Multigrid Methods and Applications in CFD

Transcription

1 Multgrd Metods and Applcatons n CFD Mcael Wurst 0 May 009 Contents Introducton Typcal desgn of CFD solvers 3 Basc metods and ter propertes for solvng lnear systems of equatons 4 Geometrc Multgrd 3 5 Algebrac Multgrd 7 6 Examples 9 7. Advantages and Dsadvantages of Algebrac Multgrd Introducton Multgrd Metods are a group of algortms n numercal analyss for solvng lnear systems of equatons. Tey are especally elgble for ellptcal problems, for example te Posson equaton. Ts type of equaton arses among oters from te dscretsaton of te pressure correcton equaton n te soluton of te Naver-Stokes equaton. Te advantage ere s te lnear dependence of te soluton tme from te number of grd ponts N wc could be stated as optmal. In ts work, two dfferent multgrd metods (geometrc multgrd and algebrac multgrd) are presented and te dfferences between tem are sown. Addtonally ter applcaton to computatonal flud dynamcs s demonstrated wt an example. Typcal desgn of CFD solvers Tere are bascally two approaces for te soluton of te Naver-Stokes equaton, te coupled solver and te segregated solver. Te Naver-Stokes equaton s a nonlnear system of equaton. It contans 4 unknowns, tree velocty components and te pressure, but conssts only of tree equatons. Terefore te conservaton of mass s used as an addtonal equaton. In te segregated solver one uses an teratve - -

2 sceme for te soluton. Frst te momentum equaton s solved wt te pressure dstrbuton from te last tme step wle n a second step te flud velocty s corrected based on te pressure from te pressure-correcton equaton. Ts pressure-correcton equaton s a Posson equaton wose coeffcent matrx s ll condtoned. Neverteless te soluton of te pressure-correcton equaton s needed to a tgt tolerance for guaranteeng mass conservaton. Ts makes a fast, robust lnear solver necessary wc can be aceved by multgrd metods. Te coupled solver dscretses te momentum and te pressure correcton equaton n suc a way tat one gets a bg coupled block equaton system wc solves te pressure and te tree velocty components n one matrx system. Ts system can become qute large wc explans te need for a fast lnear solver based on multgrd metods. 3 Basc metods and ter propertes for solvng lnear systems of equatons In te coupled as well as n te segregated approac one gets a lnear system of equatons gven by or n components A u = f au = f In ts context u s te exact soluton, A s a sparse (symmetrc) matrx of sze n n and v denotes an approxmaton to te exact soluton u. Tere are now two mportant measures of v as an approxmaton to u: Absolute error e gven by e = u v. Te problem s tat te absolute error s naccessable as te exact soluton tself. Resdual r; s te amount by wc te approxmaton v fals to satsfy te orgnal problem Au=f: r = f A v. Bot measurements are vectors and can be measured by any vector norm, lke te L or te L norm: e = max e n and e = n = e. - -

3 In general, tere are now two possbltes for te soluton of te lnear system of equatons, drect metods and teratve or relaxaton metods. Drect metods lke Gauss elmnaton solve te problem to te computatonal accuracy but ave g computatonal costs. Iteratve metods nstead solve te problem only by an approxmaton but ts could be sometmes suffcent and terefore less tme consumng. Two teratve metods are te Jacob relaxaton and te Gauss-Sedel relaxaton. Te dea of te Jacob metod s to solve te t equaton for te t unknown usng te approxmatons from te last teraton for te oter unknowns. It can be wrtten by u f a u ( n+ ) ( n) =. a Te Gauss-Sedel metod ncorporates a smple cange: components of te new approxmaton are used as soon as tey are computed: Ts means tat components of te approxmaton vector v are used as soon as tey are updated: u = f - au - au a < > (n+) (n+) (n). Applyng tese teratve metods one can observe tat te error decreases eac teraton. But after some teratons te error does not reduce anymore. One can explan ts beavour assumng te error beng a vector sum from te egenvectors of te matrx A: (0) e n = k = c w k k. Tese egenvectors are connected closely to te egen modes of te problem. If we now apply an teratve metod, t s found out tat te error components from gfrequent egenvectors (or modes) dsappear soon and te error components from low-frequent egenvectors (or modes) do not dsappear. Terefore tese teratve metods are also called smooters. 4 Geometrc Multgrd Te open queston s now ow to mprove tese teratve metods n order to reduce all error components. Te dea bend te multgrd metods s now to ave at least two dfferent grds of dfferent sze, one coarse grd and one fne grd. One can sow tat te error on te coarse grd looks more oscllatory tan on te fne grd or n oter - 3 -

4 words te error on a fne grd looks less smot tan on a coarse grd. Ts can be used now for te followng procedure. Frst oen apples an teratve sceme on a fne grd untl te error only conssts of smoot, low frequent components. Ten ts error s transformed to te coarse grd were t looks more oscllatory. If we now relax on ts grd t sould effectvely elmnate addtonal error modes. Ts could be contnued wt more tan one coarse grd but t leaves te problem f te error s stll smoot on te fne grd. Anoter possblty s te use of te resdual equaton and applyng te followng sceme, called te correcton sceme: Here Relax on A u = f on Ω to obtan an approxmaton v Compute te resdual r = f-av and relax on te resdual equaton Ω to obtan an approxmaton to te error e Correct te approxmaton obtaned on Ω Ω : v v + e wt superscrpt denotes te fne grd wle Ae = r on Ω wt te error estmate obtaned on Ω wt superscrpt denotes te coarse grd. Ts basc sceme could also be used for more tan one coarse grd wc wll be sown later. Based on ts sceme te followng formal steps ave to be defned: Selecton of te coarse grd(s) Defnton of transformaton operators for vectors and te matrx A from coarse to fne grds and vce versa In geometrd multgrd te selecton of coarse grds s based on geometrc relatons, for example n a one dmensonal problem every second fne grd pont s defned as a coarse grd pont. Wt ts relaton t s possble to desgn as many coarse grds as desred. Te defnton of transformaton operators s spltted nto two parts. Te transformaton from coarse to fne grds s denoted as nterpolaton / prolongaton and te transformaton from fne to coarse grd s called restrcton. Te prolongaton s defned as a lnear nterpolaton. In te one-dmensonal case ts s gven by v v = f te pont s bot on te fne and on te coarse grd and by ( v + = v + v + ) wt n 0-4 -

5 f te pont s only on te fne grd. Ts relaton can be seen n Fgure. Fgure : Interpolaton / Prolongaton n te one-dmensonal case Te restrcton s normally calculated by full wegtenng wc means tat te coarse grd pont s defned by an wegted average of ts negbours. For te onedmensonal case t s gven by (compare Fgure ) n v = ( v + v + v+ ) 0. 4 Fgure : Restrcton n te one-dmensonal case For te one-dmensonal case te restrcton operator and te prolongaton operator can also be wrtten n matrx form: Restrcton operator: I =

6 Prolongaton operator: I =... It can be seen tat te restrcton and te prolongaton operator are transpose to eac oter except for a constant: I ( ) T = c I. Ts property s called varatonal property. Anoter mportant concept s te calculaton of te matrx A on te coarse grd. Ts can eter be done by dscretzng te problem on te coarse problem or n general by multplyng te fne grd matrx condton): A wt te prolongaton and restrcton operator (called Galerkn A = I A I. Now ts possble to apply any multgrd sceme, for example te V-cycle wc s based on te correcton sceme. It s buld up by arbtrary coarse grds and s called V-cycle because t goes from te fnest grd down to te coarsest grd and te up agan (compare Fgure 3): Relax on u = f Contents f = A I r ν tmes wt ntal guess v Relax on Compute A u = f f = I r 4 4 ν tmes wt ntal guess v Relax on Compute A u = f f = I r ν wt ntal guess 4 v. Solve A L u L = f L (maybe possble wt drect solver)

7 Correct v v + I v Relax A u = f ν tmes wt ntal guess v 4 Correct v v + I v 4 4 Relax A u = f ν tmes wt ntal guess v Correct v v + I v Relax on A u = f ν tmes wt ntal Relax g uess v Restrcton 4 8 Prolongaton 6 5 Algebrac Multgrd Geometrc multgrd s very effectve f a problem s solved on a structured mes but for many problems an unstructured mes s necessary were t s dffcult to defne coarse grd based on geometrc relatons. Instead one wants to defne coarse grds n an algebrac sense. But for transformng a multgrd sceme lke te V-cycle from geometrc to algebrac multgrd one also needs to defne te followng tngs n an algebrac sense: Smootness of an error Fgure 3: V-cycle Transfer of vectors and te matrx A from coarse to fne grds and vce versa Applcablty of smooters (Gauss-Sedel, Jacob) - 7 -

8 In algebrac multgrd an error s defned as a smoot error f or n components: Ae 0 ae + ae 0 ae ae Tat means a smoot error can be approxmated by a wegted average of negbourng varables. An mportant concept for te selecton of te coarse grd s te defnton of strong nfluence and dependence. Bot defntons take nto account tat for dagonal domnant matrx te t row s assocated wt t unknown. Of course, t usually takes all of te equaton varables to determne any gven varable precsely but maybe t s possble to ave a preselecton of varables tat are more mportant tan oters Defnton : Gven a tresold value 0< θ, te varable (pont) u strongly depends on te varable (pont) u f a θ max{ a }. k k Ts says tat te varable strongly depends on grd a pont f te coeffcent s comparable n magntude to te largest off-dagonal coeffcent n te t equaton. We can state ts defnton from anoter perspectve. Defnton : If te varable u strongly depends on te varable u, ten te varable u strongly nfluences te varable. Two sets can be derved from tese two defntons. S : set of ponts tat strongly nfluence, tat s te ponts on wc te pont strongly depends. { : θ max { k} } S = a a k T S : set of ponts tat strongly depend on te pont. { : } T S = S Tese sets are mportant for te selecton sceme of te coarse grd. Te coarse grd sould fulfl te followng condtons: u

9 Smoot error can be approxmated accurately Good nterpolaton to te fne grd Sould ave substantally fewer ponts, so te problem on te coarse grd can be solved wt lttle expense Te selecton process s carred out lke ts: ) Defne a measure to eac pont of ts potental qualty as a coarse (C) pont: amount λ T of members of S ) Assgn pont wt maxmum λ to C-pont T 3) All ponts n S become fne (F) ponts 4) For eac new F pont : ncrease te measeure λ k for all eac unassgned pont k tat strongly nfluence : k S 5) Do )-4) untl all ponts are ass gned If te coarse grds are selected, transformaton operators must be defned between te grds. In general te component ( ) determned lke ts: e f C = e f F ( I e) ω C Te nterpolaton wegts I e of a vector on te fne grd can be ω ave nformaton about te negbourng, off-dagonal elements wc can be dvded nto te negborng coarse grd ponts tat strongly nfluence I, te negborng fne grd ponts tat strongly nfluence I and nto ponts tat do not strongly nfluence I (fne and coarse grd ponts). 6 Examples For testng te algebrac multgrd a test case (compare [3]) was set up. Te Posson equaton s solved n four dfferent varants at four dfferent domans: - 9 -

10 Te coeffcents a, b and c are dfferent on eac doman: a=000 a= c= c= b=0 b= a= a=000 c= c= b=0 b=0 Te equaton s dscretzed usng te fnte volume metod and solved on a 00 x 00 mes. Fgure 4 tll Fgure 6 sow te tree frst coarse levels wt red ponts beng te coarse varables on te next grd 'Y' 'X' Fgure 4: Grd - 0 -

11 'Y' 'X' Fgure 5: Grd 'Y' 'X' Fgure 6: Grd 8 7. Advantages and Dsadvantages of Algebrac Multgrd Advantages of algebrac multgrd are tat te soluton s fast and robust and s very effectve for segregated solvers. Dsadvantages are tat te trple matrx operaton at te Galerkn step s a very expensve step and ard to parallse. Te defnton of coarse grds leads to a g setup-pase and g storage requrements. Addtonally ts algortm s not sutable - -

12 for coupled solvers were te aggregaton based algebrac multgrd s a cure aganst t []. - -

13 Bblograpy [] M. Raw, Robustness of coupled Algebrac Multgrd for te Naver-Stokes equatons, AIAA Paper (996) [] W. Brggs, V. Henson und S. MacCormck:, A multgrd tutoral;. edton; SIAM Pladelpa (000); [3] K. Stüben, Algebrac Multgrd (AMG): An Introducton wt Applcatons; GMD Report 70 (999); - 3 -