the system of lnear equaton s modfed n the second step. Ths method s classfed n weakly mposton of the essental boundary condton. The numercal examples confrm that the two-step and the drect methods are smlar, when there are no essental boundary condtons. However, n the stuaton that the essental boundary condtons are more complex, the results for the proposed method offer more accurate soluton n comparson wth the drect method. Furthermore, comparng the results of the two-step, drect and classc fnte element methods, t can be demonstrated that the convergence of the proposed method s faster than two other methods. Also, the numercal examples show that usng the two-step method for mposton of the essental boundary condtons enable us to use coarse meshng and obtan a very accurate result. 6. References 1. Hughes, T.J.R., Cottrell, J.A.Y. and Bazlves, Y., Isogeometrc analyss: CAD, fnte elements,nurbs, exact geometry and mesh refnement, Comput. Meth. Appl. Mech. Eng., (194), pp (2005). 2. Bazlevs, Y., Hughes, T.J.R., NURBS-based sogeometrc analyss for the computaton of flows about rotatng components, Comput. Mech., (43), pp (2008). 3. Bazlevs, Y., Mchler, C., Calo, V.M., Hughes, T.J.R., Weak Drchlet boundary condtons for wall-bounded turbulent flows, Comput. Meth. Appl. Mech. Eng., (196), pp (2007). 4. Bazlevs, Y., Calo, V.M., Hughes, T.J.R., Zhang, Y.. Isogeometrc flud structure nteracton: theory, algorthms and computatons, Comput. Mech., (43), pp 3 37 (2008). 15

16 5. Hughes, T.J.R., Real, A., Sangall, G., Dualty and unfed analyss of dscrete approxmatons n structural dynamcs and wave propagaton comparson of p-method fnte elements wth k-method NURBS, Comput. Meth. Appl. Mech. Eng., (197), pp (2008). 6. Zhang, Y.J., Bazlevs, Y., Goswam, S., Bajaj, C.L., Hughes, T.J.R., Patent-specfc vascular NURBS modelng for sogeometrc analyss of blood flow, Comput. Meth. Appl. Mech. Eng., (196), pp (2007). 7. Cottrell, J.A., Real, A., Bazlevs, Y., Hughes, T.J.R., Isogeometrc analyss of structural vbratons, Comput. Meth. Appl. Mech. Eng., (195), pp (2006). 8. Benson, D.J, Bazlevs, Y., Hsu, M.C., Hughes, T.J.R., Isogeometrc shell analyss: the Ressner Mndln shell, Comput. Meth. Appl. Mech. Eng., (199), pp (2010). 9. Gómez, H., Calo, V.M., Bazlevs, Y., Hughes, T.J.R., Isogeometrc analyss of the Cahn Hllard phase-feld model, Comput. Meth. Appl. Mech. Eng., (197), pp (2008). 10. Wang, D. and Xuan, J., An mproved NURBS-based sogeometrc analyss wth enhanced treatment of essental boundary condtons, Comput. Meth. Appl. Mech. Eng., (199), pp (2010). 11. Chen, J.S., Wang, H.P., New boundary condton treatments n meshless computaton of contact problems, Comput. Meth. Appl. Mech. Eng., (187), pp (2000). 12. Shojaee, S., Izadpanah, E., and Haer, A., mposton of essental boundary condtons n sogeometrc analyss usng Lagrange multpler method Internatonal Journal of Optmzaton n Cvl Engneerng, (2), pp , (2012). 13. Bazlevs, Y., Hughes, T.J.R., weak mposton of drchlet boundary condtons n flud mechancs, computer&fluds, (36), pp (2007). 14. Pegl L, Tller W. The NURBS Book (Monographs n vsual communcaton), 2nd ed. Sprnger. New York, (1997). Saeed Shojaee obtaned hs B.S. degree from Shahd Bahonar Unversty of Kerman n 2001, and hs M.S. and Ph.D. degrees n Structural Engneerng from Iran Unversty of Scence and Technology n 2003 and 2007, respectvely. He s currently Assocate Professor n the Department of Cvl Engneerng at Shahd Bahonar Unversty n Kerman, Iran. Hs man research nterests nclude: optmal analyss and desgn of structures, metaheurstc optmzaton technques and applcatons, computatonal mechancs and sesmc desgn of structures. Ebrahm Izadpanah receved hs M.E. degree n Structural Engneerng from the Shahd Bahonar Unversty of Kerman n He s currently a Ph.D. degree canddate n Shahd Bahonar Unversty of Kerman. Hs research nterests nclude: computatonal mechancs and Isogeometrc analyss.he has publshed a number of papers n journals. Saeed Nazar obtaned hs B.S. degree from Shahd Bahonar Unversty of Kerman n He s currently a M. SC. student n Amrkabr Unversty of Technology. 16

17 Fgures Capton Fgure 1. Cubc bass functons for an open knot vector 0,0,0,0,0.25,0.5,0.75,1,1,1,1 Fgure 2. Outward unt normal vector on Drchlet boundary, (a) correct drecton of n, (b) ncorrect drecton of n Fgure 3. Drecton of Drchlet boundares n parametrc space Fgure 4. Incorrect placements of p Fgure 5. Geometry and boundary condtons of Annulus dsk u n 0 on N1 and N2 Fgure 6. Temperature dstrbuton on annulus dsk at t=0 Fgure 7. Error dstrbuton of approxmated temperature at t=0 Fgure 8. Fnal mesh of annulus dsk problem wth 360 elements n IGA Fgure 9. Convergence procedure of two-step, drect and fnte element methods for annulus dsk problem Fgure 10. Temperature dstrbuton on dsk at t=3sec 17

18 Fgure 11.Temperature varaton wthn tme at p= (1.5, 0) Fgure 12. Square plate wth crcular hole n center Fgure 13. (a) Temperature dstrbuton on square plate wth crcular hole at t=0, (b) Error dstrbuton of approxmated temperature at t=0 Fgure 14. Fnal meshng of example 2 wth 720 elements n IGA Fgure 15. Convergence procedure of two-step, drect and fnte element methods for example 2 (a) Absolute temperature, (b) Relatve temperature wth respect to fnal meshng Fgure 16. Temperature dstrbuton on annulus dsk at t=3sec Fgure 17. Temperature varaton wthn tme at p= (1.5, 0) Tables Capton Table 1. Refnement procedure of annulus dsk problem n IGA and FEM Table 2. Intal temperature and boundary condtons n each patch Table 3. Refnement procedure of square plate problem n IGA and FEM 18

19 Fgure 1. Cubc bass functons for an open knot vector 0,0,0,0,0.25,0.5,0.75,1,1,1,1 19

20 (a) (b) Fgure 2. Outward unt normal vector on Drchlet boundary, (a) correct drecton of n, (b) ncorrect drecton of n 20

21 Drchlet Boundary Fgure 3. Drecton of Drchlet boundares n parametrc space (a) (b) Fgure 4. Incorrect placements of p 21

22 ro 2 r 1 N 1 N 2 u Fgure 5. Geometry and boundary condtons of Annulus dsk on and n 0 N1 N2 Fgure 6. Temperature dstrbuton on annulus dsk at t=0 22

23 Fgure 7. Error dstrbuton of approxmated temperature at t=0 Fgure 8. Fnal mesh of annulus dsk problem wth 360 elements n IGA 23

24 (a) (b) Fgure 9. Convergence procedure of two-step, drect and fnte element methods for annulus dsk problem: (a) Absolute temperature, (b) Relatve temperature wth respect to fnal meshng Fgure 10. Temperature dstrbuton on dsk at t=3sec Fgure 11.Temperature varaton wthn tme at p= (1.5, 0) 24

25 a Patch 2 r 1 Patch1 N D Patch 3 Patch 4 a 2 Fgure 12. Square plate wth crcular hole n center. (a) (b) Fgure 13. (a) Temperature dstrbuton on square plate wth crcular hole at t=0, (b) Error dstrbuton of approxmated temperature at t=0 25

26 Fgure 14. Fnal meshng of example 2 wth 720 elements n IGA (a) (b) Fgure 15. Convergence procedure of two-step, drect and fnte element methods for example 2 (a) Absolute temperature, (b) Relatve temperature wth respect to fnal meshng 26

27 Fgure 16. Temperature dstrbuton on annulus dsk at t=3sec Fgure 17. Temperature varaton wthn tme at p= (1.5, 0) Table 1. Refnement procedure of annulus dsk problem n IGA and FEM Number of elements IGA FEM Table 2. Intal temperature and boundary condtons n each patch: Patch1 Intal temperature Essental B.C Natural B.C H( ) 50y 2 2 y h1 0 g

28 Patch2 Patch3 H y x 2 2 ( ) 50( ) H( ) 50x 2 g y x h ( ) 2 x h3 0 g3 50 Patch4 H( ) 0 g4 0 h4 0 Table 3. Refnement procedure of square plate problem n IGA and FEM Number of elements IGA FEM