Structured Grid Generation Via Constraint on Displacement of Internal Nodes

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1 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 79 Structured Grd Generaton Va Constrant on Dsplacement of Internal Nodes Al Ashrafzadeh, Razeh Jalalabad Abstract Structured grd generaton methods have been used for many years to dscretze the soluton doman n flud dynamcs smulatons. Varous dfferental methods have been employed for ths purpose but the tradtonal method, whch transforms a logcal grd n the computatonal doman to a boundary-ftted grd n the physcal doman, s employng a set of poson equatons. In ths paper a new grd generaton method s ntroduced n whch the dfferences between coordnates of boundary nodes of a smple ntal grd and fnal grd are used as the boundary condton of a set of poson equatons known as grd generaton equatons. These Poson equatons are solved on the ntal grd to obtan the dsplacement of nodal coordnates and construct the fnal grd. Two dmensonal grd generaton examples are fnally presented and the grd qualtes are compared wth the results of an avalable dfferental grd generaton method. The underlyng deas are clearly extendble to three-dmensonal problems as well. Index Term Dsplacement of Boundary Nods, Intal Grd, Structured Grd Generaton, Poson Equatons. I. INTRODUCTION Numercal soluton of equatons, whch descrbe fluds flows n practcal problems, s a usual approach n Computatonal Fluds Dynamcs (CFD). Ths approach requres powerful dscretzaton methods to dscrete the dfferental terms n equatons and the physcal doman. Grd generaton strateges are used to dscrete physcal domans and n the structured or unstructured generated grds [1,], a set of elements are generated throughout the doman regardng the boundares shapes. Accuracy and effcency of numercal soluton of equatons are strongly affected by the employed grd generaton methods [3,4]. In D structured grd generaton a physcal doman s correspondent to a logcal doman as shown n fgure 1. The ntersecton of the coordnate lnes and s known as the grd pont, n the physcal doman (fgure 1b) [5]. Tradtonal methods for mappng the unt square onto the physcal doman are algebrac and dfferental grd generaton methods. In algebrac grd generaton methods the poston of Al Ashrafzadeh s wth Khaeh Nasr Toos Unversty of Technology, Tehran, Iran. (e-mal: ashrafzadeh@kntu.ac.r). Razeh Jalalabad ss wth Khaeh Nasr Toos Unversty of Technology,Tehran, Iran. (Correspondng auth, phone: ; e- mal: ralalabad@yahoo.com). nods n logcal doman s changed to ther new poston n physcal doman usually by usng an nterpolaton technque [6]. But n dfferental grd generaton methods, some constrants are used as grd generaton equatons and when solved, the logcal grd s mapped on to the physcal grd [7]. Fg. 1. The Cartesan logcal grd and the Physcal grd. A set of wdely used dfferental grd generaton equatons proposed by Thompson, Thames and Mastn (TTM) [8] s: P, (1a) xx yy xx yy Q, (1b) P and Q, the control functons whch are used for better control of the dstrbuton of grd lnes, need to be known at all nodal ponts before the soluton. Several methods have been proposed to calculate these functons some of whch use some 1-dmensonal or mult-dmensonal nterpolaton technques to calculate these functons n the doman regardng to ther values on the boundares [9,1,11]. Boundary values are calculated usng (1a) and (1b) and pavng layers n physcal doman. Pavng layers are two layers of grd generated adacent to the boundares usng a smple algebrac method as shown n Fg.. In 1D nterpolaton technques, the followng onedmensonal formulas can be used to calculate the nternal values of the source functons usng ther correspondng boundary values: P, C PS 1 C PN Q, C Q 1 C Q W E In mult-dmensonal nterpolaton technques, the nternal values of the source functons are obtaned n the doman usng a mult-dmensonal set of formulas or through the soluton of a Drchlet boundary value problem IJBAS-IJENS August 11 IJENS

2 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 8 In order to study the presented method, the underlyng theory s dscussed frst. Then the descrpton of the method and the governng equatons are presented. The paper s followed by a secton on some grd generaton examples, a dscusson on the results and concluson secton. Fg.. Pavng layers used to calculate nodal values of P and Q near the boundares of the physcal doman. Although some schemes have been proposed to solve (1a) and (1b) drectly [11], the usual method to solve these equatons s nvertng equatons analytcally, lnearzng and then solvng them numercally n the logcal doman. When nverted, the system of equatons to be solved s: g x g1x g11x J x P x Q (3a) g y g y g y J y P y Q 1 11 (3b) The most usual approach n the orthogonal grd generaton, as another robust method n dfferental grd generaton, s a method proposed by Ryskn and Leal [1]. Based on the assumptons of contnuty and orthogonalty of the coordnate lnes and by mposng the orthogonalty condton, g x x y y, the followng grd 1 generaton equatons are obtaned: x 1 x f (4a) f y 1 y f (4b) f f s the dstorton factor and s defned as follows: f x y g (5) g11 x y Calculaton of the nodal values of dstorton factor n the doman s a maor step n the orthogonal grd generaton and s dscussed n a number of publcatons [1, 13, 14]. Most commonly, f s calculated frst at the boundares and then nterpolated nto the doman. The dea underlned n the above two classcal methods, s mappng logcal doman onto the physcal doman by two Laplace equatons. These grd generaton equatons are n fact constrants on the mappng functons x(, ), y(, ) and are solved on the logcal grd. In the new structured grd generaton method presented here, two lnear dfferental equatons are used to constrant the boundary dsplacement of an ntal grd nstead of mappng functons x (, ), y(, ). In the mult-dmensonal nterpolaton technque appled, two Poson equatons are solved to obtan the dsplacement of nodal coordnates n the physcal doman. II. THE PROPOSED GRID GENERATION METHOD A. Grd Generaton Procedure Usng smple algebrac grd generaton technques, an ntal grd, wth the nodal coordnates x, y, can be generated for a doman as shown n Fg.3. Ths ntal smple grd s located n the physcal x, y space. The dea s to fnd a way to dsplace the boundares to conform them to the boundares of target geometry n physcal doman (Fg.3b). Obvously the dsplacement vectors whch connect the ntal boundary nodes to the target boundary nodes can be calculated, so a multdmensonal nterpolaton technque s employed to fnd the dsplacements of the nternal nodes. Here two ellptc equatons are used as grd generaton equatons whch nterpolate the boundary dsplacements (nterpolants) nto the doman to generate the physcal grd. Fg. 3. A smple ntal grd and the physcal doman. By usng smple algebrac methods, several ntal grds can be generated for any doman. The shape of ntal grd can affect the computatonal cost and the method of solvng grd generaton equatons. Several possble ntal grds generated for physcal domans are ntroduced next and the dscretzaton scheme s presented afterwards. B. The Intal Grd The ntal grd shown n Fg. 3a s a smple grd wth all the smple features of the logcal grd, and s an approprate grd for a Fnte Dfference Method (FDM) soluton to the grd generaton equatons. Employng the nformaton from geometry, more approprate grds can be generated. Some of possble ntal grds for one sample geometry, here called partally adapted grds, are shown n Fg. 4. These ntal grds, respect some of the features of the target geometry, but the dstorted quadrlateral cells, may not be approprate for FDM soluton to the governng equatons. So, a more powerful dscretzaton method such as FEM or FVM s needed to solve the dfferental grd generaton equatons on a partally adapted ntal grd IJBAS-IJENS August 11 IJENS

3 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 81 Fg. 4. Partally adapted ntal grds. Adapted to the corner ponts, adapted to one boundary, adapted to two boundares and adapted to three boundares. As shown n Fg. 4, the ntal grd can share only the corner ponts wth the target geometry or both the corner ponts and some boundares. An FEM or FVM dscretzaton method can be appled to solve the dfferental grd generaton equatons for all ntal grds. These schemes can also be applcable for some specal geometres n whch a smple, not adapted ntal grd may have dstorted quadrlateral cells. Such a doman s shown n Fg. 5. Fg. 5. Two knds of smple, not adapted ntal grd for a specal geometry Usng FEM or FVM dscretzaton method to solve the dfferental grd generaton equatons, the user can solve the grd generaton problem n mult steps. In ths approach an ncrement of boundary dsplacement s used as boundary condton and an ntermedate target doman s generated n each step. Each ntermedate grd serves as the ntal grd for the next target grd and has dstorted quadrlateral cells. In the fnal step, the target doman s the target geometry. Several target domans for a smple geometry s shown n Fg. 6. As t wll be shown n the result secton, usng ths technque help us to avod foldng n complex geometres and generate a smoother grd. In ths paper an Element-Based Fnte Volume method has been used for the soluton of grd generaton equatons. Fg. 6. Usng mult step technque n a grd generaton problem Algebrac methods can be used as a robust and fast method to generate smple or partally adapted grds. In order to generate smple ntal grd, the corners of fnal geometry s connected wth straght lnes, but n partally adapted grd, the user choose some of the boundares from fnal doman as adapted boundares and generate other boundares by connectng the correspondng corners wth straght lnes. After generatng the boundares of ntal doman, the ntal grd s generated wth Transfnte Interpolaton Technque (TFI) [6]. The TFI formulaton used here for generatng s as follows: R, C, N R, N C,1R,1 C1, R1, CM, RM, (6) C R C R C R C R 1, N 1, N M, N M, N 1,1 1,1 M,1 M,1 In ths formula the coordnate of node, s generated by usng the coordnates of boundary nodes n a (M N) grd. C. Numercal Soluton of Grd generaton Equatons The equatons used to nterpolate the boundary dsplacements nto the doman, n order to obtan the coordnates of nternal nodal ponts, are two poson equatons as follows: x x x P, (7a) x y y x y y Q, (7b) y The boundary condtons for these equatons are the dsplacements calculated on the boundares. So for partally adapted grds (as shown n Fg. 4b, c, d), x, y for boundares whch should be dsplaced to conform to the target geometry and x, y for adapted boundares. P and Q are the functons used for better control of the dstrbuton of grd lnes lke n (1a) and (1b). These IJBAS-IJENS August 11 IJENS

4 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 8 parameters are calculated usng (7a) and (7b) and the pavng layers generated on the boundares n physcal doman (Fg. ), and then nterpolated nto the doman usng TFI technque. In the Element-Based Fnte Volume method used here for the soluton of (7a) and (7b), each node n the D physcal doman s consdered n a control volume as shown n Fg. 7 [15]. In ths fgure the sold lnes are the ntal grd lnes and dashed lnes show the control volume around nod P. Defnng x y q ( x), q ( y) (8) The ntegral of (7a) over the control volume wth volume V and surface S, s q dv q ds (9) V S The surface S conssts of a number of panels, and P s an ntegraton pont located at the center of each panel. Fg. 7. A D element-based fnte volume grd arrangement. The ntegrals n (9) are approxmated as follows: q p ds q p S F S p p where S p s the area of each panel contanng P and p p (1) F can p be thought of as a generalzed flow across the panel P drven by f. The scalar quantty F at the ntegraton pont p s functon of the element local coordnates and the fnal algebrac equaton usng the algebrac constrant on namng convecton n Fg. 7 s: C ( x) C ( x) C ( x) P C N SW P C ( x) N ( x) SW W S C NE W C ( x) S ( x) E C NE NW C E ( x) SE NW ( x) SE F n the p (11) Equaton (11) s a nne-pont computatonal molecule whch constrans the x n node P. Applyng a smlar dervaton on (7b), another algebrac equaton relatng y, and for each nteror node s generated. These equatons can be wrtten n the followng matrx forms: A1 ( x) b1 (1a) A y b (1b) The nodal values of ( x), y, and prescrbed for adacent boundary nodes, are used as boundary condtons of these equatons. These lnear algebrac sets of equatons can be solved by any drect or teratve lnear solvers. III. RESULTS AND DISCUSSION A number of grd generaton examples solved by TTM method and the new ellptc method proposed here are presented n ths secton. Four geometres are chosen here and a course ( ) grd s generated n two of them n order to vsualze the detals of performance of all methods better. Fner ( 1 1 ) grds are generated n two other geometres. The proposed grd generaton equatons are solved by FDM method n one step and also by FVM solver through mult steps and wth two dfferent ntal grds. In order to compare the grd qualty, skewness has been chosen as a parameter to measure the qualty of grds. Skewness of a cell whch s between and 1 measures the devaton from the orthogonalty of the coordnate lnes. The computatonal cost n ellptc grd generaton depends mostly on the cost of the soluton of the grd generaton equatons. Both TTM equatons and the proposed equatons are lnear poson equatons and so the computatonal cost s smlar. Fg. 8 shows the dscretzed boundares and the ( ) grd generated by TTM and new proposed equatons n frst doman. The boundary nodes of physcal doman are represented by * sgns and ntal grds for each solver s shown n Fg.8d,8g,8. The grd generated by TTM solver and FDM solver are smlar. FVM solver wth smple ntal grd generates a dfferent smoother grd but as shown n Fg. 8k and 8l, by usng FVM solver n mult steps wth the partally adapted ntal grd shown n Fg. 8, a smoother grd s generated. As there are 1 cells along each coordnate lne n the grds, each qualty measure dagram shows the relevant skewness quantty for all 1 cells on a three-dmensonal plot contanng 1 1 data ponts. Fg. 9 shows the second doman n whch TTM method generates a smooth grd but FDM mplementaton of proposed equatons generates a folded grd. The ntal grd for FDM solver and smple ntal grd for FVM solver are smlar for ths geometry. As t s shown n Fg. 8g to 8l, the partally adapted ntal grd, help FVM solver to generate a smoother grd. Agan the skewness parameter s shown for all 1 cells on a three-dmensonal plot. In Fg. 1 a fner ( 1 1) grd s generated n thrd sample doman. FVM solver has been used only for mult step soluton for ths doman and ntal grd for FDM and FVM soluton are smlar for ths geometry. As shown n three-dmensonal skewness plot whch contans data ponts n the doman, FVM solver generates a smoother grd compared wth FDM and TTM solver. Fg. 11 shows the ( 1 1) grds generated around an arfol. The nodes around the arfol are dstrbuted unformly and ntal grds are shown around half of the arfol and FVM solver has been used only for mult step soluton. The skewness parameter s agan shown for all 4 cells on a threedmensonal plot. As shown n Fg. 11d, h and l, TTM and FDM solver generate smoother grds for ths geometry IJBAS-IJENS August 11 IJENS

5 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 83 (e) (f) (g) (h) () () (k) (l) Fg. 8. The Physcal doman, TTM grd,qualty measures for TTM grd, the ntal grd for FDM one step soluton,the grd obtaned by the proposed method va FDM (e), qualty measures for FDM grd (f), a smple ntal grd (g), the grd obtaned by the proposed method va FVM for smple ntal grd (h), qualty measures for FVM soluton of equatons on smple ntal grd (), a partally adapted ntal grd (), the grd obtaned by the proposed method va FVM for partally adapted ntal grd (k), qualty measures for FVM soluton of equatons on partally adapted ntal grd (l) IJBAS-IJENS August 11 IJENS

6 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 84 (e) (f) (g) (h) () () (k) (l) Fg. 9. The Physcal doman, TTM grd,qualty measures for TTM grd, the ntal grd for FDM one step soluton,the grd obtaned by the proposed method va FDM (e), qualty measures for FDM grd (f), a smple ntal grd (g), the grd obtaned by the proposed method va FVM for smple ntal grd (h), qualty measures for FVM soluton of equatons on smple ntal grd (), a partally adapted ntal grd (), the grd obtaned by the proposed method va FVM for partally adapted ntal grd (k), qualty measures for FVM soluton of equatons on partally adapted ntal grd (l) IJBAS-IJENS August 11 IJENS

7 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 85 (e) (f) (g) (h) () () (k) (l) Fg. 1. The Physcal doman, TTM grd, a larger vew of a secton of north boundary, qualty measures for TTM grd, the ntal grd for FDM one step soluton(e),the grd obtaned by the proposed method va FDM (f), a larger vew of a secton of north boundary (g), qualty measures for FDM grd (h), a partally adapted ntal grd (), the grd obtaned by the proposed method va FVM for partally adapted ntal grd(), a larger vew of a secton of north boundary (k), qualty measures for FVM soluton of equatons on partally adapted ntal grd (l) IJBAS-IJENS August 11 IJENS

8 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 86 (e) (f) (g) (h) () () (k) (l) Fg.11. The Physcal doman, TTM grd, a larger vew the grd around arfol, qualty measures for TTM grd, the ntal grd for FDM one step soluton(e),the grd obtaned by the proposed method va FDM (f), a larger vew the grd around arfol (g), qualty measures for FDM grd (h), a partally adapted ntal grd (), the grd obtaned by the proposed method va FVM for partally adapted ntal grd(), a larger vew the grd around arfol (k), qualty measures for FVM soluton of equatons on partally adapted ntal grd (l). IV. CONCLUSION In ths paper, after revewng the two most commonly used classcal methods of ellptc grd generaton, a new ellptc grd generaton method was proposed. In ths method the general dea was smlar to the prevous methods; solvng a multdmensonal nterpolaton problem, but the nterpolants were dfferent parameters. In the smple dfferental method presented, an ntal grd was deformed to conform to the gven physcal boundares and the dfferences between coordnates of boundary nodes of an ntal grd and the fnal grd are used as nterpolants. Two poson equatons are ntroduced as grd generaton equatons and the boundary condtons are the nterpolants dscussed. FDM scheme n one step and FVM scheme n both one step and mult steps have been used to solve the equatons and the skewness dagram s presented to study the smoothness of fnal grds better. FVM solver generates smoother grds wth better qualty especally n complex geometres and wth a partally adapted grd as an ntal grd. As a result, t can be mentoned that the proposed method solve grd generaton problem from a dfferent vewpont and n general ths method s computatonally smlar to TTM method whle both methods provde grds wth comparable qualtes. REFERENCES [1] R. W. Noack, D. A. Anderson, Soluton-Adaptve Grd Generaton usng Parabolc Partal Dfferental Equatons, Journal of The Amercan Insttute of Aeronautcs and Astronautcs, Vol. 8, pp , 199. [] D. L. Marcum, N. P. Weatherll, Unstructured Grd Generaton Usng Iteratve Pont Inserton and Local Reconnecton, Journal of The Amercan Insttute of Aeronautcs and Astronautcs, Vol. 33, pp , [3] C. Cont, R. Morand, R. M. Sptaler, An algebrac ellptc algorthm for boundary orthogonal grd generaton, Journal of Appled Mathematcs and Computaton, Issue 16, pp. 15 7, 5. [4] Y. Zhang, Y. Ja, S. S.Y. Wang, D nearly orthogonal mesh generaton wth controls on dstorton functon, Journal of Computatonal Physcs, Issue 18, pp , 6. [5] A. Ashrafzadeh, R. Jalalabad and M. Bazargan, A New Structured Grd Generaton Method, The 11th ISGG IJBAS-IJENS August 11 IJENS

9 Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 87 Conference, Ecole Polytechnque, Montreal, Canada, May 5-8, 9. [6] Peter R. Eseman, Yung K. Choo and Robert E. Smth, Algebrac Grd Generaton wth Control Ponts, Fnte elements flud, Vol. 8, pp , 199. [7] P. Barrera-Sánchez, F.J. Domínguez-Mota, G.F. González-Flores, Longna J.Castellanos Noda y Ángel A.Pérez Domínguez, Area Functonals For Hgh Qualty Grd Generaton, 4º Congreso Internaconal, º Congreso Naconal sobre Métodos Numércos en Ingenería y Cencas Aplcadas, Méxco, 7. [8] J. F. Thompson, F. C. Thames and C. W. Mastn, Automatc Numercal Generaton of Body-Ftted Curvlnear Coordnate System for Felds Contanng Any Number of Arbtrary Two- Dmentonal Bodes, Journal of Computatonal Physcs, Vol. 15, pp , [9] P.D.Thomas and J.F.Mddlecoff, Drect Control of the Grd Ponts Dstrbuton n Meshes Generated by Ellptc Equatons, Journal of the Amercan Insttute of Aeronautcs and Astronautcs, Vol. 18, pp , 198. [1] A. Ashrafzadeh, G. D. Rathby, Drect Desgn Soluton of the Ellptc Grd Generaton Equatons, Journal of Numercal Heat Transfer, Vol. 5, pp. 17-3, 6. [11] S. P. Spekrese, Ellptc Grd Generaton Based on Laplace Equatons and Algebrac Transformatons, Journal of Computatonal Physcs, Vol. 118, pp , [1] G.Ryskn & L.G.Leal, orthogonal mappng, Journal of Computatonal Physcs, Vol. 5, pp. 71 1, [13] Andre Bourchetn and Ludmla Bourchetn, On generaton of Orthogonal Grds, Journal of Appled Mathematcs and Computng, Vol. 173, pp , 6. [14] R. duraswam and A. Prosperett, Orthogonal Mappng n Two Dmensons, Journal of Computatonal Physcs, Vol. 98, pp , 199. [15] A. Ashrafzadeh, G. D. Rathby, and G. D. Stubley, Drect Desgn of Shape, Numercal Heat Transfer, Part B, Vol. 41, pp. 51-5, IJBAS-IJENS August 11 IJENS

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