An adaptive gradient smoothing method (GSM) for fluid dynamics problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluds 2010; 62:499 529 Publshed onlne 17 March 2009 n Wley InterScence (www.nterscence.wley.com)..2032 An adaptve gradent smoothng method (GSM) for flud dynamcs problems George X. Xu 1,,,G.R.Lu 2,3, and Atus Tan 4, 1 Insttute of Hgh Performance Computng, Fusonopols, 1 Fusonopols Way, #16-16 Connexs, Sngapore 138632, Sngapore 2 Centre for Advanced Computatons n Engneerng Scence, Department of Mechancal Engneerng, Natonal Unversty of Sngapore, 10 Kent Rdge Crescent, Sngapore 119260, Sngapore 3 Sngapore-MIT Allance (SMA), E4-04-10, 4 Engneerng Drve 3, Sngapore 117576, Sngapore 4 Department of Mathematcs, Keo Unversty, Hyosh, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan SUMMARY In ths paper, a novel adaptve gradent smoothng method (GSM) based on rregular cells and strong form of governng equatons for flud dynamcs problems wth arbtrary geometrcal boundares s presented. The spatal dervatves at a locaton of nterest are consstently approxmated by ntegrally averagng of gradents over a smoothng doman constructed around the locaton. Such a favorable GSM scheme corresponds to a compact stencl wth postve coeffcents of nfluence on regular cells. The error equdstrbuton strategy s adopted n the soluton-based adaptve GSM procedure, and adaptve grds are attaned wth the remeshng technques and the advancng front method. In ths paper, the adaptve GSM has been tested for solutons to both Posson and Euler equatons. The senstvty of the GSM to the rregularty of the grd s examned n the solutons to the Posson equaton. We also nvestgate the effects of error ndcators based on the frst dervatves and second dervatves of densty, respectvely, to the solutons to the shock flow over the NACA0012 arfol. The adaptve GSM effectvely yelds much more accurate results than the non-adaptve GSM solver. The whole adaptve process s very stable and no spurous behavors are observed n all testng cases. The cosmetc technques for mprovng grd qualty can effectvely boost the accuracy of GSM solutons. It s also found that the adaptve GSM procedure usng the second dervatves of densty to estmate the error ndcators can automatcally and accurately resolve all key features occurrng n the flow wth dscontnutes. Copyrght q 2009 John Wley & Sons, Ltd. Receved 21 May 2008; Revsed 15 January 2009; Accepted 6 February 2009 KEY WORDS: numercal method; FVM; adaptve; flud dynamcs; GSM; GSD; stencl Correspondence to: George X. Xu, Insttute of Hgh Performance Computng, Fusonopols, 1 Fusonopols Way, #16-16 Connexs, Sngapore 138632, Sngapore. E-mal: xuxg@hpc.a-star.edu.sg Professor. Copyrght q 2009 John Wley & Sons, Ltd.

500 G. X. XU, G. R. LIU AND A. TANI 1. INTRODUCTION In numercal methods, a gradent at a locaton s often approxmated wth the help of Taylor-seres expanson or a predefned polynomal functon [1]. Alternatvely, t can also be approxmated by spatal averagng of gradents over a smoothng doman constructed around the locaton of nterest. Such a technque s referred as gradent smoothng operaton [2]. The use of gradent smoothng operaton can be found n many weak-form meshfree methods. It has been used by Chen et al. [3] for stablzng the nodal ntegraton n the Galerkn meshfree method. It s also used n the lnearly conformng pont nterpolaton method [4, 5] for soluton stablzaton and for obtanng upper bound solutons. In these methods, the gradent smoothng operaton was used n the approxmaton of strans. Wth an adopted pecewse constant smoothng functon, va Green Gauss theorem (known as dvergence theorem), the doman ntegraton n the orgnal form was converted nto lne ntegraton along the boundares of smoothng doman n two-dmensonal problems. It was found that the stablzed nodal ntegraton gave hgher effcency wth desred accuracy and convergent propertes. The use of smoothng operaton also avoded the evaluaton of dervatves of grd-free shape functons requred by drect nodal ntegraton and just resulted n hgher accuracy. The smoothed gradent matrx was shown to satsfy lnear exactness n the Galerkn approxmaton of a second-order partal dfferental equaton. In addton, smoothng technques have also been employed n the wdely used smoothed partcle method [6, 7] manly as a general means of functon approxmaton. Most above-mentoned research efforts relate to the use of gradent smoothng operaton for weak-form governng equatons. In such applcatons, the gradent smoothng operaton s only adopted as an auxlary component n the development of man frameworks for varous meshfree methods. A gradent smoothng method (GSM) for strong-form equatons has been most recently developed [2], n whch the gradent smoothng operaton s consstently and successvely used to approxmate the frst and second dervatves at dfferent locatons and the resultant nstantaneous equatons are solved subsequently. Varous dscretzaton schemes for approxmatng spatal dervatves have been devsed and some favorable schemes vald for varous types of grd are selected from the vewponts of effcency and accuracy. The excellent schemes for the GSM have been successfully formulated and appled for smulatng compressble flows and heat conducton problems. Our prevous efforts showed that the proposed GSM s conservatve, conformal, effcent, robust, and accurate. Our prevous GSM solutons for heat conducton and compressble flow problems, as presented by Lu and Xu [2], have been attaned usng pre-generated fxed grds wth nodes that can be rregularly dstrbuted. Such fxed grds may not be good enough to cater for the flow phenomena nvolvng abrupt changes, for example, a flow over an RAE2822 arfol [2]. Notceable dscrepancy between GSM soluton to the expermental data was stll observed across the shock regon, manly due to the nsuffcently fne cells n the shock area that s, n general, unknown when the grds are created before the analyss. Hence, t s very dffcult to generate a good set of grd before the solutons to the flow feld are attaned. The soluton-based adaptve processes [8, 9] gve the solutons to overcome such a dffculty and consequently yeld a set of optmal grd sutable for abrupt changes. The man objectve n our current study s to develop an adaptve remeshng process for the robust, stable, and accurate GSM so as to automatcally and successfully resolve the flows wth abrupt changes, such as a shock flow, n a self-contaned manner. Besdes, the senstvty of the GSM to the rregularty of grd s further examned.

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 501 In ths paper, our recent progress n the development of the adaptve GSM s presented. In the followng sectons, the prncple of the GSM s frst ntroduced. The GSM approxmatons to the gradents (frst dervatve) and the Laplace operator (second dervatve) of a feld varable are presented n detal. The adopted adaptve remeshng process usng the advancng front technque s descrbed n detals. The adaptve GSM s then used to solve the Posson equatons for benchmarkng purpose. The roles of some cosmetc postprocesses for adaptve grds, ncludng removal of redundant trangles, dagonal swappng, and grd smoothng, are also addressed. Fnally, the successful applcaton of the adaptve GSM to resolve an actual flud flow problem of an nvscd shock flow over the NANA0012 arfol s presented. 2. GRADIENT SMOOTHING METHOD (GSM) In the GSM, dervatves at varous locatons, such as at nodes, centrods of cells, and mdponts of cell-edges, are approxmated usng gradent smoothng operaton over relevant gradent smoothng domans. Ths secton brefly elucdates the fundamental prncples and mplementaton procedure n the GSM. 2.1. Gradent smoothng operaton For smplcty, a two-dmensonal problem s consdered here to llustrate the gradent smoothng operaton. Such a process was ntally proposed to approxmate a functon or ts gradent at a pont [2, 6, 10, 11]. For example, the gradents of a feld varable U at a pont of nterest at x n doman Ω can be approxmated n the form of U U(x ) U(x) w(x x )dx (1) Ω where s gradent operator and w s a smoothng functon. If we ntegrate Equaton (1) by parts or use dvergence theorem, Equaton (1) becomes U U(x) w(x x ) n ds U(x) w(x x )dv (2) Ω Ω where Ω represents the boundary of the gradent smoothng doman and n denotes the outwardpontng unt normal vector on Ω, as shown n Fgure 1. The smoothng functon, w, s chosen properly accordng to the requrement on the accuracy n approxmaton of the functon, and needs to satsfy some condtons (see, for example, [2, 7]). For smplcty, the smoothng functon n current study s set to be pecewse constant over the smoothng doman as follows: w = { 1/V, x Ω 0, x / Ω (3) where V stands for the area of the smoothng doman Ω. The proposed smoothng functon can automatcally satsfy the basc condton of partton of unty over the smoothng doman, e.g. Ω w(x x )dv =1.

502 G. X. XU, G. R. LIU AND A. TANI Fgure 1. A smoothng doman for a pont at x. Thus, the second term on the rght-hand sde n Equaton (2) vanshes, and Equaton (2) reduces to U 1 V Ω U n ds (4) Equaton (4) gves a smple way to approxmate to gradents at a pont by area-weghted ntegral along the boundary of a local smoothng doman. It s apparent that usng Equaton (4) s more effcent than usng Equaton (1) to approxmate the gradents, because numercal ntegral along the boundary s much more cost-effectve than over the surface of the smoothng doman. Analogously, by successvely applyng the gradent smoothng technque for second-order dervatves [12 14], the Laplace operator at an arbtrary pont x can be approxmated as ( U ) 1 n U ds (5) V Hence, spatal dervatves at any pont of nterest can be approxmated usng Equatons (4) and (5) over a smoothng doman that needs to be properly defned for a purpose [2]. Ω 2.2. Smoothng domans In the GSM, the values of feld functons are stored at nodes that can be rregularly dstrbuted n space. By connectng nodes, the problem doman s frst dvded nto a set of prmtve cells of any shape. Because trangular cells can be automatcally generated, they are usually preferred. Based on these cells, a smooth doman for any pont of nterest can be constructed. We devse dfferent types of smoothng domans for approxmatng frst dervatve at dfferent locatons. Fgure 2 shows three types of gradent smoothng domans, over whch spatal dervatves are approxmated wth gradent smoothng operaton. The frst type of smoothng doman s the node-assocated GSD (ngsd) for the approxmaton of dervatves at a node of nterest. As shown n Fgure 2, the ngsd s formed by connectng the centrods of relevant trangles wth mdponts of nfluenced cell-edges. The second s dentcal to a prmtve cell, whch s used for approxmatng dervatves at the centrod of the cell, as n the cell-centered FVM [15]. It s called centrod-assocated GSD

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 503 j k+1 Cjk jk+1 M j k J k nm jk C jk j k+1 nm jk C jk 1j k J C jk 1j k J k-1 node ngsd mdpont cgsd centrod mgsd Fgure 2. Illustraton of gradent smoothng domans and doman-edge vectors adopted n GSM. (cgsd) here. The thrd s named mdpont-assocated GSD (mgsd) used for the calculaton of the gradents at the mdpont of a cell-edge of nterest. The proposed mgsd, as shown n Fgure 2, s formed by connectng the end-nodes of the cell-edge wth the centrods on the both sdes of the cell-edge. As reported n [2], the adopton of gradent smoothng operaton over such a mgsd to approxmate the gradent at a mdpont, whch s needed to dscretzed Equaton (5), can successfully avod the checkerboard problem that s usually encountered by the approxmaton based on smply arthmetc average of gradents at two consttute nodes. Furthermore, the resultant scheme s remarkably robust to the rregularty of trangular cells across the doman. By vrtue of Equatons (4) and (5), spatal dervatves at any pont of nterest can be approxmated based on the correspondng smoothng doman descrbed above. As outlned by Lu and Xu [2], dfferent schemes are devsed and then evaluated on the bass of stencl analyses and numercal tests. As a result, a scheme whch s based on one-pont quadrature and consstent applcaton of gradent smoothng at all locatons of nterest s most preferable, because of ts balanced accuracy and computatonal effcency, and attractve robustness to grd rregularty. In ths paper, ths scheme s adopted n the development of adaptve GSM. 2.3. Approxmaton to spatal dervatves We now need to accurately predct the ntegrals along the boundares of varous types of GSDs. In ths work, spatal dervatves (both frst-order and second-order) at nodes are approxmated usng the one-pont quadrature (rectangular rule). As such, the ntegrand (U or U) for each domanedge takes the values at the mdpont of the connected cell-edge. Such a procedure s ndependent of the values at centrods of cells and, therefore, the approxmaton for the gradents at centrods, whch s necessarly needed n two-pont quadrature schemes (see, for example, [5]), s avoded. Two-pont quadrature s only used for the approxmaton of gradents at mdponts of cell-edges. Detals wll be elucdated below.

504 G. X. XU, G. R. LIU AND A. TANI 2.3.1. Frst dervatves at nodes. Wth the gradent smoothng operaton usng Equaton (4), the frst-order dervatves at nodes can be approxmated as x (U N ) 1 (V N ) n k=1 (ΔS x N ) j k (U M ) jk (6) where y (U N ) 1 (V N ) n k=1 (ΔS y N ) j k (U M ) jk (7) (ΔS x N ) j k =ΔS Mjk C jk j k+1 (n x ) Mjk C jk j k+1 +ΔS Mjk C jk 1 j k (n x ) Mjk C jk 1 j k (8) (ΔS y N ) j k =ΔS Mjk C jk j k+1 (n y ) Mjk C jk j k+1 +ΔS Mjk C jk 1 j k (n y ) Mjk C jk 1 j k (9) Here, denotes the node of nterest and j k s the other end-node of the cell-edge lnked to node (see Fgure 2). M jk denotes the mdpont of the cell-edge of nterest, j k. C jk j k+1 and C jk 1 j k represent the centrods of two trangular cells connected to the cell-edge j k. The total number of supportng nodes wthn the stencl of the node s denoted by n. U N, U M,andU C denote values of the feld varable at nodes, mdponts of cell-edges, and centrods of trangular cells, respectvely. Thanks to the one-pont quadrature, for an ngsd of nterest, a par of doman-edges connected wth cell-edge j k are evaluated n a lumped manner. ΔSN x and ΔSy N correspond to the two components of the pared doman-edges. n x and n y represent the two components of the unt normal vector of the doman-edge under a Cartesan coordnate system. V N s the area of an ngsd. These geometrcal parameters are calculated and stored before the resultant algebrac equatons are solved. The values of the feld varable U at non-storage locatons,.e. at mdponts and centrods, are computed by smple nterpolaton of functon values at related nodes, respectvely, n the manner of (U M ) jk = (U N ) +(U N ) jk 2 (10) (U N ) +(U N ) jk +(U N ) jk+1, 1 k<n 3 (U C ) jk j k+1 = (11) (U N ) +(U N ) jn +(U N ) j1, k =n 3 (U N ) +(U N ) jk +(U N ) jk 1, 1<k n 3 (U C ) jk 1 j k = (U N ) +(U N ) j1 +(U N ) jn, k =1 3 The values of feld functon at centrods are only needed for approxmaton of gradents at mdponts of cell-edges. (12)

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 505 2.3.2. Second dervatves at nodes. In the GSM, wth one-pont quadrature, the second dervatves n Laplace operator are approxmated usng Equaton (5) n the followng manner: ( U N ) 1 [ n Ω k=1 x (U M) jk (ΔSN x ) j k + ] y (U M) jk (ΔS y N ) j k (13) It s apparent that the gradents at mdponts are needed to complete the approxmaton of secondorder dervatves at nodes. As mentoned n the prevous secton, the two-pont quadrature s proposed to approxmate gradents at mdponts n the GSM. As a result, usng Equaton (4) over the relevant mgsd as shown n Fgure 2, the frst-order dervatves at the mdpont of cell-edge of nterest, j k, are attaned by { 1 x (U M) jk 2 (ΔSx M ) C jk j k+1 [(U N ) +(U C ) jk j k+1 ]+ 1 2 (ΔSx M ) j k C jk j k+1 [(U N ) jk +(U C ) jk j k+1 ] + 1 2 (ΔSx M ) C jk 1 j k [(U N ) +(U C ) jk 1 j k ]+ 1 } 2 (ΔSx M ) j k C jk 1 j k [(U N ) jk +(U C ) jk 1 j k ] 1 (V M ) jk (14) { 1 y (U M) jk 2 (ΔSy M ) C jk j k+1 [(U N ) +(U C ) jk j k+1 ]+ 1 2 (ΔSy M ) j k C jk j k+1 [(U N ) jk +(U C ) jk j k+1 ] + 1 2 (ΔSy M ) C jk 1 j k [(U N ) +(U C ) jk 1 j k ]+ 1 } 2 (ΔSy M ) j k C jk 1 j k [(U N ) jk +(U C ) jk 1 j k ] 1 (V M ) jk (15) where V M represents the area of an mgsd. The relevant doman-edge vectors, ΔS x M and ΔSy M, for the mgsd of nterest, are calculated as follows: (ΔS x M ) C jk j k+1 = ΔS Cjk j k+1 (n x ) Cjk j k+1, (ΔS y M ) C jk j k+1 =ΔS Cjk j k+1 (n y ) Cjk j k+1 (ΔS x M ) j k C jk j k+1 = ΔS jk C jk j k+1 (n x ) jk C jk j k+1, (ΔS y M ) j k C jk j k+1 =ΔS jk C jk j k+1 (n y ) jk C jk j k+1 (16) (ΔSM x ) C jk 1 j k = ΔS Cjk 1 j k (n x ) Cjk 1 j k, (ΔS y M ) C jk 1 j k =ΔS Cjk 1 j k (n y ) Cjk 1 j k (ΔSM x ) j k C jk 1 j = ΔS jk C jk 1 j k (n x ) jk C jk 1 j k, (ΔS y M ) j k C =ΔS (17) jk 1 j j k C jk 1 j k (n y ) jk C jk 1 j k For regular grds (cells n square and equlateral trangle shapes), the compact stencls wth postve weghtng coeffcents for the approxmated Laplace operator are obtaned, as shown n Fgure 3. As addressed n the stencl analyses n [2], such a scheme corresponds to second-order accuracy and the nvolved computaton s cost-effectve. Ths s one of the reasons why such a dscretzaton scheme for spatal dervatves s promoted. It should be mentoned that the prncple about gradent smoothng operaton can be extended easly for three-dmensonal problems. For example, n cases where tetrahedral elements are used as

506 G. X. XU, G. R. LIU AND A. TANI 1 1 4 1 2 3 2 3-4 2 3 2 3 (a) 1 (b) 2 3 2 3 Fgure 3. Stencls for the approxmaton of the Laplace operator: (a) square and (b) equlateral trangles. j 1 j 2 j 3 Node of nterest Surroundng nodes Centrod of a tetrahedra Mdpont of an edge Centrod of a surface Partal ngsd Fgure 4. Partally formed three-dmensonal node-assocated gradent smoothng doman based on a tetrahedral prmal element. prmal elements, a typcal node-assocated smoothng doman can be formed by jonng all doman faces each of whch s generated by connectng the mdpont of a common edge, the centrods of two nvolved element surfaces and the centrod of the nvolved prmal element. A partally formed three-dmensonal node-assocated gradent smoothng doman based on a tetrahedral prmal element s shown n Fgure 4, where the relevant doman faces are shaded. Hence, the boundary ntegrals occurrng n Equatons (4) and (5) would be replaced by ntegrals over doman surfaces of such a three-dmensonal ngsd. Accordngly, n three-dmensonal applcatons, the surface normals should be used nstead of boundary normals for three-dmensonal problems. 2.4. Approxmaton to temporal dervatves For a transent or pseudo-transent problem, the governng equaton can be rewrtten n the form of t (U N )= R (18)

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 507 where R represents the resdual dependent of the feld varable U and ts dervatves. In current study, the temporal term ( U N / t) s approxmated wth the followng explct fve-stage Runge Kutta (RK5) method [16]: (U N ) (0) = (U N ) n (U N ) (1) = (U N ) (0) α 1 ΔtR (0) (U N ) (2) = (U N ) (0) α 2 ΔtR (1) (U N ) (3) = (U N ) (0) α 3 ΔtR (2) (U N ) (4) = (U N ) (0) α 4 ΔtR (3) (U N ) (5) = (U N ) (0) α 5 ΔtR (4) where the resdual R (k) s evaluated wth the values of the feld functon and ts dervatves approxmated wth the kth-stage RK soluton at node for every tme-step. Δt denotes the tme-step, and the coeffcents adopted n current study are α 1 =0.0695, α 2 =0.1602, α 3 =0.2898, α 4 =0.5060, and α 5 =1.000. The RK5 method has been wdely used n the smulatons of many transent flud flow problems, because of ts satsfactory effcency and stablty. Besdes, wth the RK5 method, only the 0th- and 5th-stage solutons at nodes are needed to be stored n memory, whch saves a lot n computatonal demands. (19) 3. ADAPTIVE REMESHING TECHNIQUE The adaptve process ams at yeldng a set of optmal grd on whch an antcpated accuracy can be acheved. Such a grd, as outlned by Babuska and Rhenboldt [17], corresponds to equally dstrbuted errors across the feld. Ths mples that the optmal grd can be acheved wth the help of the heurstc error equdstrbuton strategy. Thus, n current study, an adaptve process based on the error equdstrbuton strategy s developed and used to boost the GSM solver for better accuracy. In the proposed adaptve process, a drectonal error ndcator at each node for a current grd s evaluated frst, followed by the determnaton of meshng parameters based on error equdstrbuton strategy. Once the meshng parameters are obtaned, the whole feld wll be remeshed wth the advancng front technque. The whole process s carred out n an teratve way tll the expected accuracy or number of adaptve loops s reached. Ths secton contrbutes to the bref ntroducton about the drectonal error ndcator, at length explanaton how to determne the meshng parameters requred by the advancng front technque, and the descrpton of the overall adaptve GSM procedure. 3.1. Drectonal error ndcator In an adaptve process, t s desrable to choose an approprate error ndcator whch can be used to dentfy the regons for further ether refnement or coarsenng. The drecton-orented error ndcator as outlned by Pero et al. [18] cannot only serve such a purpose but assst to determne the coordnates of nodes for a new set of grd across the feld. Therefore, t s adopted n the current

508 G. X. XU, G. R. LIU AND A. TANI study. For example, n a one-dmensonal problem, usng fnte element method procedure, the varaton of the error E wthn a cell n whch the nterested pont x s located can be expressed as E e = 1 2 ξ(h ξ) d 2 U dx 2 (20) where ξ and h, respectvely, denote a local cell coordnate and the cell length. Ths relates the error wth the cell length and the second dervatve at pont x. Furthermore, the root mean-square error over the cell can be computed as h Ee E e L2 = 2 0 h du = 1 h 2 d 2 U 120 dx 2 (21) Accordng to error equdstrbuton strategy, a set of optmal grd corresponds to the case where the errors are equally dstrbuted across the feld, mplyng that h 2 d 2 U dx 2 =C (22) where C s a postve constant. More precsely, f δ s used to denote the optmal spacng, for the set of optmal grd, the followng relaton holds: δ 2 d 2 U dx 2 =C (23) Ths equaton gves the way to determne the optmal spacng on condtons that the second dervatves on a current set of one-dmensonal grd are approxmated. Equaton (23) can be drectly extended to mult-dmensonal problems. For a 2D problem as concerned here, t can be wrtten n the quadratc form as ( 2 δ 2 α m j α α j )=C (24), j=1 where α s an arbtrary unt vector, δ α s the spacng along the drecton of α, andm j are the components of a 2 2 symmetrc matrx of second dervatves, whch s m j = 2 U N (25) x x j The meanng of Equaton (24) s graphcally llustrated n Fgure 5 whch shows how the values of the spacng n the α drecton can be obtaned as the dstance from the orgn to the pont of ntersecton of the vector α wth the boundary of an ellpse. The major and mnor axal drectons (α 1 and α 2 ), and the lengths of semmajor and semmnor axes (δ 1 and δ 2 ) for the ellpse can be readly computed, once the egenvalues and egenvectors of the matrx m are obtaned, whch wll be shown n the followng secton. As shown n Equaton (25), the second dervatves are requred to be evaluated for estmatng numercal errors. Alternatvely, Graldo [19] has used the frst dervatves for adaptve solutons

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 509 y x α 2 δ 2 δ α α δ 1 α 1 Fgure 5. Determnaton of nodal spacng δ along a drecton α. to flud dynamc problems, for ther obvous physcal meanngs. In dong so, the gradents that are necessarly approxmated n soluton process are drectly used n the error estmaton. Ths becomes partcularly attractve for nvscd flows, snce there are no second dervatves at all appearng n orgnal governng equatons. Hence, the proposed varaton s much more cost effectve. Accordngly, the components of modfed matrx m become m j = U N x U N x j (26) In current study, the both optons for approxmatng matrx m are examned n testng cases. Some nterestng results wll be dscussed n the secton about numercal examples. It should be ponted out that the attaned grd s almost optmal when the local errors are approxmately equal at all nodes. Therefore, ths approach descrbed above s an asymptotc method and demands teratve operaton. However, as addressed by Pero et al. [18], though these error ndcators have no rgorous mathematcal proof, consderable success has been acheved wth ther use n practcal stuatons [8]. In addton, the presented error ndcator s drecton orented so that t can be used to generated ansotropc grd by takng account of drecton-dependent flow phenomena, such as shocks, contact dscontnutes, etc. Such features can be most economcally represented on grds whch are stretched n approprate drectons. Besdes, we have nterests n the potental applcatons for problems pertanng to movng objects and flud structure nteractons. In such cases, large changes n geometres and relatve locatons may occur, whch sometmes volate the use of local treatment technques ncludng node moton and local enrchment. Comparatvely, the present remeshng technque has ts dstnct advantages for such types of applcatons [20]. In present study, we restrct our nterest only to sotropc grd for the fxed computatonal doman. Our contnuous efforts wll be rendered to explore the adaptve strategy wth ansotropc grd based on such a drectonal error ndcator for tme-dependent computatonal domans. Although we restrct our focus on two-dmensonal applcaton n current study, the approach descrbed above can be extended for three-dmensonal problems n a straghtforward way. 3.2. Meshng parameters Once the frst dervatves are approxmated based on the current grd and a value for constant C s specfed, Equaton (24) wth respect to optmal spacng and nodal drectons can be solved. In the proposed adaptve process, three grd parameters for each node,.e. the node spacng (δ) equal to the length of the semmnor axs, the stretchng rato (s) and the stretchng drecton (α) along

510 G. X. XU, G. R. LIU AND A. TANI the major axs, are used for grd regeneraton. Based on a set of current grd and a user-specfed value for constant C, these parameters can be computed as C δ=δ 2 = (27) λ 2 s = δ 1 λ 2 = (28) δ 2 λ 1 α=α 1 (29) where λ 1 and λ 2 denote the egenvalues of the matrx m at a node, respectvely. Snce major and mnor axes are orthogonal wth each other, the major axs drecton, α 1, s calculated and chosen as one of the grd parameters. It can be determned wth the egenvectors for the matrx m. In order to generate a set of reasonable grd, two threshold values (δ mn and δ max ) are specfed as the bounds for the commutated spacng resultng from Equaton (27) such that δ mn δ δ max. The maxmum value δ max s manly used to avod meannglessly large spacng due to a vanshng egenvalue n Equaton (27). The value of δ max s usually chosen as the cell spacng adopted n regons wth unform flow behavors. The mnmum value δ mn s used for preventng the algorthm from creatng an excessve refnement of cells n regons wth large gradents, such as n shock regon. Besdes, snce we are nterested n the sotropc grd n current study, the stretchng rato s fxed at 1.0 throughout the numercal examples presented n ths paper. It s clear that the grd parameters for new grd generaton depend strongly upon the choce of the constant, C. In the present practce, t s defned as C =λ max δ mn (30) where λ max s the maxmum egenvalue n the entre doman. Once the grd parameters are determned as descrbed above, the adaptve grd can be regenerated wth the advanced front technque [9] whch wll be descrbed n the followng secton. 3.3. Advancng front technque The advancng front technque has a dstnctve feature that cells and ponts are generated smultaneously. Ths enables the generaton of cells of varable sze and stretchng, and dffers from the Delaunay concepts [20] whch usually connect the nodes that are already dstrbuted n space. In general, the advancng front technque can result n hgh qualty grds, and also offers the flexblty n generatng ansotropc grd and the lablty n handlng movng components. For the sake of contnuous studes to some broad extents, ths technque s adopted n the development of adaptve GSM. In general, the advancng front technque s a bottom-up approach for grd generaton. Frstly, each boundary curve s dscretzed n turn. Nodes are placed on the boundary curve components and then contguous nodes are joned wth straght lne segments. These segments wll become edges of trangular cells n the later stage. The length of these segments must, therefore, be consstent wth the desred local dstrbuton of grd spacngs whch are computed as descrbed n the precedng secton. Successvely, the trangular cells are generated. For a two-dmensonal doman, all the sdes produced n the frst step are assembled as ntal front. At any gven tme, the

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 511 Fgure 6. Trangulaton process used n our adaptve GSM code based on advancng front technque. front contans the set of all the sdes whch are currently avalable to form a trangular cell. Thus, the front s a dynamc data structure whch s updated contnuously durng the generaton process. A sde s selected from the front and a trangular element s generated accordng to computed grd parameters. The trangulaton process may nvolve creatng a new node or smply connectng the sde to an exstng node. Once the trangle s formed, the front s updated by removng the old sde out of the front lst and addng the new sdes nto the front lst. Then the trangulaton proceeds untl the contents n the front become empty. Fgure 6 llustrates the grd generaton process wth the advancng front technque for a square planar doman where the ntal front and the form of the grd at varous stages are depcted. In-depth descrpton about the advancng front technque can be found n [20]. 3.4. The adaptve GSM procedure The overall adaptve GSM soluton procedure s summarzed as follows: Step 1: GSM solutons ncludng frst-order dervatves are approxmated accordng to a set of ntal grd. Step 2: The grd parameters (δ, s and α) across the whole computatonal doman are calculated based on the current grd. Step 3: Dscretzaton of boundary curves s performed usng grd parameters obtaned n Step 2. The dscretzed boundares are used as the ntal fronts for the advancng front method. Step 4: The trangulaton process s carred out wth the advancng front technque usng the grd parameters attaned n Step 2, so as to generate the new adaptve grd.

512 G. X. XU, G. R. LIU AND A. TANI START N adp = 0 Intal grd ntal grd & Intal soluton Grd & updated soluton GSM solver END No N adp Max adp Yes N adp = N adp + 1 Calculaton of regeneraton parameters:, s and Boundary dscretzaton Doman Trangulaton Post-processng of new grd quadtree-based soluton nterpolaton Adaptve grd & Interpolated soluton Fgure 7. Workflow of the overall adaptve procedure adopted n the adaptve GSM. Step 5: Cosmetc technques ncludng dagonal swappng, removal of worse cells and grd smoothng are appled to the new set of grd for the sake of mprovng grd qualty. Step 6: Interpolate the GSM soluton for the current grd onto the new adaptve grd wth the help of quad-tree searchng technque. Step 7: Compute the solutons wth GSM solver based on the new set of adaptve grd and nterpolated solutons obtaned n Step 6. Step 8: Repeat from Step 2 to Step 7, tll the expected accuracy or the maxmum number of adaptve cycles s acheved. The workflow of the overall adaptve GSM procedure as descrbed above s shown n Fgure 7. Durng the adaptve process, a background grd for a problem of nterest s needed manly for geometry defnton. It s defned n a fle wth contents about the nodes composed of the background grd, boundary segments, and supportng curves. For a smple problem as a square doman, two cells are suffcent as the background grd. If a curve boundary exsts n the problem of nterest, a suffcently large number of nodes are requred for accurately representng the shape of the boundary. The fle for the background grd s never altered durng the adaptve process. An ntal grd wth full meshng nformaton for a problem of nterest s requred at the very begnnng of the adaptve process. Usually, the ntal grd s very coarse consstng only of a small number of trangles that can be manually defned. Such an ntal grd can be replaced by a set of well-defned grd, f avalable. The ntal grd wll be used for attanng the frst set of soluton n the use of GSM solver. After that, t wll be repeated replaced by the resultant optmal grd as the adaptve process advances. Some post-processes amng at mprovng qualty of new grd, ncludng removal of undesrable cells, dagonal swappng, and grd smoothng, are carred out rght after each set of new grd s generated wth the advancng front technque. The undesrable trangular cells are not necessarly

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 513 bad n terms of qualty but may be redundant. The removal of those cells can favorably result n the reducton n the number of cells and nodes. The dagonal swappng s acheved by smply reconnectng the exstng nodes locally so that the grd s optmzed. The exstng nodes and the number of cells wll not be changed durng such a process. In the adopted grd smoothng process, an nteror node of the grd s shfted towards the centrod of the polygon formed by ts neghborng cells. As a result, the grd qualty can be mproved n a more sotropc manner. The essence and effects of these cosmetc technques wll be addressed n detals n the numercal testng cases. At the end of each adaptve cycle, the nterpolaton of GSM solutons from the current grd onto a set of adaptve grd s performed. Intally, for each node n the set of new grd, the cell that contans the node of nterest s frst needed to be found. The searchng procedure s facltated to great extents wth the use of quad-tree searchng algorthm [21]. The quad-tree data structure for each set of current grd has been well defned before the generaton of new grd. It s also used for the nterpolaton of grd parameters from the current grd to any newly created node n the processes of boundary dscretzaton and doman trangulaton. Subsequently, the nterpolaton s carred out usng weak Lagrange Galerkn procedure [19]. The nterpolated solutons wll be used as the ntal solutons for the consequent GSM computaton. 4. NUMERICAL EXAMPLES Numercal tests on two-dmensonal Posson and Euler equatons are conducted usng our adaptve GSM code as descrbed above. 4.1. Solutons to Posson equatons 4.1.1. Governng equatons. Posson equaton for a square computatonal doman s frst solved wth our GSM code. The Posson equaton governs many physcal problems, such as the heat conducton problems wth sources. In current study, the Drchlet condtons are prescrbed on all boundares. The pseudo-transent approach s adopted for pursung steady-state solutons. The governng equatons under nvestgaton take the followng form: U t = 2 U x 2 + 2 U f (x, y) (0 x 1,0 y 1) (31) y2 The source, ntal condtons, and analytcal soluton are gven as follows: f (x, y,t)=sn(πx)sn(πy) U(x, y,0)=0 (0 x 1,0 y 1) (32) Û(x, y)= 1 sn(πx)sn(πy) (0 x 1,0 y 1) (33) 2π2 The contour plots of the analytcal soluton to the problem and the magntude of ts gradents are shown n Fgure 8(a) (b). As shown n Fgure 8(b), the red spots represent the regons wth larger gradents, whle blue spots ndcate the regons wth smaller gradents. We expect to see more nodes deployed n the red spots and fewer nodes n the blue areas, when the proposed adaptve process s engaged to remesh the doman.

514 G. X. XU, G. R. LIU AND A. TANI Fgure 8. Contour plots of analytcal solutons to the Posson problem: (a) feld functon and (b) magntude of resultant gradents. 4.1.2. Evaluaton of numercal errors. Three types of numercal errors are evaluated n the study. The convergence error ndex, ε con, takes the form of ε con = nnode =1 error= (U (n+1) U (n) ) 2 / nnode =1 (U (1) U (0) ) 2 (34) where U (n) denotes the predcted value of the feld varable at node at the nth teraton, and n node s the total number of nodes n the doman. The value of ε con s montored durng teratons and used to termnate the teratve process. In most smulatons, n order to exclude the effect due to the temporal dscretzaton, computatons are not stopped untl ε con becomes stablzed, as ndcated n Fgure 9. The numercal error n a GSM soluton for the overall feld s defned usng L 2 -norm of error, n the manner of nnode =1 / nnode (U Û ) 2 =1 Û 2 (35) where U and Û are predcted and analytcal solutons at node, respectvely. Ths type of error s used to compare the accuracy among dfferent schemes. The thrd type of error s the node-wse relatve error, whch s estmated n the form of rerror = U Û / Û (36) The node-wse relatve errors dstrbuted over the computatonal doman are used to dentfy problematc regons n smulatons.

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 515 Fgure 9. Plot of the profle on convergence hstory. 4.1.3. Adaptve solutons. In ths study, we fnd that the controlled advancng front technque may result n rregular trangles wth bad qualty. The rregularty of all trangular cells n the computatonal doman can be quantfed wth the followng formula: ne (a b ) 2 +(b c ) 2 +(c a ) 2 =1 a 2 γ= +b2 +c2 (37) n e where a, b,andc, respectvely, denote the lengths of cell-edges of a trangular cell, and n e stands for the total number of cells n the overall doman. Equaton (37) s derved from the formula proposed by Stllnger et al. [22] for a sngle trangle. Accordng to Equaton (37), the rregularty vanshes for equlateral trangles and postve for all other shapes ncludng sosceles trangles. The bgger the value of γ, the more rregular the grd. Besdes, as ndcated by Lu [23], the average nodal spacng, h, for a set of rregular trangular grd may be estmated as V h = (38) nnode 1 where V and n node denote the area of the overall computatonal doman and the total number of nodes across the doman, respectvely. It s apparent that the averaged nodal spacng decreases as the number of nodes ncreases. The consequent adaptve grds for the Posson problem, whch are attaned wth the proposed adaptve remeshng procedure, are shown n Fgure 10(a) (e). The relatvely unform grd, as shown n Fgure 10(a), s chosen as the ntal grd on whch the GSM soluton s ntally pursued. The grd parameters are then computed accordng to the equdstrbuton crteron as outlned n the prevous secton. For each adaptve remeshng procedure, the threshold value of δ mn s

516 G. X. XU, G. R. LIU AND A. TANI Fgure 10. Adaptve grds at consequent steps: (a) ntal grd; (b) 1st-adaptve grd; (c) 2nd-adaptve grd; (d) 3rd-adaptve grd; and (e) fnal adaptve grd. reduced by half so that the relatvely fner grd wll be generated subsequently. As ndcated n Fgure 10(b) (e), the cells n the regons wth consderably large gradents are reasonably refned. In other words, the regons wth large gradents, as hghlghted n Fgure 8(b), are successfully

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 517 (a) (b) (c) (d) Fgure 11. Comparson of adaptve grds wthout and wth cosmetc treatment: (a) wthout cosmetc treatment; (b) wth removal of trangles only; (c) wth removal of trangles and dagonal swappng only; and (d) wth removal of trangles, dagonal swappng and grd smoothng (cells n red, blue and cyan: selected cells wth poor qualty wthout cosmetc treatment; cells n green: cells after cosmetc treatment). and accurately resolved wth fne grd deployed by the adaptve remeshng technque adopted n the current study. Some cosmetc technques, ncludng removal of redundant cells, dagonal swappng of celledges and grd smoothng, whch are wdely used n grd generaton communty for purposes to mprove the grd qualty, have also been adopted n the proposed adaptve remeshng procedure. The effects of these technques are graphcally llustrated n Fgure 11. As shown n Fgure 11(a), wthout any cosmetc treatment, there are some hghly dstorted cells occurrng n the doman. They may result from the lmted space for the last set of fronts durng the doman trangulaton by the advancng front method. As shown n Fgure 11(b), the removal of redundant trangles cannot only reduce the number of nodes as well as cells, but mprove the grd qualty to some extents. Thanks to smply dagonal swappng treatment, some neghborng cells are reconstructed, whch leads to substantal mprovement n grd qualty, as shown n Fgure 11(c). In such a process, all nodes and the number of cells are remaned nvarant whle the lnkages for affected cells are

518 G. X. XU, G. R. LIU AND A. TANI Table I. Comparson of numercal errors for grds mproved wth cosmetc technques. Case no. Cosmetc treatment No. of nodes Irregularty (γ) L2-norm error 1. No 196 0.07567 6.9575 10 3 2. Removal of trangles 193 0.07223 6.6400 10 3 3. Removal of trangles; dagonal swappng 193 0.06545 5.7165 10 3 4. Removal of trangles; 193 0.04590 5.5950 10 3 dagonal swappng; grd smoothng Table II. Numercal errors of adaptve GSM solutons wth cosmetc technques. No. of Averaged nodal L2-norm error CPU tme Stage nodes spacng (h) (error) (s) γ δ mn δ max Intal 123 9.91E 02 1.57E 02 5.30E 02 2.19E 02 1.00E 01 0.1 1st 304 6.08E 02 2.83E 03 2.14E 01 2.05E 02 5.00E 02 0.1 2nd 1099 3.11E 02 6.88E 04 2.92E+00 1.42E 02 2.50E 02 0.1 3rd 4014 1.60E 02 9.02E 05 5.07E+01 1.21E 02 1.25E 02 0.1 4th 15149 8.19E 03 4.45E 05 9.89E+02 9.90E 03 6.25E 03 0.1 Table III. Numercal errors of adaptve GSM solutons wthout cosmetc technques. No. of Averaged nodal L2-norm error CPU tme Stage nodes spacng (h) (error) (s) γ δ mn δ max Intal 123 9.91E 02 1.65E 02 5.30E 02 3.17E 02 1.00E 01 0.1 1st 309 6.03E 02 3.06E 03 6.90E 01 3.46E 02 5.00E 02 0.1 2nd 1121 3.08E 02 6.13E 04 3.93E+00 2.75E 02 2.50E 02 0.1 3rd 4027 1.60E 02 2.00E 04 5.17E+01 2.34E 02 1.25E 02 0.1 4th 15200 8.18E 03 9.36E 05 1.22E+03 2.09E 02 6.25E 03 0.1 changed only. Furthermore, as depcted n Fgure 11(d), as each nteror node s shfted towards the center of a polygon that s formed by all trangles lnked wth the node, the qualty of overall set of grd s further mproved. Such substantal enhancement can also be found n Table I n terms of number of nodes and rregularty of resultant grds. Numercal errors of the adaptve GSM solutons obtaned wth and wthout the use of cosmetc technques are summarzed n Tables II and III, respectvely. For comparson purpose, the numercal errors based on global refnement procedure are also tabulated n Table IV. As descrbed n the precedng secton about the adaptve procedure, the sets of adaptve grds are obtaned by mantanng the value for δ max and halvng the value for δ mn consequently durng the adaptve process. The evoluton of computatonal errors of adaptve solutons to the Posson equaton can also be found n Fgure 12. As shown n Table II, as the adaptve process advances, the numercal error wll decrease monotoncally, as well as the grd rregularty. Ths confrms that the successful clusterng of nodes

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 519 Table IV. Numercal errors of GSM solutons based on global refnement. No. of Averaged nodal L2-norm error CPU tme Stage nodes spacng (h) (error) (s) γ Intal 123 9.91E 02 1.57E 02 5.30E 02 2.19E 02 1st 311 6.01E 02 6.24E 03 2.26E 01 1.28E 02 2nd 1104 3.10E 02 1.44E 03 1.96E+00 6.17E 03 3rd 4048 1.60E 02 3.73E 04 3.20E+01 3.10E 03 4th 15272 8.16E 03 9.43E 05 5.39E+02 2.09E 03 Fgure 12. Evoluton of computatonal errors wth the changes n averaged cell spacng. toward the regons wth hgh gradents, thanks to the proposed adaptve procedure, gves rse to the remarkably mproved accuracy, wth comparson to cases wth the same number of nodes unformly dstrbuted across the feld. Once a relatvely larger number of nodes are generated, t s necessary to embark the cosmetc technques wth the adaptve procedure to mprove the grd qualty so that the adaptve remeshng procedure stll yelds hgher accuracy than the global refnement process. Ths can be seen by comparng the data n Tables II and III, and n Fgure 12. Fgure 12 also tells that the adaptve GSM can produce results at an expected accuracy wth much less number of nodes than the GSM solutons based on global refnement. As shown n Tables II and IV, for an antcpated accuracy, t s allowed for a much less number of nodes usng n the proposed adaptve remeshng procedure than n the global refnement procedure. Ths wll result n dramatcally reduced demand n computatonal cost, too. It should be mentoned that the proposed GSM solver always gves accurate results for all sets of grds studed here, ncludng those grds wth hghly dstorted cells. Such a behavor agan mples that the developed GSM should be very robust and stable, as already addressed by Lu and Xu [2]. Such an attractve feature wll motvate us toward the development of ansotropc adaptve GSM n mmedate future, where stable and accurate solutons are stll antcpated but more rregular cells wll have to be used.

520 G. X. XU, G. R. LIU AND A. TANI 4.2. Solutons to Euler equatons 4.2.1. Governng equatons. In Cartesan coordnate system, wthout consderaton of source terms, the two-dmensonal Euler equatons for nvscd compressble flows take the followng form: W t + F c =0 (39) where W and F c represent, respectvely, the vectors of conservatve varables, the convectve fluxes. They are n the form of ρv ρ ρu W = ρv, ρuv +n x p F c = ρvv +n y p (40) ( ρe ρ E + p ) V ρ where ρ, u, v, p, and E denote the densty, the two Cartesan velocty components, statc pressure, and the total nternal energy, respectvely. The contravarant velocty s defned as V = v n =n x u +n y v (41) To close the equatons, the equaton of state s also ncluded: p =ρrt (42) where R s the gas constant and T s the temperature of the ar. The well-known second-order ROE method [24], whch s based on the left and rght states wth the help of Barth and Jespersen scheme [25], s used to predct the convectve fluxes. Venkatakrshnan s lmter [26] s employed to avod unphyscal oscllaton n solutons. The explct 5-stage Runge Kutta method, as descrbed prevously, s used here for steady-state solutons. 4.2.2. Boundary condtons. The boundary control surfaces are used to close the smoothng domans at boundares. At a corner pont where two or more boundares meet wth each other, the boundary control surface s splt nto control surfaces for each boundary condton separately. Sold wall In nvscd flows, slp wall condtons are mposed on the sold boundares. As vscous frcton s gnored, the velocty vector must be tangental to the surface. Ths mples that the normal velocty s zero. Hence, the contravarant velocty V s zero. Consequently, the vector of convectve fluxes n the governng equaton reduces the pressure terms alone. That s to say, 0 n x p w F c = (43) n y p w 0 where p w denotes the wall pressure. Farfeld

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 521 a n b d d b n a (a) (b) Fgure 13. Defnton of farfeld boundares: (a) subsonc nflow boundary and (b) subsonc outflow boundary. The farfeld condtons are mposed to the external bounds of the computatonal domans. As addressed n Blazek [15], the numercal mplementaton of the farfeld condtons has to fulfll two basc requrements. Frst, the truncaton of the doman should have no notable effects on the flow soluton as compared wth the nfnte doman. Second, any outgong perturbatons must not be reflected back nto the flow feld. The farfeld condtons based on the concept of characterstc varables are employed n the current study. For the cases of the subsonc or transonc flows that are studed here, the farfeld condtons for subsonc nflow and outflow are appled to external bounds n the smulatons. For a subsonc nflow, as shown n Fgure 13(a), the followng formulatons are mposed at the farfeld bounds: p b = 1 2 {p a + p d ρ 0 c 0 [n x (u a u d )+n y (v a v d )]} ρ b = ρ a +(p a p d )/c 2 0 u b = u a n x (p a p b )/(ρ 0 c 0 ) v b = v a n y (p a p b )/(ρ 0 c 0 ) (44) where ρ 0 and c 0 correspond to values at the reference state that s set equal to the state at the nteror pont. The values at pont a are determned from the freestream state. For a subsonc outflow as shown n Fgure 13(b), the varables at farfeld boundares are obtaned from p b = p a ρ b = ρ d +(p b p d )/c 2 0 u b = u d +n x (p d p b )/(ρ 0 c 0 ) v b = v a +n y (p d p b )/(ρ 0 c 0 ) (45) wth p a beng the prescrbed statc pressure. In the present study, one-layer of dummy cells s generated on the farfeld boundares, whch s used to determne the values of feld varables and ther gradents on the boundares. Besdes, the vortex correcton n 2D, suggested by Usab and Murman [27], s ncluded n the calculaton at the farfeld boundares n order to shrnk the computatonal doman. 4.2.3. Adaptve GSM solutons. The adaptve GSM solver s used to solve the transonc nvscd flow over the NACA0012 arfol. For the testng case, the freestream corresponds to the followng condtons: T =288K, p =1.0 10 5 Pa, Ma=0.8, and α=1.25. Here, T, p,andma denote, respectvely, the temperature, statc pressure, and Mach number of the freestream, and α stands

522 G. X. XU, G. R. LIU AND A. TANI for the angle of attack of the NACA0012 arfol. In ths test case, a strong shock occurs on the upper surface of the arfol and another relatvely weak shock takes place on the lower surface. As shown n Fgure 14(a), based on a set of relatvely coarse grd, both the shocks are captured n the standard GSM solver. However, t s obvous that the two predcated shocks are pretty wde, whch can be attrbuted to the nsuffcent resoluton of grd near the shock regons. The proposed adaptve GSM solver descrbed n the precedng sectons s used to mprove the resolutons n both the shock regons. In ths study, error ndcators based on the frst and second dervatves of densty, respectvely, are tested for comparson and the GSM results for the two optons are presented n detals below. As reported n the secton about GSM solutons to the Posson equaton, the error ndcators based on frst dervatves are suffcently good to dentfy the regons to be refned, whch correspond to relatvely large gradents n the smooth flow feld. Snce the frst dervatves of densty are necessarly approxmated and stored n the GSM analyses of the nvscd flow over the NACA 0012 arfol, t s ntrnsc to re-use them to estmate error ndcators n the adaptve GSM solver. The consequent adaptve grds and solutons are shown n Fgure 14(a) (e). It s strkng that the grds around the strong shock are suffcently refned and the predcted shock front becomes thnner as expected. However, t should be noted that the grd resoluton around the weak shock s not mproved. The smlar observaton has ever been reported by Graldo FX [19]. Besdes, studes about ths opton also reveal that the fnal adaptve results show hgh dependence on the resoluton of boundary nodes on the arfol surfaces. It should be mentoned that the results shown n Fgure 14 have been obtaned wth the help of addtonal constrant for preservng the shape of the arfol wth suffcent enough boundary nodes. When a set of consderably coarse grd, as shown n Fgure 17(a), s used as the ntal grd and the boundary nodes on the wall surfaces are allowed to be freely adapted wth the algorthm descrbed n the precedng secton, subsequent adaptve grds, as shown n Fgure 15, are found to fal n accurately resolvng the leadng edge and thus the predcted strong shock s wrongly pushed towards the tralng edge. The manfest devaton can also be seen n Fgure 16 where the pressure coeffcents on the arfol surfaces are plotted. In summary, the two scenaros are consdered here: In the free adapton scenaro, the boundary nodes on the wall surfaces are purely adapted accordng to the estmated errors based on local frst dervatves of densty. In the second scenaro, a mnmum grd sze s mposed such that the shape of the wall geometry can be well preserved durng the adaptve process. It s notceable that the strong shock occurrng on the upper surface s always successfully captured and resolved wth fner grds n both the scenaros. Whle, the weak shock on the lower surface can be vsble only n the second scenaro where the nodes dstrbuted on the wall are manpulated so as to accurately preserve the physcal shape of the arfols. Alternatvely, the error ndcators based on the second dervatves of densty, as descrbed n Equaton (25), are also examned n ths case. Wth ths opton, a set of consderably coarse grd (see Fgure 17(a)) s used as the ntal grd and the nodes on the arfol surfaces are freely adapted. The adaptve grds as well as the correspondng GSM solutons are shown n Fgure 17. In comparson wth error ndctors usng frst dervatves of densty, relatvely broader zones are dentfed for further refnement n the very frst adaptve process, as shown n Fgure 17(b). The shock front s not suffcently refned, but ts adjacent regons are refned nstead. As shown n Fgure 17(c) (e), as the adapton progresses, the narrower regons across the shocks are successvely refned and thus the shock front s necessarly refned, too. We fnd out that the multple key features n the transonc flow over the NACA0012 arfol, whch occur at the two shock regons, the leadng, and

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 523 Fgure 14. Plots of adaptve grds and contours of densty obtaned wth the adaptve GSM solver based on the frst-dervatves of densty: (a) ntal grd (boundary preservng, n node =8346); (b) 1st-adaptve grd (n node =9381); (c) 2nd-adaptve grd (n node =12788); (d) 3rd-adaptve grd (n node =14139); and(e) fnal adaptve grd (n node =17955). tralng edges, respectvely, are all well resolved n the adaptve analyses. In addton, the nodes dstrbuted on the wall surfaces are freely and necessarly refned so that the shape of the wall geometres s well captured. In summary, the opton for error ndcators based on second dervatves s more accurate to capture all the key features than the opton based on frst dervatves.

524 G. X. XU, G. R. LIU AND A. TANI (a) (b) Fgure 15. Effects of resolutons of ntal meshes on the accuracy of adaptve solutons: (a) close-up vew of poor grd resoluton at the leadng edge and (b) global vew of the adaptve grd and contour plot of densty. -1.5-1 -0.5 C p 0 0.5 1 ntal grd (free adapton on boundary) adaptve grd (free adapton on boundary ntal grd (boundary preservng) adaptve grd (boundary preservng) 1.5 0 0.2 0.4 0.6 0.8 X/C Fgure 16. Predcted pressure coeffcents wth the adaptve GSM solver based on frst dervatves of densty. Fgure 18 shows the predcted pressure coeffcents on the surface of the arfol for both the optons tested n the current study. Apparently, based on the second dervatves of densty, the weak-shock regon s precsely resolved as well as the strong shock regon, leadng edge and

AN ADAPTIVE GSM FOR FLUID DYNAMICS PROBLEMS 525 (a) (b) (c) (d) (e) Fgure 17. Plots of adaptve grds and contours of densty obtaned wth the adaptve GSM solver based on the second dervatves of densty: (a) ntal grd (free adapton on boundares, n node =3927); (b) 1st-adaptve grd (n node =8718); (c) 2nd-adaptve grd (n node =10148); (d) 3rd-adaptve grd (n node =25665); (e) fnal adaptve grd (n node =34079).