Isogeometrc Analyss wth Geometrcally Contnuous Functons on Mult-Patch Geometres Maro Kapl Floran Buchegger Mchel Bercover Bert Jüttler G+S Report No 35 August 05
Isogeometrc Analyss wth Geometrcally Contnuous Functons on Mult-Patch Geometres Maro Kapl a Floran Buchegger a Mchel Bercover b Bert Jüttler a a Johannes Kepler Unversty Lnz Austra b The Hebrew Unversty of Jerusalem Israel Abstract We present a method for generatng a bass of the space of bcubc and bquartc C -smooth geometrcally contnuous sogeometrc functons on blnear mult-patch domans Ω R The resultng bass functons are constructed from C -smooth geometrcally contnuous sogeometrc functons on blnearly parameterzed two patch domans (cf [0]) and are descrbed by explct formulas for the Bézer coeffcents of ther splne segments These C - smooth sogeometrc functons possess potental for applcatons n sogeometrc analyss whch s demonstrated by several examples (eg solvng the bharmonc equaton) The numercal results ndcate optmal rates of convergence Keywords: Isogeometrc Analyss C -smooth sogeometrc functons geometrcally contnuous sogeometrc functons mult-patch doman bharmonc equaton Introducton Isogeometrc Analyss (IgA) s a promsng framework for performng numercal smulaton whch uses the same (ratonal) splne functon space for representng the geometry of the physcal doman and descrbng the soluton space [4 9] One possblty n IgA to deal wth domans of general topology s the use of mult-patch parameterzatons Several methods for couplng the sngle patches exst eg [ 3 7 3 6 9] but most of these technques do not provde globally C -smooth functons A recent strategy to overcome ths lmtaton s the use of the concept of geometrc contnuty [7] Ths concepts represents a well known approach n Computer Aded Geometrc Desgn for generatng smooth mult-patch surfaces possessng extraordnary vertces [5 8] The constructon of C -smooth functons (or even functons wth hgher smoothness) s based on the observaton whch has been formalzed frstly by Grosser and Peters [6] that the C s -smoothness of an sogeometrc functon s equvalent to the geometrc smoothness of order s (G s -smoothness) of ts graph surface where s s a postve nteger Ths fact Correspondng author Emal addresses: marokapl@jkuat (Maro Kapl) floranbuchegger@jkuat (Floran Buchegger) berco@cshujacl (Mchel Bercover) bertjuettler@jkuat (Bert Jüttler) Preprnt submtted to Elsever August 3 05
motvated us to denote the C s -smooth functons on a mult-patch doman as C s -smooth geometrcally contnuous sogeometrc functons [0] For the case s = two dfferent strateges followng the concept of geometrc smoothness have been explored One approach derves the C -smooth functons from methods for constructng G -smooth mult-patch surfaces Examples are the technques [ 4 5] whch are based on dfferent G -smooth mult-patch splne surface constructons or the method [8] whch generates the C -smooth functons by means of a T-splne representaton (cf [0 ]) In contrast the second approach generates a bass of the entre space of C -smooth functons on a partcular class of mult-patch geometres cf [ 0] Eg n [0] the space of bcubc and bquartc C -smooth geometrcally contnuous sogeometrc functons on blnearly parameterzed two-patch domans has been analyzed Furthermore a smple framework for the constructon of a bass for ths space has been developed The present work extends the results of [0] to blnearly parameterzed mult-patch domans Analogous to [0] the numercal experments ndcate that the space of C -smooth geometrcally contnuous sogeometrc functons provdes the full approxmaton power of splnes for these mult-patch domans Among other examples we numercally solve the bharmonc equaton on dfferent blnear mult-patch geometres where the sogeometrc analyss s smplfed by usng the C -smoothness of the dscretzaton space The paper s organzed as follows Secton descrbes the class of blnearly parameterzed mult-patch domans Ω R whch s consdered throughout ths paper In addton we present the concept of bcubc and bquartc C -smooth geometrcally contnuous sogeometrc functons whch was ntroduced for a more general settng n [0] The explct constructon of a bass of the space of C -smooth geometrcally contnuous sogeometrc functons defned on blnearly parameterzed two-patch domans for bquartc functons from [0] s recalled n Secton 3 In addton we present a smlar new constructon for bcubc functons Moreover we propose a dfferent choce of the bass functons n the vcnty of patch vertces of valence m 3 snce ths wll be advantageous for analyzng the full mult-patch case Explct formulas for the Bézer coeffcents of the splne segments of all functons are specfed n Appendx A We then use the dfferent functons for the two patch case to generate a bass of the space of C -smooth geometrcally contnuous sogeometrc functons on blnear mult-patch domans n Secton 4 The dmenson of the resultng space of the C -smooth sogeometrc functons s also analyzed Fnally Secton 5 shows several examples that demonstrate the potental of our constructon for sogeometrc analyss More precsely we use C -smooth bcubc and bquartc functons for performng L -approxmaton and for solvng Posson s equaton and the bharmonc equatons on dfferent mult-patch domans The numercal results ndcate optmal rates of convergence Prelmnares The noton of C s -smooth geometrcally contnuous sogeometrc functons on general mult-patch domans was ntroduced n [0 Secton ] In ths secton we frst present the partcular class of blnear mult-patch domans Ω R whch wll be consdered
throughout the paper Then we recall the concept of geometrcally contnuous functons for s = Blnearly parameterzed mult-patch domans For a gven degree d {3 4} and a number of nner knots k satsfyng k 5 d we denote wth Skp d the tensor-product splne space of degree (d d) whch s defned on [0 ] by choosng k unform nner knots of multplcty p 0 n both parameter drectons We consder a computatonal doman Ω = n Ω (l) R l= whch s the unon of n quadrlateral patches Ω (l) l { n} Each patch s the mage Ω (l) = G (l) ([0 ] ) of the unt square under a bjectve regular geometry mappng G (l) S d kd Sd kd : [0 ] R ξ (l) = (ξ (l) ξ(l) Each geometry mappng G (l) s represented n the form ) (G(l) G(l) ) = G(l) (ξ (l) ) G (l) (ξ (l) ) = Î d (l) ψ (ξ (l) ) d (l) wth splne control ponts R and tensor-product B-splnes ψ spannng the splne space Skd d where Î = {( ) 0 j d + kd k j = } s the ndex set Each geometry mappng G (l) has a Bézer representaton G (l) (ξ (l) ) = Ĩ d (l) ψ (ξ (l) ) where Ĩ = {( ) 0 j d + k + kd j = } s the ndex set R are the Bézer control ponts and ψ are the tensor-product B-splnes of the space Skd+ d Let I be the ndex space I = {l} Ĩ l {n} For a common nterface e of two neghborng splne patches G (l) we denote by I e I the ndex space I e = {(l ) I d (l) s Bézer control pont of the nterface e or of one of the two neghborng columns of control ponts on each sde of e} Throughout the paper we wll make the followng addtonal assumptons concernng the mult-patch doman Ω: d (l) 3
(p q ) (03) (p 3q 3) (p q ) (p 3q 3) (0) (p 0q 0) (00) (00) (p q ) (p 0q 0) d = 3 d = 4 (p q ) Fgure : The local geometry of the frst three (for d = 3) and frst two (for d = 4) pars of neghborng splne segments of a two-patch doman s determned by the 6 patch vertces (0 0) (0 3)/(0 ) and (p q ) = 0 3 I All geometry mappngs G (l) l { n} are also defned and regular on a neghborhood of [0 ] and the nterors of all patches Ω (l) l { n} are mutually dsjont e G (l) ((0 ) ) G (l ) ((0 ) ) = for l l { n} wth l l II Two neghborng patches Ω (l) and Ω (l ) l l { n} wth l l always share the whole common edge e the mult-patch doman Ω has no T -jonts We denote the resultng two-patch doman Ω (l) Ω (l ) by Ω (ll ) and denote ther common nterface by e (ll ) III All two-patch domans Ω (ll ) of two neghborng patches Ω (l) and Ω (l ) l l { n} wth l l satsfy the so-called genercty condton (cf [0 Eq (0)]) Assume that the two-patch doman Ω (ll ) s determned by the 6 patch vertces (0 0) (0 ) and (p q ) = 0 3 see [0 Fg ] and that the correspondng geometry mappngs G (l) and G (l ) satsfy G (l) ( ξ ) = G (l ) (0 ξ ) Then the condton p p p 0 p 3 s fulflled and for j { k} the 3 dfferent Bézer control ponts d (l) of the set { d (p) (p ) I e (ll ) and d (p) s a Bézer control pont that corresponds to the d + knots ( j k+ j k+ j k+ are not collnear ) n ξ(l) -drecton} IV We make the followng techncal assumpton for each boundary vertex of valence m = 3 of the mult-patch doman Ω After transformng the coordnates of the 4
vertces of the frst 6 d pars of splne segments nto the canoncal coordnate system shown n Fg where (0 0) s the correspondng boundary vertex the coordnates satsfy the condton p 0 ((5 d)q + q 3 (6 d)) ((5 d)p + p 3 )(q 0 ) Ths condton means that the three ponts (0 ) (p 0 q 0 ) and ( (5 d)p +p 3 are not collnear (5 d)q +q 3 ) 6 d 6 d V We make the followng techncal assumpton for each nner vertex of arbtrary valence (e m 3) or boundary vertex of valence m 4 of the mult-patch doman Ω After transformng the coordnates of the vertces of the frst 6 d pars of splne segments nto the canoncal coordnate system shown n Fg where (0 0) s the correspondng vertex the coordnates satsfy the two condtons and p 0 q p q 0 p 0 ( q ) p ( q 0 ) The frst condton means that the three ponts (0 0) (p 0 q 0 ) and (p q ) are not collnear and the second condton means that the three ponts (0 ) (p 0 q 0 ) and (p q ) are not collnear Examples of such mult-patch domans are shown n Fg (frst row) Space of C -smooth functons on blnear mult-patch geometres The space Ṽ of sogeometrc functons on a mult-patch doman Ω s gven by { } Ṽ = v Ω : v Ω (l) Skd d (G (l) ) for all l { n} An sogeometrc functon w Ṽ s represented on each patch Ω(l) l { n} by ) (w Ω (l))(x) = w (l) (x) = (W (l) (G (l) ) (x) x Ω (l) wth W (l) S d kd and the assocated graph surface F (l) of w (l) s gven by ( ) T F (l) (ξ (l) ) = G (l) (ξ (l) ) G (l) (ξ (l) ) W (l) (ξ (l) ) }{{} =G (l) (ξ (l) ) Each functon W (l) l { n} has a local splne representaton W (l) (ξ (l) ) = Î b (l) ψ (ξ (l) ) 5
wth b (l) R Consequently W (l) has also a local Bézer representaton W (l) (ξ (l) ) = Ĩ a (l) ψ (ξ (l) ) wth a (l) R For a functon w Ṽ we denote by supp BB(w) I the support of w n the Bézer coeffcent space e supp BB (w) = {(l ) I a (l) s Bézer coeffcent of W (l) wth a (l) 0} In the followng we are nterested n the space V = Ṽ C (Ω) e { } V = v C (Ω) : v Ω (l) Skd d (G(l) ) for all l { n} that contans the globally C -smooth sogeometrc functons defned on the mult-patch doman Ω Accordng to [0 Theorem ] an sogeometrc functon w Ṽ belong to the space V f and only f for all neghborng patches Ω (l) and Ω (l ) l l { n} wth l l the assocated graph surfaces F (l) and F (l ) are G -smooth along the common nterface e (ll ) Ths equvalence of the C -smoothness of an sogeometrc functon and the G -smoothness of ts graph surfaces s the reason to call the functons w V also C -smooth geometrcally contnuous sogeometrc functons As descrbed n [0 Subsecton 3] by mposng the G -smoothness between the graph surfaces of all neghborng patches Ω (l) Ω (l ) l l { n} wth l l we obtan lnear constrants on the splne coeffcents b (l) or on the Bézer coeffcents a (l) Ths can be formulated as a homogeneous lnear system or ( Ĥb = 0 b = b (l) ( Ha = 0 a = ) l {n} Î a (l) ) (l) I () () respectvely Then a bass of the nullspace of Ĥ or H defnes va the splne coeffcents b (l) or the Bézer coeffcents a (l) respectvely a bass of C -smooth geometrcally contnuous sogeometrc functons for the space V A constructon of a bass of V for blnearly parameterzed two-patch domans s explaned n [0 Secton 3] where the generated bass conssts of two dfferent knds of functons In the followng secton we wll present these two dfferent knds of bass functons Ths provdes us a possblty to generate a bass of V for blnearly parameterzed mult-patch domans Ω see Secton 4 Thereby the bass functons wll be generated by means of a sutable choce of the Bézer coeffcents a (l) 3 C -smooth functons on blnear two-patch geometres In ths secton we restrct ourselves to two-patch domans Ω = Ω () Ω () that satsfy Assumptons I-IV As explaned n [0 Secton 3] a possble bass of the space V for 6
blnearly parameterzed two-patch domans conssts of two dfferent knds of C -smooth geometrcally contnuous sogeometrc bass functons called bass functons of the frst knd and bass functons of the second knd In ths paper we wll refer to these two dfferent knds of bass functons as patch bass functons and edge bass functons respectvely We summarze them especally wth respect to the choce of the correspondng Bézer coeffcents a (l) In the case of the edge bass functons we recall an explct constructon of these functons for d = 4 from [0] and present a new smlar constructon for d = 3 In addton we present for both degrees a further explct constructon of the edge bass functons n the vcnty of the boundary vertces of a common nterface All the presented functons are used n Secton 4 to generate C -smooth bass functons defned on the blnear mult-patch domans Ω 3 Patch bass functons These functons are obtaned by composng a tensor-product B-splne functon ψ of one patch Ω (l) that has functon values and frst partal dervatves equal to zero along a common nterface wth another patch wth the nverse of the geometry mappng G (l) e { ( x ψ (G (l) ) )(x) f x Ω (l) l { } (3) 0 otherwse The support of each functon s contaned n only one of the two patches Ω () and Ω () In addton exactly one splne coeffcent b (l) s non-zero whch has the value one All coeffcents b (l) that correspond to the splne control ponts of the common nterface or one of the neghborng columns of the two patches G () and G () are zero That means for each splne control pont of the geometry mappngs G () and G () that do not belong to the common nterface of the two patches and to the neghborng column of splne control ponts we have exactly one patch bass functon Therefore the number of such functons for a two-patch doman s gven by (d + k(d ))(d + + k(d )) Snce the support of the tensor-product functon ψ can be obtaned n one two or four splne segments of the correspondng geometry mappng G (l) more than one Bézer coeffcent a (l) can be non-zero wth a (l) { } and these values are ndependent of the 4 geometry mappngs Analogous to the splne coeffcents b (l) all Bézer coeffcents a (l) that correspond to the Bézer control ponts of the common nterface or one of the neghborng columns of the correspondng splne segments of G () and G () have to be zero e a (l) = 0 for (l ) I e () Fg and 3 shows all possble supports of a patch bass functon w n the Bézer coeffcent space (e supp BB (w)) wth ther correspondng values of the Bézer coeffcents a (l) for d = 3 and d = 4 respectvely 3 Edge bass functons These functons w possess a support that s contaned n both patches Ω () and Ω () where supp BB (w) conssts only of Bézer coeffcents a (l) wth (l ) I e () In contrast to 7
(a) (b) (c) 4 4 4 4 (g) (h) (k) Fgure : All possble supports supp BB (w) (red squares) of the patch bass functons w for d = 3 wth the correspondng values of the Bézer coeffcents a (l) up to symmetres The shown splne segments are part of a splne patch Ω (l) where the blue edges have to be a part of the boundary Ω and the green edges can be a part of the boundary Ω 8
(a) (b) (c) (d) (e) (f) (g) (h) (j) 4 4 4 4 (k) Fgure 3: All possble supports supp BB (w) (red squares) of the patch bass functons w for d = 4 wth the correspondng values of the Bézer coeffcents a (l) up to symmetres The shown splne segments are part of a splne patch Ω (l) where the blue edges have to be a part of the boundary Ω and the green edges can be a part of the boundary Ω 9
the patch bass functons the Bézer coeffcents a (l) of the edge bass functons depend on the geometry mappngs The number of these functons was nvestgated n [0] Lemma ([0] Lemma 3 and Theorem 4) The number of lnearly ndependent edge bass functons s equal to (d + ) + (d 4)k For d = 4 a possble choce of these 9 + 4k edge bass functons w was presented n [0] where the functons are categorzed nto 4 dfferent types (A B L U) wth several subtypes The classfcaton s based on the sze and locaton of ther supports supp BB (w) see [0 Subsecton 33 and Appendx] The functons of type L and U are defned on the lower and upper boundary of the common nterface respectvely and the functons of type A and B are defned on nner parts of the common nterface The graphs of the dfferent types of functons are vsualzed n [0 Fg 6] Snce the functons of type L and U concde wth respect to swappng the lower boundary wth the upper boundary of the two-patch doman and vce versa t suffces to consder the functons of type L for the lower and upper boundary of the common nterface Fg 5 shows the supports supp BB (w) of the functons w of type A B and L for d = 4 A smlar choce of the 7 + k edge bass functons w for d = 3 s possble too where the functons are classfed agan wth respect to ther supports supp BB (w) see Fg 4 In contrast to d = 4 the resultng functons of type A B and L have slghtly ncreased supports but are located agan on the boundary of the common nterface for type L and on nner parts of the common nterface for type A and B For both degrees we present n Appendx A more detals about these dfferent types of bass functons ncludng explct formulas for ther Bézer coeffcents The followng lemma guarantees that the bass functons of type A B and L are a sutable choce of the (d + ) + (d 4)k edge bass functons Lemma The (d 9) + (d 4)k bass functons of type A and B for the nner part of the edge and the 0 bass functons of type L for the two boundares of the edge (see Appendx A) are lnearly ndependent Proof By usng symbolc computaton t can be shown that the Bézer coeffcent vector of each functon s a soluton of the homogeneous lnear system () The lnear ndependence of the functons can be proven by analyzng ther Bézer coeffcent vectors The patch bass functons and the bass functons of type A B and L form a bass of V Theorem 3 ([0] Theorem 6 for d = 4) The (d 9) + (d 4)k bass functons of type A and B for the nner part of the edge and the 0 bass functons of type L for the two boundares of the edge (see Appendx A) combned wth the patch bass functons defned n (3) form a bass of the space of C -smooth geometrcally contnuous sogeometrc functons Proof The number of the patch bass functons plus the number of the edge bass functons s equal to the dmenson and all the functons are lnearly ndependent compare [0 Theorem 4] 0
(p q ) (04) (p 3q 3) (p q ) (03) (p 3q 3) (p q ) (03) (p 3q 3) a 30 a 7 a 8 a 8 a 9 a 7 a 8 a 8 a 9 a a a a a 3 a 4 a 5 a 5 a 6 a 8 a 9 a 9 a 0 a 8 a 9 a 9 a 0 a a a a 3 a 8 a 9 a 9 a 0 a 8 a 9 a 9 a 0 a 8 a 9 a 9 a 0 a 5 a 6 a 6 a 7 a 5 a 6 a 6 a 7 a 8 a 9 a 9 a 0 a a 3 a 3 a 4 a a 3 a 3 a 4 a 5 a 6 a 6 a 7 a 9 a 0 a 0 a a 9 a 0 a 0 a a a 3 a 3 a 4 a 9 a 0 a 0 a a 9 a 0 a 0 a a 9 a 0 a 0 a a 6 a 7 a 7 a 8 a 6 a 7 a 7 a 8 a 9 a 0 a 0 a a 3 a 4 a 4 a 5 a 8 a 0 a a a (p 0q 0) (00) (p q ) (p 0q 0) (00) (p q ) (p 0q 0) type L L Y type A type B (00) (p q ) Fgure 4: The maxmal possble support n Bézer coeffcent space supp BB (w) (red and blue squares) of the edge bass functons w of type A B L L and Y for d = 3 The shown splne segments are part of a two-patch doman Ω = Ω () Ω () The red edges are part of the common nterface of Ω () and Ω () the blue edges have to be a part of the boundary Ω and the green edges can be a part of the boundary Ω (See Appendx A for explct formulas for the values a of the correspondng Bézer coeffcents wth respect to the gven local geometry)
(p q ) (03) (p 3q 3) (p q ) (0) (p 3q 3) (p q ) (0) (p 3q 3) a 7 a 5 a a 3 a 3 a 4 a a 3 a 3 a 4 a 9 a 0 a 0 a a 6 a 7 a 7 a 8 a 3 a 4 a 4 a 5 a 8 a 9 a 9 a 0 a 5 a 6 a 6 a 7 a a 3 a 3 a 4 a a 3 a 3 a 4 a 9 a 0 a 0 a a 6 a 7 a 7 a 8 a 4 a 5 a 5 a 6 a 4 a 5 a 5 a 6 a a a a 3 a 8 a 9 a 9 a 0 a 5 a 6 a 6 a 7 a a 3 a 3 a 4 a a 3 a 3 a 4 a a 0 a a a (p 0q 0) (00) (p q ) (p 0q 0) (00) (p q ) (p 0q 0) type L L Y type A type B (00) (p q ) Fgure 5: The maxmal possble support n Bézer coeffcent space supp BB (w) (red and blue squares) of the edge bass functons w of type A B L L and Y for d = 4 The shown splne segments are part of a two-patch doman Ω = Ω () Ω () The red edges are part of the common nterface of Ω () and Ω () the blue edges have to be a part of the boundary Ω and the green edges can be a part of the boundary Ω (See Appendx A for explct formulas for the values a of the correspondng Bézer coeffcents wth respect to the gven local geometry) Instead of the 5 bass functons of type L on the two boundares of the common nterface we construct 5 dfferent edge bass functons on these boundares as follows We select the 5 values a 0 a a 3 a 4 and a 5 of the correspondng Bézer coeffcents n such a way that one of them possesses the value one and the remanng values are zero see Fg 4-Fg 6 We refer to the resultng 5 bass functons as functons of type L For d = 4 the 5 dfferent functons of type L are vsualzed n Fg 7 In contrast to the functons of type L these functon of type L are not well-defned f Assumpton V s not fulflled for the correspondng vertex of the common nterface The functons of type L can be obtaned by lnearly combnng the functons of type L see Appendx A We thus obtan: Lemma 4 Let v 0 be one of the two vertces of the common nterface of the two-patch doman Ω If Assumpton V s satsfed for the vertex v 0 then the bass functons of type L n the vcnty of v 0 span the same space as the correspondng bass functons of type L Proof If Assumpton V s satsfed for the common vertex v 0 then all factors of the n Appendx A presented lnear combnatons for the functons of type L are well-defned Snce the bass functons of type L are lnearly ndependent due to ther canoncal constructon the correspondng bass functons of type L span the same space as the correspondng bass functons of type L Remark 5 Assumpton IV s only needed to ensure that the functons of type AB and L are lnearly ndependent whch would be otherwse volated by the fact that L = L When
usng the functons of type L nstead of the functons of type L the lnear ndependence s stll guaranteed n the case that Assumpton IV s not fulflled but Assumpton V s satsfed 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a a 0 0 a a 0 a a 0 0 a a 0 0 a a 0 type L type L type L 3 type L 4 type L 5 Fgure 6: The 5 functon of type L are obtaned by choosng the values a of the correspondng Bézer coeffcents n the blue squares n Fg 4 and 5 n the canoncal way as shown The value a s equal to zero for the functons of type L 3 and L 5 and unequal to zero for the functons of type L L and L 4 type L type L type L 3 type L 4 type L 5 Fgure 7: All dfferent edge bass functons of type L for d = 4 The formulas for the values of ther correspondng Bézer coeffcents are presented va lnear combnatons of the edge bass functons of type L n Appendx A 4 C -smooth functons on blnear mult-patch geometres We present the constructon of a bass of the space V of C -smooth geometrcally contnuous sogeometrc functons on blnearly parameterzed mult-patch domans Ω The obtaned bass functons wll be based on the dfferent bass functons of two-patch domans from the prevous secton Some of the functons concde wth a patch bass functon or wth a edge bass functon of type A B or L and some of the functons are bult from edge bass functons of type L In the followng we only consder mult-patch domans Ω that satsfy Assumptons I-V 3
4 Bass functons from the two-patch case For each two-patch doman Ω (ll ) l l { n} wth l l we can extend the resultng bass functons w from the prevous secton to the whole mult-patch doman Ω by settng all Bézer coeffcents a (p) of the addtonal patches Ω (p) p { n} wth p l and p l to zero For ths we defne the followng operator Q: Defnton 6 Let w be an sogeometrc functon on a two-patch doman Ω (ll ) l l { n} wth l l possessng a Bézer coeffcent vector a = (a (p) ) p {ll } Ĩ The operator Q generates the Bézer coeffcent vector Qa = ā wth ( ) ā = and for (p ) I a (p) (p) I { ā (p) (p) a = f p {l l } 0 else In the followng we defne wā to be the sogeometrc functon on Ω that s determned by the Bézer coeffcent vector ā = (ā l ) (l) I Let us consder the C -smooth geometrcally contnuous sogeometrc bass functons w for two-patch domans from the prevous secton whch possess the Bézer coeffcent vectors a for the correspondng two-patch domans Clearly w Qa L (Ω) but not all functons w Qa are C -smooth on the whole mult-patch doman Ω Lemma 7 For each two-patch doman Ω (ll ) l l { n} wth l l we compute the Bézer coeffcent vectors a of all patch bass functons and edge bass functons of type A B L and L Then an extended functon w Qa s C -smooth on Ω f and only f the ntal sogeometrc functon on the two-patch doman Ω (ll ) e w Qa Ω (ll ) s one of the patch bass functons (f) (j) and (k) from Fg and 3 or one of the patch bass functons (a)-(e) and (g)-(h) from Fg and 3 for whch all boundary edges of Ω (ll ) stll are the boundary edges of Ω or an edge bass functon of type A or B or an edge bass functons of type L or L wth a support n the vcnty of a boundary vertex of valence m = 3 Proof Ths can be easly verfed by analyzng the smoothness of the resultng functons on Ω We notce that we do not get any C -smooth functon on Ω possessng non-zero Bézer coeffcents a (l) n the vcnty of an nner vertex or a boundary vertex of valence m 4 The followng subsecton explans the constructon of such C -smooth geometrcally contnuous sogeometrc bass functons on Ω 4
v j+ w j+ Ω (j+) v 0 v j Ω (j) a 6 a7 a 8 Ω (j+) v j+ Ω (j) Ω (j ) w j v j a 3 a4 a 5 a 0 a a w j v 0 Fgure 8: Left: An nner vertex v 0 of valence m 3 wth the m neghborng patches Ω (j) n clockwse order around the vertex v 0 Rght: The 6 values a {0 5} of the Bézer coeffcents (blue vertces) of the two-patch doman Ω (j((j+) mod m)) whch correspond to the nner vertex v 0 or to the Bézer control ponts n the one-rng neghboorhood of v 0 (Here the Bézer coeffcents along the common nterface (red edge) are drawn only once for both patches snce ther values are equal) 4 Vertex bass functons The edge bass functons of type L wll be used to generate C -smooth geometrcally contnuous sogeometrc bass functons wth non-zero Bézer coeffcents a (l) n the vcnty of the nner vertces and boundary vertces of valence m 4 We frst descrbe the constructon of these functons n the case of nner vertces and wll later slghtly adapt ths constructon to the boundary vertces For both cases we wll call the resultng C - smooth functons vertex bass functons Let us consder an nner vertex v 0 of the mult-patch doman Ω of valence m 3 and assume wthout loss of generalty that the m neghborng patches Ω (j) j { m} are gven n a clockwse order around the vertex v 0 see Fg 8 (left) For each two patch doman Ω (j((j+) mod m)) j { m} we generate 5 new edge bass functons denoted by type Y -Y 5 Ther values a {0 5} of the possbly non-zero Bézer coeffcents that correspond to the nner vertex v 0 or to the Bézer control ponts n the one-rng neghborhood of v 0 see Fg 4 5 and 8(rght) are determned as follows: Y : a 0 = a = a = a 3 = 0 a 4 = a 5 = 0 Y : a = 0 a 3 = 0 a 5 = 0 The remanng values a 0 a and a 4 are obtaned by satsfyng the condtons w(v 0 ) = 0 and w(v 0 ) = ( 0) T wth respect to the global coordnates for the correspondng sogeometrc functon w Y 3 : a = 0 a 3 = 0 a 5 = 0 The remanng values a 0 a and a 4 are obtaned by satsfyng the condtons w(v 0 ) = 0 and w(v 0 ) = (0 ) T wth respect to the global coordnates for the correspondng sogeometrc functon w Y 4 : a 0 = 0 a = 0 a = 0 a 3 = a 4 = 0 a 5 = 0 Y 5 : a 0 = 0 a = 0 a = 0 a 3 = 0 a 4 = 0 a 5 = 5
The resultng 5 edge bass functons of type Y can be represented as a lnear combnaton of the 5 edge bass functons of type L see Appendx A3 Lemma 8 Let Ω (j((j+) mod m)) j { m} be a two-patch doman wth the common vertex v 0 The edge bass functons of type Y span the same space as the edge bass functons of type L Proof It suffces to show that the bass functons of type Y are lnearly ndependent whch s trvally satsfed due to the selecton of the values a {0 5} of the correspondng Bézer coeffcents Let us denote the Bézer coeffcent vectors of the sogeometrc functons of type Y by a (j) { 5} j { m} Clearly the functons w Qa (j) are C -smooth wthn all sngle patches Ω (l) l { n} but cannot be C -smooth at some common nterfaces of neghborng patches of Ω We nvestgate these nterfaces but frst we defne: Defnton 9 Let w be an sogeometrc functon on Ω wth a Bézer coeffcent vector a We defne D a to be the collectons of those common nterfaces of neghborng patches of Ω where w s not C -smooth e D a = {all patch nterfaces e of Ω w s not C -smooth at e} Lemma 0 Let j { m} The sets D Qa (j) { 5} are gven by D Qa (j) = {e (j(j ) mod m) e ((j+) mod m(j+) mod m) } for { 4} and by D Qa (3j) = {e (j(j ) mod m) } D Qa (5j) = {e ((j+) mod m(j+) mod m) } Proof Ths can be easly checked by analyzng the dfferent functons w Qa (j) { 5} wth respect to C -smoothness across all common nterfaces for We wll generate the vertex bass functons for the nner vertex v 0 by assemblng them from dfferent functons w Qa (j) For ths we frst defne the operator whch wll be used later to construct the Bézer coeffcents of the vertex bass functons Defnton Let ( ã = ã (l) ) (l) I ( and â = â (l) ) (l) I be the Bézer coeffcent vectors of two sogeometrc functons on Ω generates the Bézer coeffcent vector The operator ã â = a wth ( a = a (l) ) (l) I 6
and a (l) = ã (l) â (l) ã (l) undefned f â (l) = 0 f ã (l) = 0 f ã (l) = â (l) else for (l ) I Note that f all coeffcents a (l) are well-defned then the Bézer coeffcent vector a defnes an sogeometrc functon w on Ω In addton the operator s commutatve and assocatve We wll need later the followng lemma Lemma Let ã and â be the Bézer coeffcent vectors of the two sogeometrc functons w and ŵ respectvely both defned on Ω and assume that all Bézer coeffcents a (l) of a = ã â are well-defned If e Dã e Dâ and supp BB ( w) I e supp BB (ŵ) I e then e D a Proof For (l ) I e the Bézer coeffcents a (l) concde wth the Bézer coeffcents â (l) and therefore the functon w a s also C -smooth at e We generate now 3 + m vertex bass functons for the nner vertex v 0 where the Bézer coeffcent vectors a of 3 functons are defned by a = m Qa (j) (4) j= for { 3} and the Bézer coeffcent vectors a of m further functons are defned by a = Qa (j5) Qa ((j+) mod m4) (5) for j { m} The resultng m + 3 vertex bass functons are C -smooth on Ω Lemma 3 All sngle Bézer coeffcents a (l) of the Bézer coeffcent vectors a n (4) and (5) are well defned and the resultng vertex bass functons w a are C -smooth on Ω Proof We frst show that all resultng Bézer coeffcents a (l) of a vertex bass functon are well-defned We observe that the ntersecton of the supports supp BB (w) of any two dfferent nvolved and to the mult-patch doman Ω extended functons w of type Y can only contan Bézer coeffcents that correspond to the vertex v 0 and to the neghborng Bézer control ponts Consequently each resultng Bézer coeffcent a (l) of a vertex bass functon that do not correspond to the vertex v 0 or to a neghborng Bézer control pont s trvally well-defned It remans to be shown that the remanng 4m Bézer coeffcents a (l) of a vertex bass functon are well-defned too Ths s guaranteed by the choce of the values of the correspondng Bézer coeffcents of the nvolved and to the mult-patch doman Ω extended 7
functons of type Y If a Bézer coeffcent a (l) of a vertex bass functons s contaned n the ntersecton of the supports supp BB (w) of any two dfferent nvolved and extended functons w of type Y then the value of ths Bézer coeffcent s equal for both extended functons of type Y To show that a vertex bass functons w wth the Bézer coeffcent vector a s C - smooth on Ω we subsequently apply Lemma and obtan D a = whch concludes the proof We wll also call the frst three vertex bass functons proper vertex functons and the remanng m vertex bass functons twst functons An example of these 8 bass functons for an nner vertex v 0 of valence 5 s vsualzed n Fg 9 proper vertex functons twst functons twst functons Fgure 9: An example of all dfferent vertex bass functons (e 3 proper vertex functons and 5 twst functons) of an nner vertex of valence 5 for d = 4 Slghtly adapted we can construct the vertex bass functons for a boundary vertex of valence m 4 Let us consder such a boundary vertex v 0 of the mult-patch doman Ω and assume wthout loss of generalty that the m neghborng patches Ω (j) j { m } are gven n a clockwse order around the vertex v 0 as vsualzed n Fg 0 Agan we 8
v m Ω (m ) v 0 v w m Ω () v m v w Fgure 0: A boundary vertex v 0 of valence m 4 wth the m neghborng patches Ω (l) l { m } n clockwse order around the vertex v 0 The red edges are the common edges between two neghborng patches and the blue edges are boundary edges of the mult-patch doman Ω compute frst the functons of type Y for all two-patch domans Ω (j(j+)) j { m } and denote the resultng Bézer coeffcent vector of the functon of type Y for the twopatch domans Ω (j(j+)) by a (j) We construct now 3 + (m ) vertex bass functons for the boundary vertex v 0 where the Bézer coeffcent vectors a of 3 functons are defned by m a = Qa (j) (6) for { 3} the Bézer coeffcent vectors a of m 3 functons are defned by j= a = Qa (j5) Qa (j+4) (7) for j { m 3} and the Bézer coeffcent vectors a of functons are only defned by a = Qa (4) and a = Qa (m 5) (8) Lemma 4 All sngle Bézer coeffcents a (l) of the Bézer coeffcent vectors a n (6) (7) and (8) are well defned and the resultng sogeometrc functons w a are C -smooth on Ω Proof Smlar to the proof of Lemma 3 one can show that all sngle Bézer coeffcents a (l) of the resultng Bézer coeffcent vectors a are well defned and that all functons are C - smooth on Ω We wll call agan the frst three vertex bass functons proper vertex functons and the remanng m vertex bass functons twst functons 43 Bass of the space In the prevous two subsectons we have explaned the constructon of dfferent knds of C -smooth geometrcally contnuous sogeometrc functons on Ω Let us now subdvde these functons nto four sets H and nvestgate the cardnalty ν of each resultng set H { 4} The set H conssts of the patch bass functons extended to Ω whch are obtaned for each splne control pont of a geometry mappng G (l) that does not belong to a common 9
nterface wth a neghborng patch and to the neghborng column of splne control ponts For each geometry mappng G (l) the number of these functons depends on the number of common nterfaces of the consdered patch wth ts neghborng patches In detal ths number ν (l) s gven by (d + k(d ))(d + + k(d )) for one nterface (d + k(d )) for two nterfaces wth common vertex ν (l) = (d 3 + k(d ))(d + + k(d )) for two nterfaces wthout common vertex (d 3 + k(d ))(d + k(d )) for three nterfaces and (d 3 + k(d )) for four nterfaces Consequently ν s then obtaned by summng up the resultng numbers of functons for the sngle patches e n ν = ν () = The set H contans for each common nterface of two neghborng patches Ω (l) and Ω (l ) l l { n} wth l l the (d 9) + (d 4)k edge bass functons of type A and B extended to Ω Therefore ν s equal to ν = r ((d 9) + (d 4)k) where r s the number of dfferent such two-patch domans Ω (ll ) The set H 3 contans for each boundary vertex of valence m = 3 the 5 edge bass functons of type L extended to Ω Consequently ν 3 s equal to ν 3 = 5r 3 where r 3 s the number of dfferent vertces of valence m = 3 The set H 4 conssts of the 3+m vertex bass functons for each nner vertex of valence m and of the 3 + (m ) vertex bass functons for each boundary vertex of valence m 4 Therefore ν 4 s gven by ν 4 = (3 + m v ) + (3 + m v ) v:type I vertex (nner) v:type I vertex (boundary) where m v s the valence of the vertex v The unon of these sets e H = 4 = H s a bass of V Theorem 5 The set H s a bass of the space V of C -smooth geometrcally contnuous sogeometrc functons defned on Ω proof We wll show that each functon w V can be unquely represented as a lnear combnaton of the functons of H e w(x) = 4 ν λ jzj(x) x Ω = j= 0
where z j are the functons of the set H and λ j are the correspondng coeffcents Clearly each functon w V possesses a unque Bézer representaton Let us consder two dfferent types of sets of Bézer coeffcents denoted by A (l) and A (ll ) where and A (l) = {a (l) (l ) I e (ll ) for l { n} wth l l} A (ll ) = {a (l) (l ) I e (ll )} {a (l ) (l ) I e (ll )} Consequently the unon of all sets A (l) and A (ll ) s an overlappng decomposton of the set A of all Bézer coeffcents a (l) of w In the followng we consder only non-empty sets A (l) and A (ll ) For a set F of functons on Ω we denote by F A (l) and F A (ll ) the set of those non-zero functons whch are obtaned by restrctng the functons of F to the Bézer coeffcents a (l) from the set A (l) and A (ll ) respectvely Our constructon of the functons of H ensures that each set of functon H A (l) and H A (ll ) s a bass of V A (l) and V A (ll ) respectvely In case of each set A (l) the set of resultng functons conssts of ν (l) lnearly ndependent patch bass functons whch span the space V A (l) and n case of each set A (ll ) the set of resultng functons conssts of (d+)+(d 4)k lnearly ndependent edge bass functons whch span the space V A (ll ) compare Lemma 4 and 8 Therefore a functon w V has a unque representaton w Ā = 4 ν λ j z j Ā (9) = for each set Ā equal to a set A(l) or A (ll ) (or more precsely unque coeffcents λ j for the non-zero functons zj Ā ) It remans to be shown that each coeffcent λ j has to be unquely determned for all representatons (9) where the correspondng functon zj s not the zero functon For the coeffcents λ j wth { 3} ths s trvally satsfed snce each functon z j Ā { 3} s a non-zero functon exactly for one set Ā (A (l) for = and A (ll ) for = 3) To show that the coeffcents λ 4 j are also unquely determned we use the fact that each functon zj 4 s generated n such a way that each Bézer coeffcent a(l) of zj 4 s unquely determned for all functons z4 A (ll ) wth a (l) A (ll ) and that the functons z4 A (l) are zero for all sets A (l) Proposton 6 The dmenson of the space V of C -smooth geometrcally contnuous sogeometrc functons defned on Ω s equal to j= dm V = 4 ν = proof Snce the cardnalty of each set H { 4} s gven by the values ν we drectly obtan the dmenson of V as the sum of these values
Note that a functon w V belong to the space H (Ω) snce they are globally C - smooth and pecewse C -smooth Ths provdes us a possblty to use them to solve amongst others the bharmonc equaton over dfferent mult-patch domans whch wll be demonstrated on the bass of some examples n the followng secton 5 Examples We present several examples of usng C -smooth geometrcally contnuous sogeometrc functons on dfferent mult-patch domans to perform L approxmaton to solve Posson s equaton or the bharmonc equaton over these domans For all three applcatons we consder the same model problems as descrbed n [0 Secton 4-43] for two-patch domans and use the theren explaned sogeometrc approaches for solvng the partcular problems The frst example numercally analyzes the approxmaton power of C -smooth geometrcally contnuous sogeometrc functons on mult-patch domans Example 7 We consder three dfferent mult-patch domans Ω see Fg (frst row) whch consst of three four and fve quadrlateral patches Ω (l) respectvely The correspondng geometry mappngs G (l) are blnear parameterzatons whch are represented as Bézer patches of degree (d d) for d {3 4} For each mult-patch doman Ω the resultng geometry mappngs G (l) determne a space of C -smooth geometrcally contnuous sogeometrc functons on Ω Analogous to the numercal results for two-patch domans n [0] we generate a sequence of nested spaces of C -smooth geometrcally contnuous sogeometrc functons By consderng the B-splnes patches G (l) S d λ d Sd λ d λ N 0 we get a refned space of C -smooth geometrcally contnuous sogeometrc functons on Ω The resultng space wll be denoted agan by V h where h = O( λ ) and λ s the level of refnement We construct a bass for these spaces by usng the method descrbed n the prevous sectons Thereby we slghtly modfy the constructon of some functons for low levels e for λ = 0 for d = 3 and for λ = 0 for d = 4 snce our general method does not work for these levels More precsely we construct for these levels a bass of the nullspace of H or of H of the homogeneous system () or () respectvely n a dfferent way By addtonally requrng that the functons w V h have to satsfy the homogeneous boundary condtons w (x) = 0 on Ω (0) and w (x) = w (x) = 0 on Ω () n we can generate a sequence of nested spaces whch can be used for solvng Posson s equaton and the bharmonc equaton respectvely These spaces wll be denoted agan by V 00h and V 0h respectvely A bass of these spaces can be obtaned from the correspondng bass of V h by removng those boundary bass functons w whch do not satsfy the approprate boundary condtons Note that for d = 3 the coarsest level for the space V 0h start wth
λ = snce there do not exst C -smooth geometrcally contnuous sogeometrc functons whch satsfy the boundary condtons () for λ = 0 We use the resultng C -smooth geometrcally contnuous sogeometrc functons to solve three dfferent applcatons on the three mult-patch domans (e three- four- and fve patch doman) Frst we approxmate for all three domans the functon z(x x ) = cos(x ) sn(x ) shown n Fg (second row) Second we numercally solve Posson s equaton over the three domans The dfferent rght-hand sde functons f of Posson s equaton (cf [0 Eq (6)]) are obtaned by dfferentatng the functons u(x x ) = C n = e (x x ) where C s some dfferent constant for each doman and e are the lnes defned by the boundary edges of the domans Thrd we numercally solve the bharmonc equatons over the three domans where the dfferent rght-hand sde functons f of the bharmonc equaton (cf [0 Eq (8)]) are obtaned by dfferentatng the correspondng functons ũ = u The functons u and ũ satsfy the homogeneous boundary condtons (0) and () respectvely and are vsualzed n Fg (thrd row) and Fg (fourth row) respectvely We present the resultng relatve H -errors ( = 0 for L -approxmaton = 0 for Posson s equaton and = 0 for bharmonc equaton) for the dfferent level λ of refnement n Fg For all three applcatons the numercal results ndcate an optmal convergence rate of O(h d+ ) n the assocated H -norms = 0 The followng example explores two possbltes from [0 Secton 34] to deal wth more general domans Example 8 As frst example we consder a doman wth hole consstng of four quadrlateral patches see Fg 3 (frst row) We have relaxed the requrement of blnear patches by modfyng some control ponts whch do not affect the edge bass functons and vertex bass functons Ths allows us to construct the shown computatonal doman wth a curved boundary around the hole Smlar to Ex 7 we solve Posson s equaton wth the rght-hand sde functon f obtaned by dfferentatng the functon u(x x ) = 000 (x 5)(x + 5)(x 5)(x + 5)(x + x 9 4 ) and wth the non-homogeneous Drchlet boundary condtons derved from ths functon u Fg 3 shows the exact soluton u (second row) and the resultng relatve H -errors = 0 (thrd row) by usng bquartc functons for solvng Posson s equaton on a refned mesh The second computatonal doman whch s modeled after a car part see Fg 3 (frst row) was constructed by followng the second generalzaton from [0 Secton 34] Frst we choose a blnear reference geometry Ḡ of the target geometry (e the desred car part) where the geometry mappng Ḡ conssts of fve patches Ḡ(l) S33 4 S4 33 Second 3
Three-patch doman Computatonal domans Ω e Four-patch doman e e Fve-patch doman e e e3 e3 e8 e e4 e0 e6 e5 e3 e7 e4 e4 e9 e6 e5 e8 e6 e5 e7 Graphs of functons z on Ω Graphs of functons u on Ω wth the property u = 0 on Ω Graphs of functons u on Ω wth the property u = u n = 0 on Ω Fgure : Three dfferent mult-patch domans as computatonal domans (frst row) wth the exact solutons z for L -approxmaton (second row) the exact solutons u for solvng Posson s equaton (thrd row) and the exact solutons u for solvng the bharmonc equatons (fourth row) we compute for ths reference geometry a bass of the correspondng space V of C -smooth geometrcally contnuous sogeometrc functons Thrd we ft the target geometry by usng L -approxmaton for each coordnate functon to acheve a geometry mappng G V V whch descrbes the car part and where the common nterfaces do not resemble solely 4
d = 3 three patch four patch fve patch Log H 0 error 00 0000 4 Log H 0 error 00 0000 0 6 4 Log H error 00 0000 0 6 3 0 6 0 8 0 8 0 3 4 5 0 3 4 5 0 3 4 5 Level Level Level L approxmaton H 0 -error Posson s equaton H 0 -error Posson s equaton H -error Log H 0 error 00 0000 4 Log H error 00 0000 3 Log H error 00 0000 0 6 0 6 0 6 0 3 4 5 Level 0 3 4 5 Level 0 3 4 5 Bharmonc equaton H 0 -error Bharmonc equaton H -error Bharmonc equaton H -error d = 4 Level 00 00 00 Log H 0 error 0000 0 6 0 8 5 Log H 0 error 0000 0 6 0 8 0 0 5 Log H error 0000 0 6 0 8 0 0 4 0 0 3 4 5 0 3 4 5 0 3 4 5 Level Level Level L approxmaton H 0 -error Posson s equaton H 0 -error Posson s equaton H -error 00 00 00 Log H 0 error 0000 0 6 5 Log H error 0000 0 6 4 Log H error 0000 0 6 3 0 8 0 8 0 8 0 3 4 5 0 3 4 5 0 3 4 5 Level Level Level Bharmonc equaton H 0 -error Bharmonc equaton H -error Bharmonc equaton H -error Fgure : The relatve H -errors by performng L approxmaton ( = 0) solvng Posson s equaton ( = 0 ) and solvng the bharmonc equaton ( = 0 ) over the three dfferent mult-patch domans gven n Fg The estmated convergence rates (e the dyadc logarthm of the rato of two consecutve relatve errors) are demonstrated wth the help of the correspondng slope trangles 5
straght lnes Ths constructon of the computatonal doman ensures that the assocated bass functons possess the same Bézer coeffcents as the bass functons for the reference geometry Smlar to the doman wth hole we solve Posson s equaton wth the rght-hand sde f obtaned by the exact soluton u(x x ) = 5000 (4 x )(4+ 5x 3 x )( 3 x )(x +5)( 7 x )(x +(x + 7 ) 4)(x 5) see Fg 3(second row) and wth the correspondng non-homogeneous Drchlet boundary condtons The resultng relatve H -errors = 0 by usng bquartc functons for solvng Posson s equaton over the ftted doman are vsualzed n Fg 3 (thrd row) 6 Concluson We constructed a bass of the space V of bcubc and bquartc C -smooth geometrcally contnuous sogeometrc functons on blnearly parameterzed mult-patch domans Ω R The resultng bass functons are based on C -smooth functons from the two patch case (cf [0]) and can be easly obtaned by means of explct formulas for the Bézer coeffcents of ther splne segments We also presented numercal experments of usng the obtaned bcubc and bquartc C -smooth sogeometrc functons for dfferent applcatons whch showed optmal rates of convergence The paper leaves several open ssues for possble future research One such topc conssts n the theoretcal nvestgaton of the approxmaton power of the space of C -smooth geometrcally contnuous sogeometrc functons defned on blnear mult-patch domans We expect them to possess the full aproxmaton power snce the space of these functons contans cubc and quartc polynomals on the doman Another challengng topc s the constructon of geometrcally contnuous sogeometrc functons of hgher degree and/or smoothness We have already demonstrated the potental of the geometrcally contnuous sogeometrc functons by solvng the bharmonc equaton The exploraton of further possble applcatons whch requre these functons of hgher smoothness s of nterest too Fnally the extenson of the framework to the threedmensonal case wll be consdered Appendx A Edge bass functons In [0] we specfed for d = 4 smple explct formulas for the Bézer coeffcents of the edge bass functons for blnearly parameterzed two-patch domans Ω In ths secton we recall these formulas n a slghtly more compact way and present smlar ones for d = 3 Moreover we add for both degrees two further possble choces of the Bézer coeffcents of the edge bass functons on the boundares of the common nterface of the two-patch doman whch are obtaned as lnear combnatons of the ntal boundary functons These functons were called n [0] bass functon of the second knd 6
Doman wth hole Computatonal domans Car part Exact solutons Relatve H -errors = 0 #ptchs #fcts rel H 0 -error rel H -error 64 644 00963e-04 404480e-04 #ptchs #fcts rel H 0 -error rel H -error 80 808 6457694e-04 44707e-03 Fgure 3: Two dfferent computatonal domans (frst row) wth the exact solutons u for solvng Posson s equatons (second row) and the relatve H -errors = 0 (thrd row) 7
Appendx A Bass functons of type A B and L We present for both degrees d {3 4} a possble choce of the (d + ) + (d 4)k edge bass functons for a two patch doman Ω As explaned n Secton 3 only Bézer coeffcents a (l) that correspond to the Bézer control ponts of the common nterface of the two-patch doman or to the neghborng column of Bézer control ponts of the correspondng splne segments can be non-zero for these functons Consequently ther supports can be only contaned n splne segments along the common nterface more precsely on one to four pars of neghborng splne segments Due to the sze and locaton of these supports we can classfy the functons nto dfferent types (A B and L) wth possble subtypes see Fg 4 and 5 and Table A In the followng we specfy smple explct formulas for the non-zero values a of the correspondng Bézer coeffcents of the dfferent types of functons wth respect to the local geometry gven by the patch vertces (0 0) (0 )/(0 3) (p 0 q 0 ) (p q ) (p q ) (p 3 q 3 ) vsualzed n Fg 4 and Fg 5 for d = 3 and d = 4 respectvely (Compare [0 Appendx] for d = 4) To present the formulas n a short and compact way we wll use the followng auxlary terms: and α j = p q j p j q β j = p j p γ j = α j + β j δ j = α j + β j ε j = α j + 3β j η j = α j + 4β j for j {0 3} wth < j Table A: The dfferent types of functons (A B and L) for d {3 4} possess supports that are contaned n one to four pars of neghborng splne segments along the common nterface see Fg 4 and 5 Whereas the functons of type L are only defned on the lower and upper boundary of the common nterface the functons of type A and B are defned on each such possble pars of neghborng splne segments along the common nterface Type # pars of segments locaton subtypes total number d = 3 A 3 pars on each such pars A k B 4 pars on each such pars B k L pars both boundares of common nterface L L 5 0 d = 4 A pars on each such pars A A 3 3k B 3 pars on each such pars B k L pars both boundares of common nterface L L 5 0 Bass functons of type A B and L for d = 3 Type A: a 6 = p 0 3p +p 3 a 8 = p 3p +p 3 a 9 = p 0+p 3p +p 3 a = p +p 3 3p +p 3 a = 3p 0+p 3p +p 3 a 4 = a 5 = p 0+3p 3p +p 3 a 7 = p +3p 3 3p +p 3 a 8 = p 0+p 3p +p 3 a 0 = p +p 3 3p +p 3 a = p 3p +p 3 a 3 = p 3 3p +p 3 8