Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 54
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15 are symmetricalleading to symmetrical systems that can be solved using the CG method discussedabove. However, the matrix A obtained from the discretization of the generalconservation equation arising in CFD applications is unsymmetrical yielding anunsymmetrical system of equations. To be able to solve this system using the CGmethod, it should be transformed into a symmetrical one [20].
One way to do that isto rewrite Eq. (10.1) as0ATA0"_//#¼ b0ð10:121Þ_where / is a dummy variable added in order to convert the original unsymmetricalsystem into a symmetrical one amenable to a solution by the CG method. Whenapplied to this system, the CG method results in two sequences of CG-like vectors,the ordinary sequence based on the original system with the coefficient matrix A,from which ϕ is calculated, and the shadow sequence for the unneeded system with_the coefficient matrix AT , from which / can be calculated if desired. Because of thetwo series of vectors the name bi-conjugate gradient (BiCG) is coined to themethod.
The same terminology used with the CG method is used here. The series ofordinary vectors for the residuals and search directions _are denoted by r and d,_respectively, and their shadow equivalent forms by r and d. The bi-orthogonality ofthe residuals is guaranteed by forming them such that_ðmÞTr ðnÞ T_rðnÞ ¼ rrðmÞ ¼ 0m\nð10:122Þand the bi-conjugacy of the search directions is fulfilled by requiring that_ ðnÞTd T _ ðmÞAdðmÞ ¼ dðnÞ AT d ¼ 0m\nð10:123ÞMoreover, the sequences of residuals and search directions are constructed such thatthe ordinary form of one is orthogonal to the shadow form of the other.Mathematically this is written as_ðnÞrT T _ðmÞdðmÞ ¼ rðnÞ dm\nð10:124ÞSeveral variants of the method, which has irregular convergence with the possibilityof breaking down, have been developed and the algorithm described next is due toLanczos [21, 22].34210Solving the System of Algebraic EquationsThe BiCG algorithm of Lanczos can be summarized as follows:d ( 0 ) = r ( 0 ) = d( 0 ) = r ( 0 ) = b A ( 0 ) (choose starting directions)iterate starting at ( n ) until convergence( r( ) ) r( )( d( ) ) Ad( )Tn(n)=( n+1)=( n+1)Tn(n)+(n)(n) (n)d(n)( r( ) )=( r( ) )n+1n(Choose factor inn(n)r=rr ( n+1) = r ( n )( n+1)n(Obtain newd direction))(n)Ad (calculate new r residual)A Td ( n ) (calculate new r residual)Tr ( n+1)Tr(n)(Calculate coefficient to conjugate residual)d( n+1) = r ( n+1) +( n+1) ( n )(obtain new searchd direction)d ( n+1) = r ( n+1) +( n+1) ( n )(obtain new searchd direction)ddThe BiCG method necessitates a multiplication with the coefficient matrix andwith its transpose at each iteration resulting in almost double the computationaleffort required by the CG method per iteration.Preconditioning may also be used with the BiCG method.
For the same terminology as for the preconditioned CG method, a robust variant of the methoddeveloped by Fletcher [23] is summarized next.The preconditioned algorithm for the BiCG of Fletcher is as follows(P representing the preconditioning matrix):r(0) = r(0) = b A= P 1r ( 0 ) , d( 0 ) = P Tr ( 0 ) (choose starting directions)iterate starting at ( n ) until convergence( r( ) ) P r( )=( d( ) ) Ad( )Tn(n)( n+1)=(n)+r ( n+1) = r ( n )r ( n+1) = r ( n )( r( ) )=( r( ) )n(Choose factor inn(n) (n)d(Obtain newd direction))Ad ( n ) (calculate new r residual)(n) T (n)A d (calculate new r residual)TP 1r ( n+1)TP 1r ( n )d( n+1) = P 1r ( n+1) +( n+1)(0)(n)n+1( n+1),dn1Tnd(0)=P rT( n+1)+(Calculate coefficient to conjugate residual)( n+1) ( n )d(obtain new searchd direction)( n+1) ( n )(obtain new searchd direction)d10.3Iterative Methods343Other variants of the BiCG method that are more stable and robust have beenreported such as the conjugate gradient squared (CGS) method of Sonneveld [24], thebi-conjugate gradient stabilized (Bi-CGSTAB) method of Van Der Vorst [25] and thegeneralized minimal residual method GMRES [13, 26–29].
These methods are usefulfor solving large systems of equations arising in CFD applications as they areapplicable to non-symmetrical matrices and to both structured and unstructured grids.10.4 The Multigrid ApproachThe rate of convergence of iterative methods drastically deteriorates as the size ofthe algebraic system increases, with the drop in convergence rate even observed inmedium to large systems after the initial errors have been eliminated.
This hasconstituted a severe limitation for iterative solvers. Luckily it was very quicklyfound that the combination of multigrid and iterative methods can practicallyremedy this weakness.Developments in multigrid methods started with the work of Fedorenko [30](Geometric Multigrid), Poussin [31] (Algebraic Multigrid), and Settari and Azziz[32], and gained more interest with the theoretical work of Brandt [33]. Whilehigh-frequency or oscillatory errors are easily eliminated with standard iterativesolvers (Jacobi, Gauss-Seidel, ILU), these solution techniques cannot easily removethe smooth or low frequency error components [34].
Because of that these solutionλ1λ2λ3λ4λ5Fig. 10.4 Schematic of different error modes in a one dimensional grid34410Solving the System of Algebraic Equationsmethods are denoted by smoothers in the context of multigrid methods. An illustration of error frequency is shown in Fig. 10.4 where the variations of errorfrequency modes for a one dimensional problem are plotted.The error modes shown in Fig. 10.4 vary from high frequency of short wavelength k1 to low frequency of long wavelength k5 and are plotted collectively on thetop of the figure. The one dimensional domain is discretized using the onedimensional grid shown and the various modes are separately plotted over the samegrid. As can be seen, the high frequency error appears oscillatory over an elementand is easily sensed by the iterative method.
As the frequency of the error decreasesor as the wavelength ðkÞ increases, the error becomes increasingly smoother overthe grid as only a small portion of the wavelength lies within any cell. This getsworse as the grid is further refined, leading to a higher number of equations andexplaining the degradation in the rate of convergence as the size of the systemincreases.Multigrid methods improve the efficiency of iterative solvers by ensuring that theresulting low frequency errors that arise from the application of a smoother at anyone grid level are transformed into higher frequency errors at a coarser grid level.By using a hierarchy of coarse grids (Fig. 10.5), multigrid methods are able toovercome the convergence degradation.Generally the coarse mesh can be formed using either the topology and geometryof the finer mesh, this is akin to generating a new mesh for each coarse level on topof the finer level mesh or by direct agglomeration of the finer mesh elements[35–40]; this approach is also known as the Algebraic MultiGrid Method (AMG).In the AMG no geometric information is directly needed or used, and theagglomeration process is purely algebraic, with the equations at each coarse levelreconstructed from those of the finer level, again through the agglomeration process.
This approach can be used to build highly efficient and robust linear solversFig. 10.5 A schematic of thehierarchy of grid systems usedwith the multi grid approach10.4The Multigrid Approach345for both highly anisotropic grids and/or problems with large changes in thecoefficients of their equations.In either approach, a multigrid cycling procedure is used to guide the traversal ofthe various grid hierarchies. Each traversal from a fine grid to a coarse one involves:(i) a restriction procedure, (ii) the setup or update of the system of equations for thecoarse grid level, and (iii) the application of a number of smoother iterations.A traversal from a coarse grid to a finer one requires: (i) a prolongation procedure,(ii) the correction of the field values at the finer level, and (iii) the application of anumber of smoother iterations on the equations constructed during restriction.
Thevarious steps needed are detailed next.10.4.1 Element Agglomeration/CoarseningThe first step in the solution process is to generate the coarse/fine grid levels by anagglomeration/coarsening algorithm. Three different approaches can be adopted forthat purpose. In the first approach, the coarse mesh is initially generated and the finelevels are obtained by refinement [41, 42].
This facilitates the definitions of thecoarse-fine grid relations and is attractive in an adaptive grid setup [41–43].A major drawback however, is the dependence of the fine grid distribution on thecoarse grid. In the second method, non-nested grids are used [44] rendering thetransfer of information between grid levels very expensive. In addition, bothapproaches do not allow good resolution of complex domains. In the thirdapproach, recommended here, the process starts with the generation of the finestmesh that will be used in solving the problem. Then coarse grid levels are developed through agglomeration of the fine-grid elements [45, 46], as shown inFig.
10.6, with the agglomeration process based either on the elements geometry oron a criterion to be satisfied by the coefficients of neighboring elements. Thediscussions to follow are pertinent to the third approach.Coarse grid levels are generated by fusing fine grid elements through anagglomeration algorithm. For each coarse grid level, the algorithm is repeatedlyapplied until all grid cells of the finer level become associated with coarse grid cells.Agglomerationi2i6i5i4i1Ii3Fig. 10.6 Agglomeration of a fine grid level to form a coarse grid level34610Solving the System of Algebraic EquationsDuring this heuristic agglomeration process, fine grid points are individually visited.
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