a fully coupled newton-krylov solver for turbulent

ICAS 2002 CONGRESS

A FULLY COUPLED NEWTON-KRYLOV SOLVER FOR TURBULENTAERODYNAMIC FLOWS

Todd Chisholm and David W. ZinggUniversity of Toronto Institute for Aerospace Studies

4925 Dufferin Street, Toronto, ONCanada, M3H 5T6

[email protected]

Keywords: airfoil, Navier–Stokes,high-lift, Newton’smethod,GMRES

Abstract

A fast Newton-Krylov algorithm is presentedfor solvingthecompressibleNavier-Stokesequa-tionson structuredmulti-block grids with appli-cationto turbulentaerodynamicflows. Theone-equationSpalart-Allmarasmodelis usedto pro-vide theturbulentviscosity. Theoptimizationofthe algorithmis discussed.ILU(4) is suggestedfor a preconditioner, operatingon a modifiedJa-cobian matrix. An efficient startupmethodtobring the systeminto the region of convergenceof Newton’smethodis given.Threetestcasesareusedto demonstrateconvergencerates. Single-elementcasesaresolvedin lessthan100secondsonanengineeringworkstation,while thesolutionof a multi-elementcasecanbefoundin lessthan25minutes.

1 Introduction

Recently, Newton-Krylov methods have beenshown to be very effective in reducing the timerequired to compute numerical solutions to theNavier-Stokes equations. Blanco and Zingg [4]studied the solution of the Euler equations onunstructured grids with a matrix-free Newton-Krylov method. Geuzaine [7] used a simi-lar method with the compressible Navier-Stokes

Copyright c© 2002 by T. Chisholm and D.W. Zingg.Published by the International Council of the AeronauticalSciences, with permission.

equations, modeling turbulence with the Spalart-Allmaras model. Barth and Linton [3] stud-ied a parallel implemention of a Newton-Krylovsolver on unstructured grids for two- and three-dimensional flows. Pueyo and Zingg [14] solvedthe turbulent, compressible Navier-Stokes equa-tions on structured grids. They have demon-strated that this approach is competitive withstate of the art multigrid methods. However,their work was limited to representing turbulencewith the algebraic Baldwin-Lomax model on sin-gle block grids. Here we discuss the solution ofthe Navier-Stokes equations on single- and multi-element airfoils, using the one-equation Spalart-Allmaras turbulence model [17].

2 Algorithm Description

2.1 Governing Equations

We study the solution of the steady compress-ible thin-layer Navier-Stokes equations on struc-tured grids. A generalized curvilinear coordinatetransformation is used to map the physical spaceto a rectangular computational domain. The useof multiple blocks allows for complex geome-tries such as multi-element airfoils. A circula-tion correction is used to reduce the effect of thefarfield boundary. The Spalart-Allmaras turbu-lence model, including trip terms, is implementedas described in [8], with a small change in thecalculation of the modified voriticity factor, first

1

TODD CHISHOLM AND DAVID W. ZINGG

used by Ashford [1]

S̃= S fv3 +ν̃T

κ2d2 fv2 (1)

fv2 =(

1+χ

cv2

)−3

(2)

fv3 =(1+χ fv1)(1− fv2)

χ(3)

with cv2 = 5.0. The original form, which al-lowed S̃ to become negative, introduced a localminimum quite close to the solution root at somenodes at the edge of recirculation bubbles. Thiswould cause the residual to hang, despite the ma-jority of the flow being converged. The new formseems to avoid this problem.

2.2 Spatial Discretization

The spatial discretization follows that used byNelson, et al. [11]. Second-order centred differ-ences are used to approximate derivatives. BothJameson’s [9] scalar and Swanson and Turkel’s[18] matrix second- and fourth-difference dissi-pation models can be used to stabilize the centreddifference scheme. A pressure switch is used tocontrol the activation of second-difference dissi-pation. The matrix dissipation model uses twoswitchesVl andVn to avoid the effect of overlysmall eigenvalues in the flux Jacobian matrix. WeuseVl = Vn = 0.1 for subsonic cases, andVl =0.025 andVn = 0.25 for transonic cases. The tur-bulent viscosity convection and diffusion termsare discretized using first-order upwinding andsecond-order centred differencing, respectively,as suggested in [17], with two differences. Theturbulence equation is scaled byJ−1, the grid-metric Jacobian, and the state variableν̃ is re-placed withJ−1ν̃. These changes help keep theentries in each block of the Jacobian of similarmagnitude. This improves the conditioning of thesystem, making the Krylov solver more efficient.

Since a Newton solver is used to solve theresulting nonlinear system, it is important thatall of the boundaries be handled fully implicitly.This includes the interfaces between blocks. TheNavier-Stokes equations are solved on these in-terfaces in the same manner as the interior nodes.

2.3 Newton-Krylov Algorithm

The Nonlinear System.After spatial discretiza-tion, we have a system of the form

R(Q̂∗) = 0 (4)

where each block ofQ̂, the conservative statevariables with the turbulence variable, is

Q̂i = J−1i Qi = J−1

i [ρi ,ρui ,ρvi ,ei , ν̃i ]T

To find Q̂∗ which satisfies Eq. 4, we applythe implicit Euler method repeatedly until someconvergence criterion, typically‖R‖2 < 10−12, isreached:

[I

∆tn− ∂Rn

∂Q̂n

]∆Q̂n = Rn

Q̂n+1 = Q̂n +∆Q̂n

We call these theouter iterations. When thetime step is increased towards infinity, Newton’smethod is approached. If∆t is increased appro-priately as‖R‖ decreases, the quadratic conver-gence characteristic of Newton’s method can beachieved, while dramatically increasing the re-gion of convergence. Note that, in order for New-ton’s method to converge quadratically,∂R

∂Q mustbe accurate. This requires that the equations befully coupled.

The Linear System. In order for∆Q̂n to befound, a linear system needs to be solved. Thissystem tends to be very large, so that direct so-lution is prohibitive in both memory and time.Fortunately, finding the exact∆Q̂ is not neces-sary, and we may settle for finding an approxima-tion. This is an inexact-Newton method. Thereare a number of popular methods of finding theapproximate solution of the linear system. Theproper selection and use of this method is crucialto the success of the overall solver [2]. The mostsuccessful class are the Krylov iterative meth-ods. Specifically, the preconditioned GeneralizedMinimum Residual (GMRES)[16] has proven tobe effective for aerodynamic systems. We callthese linear iterations theinner iterations.

Over-solving the linear system needs to beavoided for efficiency. A stopping criterion is

2

A Fully Coupled Newton-Krylov Solver for Turbulent Aerodynamic Flows

needed for the inner iterations. There are twoconsiderations. First, we use a target reduction inthe inner residual. Pueyo [13] found a one orderof magnitude reduction ideal to balance outer andinner iteration efficiency. This is appropriate inthe turbulent case during the final Newton stage,but not in the startup. This will be discussed inthe next section. The second consideration is set-ting the maximum number of iterations of GM-RES. The amount of memory and CPU time in-creases with each GMRES iteration, so a limit isrequired. GMRES may be restarted, which keepsthe memory requirements lower, while allowingfurther solution of the linear system. However,this can significantly slow the linear system con-vergence, due to the very poor conditioning seenin these systems. Typically, we do not use restart-ing for this reason.

The convergence rate of GMRES is very sen-sitive to the condition number of the matrix.Since the Jacobian of the equations being solvedis typically extremely ill-conditioned, a good pre-conditioner is required to limit the number of in-ner iterations. Pueyo and Zingg [14] have shownthat an incomplete LU preconditioner (ILU) [5]with two levels of fill minimizes solution time.They also found that a preconditioner based ona first-order Jacobian is more efficient than theexact Jacobian, both in saving memory and CPUtime. The first-order Jacobian is formed by usingonly second-difference dissipation. This reducesthe number of entries per equation to five insteadof nine. It tends to give a better conditioned ma-trix, which leads to a more stable LU factoriza-tion. The coefficient of the second-difference dis-sipation used in the Jacobian matrix is found by

εl2 = εr

2 +σεr4

whereσ is found empirically.Pueyo and Zingg used scalar artificial dis-

sipation. The use of matrix dissipation signifi-cantly worsens the conditioning of the precondi-tioner, due to a reduction in diagonal dominance.This requires further modifications. Two meth-ods have been investigated. The most obviousis to include a time step. In order to achievepreconditioner stability, a sufficiently small time

step needs to be taken. This is due to the exis-tence of negative entries on the diagonal. Largetime steps (with correspondingly small additionsto the diagonal) have the possibility of worseningthe condition of the matrix by reducing the diago-nal. Small time steps are obviously not desirable,as they dramatically slow outer residual conver-gence. For this reason, we add the time step onlyto the matrix used for the preconditioner.

Another method of increasing diagonal dom-inance is to increase the value of the matrix dissi-pation switches (Vn andVl ) used in the first-orderJacobian matrix the factorization is based on.These switches set the minimum level of dissipa-tion, making them a natural choice for controllingsmall diagonals. As they are increased towardunity, scalar dissipation is approached. Note thatthe switches in the residual evaluation remain thesame, so that the final solution is unaffected. Theswitches in the Jacobian used for the outer iter-ations are also unchanged, as inaccuracies therecan adversely affect convergence. There are twoconflicting effects that have to be balanced to op-timize the preconditioner settings. The precondi-tioner must be conditioned well enough to be sta-ble, but remain close enough to the true Jacobianto provide adequate clustering of the eigenval-ues. A useful tool in evaluating a preconditioneris its condition number estimate, as discussed byChow and Saad [5]. This is simply theL2 normof the solution toLU ·~c=~1. If a preconditioner isperforming poorly, and has a high condition num-ber estimate of the order of 107 or greater, morediagonal dominance is needed. It is usually betterto err on the side of being too well conditioned.While this will slow convergence somewhat, thelinear solves are much more robust.

To help reduce the effect of entries droppedfrom the preconditioner, reverse Cuthill-McKeereordering is used [6]. Good reordering is espe-cially important in the multiblock case, due to theincreased number of far off-diagonal entries re-sulting from the block boundaries.

The GMRES algorithm only requires matrix-vector multiplies, and does not explicitly requirethe matrix, except in forming the preconditioner.A Jacobian-free implementation of GMRES may

3


be used, which has been found by Pueyo [13]to be faster, as well as resulting in significantmemory savings. Unfortunately, the necessarychanges in the Jacobian during startup eliminatethis benefit when the Spalart-Allmaras turbulencemodel is used.

Startup. Grid sequencing is used to helprapidly eliminate initial transients. This involvespartially solving on one or more coarse grids,each formed by removing every other grid linein each direction from the next finer grid. Thesolution is interpolated from the coarse grid tothe fine grid. This helps to bring the solution onthe fine grid closer to the region of convergenceof Newton’s method, allowing higher initial timesteps. Grid sequencing is also particularly help-ful in initializing the turbulence quantities. Themodel will tend to take a number of iterationsat a low time step before tripping occurs. Theseiterations are very expensive to perform on thefine grid, but can be rapidly done on a twice-coarsened grid. It is important to ensure that theflow on the coarsened grids are tripped beforepassing to the fine grid. If a region is not prop-erly tripped when the Newton’s stage is started,divergence is likely. Raising the trip coefficientct1 in the turbulence model help encourage tran-sition on the coarse grids.

Examination of the production and destruc-tion terms of the turbulence model reveals thatthese terms are unstable with negativeν̃. There-fore, it is crucial to take steps to ensure thatν̃ re-main positive. This is a particular problem duringthe early iterations, when the solution is rapidlychanging. There are a number of strategies whichcan be used to avoid negative values. Spalart andAllmaras [17] recommend that a modified lin-earization of the equations be used during startup.This modifies the turbulence model Jacobian sothat it becomes anM matrix. The flow por-tion of the matrix is unchanged. While this pre-vents quadratic convergence, it was found thatthis modification is only necessary during the im-plicit Euler stages. This modification will be ef-fective only if the linear system is solved suffi-ciently well. If the same tolerance appropriatefor the laminar equations is used, large negative

turbulence viscosities show up very quickly, de-spite using a small time step. Inner tolerancesof approximately 10−4 are necessary to see theadvantage of the modified turbulence model Ja-cobian. This is not nearly as detrimental as itwould seem at first glance. The first few ordersof magnitude reduction of the linear residual hap-pen much faster than in the laminar case, often inonly two or three iterations. This phenomenononly seems to occur when the modified Jacobianis used. Note that during startup, Jacobian-freeGMRES cannot be used, since non-negative tur-bulent viscosities cannot be guaranteed.

Nemec and Zingg [12] used approximatefactorization with the modified Jacobian duringstartup with good results. This is an inexpen-sive approach to approximately solving the linearsystem. The turbulence quantities are decoupledfrom the mean flow equations, and the linear sys-tem for each block is solved separately.

The approach followed by Geuzaine [7] wasto use a variable time step based on the switchedevolution relaxation method of Mulder and vanLeer [10]. The time step in the early iterationsis limited, so that the updates to the turbulencequantity are well bounded.

We have found that a combination of thesemethods works well. Starting with the modifiedJacobian allows larger time steps while maintain-ing stability. Variable time steps progress the so-lution rapidly. The method of adjusting the ref-erence time step is somewhat different than SER,though similar in spirit:

∆tre f = α[

1||R||2

]β(5)

Choosingα = 10.0 andβ = 1.0 gives good re-sults. The time step of the both the Navier-Stokesand turbulence model equations also vary spa-cially. The geometric time step following Pul-liam [15] is used for the flow equations:

∆t =∆tre f

1+√

J(6)

whereJ is the Jacobian of the metric of the curvi-linear coordinate transformation. The turbulence

4


model uses a time step based on the distance tothe closest wall:

∆t = ∆tre f · τ ·d2 (7)

This is inspired by the destruction term of themodel, which is proportional to the inverse of thesquare of the distance. Virtually all instabilitiesoccur in the region close to the body, where thedestruction term is strong.τ allows us to set anappropriate scaling for the turbulence model timestep. 10.0 gives good convergence while keepingthe model stable.

Even with these changes, negative values ofν̃ are likely to be found, especially after interpo-lation between grids. These values are clipped tozero after each update. Experiments have beencarried out with other techniques including mod-ifying the production and destruction terms fornegative values, with varying degrees of success.Simple clipping seems to be the most robust, andallows more aggressive time steps.

Matrix dissipation tends to be somewhatmore unstable than scalar, especially duringstartup. Scalar dissipation is used on the coarsegrids for this reason. The interpolation error alsoseems to eliminate any gains which could be hadby using matrix dissipation on the coarse grids.

3 Test Cases

Three test cases are presented. Two are single-element, one subsonic, the other transonic. Thethird case is an airfoil with a detached flap andslat, at low Mach number and high angle of at-tack. The flow conditions are shown in Table 1,grid details in Table 2. Off-wall spacing is givenrelative to chord length. Figure 1 shows the gridaround the multi-element airfoil.

The GMRES iterations for the single-elementcases are limited to thirty search directions, withone restart. This is not sufficient for the larger,multiblock case. Sixty search directions with norestarts allow the linear solver to converge suffi-ciently for this case.

Case Mach Alpha Re·106 Airfoil

1 0.3 6.0◦ 9.0 NACA00122 0.729 2.31◦ 6.5 RAE28223 0.197 20.18◦ 3.52 A2

Table 1 Flow conditions

Case Dimensions Nodes Offwall Spacing

1 305× 57 17385 10−6

2 257× 57 14619 2·10−6

3 – 71868 10−6

Table 2 Grids

Fig. 1 Case 3 grid

5


Inner iterationsσ Case 1 Case 2

4 430 div5 323 2636 334 2917 379 3088 401 3219 483 345

Table 3 Inner iterations vs.σ

4 Algorithm Optimization

Choosing a proper startup sequence is crucialto obtaining maximum efficiency. The Newtonsteps will zero the residual much more quicklythan the implicit Euler steps. We have had suc-cess with the following. Use∆tre f = 50 on thecoarsest grid, until the turbulence model fullytrips. This is indicated by a peak in the turbu-lent residual. While the maximum value variesstrongly by case, a minimum number of itera-tions of 15 should ensure tripping. At this point,the turbulence has stabilized enough to use a timestep of 500. When||R||2 < 0.01, the solution isinterpolated to the next grid. One iteration at atime step of 50 is used to smooth the interpo-lation error. After this∆tre f = 500 is used un-til ||R||2 < 0.01, and the solution is interpolatedto the final grid. At this point, the variable timestep is used along with the modified Jacobian, un-til ||R||2 < 0.01. The unmodified Jacobian andvariabled time step are used until convergence.Note that the residual vector is comprised of boththe four mean flow equations and the turbulenceequation. The norm, and therefore the transitionbetween stages, is typically dominated by the tur-bulence residual. This is appropriate, consider-ing that the turbulence model is less stable thanthe flow equations. This method seems to be rea-sonably robust for the aerodynamic flows underconsideration. Further convergence of the coarsegrids does not appear to be beneficial. This seemsto result from the turbulent viscosity not interpo-lating well. Three grids are used in all cases.

Table 3 shows the effect of varyingσ on the

total number of inner iterations needed to reach‖R‖2 < 10−12 on the fine grid for cases 1 and2. div indicates that the case diverged. Basedon these results,σ = 5 was chosen.

The transonic case is the best conditioned ofthe tests. This is likely due to the higher Machnumber, increased dissipation, and relatively lownumber of grid nodes. It is appropriate to sim-ply apply a time step of 100.0 to the precon-ditioner to stabilize the LU factorization whensolving transonic cases. Cases 1 and 3 demandmore attention, due to the low Mach numbers,and in case 3, the large number of nodes. In thesecases a combination of modified switch valuesand a time step is appropriate.Vl =Vn = 0.4 with∆t = 100.0 is a good choice. Note that the switchvalues in the residual and exact Jacobian remainunchanged.

As mentioned previously, GMRES does notrequire the matrix to be explicitly formed. Thereis a trade-off in speed between matrix-free andmatrix-explicit GMRES, which depends on thenumber of linear iterations. The former requiresone residual evaluation per iteration. The lat-ter requires a matrix construction when begin-ning the linear solve, plus a matrix-vector mul-tiply per iteration. Since the matrix-vector multi-ply is cheaper than a residual evaluation, matrix-free GMRES becomes less efficient with moredifficult systems. Figure 2 compares these twomethods. The matrix-free solver requires 25%more time. This case, which has a particularlyill-conditioned matrix, takes roughly 30 inner it-erations for each Newton step. This makes thematrix-explicit GMRES more attractive. Figure3 shows a different situation for case 2, the tran-sonic case. The matrix here is better conditioned,and needs about 15 inner iterations per Newtonstep. Due to the presence of the shock, the pres-sure switch has activated second-difference dis-sipation. Adding the linearization of the switchand the dissipation coefficient requires signifi-cant amounts of time, and slows the linear solves.However, without these terms in the Jacobian ma-trix, the convergence of the outer iterations is af-fected. Figure 3 compares matrix-free GMRESwith matrix-explicit without the differentiation of

6


1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 20 40 60 80 100 120 140

L2 n

orm

of r

esid

ual

�

CPU time - seconds

Mat-expMat-free

Fig. 2 Case 1 matrix-free vs. matrix-explicitGMRES

the switch and dissipation coefficient. Matrix-free is clearly the best choice for transonic cases.

The choice of level of fill in the precondi-tioner is important in balancing the memory useand CPU time. There is an optimum level offill to minimize CPU time. Higher fills producemore powerful preconditioners, which reduce thenumber of inner iterations, but require signifi-cantly more time to factorize and apply. Table 4shows residual convergence times for cases 1 and2. Both cases show that ILU(4) is a good choicefor minimizing CPU time. This adjustable pa-rameter has the advantage of allowing a trade-offbetween memory and speed. The matrix formedwhen a finite time step is used is quite well condi-tioned, so that a preconditioner with a fill of two

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 20 40 60 80 100 120 140 160 180 200

L2 n

orm

of r

esid

ual

�

CPU time - seconds

Mat-expMat-free

Fig. 3 Case 2 matrix-free vs. matrix-explicitGMRES

ILU fill Case 1 Case 2

2 168 1293 94 1124 89 925 93 143

Table 4 Convergence time (seconds) vs. ILU fill

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

0 100 200 300 400 500 600

L2 n

orm

of r

esid

ual

�

CPU time - seconds

NK - densityNK - SA

AF - densityAF - SA

Fig. 4 Case 1 residual history

is sufficient during the startup phase.

5 Results

Figures 4, 5, and 6 compare the convergencehistories of the Newton-Krylov solver with anapproximately-factored solver. All cases used anILU(4) preconditioner, and were run on an Al-pha 667MHz EV67 CPU. Cases 1 and 3 are muchfaster than the approximate factored solver, whilecase 2 shows only a marginal improvement. Thisis due to the difficulties with the dissipation termsin the Jacobian, and the resulting use of matrix-free GMRES. Case 3 is not fully converged bythe approximately factored solver. There are re-circulation bubbles behind the slat, the cove onthe main body, and at the trailing edge of theflap. Difficulties arise at the edges of these bub-bles with the highly nonlinear destruction termof the turbulence model. Outside of these fewnodes, the model is converged.

7


1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

0 20 40 60 80 100 120 140 160 180

L2 n

orm

of r

esid

ual

�

CPU time - seconds

NK - densityNK - SA

AF - densityAF - SA


1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

10000

0 500 1000 1500 2000 2500 3000 3500 4000 4500

L2 n

orm

of r

esid

ual

�

CPU time - seconds

NK - densityNK - SA

AF - densityAF - SA


6 Conclusions

An efficient Newton-Krylov solver has been pre-sented for the steady compressible Navier-Stokesequations governing turbulent flows over multi-element airfoils. Proper optimization is essen-tial. This includes using grid sequencing and theimplicit Euler method for startup. During thisphase, the Jacobian matrix must be modified toensure that the turbulent viscosity remains posi-tive. Incomplete LU preconditioning is used. Alevel of fill of four was found to be optimal withrespect to CPU time. The ILU factorization is ap-plied to the first-order Jacobian matrix with mod-ified second-difference dissipation. Time stepselection and modified dissipation switches areimportant to stabilize the preconditioner. Thesingle-element test cases can be solved to ma-chine zero in less than 100 seconds, while thecomplex flow on the multi-element case can befound in less than 25 minutes. The subsonic casesconverge three to five times faster than an approx-imately factored algorithm.

7 Acknowledgments

This research was supported by BombardierAerospace, and an OGSST grant from the Gov-ernment of Ontario.

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[4] Blanco M and Zingg D. W. Fast Newton-Krylovmethod for unstructured grids.AIAA Journal,Vol. 36, No 4, pp 607–612, April 1998.

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[5] Chow E and Saad Y. Experimental study of ILUpreconditioners for indefinite matrices.J. ofComp. and Appl. Mathematics, Vol. 87, pp 387–414, 1997.

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a fully coupled newton-krylov solver for turbulent

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