Boundary Orthogonality in Elliptic Grid Generation

Transcription

1 6 Boundary Orthoonality in Elliptic Grid Generation Ahed Khaayseh Andrew Kuprat C Wayne Mastin 61 Introduction 6 Boundary Orthoonality for Planar Grids Neuann Orthoonality Dirichlet Orthoonality 63 Boundary Orthoonality for Surface Grids Neuann Orthoonality Dirichlet Orthoonality 64 Boundary Orthoonality for Volue Grids Neuann Orthoonality Dirichlet Orthoonality 65 Suary 61 Introduction Eperience in the field of coputational siulation has shown that rid quality in ters of soothness and orthoonality affects the accuracy of nuerical solutions It has been pointed out by Thopson et al [8] that skewness increases the truncation error in nuerical differentiation Especially critical in any applications is orthoonality or near-orthoonality of a coputational rid near the boundaries of the rid If the boundary does not correspond to a physical boundary in the siulation orthoonality can still be iportant to ensure a sooth transition of rid lines between the rid and the adacent rid presued to be across the nonphysical boundary If the rid boundary corresponds to a physical boundary then orthoonality ay be necessary near the boundary to reduce truncation errors occurrin in the siulation of boundary layer phenoena such as will be present in a Navier Stokes siulation In this case fine spacin near the boundary ay also be necessary to accurately resolve the boundary phenoena In elliptic rid eneration an initial rid (assued to be alebraically coputed usin transfinite interpolation of specified boundary data) is relaed iteratively to satisfy a quasi-linear elliptic syste of partial differential equations (PDEs) The ost popular ethod the Thopson Thaes Mastin (TTM) ethod incorporates user-specifiable control functions in the syste of PDEs If the control functions are not used (ie set to zero) then the rid produced will be soother than the initial rid and rid foldin (possibly present in the initial rid) ay be alleviated However nonuse of control functions in eneral leads to nonorthoonality and loss of rid point spacin near the boundaries Iposition of boundary orthoonality can be effected in two different ways In Neuann orthoonality no control functions are used but boundary rid points are allowed to slide alon the boundaries until boundary orthoonality is achieved and the elliptic syste has iterated to converence This ethod which is taken up in this chapter is appropriate for nonphysical (internal) rid boundaries since rid spacin present in the initial boundary distribution is usually not aintained Previous ethods for

2 ipleentin Neuann orthoonality have relied on a Newton iteration ethod to locate the orthoonal proection of an adacent interior rid point onto the boundary The Neuann orthoonality ethod presented here uses a Taylor series to ove boundary points to achieve approiate orthoonality Thus there is no need for inner iterations to copute boundary rid point positions In Dirichlet orthoonality also taken up in this chapter control functions (called orthoonal control functions) are used to enforce orthoonality near the boundary while the initial boundary rid point distribution is not disturbed Early papers usin this approach were written by Sorenson [3] and Thoas and Middlecoff [6] In Sorenson s approach the control functions are assued to be of a particular eponential for Orthoonality and a specified spacin of the first rid line off the boundary are achieved by updatin the control functions durin iterations of the elliptic syste Thopson [7] presents a siilar technique for updatin the orthoonal control functions This technique evaluates the control functions on the boundary and interpolates for interior values A user-specified rid spacin noral to the boundary is required The technique of Spekreise [5] autoatically constructs control functions solely fro the specified boundary data without eplicit user-specification of rid spacin noral to the boundary Throuh construction of an interediate paraetric doain by arclenth interpolation of the specified boundary point distribution the technique ensures accurate transission of the boundary point distribution throuhout the final orthoonal rid Applications to planar and surface rids are iven in [5] In this chapter we present a technique siilar to [7] for updatin of orthoonal control functions durin elliptic iteration However our technique does not require eplicit specification of rid spacin noral to the boundary but as in [5] eploys an interpolation of boundary values to supply the necessary inforation However unlike [5] this interpolation is not constructed in an auiliary paraetric doain but is siply the initial alebraic rid constructed usin transfinite interpolation Althouh this rid is very likely skewed at the boundary the first interior coordinate surface is assued to be correctly positioned in relation to the boundary which is enouh to ive us the required noral spacin inforation for iterative calculation of the control functions Ghost points eterior to the boundary are constructed fro the interior coordinate surface leadin to potentially soother rids since central differencin can now be eployed at the boundary in the direction noral to the boundary Since our technique does not eploy the auiliary paraetric doain of [5] theory and ipleentation are sipler The ipleentation of this technique for the case of volue rids is straihtforward and indeed we present an eaple We ention here that Soni [] presents another ethod of constructin an orthoonal rid by derivin spacin inforation fro the initial alebraic rid However unlike our ethod which uses host points at the boundary this ethod does not ephasize capture of rid spacin inforation at the boundary Instead the alebraic rid influences the rid spacin of the elliptic rid in a unifor way throuhout the doain With no special treatent of spacin at the boundary considerable chanes in noral rid spacin can occur durin the course of elliptic iteration This ay be unacceptable in applications where the ost nuerically challenin physics occurs at the boundaries In Section 6 we present Neuann and Dirichlet orthoonality as applied to planar rid eneration We also present a control function blendin technique that allows for preservation of interior rid point spacin in addition to preservation of boundary rid point spacin In Section 63 we present analoous techniques for construction of orthoonal surface rids and in Section 64 we present the analoous techniques for volue rids To deonstrate these techniques eaples are presented in these sections We present our conclusions in Section 65 6 Boundary Orthoonality for Planar Grids We assue an initial appin (η) = ((η) y(η)) fro coputational space [ ] [ n] to the bounded physical doain Ω IR Here n are positive inteers and rid lines are the lines = i or η = with i or n bein inteers The initial appin (η) is usually obtained usin alebraic rid eneration ethods such as linear transfinite interpolation

3 Given the initial appin a eneral ethod for constructin curvilinear structured rids is based on partial differential equations (see Thopson et al [8]) The coordinate functions (η) and y(η) are iteratively relaed until they becoe solutions of the followin quasi-linear elliptic syste: where ( + P 1 11 ) + ( η ηη + Q ) η = 11 1 = = + y = = + yy η η η = = + y η η η η (61) The control functions P and Q control the distribution of rid points Usin P = Q = tends to enerate a rid with a unifor spacin Often there is a need to concentrate points in a certain area of the rid such as alon particular boundary seents in this case it is necessary to derive appropriate values for the control functions To coplete the atheatical specification of syste Eq 61 boundary conditions at the four boundaries ust be iven (These are the = = η = and η = n or left riht botto and top boundaries) We assue the orthoonality condition η = on = and η= n (6) We assue that the initial alebraic rid neither satisfies Eq 61 nor Eq 6 Nevertheless the initial rid ay possess rid point density inforation that should be present in the final rid If the alebraic rid possesses iportant rid density inforation such as concentration of rid points in the vicinity of certain boundaries then it is necessary to invoke Dirichlet orthoonality wherein we use the freedo of specifyin the control functions P Q in such a fashion as to allow satisfaction of Eq 61 Eq 6 without chanin the initial boundary point distribution at all and without reatly chanin the interior rid point distribution If however the alebraic rid does not possess relevant rid density inforation (such as ay be the case when the rid is an interior block that does not border any physical boundary) we attept to solve Eq 61 Eq 6 usin the siplest assuption P = Q = Since we are not usin the derees of freedo afforded by specifyin the control functions we are forced to allow the boundary points to slide to allow satisfaction of Eq 61 Eq 6 This is Neuann orthoonality The coposite case of havin soe boundaries treated usin Dirichlet orthoonality soe treated usin Neuann orthoonality and soe boundaries left untreated will be clear after our treatent of the pure Neuann and Dirichlet cases 61 Neuann Orthoonality As is typical let us assue that the boundary seents are iven to be paraetric curves (e B- splines) If we set the control functions P Q to zero then it will be necessary to slide the boundary nodes alon the paraetric curves in order to satisfy Eq 61 Eq 6 A standard discretization of our syste is central differencin in the and η directions The syste is then applied to the interior nodes to solve for i = ( i y i ) usin an iterative ethod With reard to the ipleentation of boundary conditions suppose alon the boundary seents = and = the variables and y can be epressed in ters of a paraeter u as = (u) and y = y(u) For the = and = boundaries let ( η ) i denote the central difference (1/( i+1 i 1 )) alon the boundaries (i = or i = ) Usin one-sided differencin for Eq 6 is discretized as ( i+ 1 i ) ( η) = alon = i = (63)

4 FIGURE 61 Chane in when boundary point is repositioned in Neuann orthoonality ( i i1 ) ( ) = alon η = i= (64) Solution of Eq 63 or Eq 64 for i = ( i y i ) in effect causes the slidin of i alon the boundary so that the rid seent between i and its neihbor on the first interior coordinate curve ( = 1 or = 1) is orthoonal to the boundary curve (See Fiure 61) To solve for i the old paraeter value u is used to solve for the new u to copute the new i Usin the Taylor epansion of (u) about u to ive i = ( u) ( u) + u( u)( u (65) substitutin Eq 65 in Eq 63 iplies that ( ) ( ) η u 1 ( ) u= u + u ( η) u( ) (66) to ive i = (u) alon the boundary = Whereas substitutin Eq 65 in Eq 64 iplies that ( ) ( ) η u 1 ( ) u= u + u ( η) u( ) (67) to ive i = (u) alon the boundary = Consider net the case where the boundaries are η = and η = n Orthoonality Eq 6 with central differencin in the direction and one-sided differencin in the η direction iplies ( ) ( ) i 1 u ( ) i u= u + u ( ) u( ) i (68)

5 FIGURE 6 An alebraic planar rid on a bicubic eoetry which ives i = (u) alon the boundary η = and ( ) ( ) in 1 u ( ) in u= u + u ( ) u( ) in (69) to ive i = (u) alon the boundary η = n These boundary condition equations are to be evaluated for each cycle in the course of the iterative procedure Note that a periodic boundary condition is used in the case of doubly connected reions Also note that durin the relaation process uards ust be used to prevent a iven boundary point fro overtakin its neihbors when slidin alon the boundaries Indeed near obtuse corners there is a tendency for rid points to try to slide alon the boundary curves past the corners in order to satisfy the orthoonality condition An appropriate uard would be to liit oveent of each rid point so that its distance fro its two boundary-curve neihbors is reduced by at ost 5% on a iven iteration down to a user-specified iniu lenth δ in physical space As an application of Neuann orthoonality consider Fiure 6 which is an initial alebraic planar rid on a bicubic eoetry The esh is hihly nonorthoonal at certain points alon the boundaries and it possesses an undesirable concentration of points in the interior of the rid In fact there is foldin of the alebraic rid in this central reion Fiure 63 shows an elliptically soothed rid usin Neuann orthoonality The rid is clearly seen to be sooth boundary-orthoonal and no loner folds in the interior For certain applications this rid ay be entirely acceptable However if the botto boundary of the rid corresponded to a physical boundary then the results of Fiure 63 iht be deeed unacceptable This is because althouh orthoonality has been established rid point distribution (both alon the boundary and noral to the boundary) has been sinificantly altered In this case the Dirichlet orthoonality technique will have to be eployed 6 Dirichlet Orthoonality The above discussion shows how orthoonality can be iposed without use of control functions by slidin rid points alon the boundary Orthoonality can also be iposed by adustin the control

6 FIGURE 63 An elliptic planar rid on a bicubic eoetry with Neuann orthoonality functions near the boundary and keepin the boundary points fied This approach was oriinally developed by Sorenson [3] for iposin boundary orthoonality in two diensions Sorenson [4] and Thopson [7] have etended this approach to three diensions However as entioned in the introduction our approach does not require user specification of rid spacin noral to the boundary Instead our technique autoatically derives noral rid spacin data fro the initial alebraic rid Assuin boundary orthoonality Eq 6 substitution of the inner product of and η into Eq 61 yields the followin two equations for the control functions on the boundaries: P = ηη 11 Q = η ηη η 11 (61) These control functions are called the orthoonal control functions because they were derived usin orthoonality considerations They are evaluated at the boundaries and interpolated to the interior usin linear transfinite interpolation These functions need to be updated at every iteration durin solution of the elliptic syste We now o into detail on how we evaluate the quantities necessary in order to copute P and Q on the boundary usin Eq 61 Suppose we are at the left boundary = but not at the corners (η and η n) The derivatives η ηη and the spacin = η are deterined usin centered difference forulas fro the boundary point distribution and do not chane However the 11 and ters are not deterined by the boundary distribution Additional inforation aountin to the desired rid spacin noral to the boundary ust be supplied A convenient way to infer the noral boundary spacin fro the initial alebraic rid is to assue that the position of the first interior rid line off the boundary is correct Indeed near the boundary it is usually the case that all that is desired of the elliptic iteration is for it to swin the intersectin rid lines so that they intersect the boundaries orthoonally without chanin the positions of the rid lines parallel to the boundary This is shown raphically in Fiure 64 where we see a rid point fro the first interior rid line swun alon the rid line to the position where orthoonality is established The

7 FIGURE 64 Proection of interior alebraic rid point to orthoonal position effect of forcin all the rid points to swin over in this fashion would thus be to establish boundary orthoonality but still leave the alebraic interior rid line unchaned The siilarity of Fiure 61 and Fiure 64 sees to indicate that this process is analoous to and hence ust as natural as the process of slidin the boundary points in the Neuann orthoonality approach with zero control functions Unfortunately this precedin approach entails the direct specification of the positions of the first interior layer of rid points off the boundary This is not perissible for a couple of reasons First since they are adacent to two different boundaries the points n 1 and 1n 1 have contradictory definitions for their placeent Second and ore iportantly the direct specification of the first layer of interior boundary points toether with the elliptic solution for the positions of the deeper interior rid points can lead to an undesirable kinky transition between the directly placed points and the elliptically solvedfor points (This kinkiness is due to the fact that a perfectly sooth boundary-orthoonal rid will probably ehibit soe sall deree of nonorthoonality as soon as one leaves the boundary even as close to the boundary as the first interior line Hence forcin the rid points on the first interior line to be eactly orthoonal to the boundary cannot lead to the soothest possible boundary-orthoonal rid) Nevertheless our natural approach for derivin rid spacin inforation fro the alebraic rid can be odified in a siple way as depicted in Fiure 65 Here the orthoonally-placed interior point is reflected an equal distance across the boundary curve to for a host point Repeatedly done this procedure in effect fors an eterior curve of host points that is the reflection of the first (alebraic) rid line across the boundary curve The host points are coputed at the beinnin of the iteration and do not chane They are eployed in the calculation of the noral second derivative at the boundary and the noral spacin 11 off the boundary; the fiedness of the host points assures that the noral spacin is not lost durin the course of iteration as it soeties is in the Neuann orthoonality approach Conversely all of the interior rid points are free to chane throuhout the course of the iteration and so soothness of the rid is not coproised More precisely aain at the left = boundary let ( η ) denote the centrally differenced derivative 1/( +1 1 ) Let ( o ) denote the one-sided derivative 1 evaluated on the initial alebraic rid Then condition Eq 6 iplies that if a is the unit vector noral to the boundary then a = ( η) y η y η η = + y η η

8 FIGURE 65 Reflection of orthoonalized interior rid point to for eternal host point Now the condition fro Fiure 64 is = Ρ ( a ) (611) where P a = aa T is the orthoonal proection onto the one-diensional subspace spanned by the unit vector a Thus we obtain = a( a )= ( yη η) ( y η y η ) (61) Finally the reflection operation of Fiure 65 iplies that the fied host point location should be iven by 1 = ( ) This can also be viewed as a first-order Taylor epansion involvin the orthoonal derivative ( ) : with = 1 The orthoonal derivative ( ) is coputed in Eq 61 usin only data fro the boundary and the alebraic rid Now in Eq 61 the control function evaluation at the boundary the second derivative is coputed usin a centered difference approiation involvin a host point a boundary point and an iteratively updated interior point The etric coefficient 11 describin spacin noral to the boundary is coputed usin Eq 61 and is iven by 1 = + ( ) ( ) = ( ) ( ) 11

9 Finally note that the value for ( ) used in Eq 61 is not the fied value iven by Eq 61 but is the iteratively updated one-sided difference forula iven by Evaluation of quantities at the = boundary is siilar Note however that the host point locations are iven by where ( ) is evaluated in Eq 61 which is also valid for this boundary On the botto and top boundaries η = and η = n it is now the derivatives η ηη and the spacin 11 that are evaluated usin the fied boundary data usin central differences Usin siilar reasonin to the left and riht boundary case we obtain that for the botto boundary the host point location is fied to be where we use ( ) = i 1 + = +( ) i 1 i = ( ) η i ( ) + = y η 11 ( y η y η ) (613) Here ( y η η ) 11 is evaluated usin central differencin of the boundary data and ( o η y o η) represents a one-sided derivative i1 i evaluated on the initial alebraic rid The etric coefficient ( ) i = ( η ) i ( η ) i is now coputed usin Eq 613 and ηη is coputed usin a host point a boundary point and an iteratively updated interior point The value of ( η ) i used in Eq 61 is not the fied value iven in Eq 613 but is the iteratively updated one-sided difference forula iven by ( η) = i i 1 i Finally the upper η = n boundary is siilar and we note that the host-point locations are iven by in 1 in η in + = +( ) with ( η ) in evaluated usin Eq 613 Quantities for the four corner points n and n are coputed soewhat differently in that no orthoonality considerations or host points are used Indeed the values η ηη 11 are all evaluated once usin one-sided difference forulas that use the specified boundary values and do not chane durin the course of iteration We foreo iposition of orthoonality at the corners because at the corners conforality is ore iportant than orthoonality In other words orthoonality at the corners should be sacrificed in order to ensure that the resultin rid does not spill over the physical boundaries in the neihborhood of the corners For the case of hihly obtuse or hihly acute corners it ay in fact be necessary to rela orthoonality in the reions that are within several rid lines of the

10 corners One way to do this is to construct host points near the corners with the orthoonal proection operation Eq 611 oitted (ie constructed by siple etrapolation) and to use a blend of these host points and the host points derived usin the orthoonality assuption To further ensure that the elliptic syste iterations do not cause rid foldin near the boundaries uards ay be eployed siilar to those entioned in the previous section on Neuann orthoonality In practice however we have found these to be unnecessary for Dirichlet orthoonality 61 Blendin of Orthoonal and Initial Control Functions The orthoonal control functions in the interior of the rid are interpolated fro the boundaries usin linear transfinite interpolation and updated durin the iterative solution of the elliptic syste If the initial alebraic rid is to be used only to infer correct spacin at the boundaries then it is sufficient to use these orthoonal control functions in the elliptic iteration However note that the orthoonal control functions do not incorporate inforation fro the alebraic rid beyond the first interior rid line Thus if it is desired to aintain the entire initial interior point distribution then at each iteration the orthoonal control functions ust be soothly blended with control functions that represent the rid density inforation in the whole alebraic rid These latter control functions we refer to as initial control function and their coputation is now described The elliptic syste Eq 61 can be solved siultaneously at each point of the alebraic rid for the two functions P and Q by solvin the followin linear syste: y y η η P R1 = Q R (614) where R1 = 1 11 and η ηη R = y y y 1 η 11 ηη The derivatives here are represented by central differences ecept at the boundaries where one-sided difference forulas ust be used This produces control functions that will reproduce the alebraic rid fro the elliptic syste solution in a sinle iteration Thus evaluation of the control functions in this anner would be of trivial interest ecept when these control functions are soothed before bein used in the elliptic eneration syste This soothin is done by replacin the control function at each point with the averae of the nearest neihbors alon one or ore coordinate lines However we note that the P control function controls spacin in the -direction and the Q control function controls spacin in the η-direction Since it is desired that rid spacin noral to the boundaries be preserved between the initial alebraic rid and the elliptically soothed rid we cannot allow soothin of the P control function alon -coordinate lines or soothin of the Q control function alon η-coordinate lines This leaves us with the followin soothin iteration where soothin takes place only alon allowed coordinate lines: 1 Pi = ( Pi + 1+ Pi 1) 1 Qi = ( Qi+ 1 + Qi 1 ) (615) Soothin of control functions is done for a sall nuber of iterations

11 FIGURE 66 An elliptic planar rid on a bicubic eoetry with Dirichlet orthoonality Finally by blendin the soothed initial control functions toether with orthoonal control functions we will produce control functions that will result in preservation of rid density inforation throuhout the rid alon with boundary orthoonality An appropriate blendin function for this purpose is i in i= 1 δ n n b e where δ is soe positive nuber chosen such that the eponential decays soothly fro unity on the boundary to nearly zero in the interior δ can be considered to be the characteristic lenth of the decay of the blendin function in the (η ) doain So for eaple if δ = 5 the orthoonal control functions heavily influence a reion consistin of 5% of rid lines which are nearest to each boundary Now the new blended values of the control functions are coputed as follows: Pi ( ) = b P( i ) + ( 1b ) P( i ) i o i I Qi ( ) = b Q( i ) + ( 1b ) Q( i ) i o i I (616) where P O and Q O are the orthoonal control functions fro Eq 61 P I and Q 1 are the soothed initial control functions coputed usin Eqs 614 and 615 As an application of Dirichlet orthoonality in Fiure 66 we show the results of soothin the alebraic rid of Fiure 6 usin orthoonal control functions only Like the rid produced usin Neuann orthoonality the rid is sooth boundary-orthoonal and no loner folds in the interior However unlike the rid of Fiure 63 we see that the rid of Fiure 66 preserves the rid point density inforation of the alebraic rid at the boundaries The effect of soothin near the boundaries has been essentially to slide nodes alon the coordinate lines parallel to the boundaries without affectin the spacin between the coordinate lines noral to the boundary We note that if the user for soe reason wished to preserve the interior clusterin of rid points in the alebraic rid then the above schee iven for blendin initial control functions with orthoonal control functions would have to be slihtly odified This is because the fact that the alebraic rid is actually folded in the interior akes the evaluation of the initial control functions usin Eq 614 illdefined This is easily reedied by evaluatin the initial control functions usin Eq 614 at the boundaries

12 only usin one-sided derivatives and then definin the over the whole esh usin transfinite interpolation Since there is no foldin of the alebraic rid at the boundaries this is well-defined (The interpolated initial control functions will reflect the rid density inforation in the interior of the initial rid because the interior rid point distribution of the initial rid was coputed usin the sae process transfinite interpolation of boundary data) Then we proceed as above soothin the initial control functions and blendin the with the orthoonal control functions Finally we note that if the alebraic initial rid possesses foldin at the boundary then usin data fro the alebraic rid to evaluate either the initial control functions or the orthoonal control functions at the boundary will not work In this case one could reect the alebraic rid entirely and anually specify rid density inforation at the boundary This would however defeat the purpose of our approach which is to siplify the rid eneration process by readin rid density inforation off of the alebraic rid Instead we suest that in this case the eoetry be subdivided into patches sufficiently sall so that the alebraic initial rids on these patches do not possess rid foldin at the boundaries 63 Boundary Orthoonality for Surface Grids Now we turn our attention to applyin the sae principles of the previous section to the case of surface rids Our surface is assued to be defined as a appin (uv): IR IR 3 The (uv) space is the paraetric space which we conveniently take to be [1] [1] The paraetric variables are theselves taken to be functions of the coputational variables η which live in the usual [ ] [ n] doain Thus = ( yz ) = ( uv ( ) yuv ( ) zuv ( )) and ( uv ) = ( u( η ) v( η )) (617) The appin (uv) and its derivatives u v etc are assued to be known and evaluatable at reasonable cost It is the ai of surface rid eneration to provide a ood appin (u(η ) v(η )) so that the coposite appin (u(η ) v(η )) has desirable features such as boundary orthoonality and an acceptable distribution of rid points A eneral ethod for constructin curvilinear structured surface rids is based on partial differential equations (see Khaayseh and Mastin [1] Warsi [9] and Chapter 9) The paraetric variables u and v are solutions of the followin quasi-linear elliptic syste: where ( u + Pu ) u + ( u + Qu ) = J u and 1 η 11 ηη η ( v + Pv ) v + ( v + Qv ) = J v 1 η 11 ηη η = u + u v + v = uu + ( uv + uv) + vv 1 11 η 1 η η η = u + u v + v 11 η 1 η η η 1 = u J u J v J v= J 11 1 v J u J = = = 11 u u 1 u v v v J = J = u v u v and 11 1 = () uv uv 1 η η (618) (619) (6)

13 Note that if u y v z then 11 = 1 1 = = 1 J = 1 and u = v = akin Eqs identical to the hooeneous elliptic syste for two-diensional rid eneration Eq 61 presented in the previous section As in the previous section the control functions P and Q can be set to zero and Neuann orthoonality can be iposed by slidin points alon the left riht botto and top boundaries These four boundaries are respectively ( v(η )) (1v(η )) (u( )) (u(n) 1) in paraetric space which are apped to the boundaries (v) (1v) (u) and (u1) in physical space Of course orthoonality ust be established in physical space As before if there is a need to respect the rid point concentration in the initial alebraic rid we ipleent Dirichlet orthoonality derivin appropriate values for P and Q 631 Neuann Orthoonality We require the condition of orthoonality in physical space: η = on = and η= n (61) Sybolically this is identical to Eq 6 but here we understand that is a coposite function Eq 617 which takes on values in IR 3 Epandin Eq 61 usin the chain rule yields the equation uu + vv + ( uv + uv) = 11 1 η η η η This orthoonality condition is used to forulate derivative boundary conditions for the elliptic syste If the left and riht boundary curves u = and u = 1 are considered we have u η = and the orthoonality condition reduces to v + 1u = (6) Siilarly alon the botto and top curves v = and v = 1 v = and orthoonality is iposed by 11uη + 1vη = (63) When solvin the elliptic syste Eq 6 deterines the values of v on the boundary seents u = and u = 1 and Eq 63 deterines the values of u on the boundary seents v = and v = 1 To ipleent this nuerically we use forward differencin on the boundaries u = and v = and backward differencin on the boundaries u = 1 and v = 1 to copute the new values for u i and v i : 1 v = ( u1 u ) + v1 < < n 1 v = ( u u 1 ) + v 1 1 ui = ( vi 1 vi ) + ui 1 11 < i< 1 uin = ( vin vin 1) + u in 1 11

14 FIGURE 67 An alebraic surface rid on a bicubic eoetry Since the boundary points are peritted to float with the solution as a eans to achieve orthoonality (Fiure 63) the values of i ust of course be reevaluated after each cycle usin the definition of the eoetry (uv) Also as in the last section uards ust be used to prevent a iven boundary point fro overtakin its neihbors when slidin alon the boundaries Fiure 67 shows an initial alebraic rid on a bicubic surface eoetry The rid was obtained usin linear transfinite interpolation and is the startin iterate for our elliptic soothin Clearly the initial rid is not orthoonal at the boundaries where orthoonality is often desired especially for Navier Stokes coputation Fiure 68 shows the elliptically soothed surface rid on the sae eoetry Neuann orthoonality was applied to allow the boundary points to float so that the rid is orthoonal on the boundary Sinificant chanes in boundary rid spacin occur near soe of the corners 63 Dirichlet Orthoonality For the case of Dirichlet orthoonality for surface rids we essentially follow the sae technique as that used in Section 6 Epressions for the control function P and Q are derived at the boundary usin the assuption of orthoonality and then to facilitate evaluation of these epressions host points are placed orthoonally off the boundary with noral spacin derived fro the initial rid (Fiure 65) We rewrite the elliptic syste Eqs in vector for: ( u + Pu ) u + ( u + Qu ) = J u 1 η 11 ηη η (64)

15 FIGURE 68 An elliptic surface rid on a bicubic eoetry with Neuann orthoonality where u = (uv) For u 1 = (u 1 v 1 ) and u = (u v ) define u o u = uu + ( uv + uv)+ vv Note that u 1 u =_ u T 1 G u which is the inner product in paraetric space induced by the etric tensor 11 1 G = Orthoonality in this inner product is equivalent to orthoonality in physical space 1 Suppose that the rid lines are orthoonal ie η = u u η vanishes Applyin u to Eq 64 yields ( u ou + Pu ou ) + u ou = J u ou 11 ηη In the sae anner applyin u to Eq 64 yields the followin equation for the second control function on the boundaries: u ou + ( u ou + Qu ou ) = J u ou η 11 ηη η η η η The values of P and Q can be deterined fro the coplete epansion of the above equations as follows:

16 P= J Q= J uu + vv + ( uv + vu ) uηηu + v v + 1 u v + v u ηη ( ηη ηη ) 11u u + v v + 1( u v + v u ) 11 η 11 uu + vv + ( uv + vu ) η 1 η η 11 11uηηuη + vηηvη + 1( uηηvη + vηηuη ) 11u u + v v + 1 u v + v u η η ( η η) 11 (65) As in the previous section these control functions derived usin orthoonality considerations are called orthoonal control functions are interpolated to the interior usin linear transfinite interpolation and are updated at every iteration durin solution of the elliptic syste We now o into soe detail about the eact way these control functions are evaluated at the boundary The ters 11 1 u v are evaluated at the boundary fro the eoetry definition (u) and do not chane durin the course of iteration At non-corner points on the left u = and riht u = 1 boundaries as in Section 6 we have that the derivatives u η u ηη and the spacin = η are deterined usin centered difference forulas fro the boundary point distribution and do not chane The noral derivative u off the boundary is coputed usin one-sided difference forulas that involve one boundary point and the adacent interior point Dependence on the interior point iplies that this value ust be updated durin the course of iteration Also updated durin the course of iteration is u which is coputed usin a centered difference forula involvin an interior point a boundary point and a host point u 1 or u +1 off the boundary The host point value is derived once at the beinnin of iteration by doin an analysis of the correct rid spacin off the boundary and by iposin physical orthoonality We now derive the location of the host points at the left u = boundary Siilar to Section 6 let (u η ) denote the centrally differenced derivative 1/(u +1 u 1 ) and let (u o ) denote the initial one-sided derivative u o 1 u where u o 1 u 1 on the initial alebraic rid and u is the unchanin boundary value Now to define u used in the definition of host points and rid spacin off the boundary we aain ake the assuption of Fiure 64 that in physical space is the proection of o (= u u o + v v o ) onto the direction a physically orthoonal to the boundary This is equivalent to Eq 611 or in ters of the rid spacin off the boundary this is equivalent to = (66) Cobinin Eq 66 with the paraetric space orthoonality condition Eq 6 we obtain

17 u u 1 u = (67) The rid point locations are then defined by the reflection operation in physical space shown in Fiure 65 or equivalently the first order Taylor epansion in paraetric space involvin the orthoonal boundary derivative: u 1 = u + ( u ) o = u ( u ) This leads to host point locations at the left boundary iven by and u u u 1 = ( ) = u ( u ) = u u u =u The last quantity required for coputation of the control functions at the u = boundary usin Eq 65 is the rid spacin orthoonal to the boundary 11 = orthoonal to the boundary We have that ( ) 1 1 v 1 v v 1 = v + ( u ) 1 = v + u 1 u 1 = v + u1 = ( ) ( ) = u + u v + v Substitutin Eq 67 into this forula we easily obtain 11 = ( ) u (68) where 11 1 Since the boundary points are fied this quantity is constant at each boundary point throuhout the iteration

18 Coputation of the control functions at the u = 1 boundary is done in the sae way as that for the u = boundary We note that Eq 67 is still valid and usin the first-order Taylor epansion the host point locations are iven by and u + = u + ( u ) 1 = u + ( u ) u u u + 1 = +( ) = u +( u ) = u + u u = u 1 1 ( ) v+ 1 v v 1 = v ( u ) 1 = v u u 1 = v 1u 1 = +( ) ( ) 1 ( ) Also note that the epression for rid spacin off the left boundary Eq 68 is still valid for the riht boundary For the non-corner botto and top boundaries we have that u u 11 = are coputed once usin centered difference forulas u η is coputed repeatedly usin a one-sided difference forula and u ηη is coputed repeatedly usin a centered difference forula involvin a host point value u i 1 or u in+1 that is coputed once usin rid spacin and physical orthoonality considerations In fact analoous to the orthoonal boundary derivative Eq 67 which is valid for the left and riht boundary we can derive with siilar reasonin that for the botto and top boundaries we should have 1 u η = v η v η 11 where v o η is a one-sided difference coputed usin the initial alebraic rid This corresponds to the orthoonal proection in physical space shown in Fiure 65 By siilar reasonin as that used for the left and riht boundaries this leads to fictitious boundary point locations 1 ui 1 = ui + vi v =v i 1 i

19 on the botto boundary and 1 u 1 = u 1v v in + in in 1 11 = v in + 1 in 1 ( ) for the top boundary Siilar to Eq 68 the rid spacin off the botto and top boundaries is iven by Usin the sae rationale as used in Section 6 quantities for the four corner points = ( ) 11 v η ( )( ) ( n) ( n) are coputed without orthoonality considerations or host points The values u u u η u ηη 11 are all evaluated once usin one-sided difference forulas usin the specified boundary values and do not chane durin the course of iteration If blendin of orthoonal and initial control functions is desired to aintain the initial interior point distribution we follow the sae prora followed in Section 6 which is to copute the initial control functions that would reproduce the alebraic rid sooth the and then blend the with orthoonal control functions usin Eq 616 However now the blendin is done in the paraetric doain so that the blendin function is iven by 1 ui vi ( 1ui ) ( 1vi ) δ b = e u v 1 i i i and δ can be considered to be the characteristic lenth of the decay of the blendin function in the (uv)-paraetric doain Fiure 69 ehibits an elliptically soothed orthoonal rid on the surface eoetry depicted in Fiure 67 The elliptic rid was enerated usin control functions coputed fro an initial alebraic rid that had been blended with orthoonal control functions coputed on the boundaries usin Eq 65 We see that initial spacin is preserved throuhout the rid and the rid near the boundaries is alost perfectly orthoonal 64 Boundary Orthoonality for Volue Grids The elliptic syste of partial differential equations for eneratin curvilinear coordinates in volues is iven by (see Chapter 4 and Thopson [7]) 3 3 n nn n+ Pn n= = 1 n=1 3 n= 1 (69) where i i = 1 3 are the curvilinear coordinates and = ( 1 3 ) is the vector of physical coordinates The construction of a three-diensional rid on a iven eoetry in physical space ( 1 3 ) ay be viewed as construction of a appin ( ) to physical space fro a convenient coputational space ( 1 3 ) which we take to be the brick [ 1 in 1 a] [ in a] [ 3 in 3 a]

20 FIGURE 69 An elliptic surface rid on a bicubic eoetry with Dirichlet orthoonality The P n are the three control functions that serve to control the spacin and the orientation of the rid lines in the field The eleents of the contravariant etric tensor n and the eleents of the covariant etric tensor n are epressed by = n n = n n Moreover the contravariant and covariant etrics are atri inverses of each other and are related as n = ( ik l il k ) ( i )( nkl ) cyclic where the square of the Jacobian of the appin ( ) is iven by = det [ ]= ) n 1 3 The elliptic eneration syste in Eq 69 is the one used in soothin volue rids The first step in solvin the syste in Eq 69 is to enerate rids on the si surfaces boundin the physical subreion Then the initial alebraic volue rid is enerated between si faces usin transfinite interpolation The initial rid is considered to be the initial solution to the elliptic syste Eq 69 and the faces of the rid provide boundary conditions for ( 1 3 ) The concept of volue orthoonality proceeds in the sae spirit as the surface case (

21 641 Neuann Orthoonality The first technique of achievin boundary orthoonality requires ovin the physical coordinates on the surface (face) (or ) so that the orthoonality conditions S l S l in a l l = = n (63) are satisfied with (l n) cyclic Assue for the oent that our obective is to ove the node ik on the surface S l in represented paraetrically by (u v ) to a new location (uv) on the surface To deterine the position of the new node we need to solve for u and v Denotin the node off the surface by usin one-sided differencin we can write on S l l in Thus the orthoonality conditions in Eq 63 are epressed as ( ) = ( ) n = (631) Taylor epansion of (uv) about (u v ) ives ( uv ) o + o ( uu)+ o ( vv) u v (63) where o = (u v ) o u = u (u v ) and o v = v (u v ) Substitutin Eq 63 in the syste Eq 631 yields ( ) ( ) ( ) ( ) o o o u ( u u)+ v ( v v)= ( ) o o o u ( u u )+ v ( v v )=( ) n n n (633) Usin the chain rule of differentiation on and η o o = u + u u v o o = u + u n u n n v and substitutin in Eq 633 we obtain the linear syste Aw = b

22 FIGURE 61 A cross section of an alebraic volue rid eterior to a booster where A = ( ) + ( ) ( ) + ( ) ( ) + ( ) ( ) + ( ) o o o o o o o o u v u v u u u v u v v v o o o o o o o o u v u v u u n u v n u u n u v n w1 u u w = = w v v o o o o ( ) uu ( ) vv b = o o o o ( ) uu n ( ) vv n Solvin the above syste for w 1 and w we then copute u = u + w 1 and v = v + w Finally we copute new coordinates (uv) to et the location of the rid point on the surface Fiure 61 shows the cross section of an alebraic volue rid on a booster eoetry Clearly the rid is hihly nonorthoonal at various points on the booster surface Fiure 611 shows the sae rid after elliptic soothin with iposed Neuann orthoonality The rid points successfully oved alon the booster surface to achieve orthoonality but with the unfortunate side effect of soe deradation of the initial boundary node distribution 64 Dirichlet Orthoonality As in the case of planar or surface rids an alternative way of constructin orthoonal volue coordinates is to keep the surface nodes fied and to allow the interior values in the array ik to ove This type of orthoonality can be enforced usin the control functions P 1 P and P 3 coputed on the surfaces An iterative solution procedure for the deterination of the three control functions for the eneral three-diensional case was initially developed by Sorenson [4] Epressions for the control functions on a coordinate surface on which l is constant can be obtained fro the two coordinate lines lyin on the surface ie the lines on which and n vary (ln) bein cyclic The developent presented here follows that of Thopson [7] S 1 in

23 FIGURE 611 A cross section of an elliptically soothed volue rid eterior to a booster with iposed Neuann orthoonality at the surface The inner product of l and n with Eq 69 and usin the orthoonality condition Eq 63 yields the followin three equations for P l P and P n on the surfaces l = const P l 1 = ll l l l ( ) l nn + n nn n nn n (634) P = - 1 ll n nn n l l n n nn nn n + ( nn n n n n ) (635) P n = - 1 ll n n l l n n nn n + ( nn n n n n ) (636) Proceedin as in the planar case we construct host points for the evaluation of boundary we define the unit vector orthoonal to the boundary l l At the = in

24 a l = l n n The fied derivative orthoonal to the boundary is then defined by o l l = P o a l where is the one-sided derivative obtained fro the initial alebraic rid and P a = aa T is the orthoonal proection onto the one-diensional subspace by the unit vector a Thus we obtain l o ( l ) = a a = So for the l in surface (ie i = ) our host point locations would be iven by o ( n l ) n n 1 k = k ( l ) (637) where ( l ) k was coputed usin Eq 637 and is fied since it depends only on fied boundary data and data fro the initial rid For the l a surface (ie i = ) our host point locations would be iven by 1 k k k + = +( ) l k aain usin the fied orthoonal derivative Eq 637 The host points for the in a 3 in and 3 a surfaces are siilarly coputed Note that for coputed by Eq 637 we have that l = o n l n l This eans that the host points will for cells with the sae volue as the first layer of cells in the alebraic rid This is epected because as in Fiure 65 for the planar case the host points have been constructed to for a surface that is the reflection of the first interior coordinate surface and so cell volue ust be conserved Of course the host points will for cells which are orthoonal to the boundary while the first layer of cells fro the alebraic rid are probably not Now siilar to the planar case the ters in Eqs are coputed usin a host point l l a boundary point and an iteratively updated interior point while ll = l coputed usin Eq 637 and is fied for the whole iteration The l ters appearin in Eq 634 are evaluated usin one-sided differencin involvin a boundary point and an iteratively updated interior point The reainin ters in Eqs are coputed usin central differencin on the fied boundary data At the 8 corners and the 1 edes the ters in Eqs are evaluated usin all one-sided differences (for the corners) or a cobination of one-sided and central differences (for the edes) As in the planar case no orthoonality inforation is incorporated into the calculation of the orthoonal control functions at these

25 FIGURE 61 A cross section of an elliptically soothed volue rid eterior to a booster with iposed Dirichlet orthoonality at the surface points that are at the boundaries of the boundary surfaces Finally the orthoonal control functions coputed usin Eqs are interpolated to the interior by linear transfinite interpolation If blendin of orthoonal and initial control functions is desired to aintain the initial interior point distribution we follow the sae prora followed in Section 6 which is to copute the initial control functions that would reproduce the alebraic rid sooth the and then blend the with orthoonal control functions usin Eq 616 However now the blendin is done on a brick rather than on a rectanle and so the blendin function is iven by where b ik ui kvi kwi k ui k vi k wi k = e 1 ( )( )( δ ) u v ik ik w ik 1 i in = 1 1 a in in = a a in 3 k in = 3 3 in As in the planar case δ is soe positive nuber that can be considered to be the characteristic lenth of the decay of the blendin function in the coputational doain In Fiure 61 we show the cross section of the rid of Fiure 61 after elliptic soothin usin Dirichlet orthoonality Clearly the rid is orthoonal at the surface and the effect of soothin has been to slide nodes alon the coordinate surfaces parallel to the boundary without affectin the spacin of the coordinate surfaces noral to the boundary

26 65 Suary A coprehensive developent has been presented for the ipleentation of boundary orthoonality in elliptic rid eneration for planar doains surfaces and volues For each of these three cases two techniques have been presented One technique Neuann orthoonality involves slidin points alon the boundaries to establish orthoonality Our ipleentation of the other technique Dirichlet orthoonality involves slidin points alon the first interior coordinate surface of the initial rid and then reflectin the across the boundary to for the host points which will be used in the coputation of the orthoonal control functions in the elliptic syste The forer technique is appropriate for interior boundaries between different rid patches while the latter technique is appropriate for physical boundaries where rid point density ust be preserved under elliptic iteration These techniques can be applied at all or selected boundaries In the case of Dirichlet orthoonality orthoonal control functions can be blended with initial control functions if preservation of interior rid point distribution is desired These orthoonality techniques have proven to be reliable and efficient in the construction of planar surface and volue rids References 1 Khaayseh A and Mastin C W Coputational conforal appin for surface rid eneration J Coput Phys pp Soni BK Elliptic rid eneration syste: control functions revisited-i Appl Math Coput pp Sorenson RL A coputer prora to enerate two-diensional rids about airfoils and other shapes by the use of Poisson s equations NASA TM NASA Aes Research Center Sorenson RL Three-diensional elliptic rid eneration about fihter aircraft for zonal finite difference coputations AIAA AIAA 4th Aerospace Science Conference Reno NV Spekreise SP Elliptic rid eneration based on laplace equations and alebraic transforations J Coput Phys pp Thoas PD and Middlecoff JF Direct control of the rid point distribution in eshes enerated by elliptic equations AIAA J pp Thopson JF A eneral three-diensional elliptic rid eneration syste on a coposite block structure Cop Meth Appl Mech and En pp Thopson JF Warsi ZUA and Mastin CW Nuerical Grid Generation: Foundations and Applications North-Holland New York Warsi ZUA Nuerical rid eneration in arbitrary surfaces throuh a second-order differential eoetric odel J Coput Phys pp 8 96