Development of Irregular-Grid Finite Difference Method (IFDM) for Governing Equations in Strong Form

Transcription

1 Developent of Irregular-Grd Fnte Dfferene Method (IFDM for Governng Equatons n Strong For GEORGE X Insttute of Hgh Perforane Coputng Sene Park Road, #0-0 The Caprorn, Sngapore Sene Park II SINGPAORE 5 xuxg@hp.a-star.edu.sg and G. R. LI Dept. of ME Natonal nversty of Sngapore Lower ent Rdge Road SINGAPORE 0 pelugr@nus.edu.sg Abstrat: - In ths paper, an rregular-grd fnte dfferene ethod (IFDM wth the use of Green-Gauss theore was developed to nuerally deal wth probles subjet to arbtrary geoetral boundares. The fous s to eludate the prnple of IFDM and the nueral proedure to solve partal dfferental equatons. Attenton s pad to the dsretzaton of spatal ters n partal dfferental forat. Totally sx types of dsretzaton shees are proposed and assessed fro ponts of vews of effeny and auray. Theoretal analyses are onduted wth regard to the opatness of stenl and postvty of the oeffents of supportng nodes. pon the analyses, two shees, II and VI, are seleted for further study. Shee II s based on onepont quadrature rule whle shee VI orresponds to -pont quadrature rule. Nueral exses by usng the two shees are arred out to predt the solutons n a square doan that s governed by a Posson equaton. The effets of rregularty of grds are also studed. Nueral results ndate that the both shees gve satsfatory predton. In addton, the shee VI gves better auray, espeally on rregular grds, but at slghtly hgher ost n oputaton. Efforts here deonstrate that the proposed IFDM ethod s good to be used n nueral sulatons wth arbtrarly geoetral bounds. ey-words: - IFDM, Quadrature, Gradent soothng doan, Stenl, Grd edge, Doan fae. Introduton Apart fro well-known fnte eleent ethod (FEM and fnte volue ethod (FVM, fnte dfferene ethod (FDM s another esh-based tradtonal nueral ethod wdely adopted n nueral sulatons, anly beause t s straghtforward and hghly effent. In onventonal FD ethods, sngle- or ultple-blok strutured grds are used. The dsretzaton of governng equatons to strutured grds results n a syste of algebra equatons wth banded atrx of oeffents. Qute a nuber of effent nueral ethods an be used to qukly get the solutons to suh a syste of algebra equatons. However, t s not trval proess to generate strutured grds for arbtrary geoetres, espeally for the topologal generaton of ultple bloks []. Ths lts the FDM only for sple geoetres. Besdes, n onventonal FDM ethod, the Jaoban atrx, resultng fro the transforaton of governng equatons fro physal doan to a oputatonal doan on a urvlnear oordnate syste, s needed to be predted. Ths poses addtonal oputatonal ost and ay gve ore errors n predton. Researhers have attepted to overoe these drawbaks of FDM entoned above. More reently, soe eshfree ethods, whh are pleented on nodes at frst plae, are developed for ths purpose. A survey paper wrtten by Babuska et al [] ephaszed on the atheatal foundaton of varous eshfree ethods. Overvew of oputatonal and pleentaton ssues related to eshfree ethods an be found n onographs about weak for [] and strong for [].

2 In general, any eshfree ethods for the soluton of partal dfferental equatons n weak for are oprehensvely studed. When governng equatons n strong for are adopted, ost of these eshfree ethods show notorous stablty, prarly beause of the senstvty of the soluton on the supportng nodes used n dsretzaton. Besdes, due to large overlap n the supportng doans, dsretzaton shees are not onservatve. Ths also poses the deand of a great deal of oputaton. A new and effent generalzed fnte dfferene ethod, rregular-grd fnte dfferene ethod (IFDM, developed for the purpose of generalty and effeny. The IFDM s derved n lght of Green-Gauss theore rather than Taylor-seres expanson theore that s usually used n any other fnte dfferene ethod. In pleentaton, sne the governng equatons are dretly dsretzed on the physal doan, the predton for Jaoban atrx ndued by oordnate transforaton s avoded. In addton, the IFDM an be appled for arbtrary geoetres. In the followng setons, the n-depth desrpton of soluton algorth of IFDM s gven frst. The approxatons to gradents (frst-order dervatve and Laplae operator (seond-order dervatve of a salar are ntrodued wth the IFDM. The fous s on the analyses of weghtng oeffents on a stenl of supportng nodes. Consequently, nueral results for a square doan governed by a Posson equaton are dsussed. The oputatonal effeny and auray of the IFDM ethod are addressed. At the end, bref onludng rearks are gven upon urrent efforts.. Soluton Algorth. Theoretal bakground Taylor-seres expanson theore s used to dsretze the dervatves n tradtonal FD ethods. Coparatvely, the IFDM ethod s derved on the bass of the Green-Gauss theore. For splty, let us take a -denson proble as an exaple to llustrate the prnples of IFDM ethod. Aordng to Green-Gauss theore [5], for a gener varable, t holds dv nds ( where s gradent operator, and V represents the area of gradent soothng doan of, ds s the length of doan fae and n s the unt noral vetor to the doan fae. Fg. Illustraton of gradent soothng doan and fae vetors (pontng outward Matheatally, on a gradent soothng doan (GSD as shaded n Fg., the gradents an be approxated as nds ( Slarly, the seond-order dervatves (Laplae operator an be approxated as ( n ds ( ( j n L j ( n As shown n Fg., the grds are used to onsttute gradent soothng doans whh are used for approxaton of dervatves ourrng n governng equatons. In urrent study, two types of GSD are adopted: one s naed edan-based GSD adopted for the approxaton of dervatves at any node of nterest. It s fored by onnetng relevant entrods of trangles wth dponts of relevant edges. The other s just the grd tself that s eployed n soe shees for predton of dervatves at any entrod. It s alled grd-based GSD here.. Dsretzaton of dervatves Totally sx shees, as suarzed n Table, were proposed and studed. For the three bas shees, I, III and V, they dffer wth one another n the order of quadrature adopted for ntegral approxaton and the approxaton of gradents at entrods. In one-pont based shees, the ntegraton over any doan fae s approxated wth retangular rule by usng the values at dpont of any edge. In two-pont based shees, trapezodal rule based on values at dponts of R j dpont entrod gradent soothng doan

3 relevant edges and entrods s adopted. Both the frst- and seond-order dervatves are obtaned by suessvely applyng Green-Gauss theore to the sae MGSD. In Shee I and III, the gradents at any entrod are obtaned by arthet averagng of gradents at the relevant grd nodes, whle they are alulated by applyng Green-Gauss theore onto grd-based GSD n shee V. Applaton of dretonal orreton to these bas shees gves another three shees, II, IV and VI, respetvely. Detals about dretonal orreton wll be llustrated n the followng seton. Table Spatal dsretzaton shees Shee Quadrature Type of GSD Dretonal orreton I One-pont MGSD no II One-pont MGSD yes III Two-pont MGSD no IV Two-pont MGSD yes V Two-pont MGSD,GGSD no VI Two-pont MGSD,GGSD yes For splty, the spatal dsretzatons based on shee I and III are desrbed here. In Shee I, dervatves at node of nterest an be approxated, respetvely, as x and SXjj, SYj j j y j ( ( ( j j SX ( j SYj j x x y y (5 where ( L ( R j j j ( j SX SX SX, SY SY SY ( L ( R ( L ( R j j j j j j SX S ( n, SY S ( n Here, ( L ( L ( L ( R ( R ( R j j x j j j y j SX j and SY j, and n x and n y, are, respetvely, the two oponents of a fae vetor and a unt noral vetor wth respet to x and y dretons. They are evaluated and stored before the ntensve alulaton s started. denotes the total nuber of nodes n the stenl of the node. In Shee III, the values of funtons and gradents at the entrods of grds are alulated by sple arthet averagng of values at onsttutve nodes. Thus, the dervatves are approxated as follows: ( L ( L ( R ( R SX j ( SX j ( x j ( L ( L ( R ( R SY j ( SY j ( y j (6 ( L ( L ( L ( L ( SX j ( SY j x x y y ( R ( R x y j ( R ( R ( SX ( SY j j where x x y y ( L ( R j ( L j k ( R j k,, L ( L ( R,, ( ( R j j k j k ( In oparson to shees based on one-pont quadrature, these shees based on two-pont quadrature pose addtonal oputatonal deands and storage for values at entrods and fae vetors assoated wth ther oon grd edge. The edge-based data struture s adopted n the study, together wth satter-gather approah, as desrbed by Barth [6][].. Analyses of Stenl of Supportng Nodes For onvenene and splty, the dsretzatons of gradents and Laplae operator are perfored onto strutured quad grds and equlateral trangles, respetvely. The oeffents of supportng nodes are analysed aordngly. The stenls for gradent approxaton wth dfferent shees are shown n Fg.. As addressed by Barth [6], for good dsretzaton shees, the weghtng oeffents for Laplae operator are expeted to be postve and proportonally vared wth dstane to the node of nterest. Besdes, for better oputatonal effeny, opat stenl orrespondng to supportng nodes should be reated. In-depth dsusson about ths ssue an be found n Haselbaher and Blazek []. As ndated n Fg. (a, ( and (e, shees I, III and V result n wde stenls wth unfavorable weghtng oeffents on strutured quad grds. Wth suh shees, as addressed n Blazek [], unexpeted deouplng solutons ay be produed, whh wll be further llustrated n the followng seton. Crupton et al [0] proposed to use odfed gradents for the approxaton of seond-order dervatves, wth the help of the followng dretonal orreton along grd edges: t t j j j j j l j ( where

4 (, j j j l j l j rj tj, rj X j X, l j X j X l j and X and X j denotes the postons of node and j, respetvely. As depted n Fg. (b and (e, relatvely opat stenl wth favorable oeffents are fored n Shee II and VI. Shee II has 5-pont based stenl and Shee VI orresponds to -node based opat stenl. However, as shown n Fg. (d, Shee IV stll results n unfavorable stenl. The analyses were also onduted on equlateral trangles. Fg. shows the stenls and orrespondng weghtng oeffents of supportng nodes for a dsretzed Laplae operator. It s obvous that shees II, IV, V and VI, all produe stenls wth favorable oeffents on equlateral trangles, exept shees I and III. Besdes, t s nterestng to note that both shees II and VI produe dental stenl, so do shees IV and V. Wth addtonal onerns about oputatonal effeny, shees II and VI are superor to shees IV and V, beause of opatness n stenl. In suary, upon the stenl analyses, both shee II and shee VI are proposed to be used n the IFDM, beause both of the produe opat stenl wth favorable oeffents on both types of regular grds of onern (quadrlateral and trangular. (, 0 (0, ( 0, (, 0 (, 6 6 (a I and II (b III, IV,V and VI ( I, II, III, IV,V and VI Fg. Stenls and weghtng oeffents for the approxaton of ( u, u on dfferent types of grds x y (, 6 6 (0, (,0 (,0 (, (, (0, (, (, (,0 (,0 (, 6 (, (b II (d IV (f VI Fg. Stenls and weghtng oeffents for the approxaton of u u on quadrlateral grds x y (a I and III (b IV and V ( II and VI Fg. Stenls and weghtng oeffents for the approxaton of u u on equlateral trangles x y. Applaton of IFDM for Soluton of Posson Equaton. Posson equaton and ts dsretzaton The IFDM s used to predt the soluton to Posson equatons on a square doan. In urrent study, the Drhlet ondton s appled to the boundares,.e., the values at the boundares are presrbed. The pseudo-transent approah s used n urrent study for pursung steady-state solutons. The governng equatons under nvestgaton take the followng for: (, f x y t x y f ( x, y, t exp( x y ( x, y,0 0 (0x, 0 y ( (a I ( III (e V The exat soluton to ths Posson equaton s ( ˆ(, x y x y e ( x, y (0 As desrbed above, the funton values at the boundares are spefed aordngly. The spatal dervatves are dsretzed as shown n prevous seton and the teporal ter s dealt wth the explt fve-stage Runge-utta (R5 ethod n urrent study. The onvergene ndex n the for of

5 nnodes nnodes ( n ( n ( (0 ( error _ nor ( / ( s evaluated at eah te-step. For exludng the effet due to the teporal dsretzaton, oputatons are ternated when error_nor approahes ahne error. In urrent study, nueral errors for the overall feld are predated n the for of nnode nnode ˆ ˆ ( / ( error where nnode s the total nuber of nodes n the doan, and and ˆ are predted and exat funton values at node, respetvely. The nodewse relatve error s also evaluated n the fashon of rerror ˆ / ˆ (. Results and Dsusson In urrent study, three types of grds,.e. strutured quadrlateral grds, unstrutured rght trangular and rregular trangles, are nvestgated, as shown n Fg. 5. The unstrutured rght trangles are generated by sply splttng the quad along one of ts dagonals. (a strutured quad (b rght trangle ( rregular trangle Fg. 5 Representatve grds under nvestgaton As shown n Fg. 6 (a, t s obvous that the deoupled soluton s predted by usng Shee I when t s appled onto strutured quadrlateral grds. Wth the help of dretonal orreton, ths proble s overoe n Shee II, as shown n Fg. 6(b. re rror (a Shee I (b Shee II Fg. 6 Contour plots of relatve errors on strutured quadrlateral grds ( nodes rerror Besdes, wth dretonal orretons, the overall nueral error s draatally redued, as shown n Table below. However, wth Shee II, saller te step s needed for stablty requreent at the ost of ore ntensve oputaton n ters of the nuber of teratons. Table Coparson of auray n predton by usng Shee I and Shee II No. of Shee I Shee II nodes error teraton error teraton 6.6e e-.5e-.66e-.6e- 0.e e-.e e- 6.e-5 Er ror.00e+00.00e-0.00e-0.00e-0.00e-0.00e-05.00e-06 Shee II (quad Shee VI (quad Shee II (rght-t Shee VI (rght-t No. of nodes Fg. Profles of auray for shees II and VI on strutured quadrlateral and rght trangular grds In general, Shee II gves ore aurate predton than Shee VI on strutured quadrlateral grds, whle the both shees result n ore aurate predton on rght trangles. More presely, the Shee VI gves slghtly better predton than Shee II when rght trangles are used. It ples that the shee VI ay gve ore aurate predton on rregular grds. Ths s proved to be true, as shown n Table. The better auray ay attrbute to the aurate predton of ntegraton over doan faes. Table Overall errors predted based on rregular trangles No. of error nodes Te-step Shee II Shee VI e-.6e e- 6.6e e-.e e-5.e e-5.e-5

6 5. Conluson In urrent study, a new rregular-grd fnte dfferene ethod (IFDM for governng equatons n strong for s developent. The IFDM an be used for general applaton subjeted to arbtrary geoetres. In ths paper, the prnple of IFDM, based on Green-Gauss theore, s ntrodued and the nueral proedure for predtons of spatal dervatves s eludated. Analyses of the stenls of supportng nodes for dsretzed spatal dervatves, wth respet to totally eght types of dsretzaton shees, are perfored. Assessent s ade upon the opatness of stenl and postvty of weghtng oeffents. Two shees wth better effeny and auray are seleted n further study. Nueral exses by usng the two shees are arred out for solutons to a Posson equaton on a square doan. It further proves that the both shees ndeed gve satsfatory predton. Besdes, the Shee VI s superor to Shee II n ters of auray, espeally on rregular grds, at the ost of slghtly hgher oputatonal deand. Nuber Flows on Hghly-Strethed Grds, 0 th Int. Conf. Nu. Meth. For Lanar and Turbulent Flows, Swansea, England,. Referenes: [] J.F. Thopson, B.. Son and N.P. Weatherll, Handbook of Grd Generaton, CRC Press, [] I. Babuska,. Banerjee and J.E. Osbor, Survey of eshless and generalzed fnte eleent ethods: A unfed approah, TICAM Report 0-0, nversty of Texas, Austn, 00. [] G.R. Lu, Mesh Free Methods: Movng beyond the Fnte Eleent Method, CRC Press, 00 [] G.R. Lu, Y.T. Gu, An Introduton to Meshfree Methods and ther Prograng, Sprnger, 005 [5] M.D. Greenberg, Advaned Engneerng Matheats, nd Edton, Prente Hall, [6] T. Barth, Apsets of unstrutured grds and fnte-volue solvers for the Euler and Naver- Stokes equatons, VI Leture Seres, 5. [] T. Barth, Nueral ethods for onservaton laws on strutured and unstrutured eshes, VI 00 Leture Seres, 00. [] A. Haselbaher and J. Blazek, Aurate and Effent Dsretzaton of Naver-Stokes Equatons on Mxed Grds, AIAA Journal, Vol,, No., 000, pp [] J. Blazek, Coputatonal Flud Dynas: Prnples and Applaton, ELSEVIER Press, Frst-Edton, 00 [0] P.I. Crupton, P. Moner, and M.B. Gles, An nstrutured Algorth for Hgh Reynolds