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Part 15 Meshing Technology
Part 15 Meshing Technology
Adjoint-Based Adaptive Meshing in a ShapeTrade Study for Rocket Ascent
Marshall Gusman, Jeffrey Housman, and Cetin Kiris
Abstract This paper investigates the drag performance of six axisymmetric shroudshapes over the ascent trajectory of the Saturn V launch vehicle. An adjoint-basedadaptive meshing algorithm is used to generate Cartesian meshes with refinement inregions of the flow that have a critical influence on drag prediction. In combinationwith an efficient parallel flow solver, this approach provides accurate predictions ofaerodynamic performance at minimal computational cost. A metric known as ‘dragloss’ is integrated over the flight trajectory to determine the best shroud shape forthe application. These methods are shown to be an effective aid to the aerodynamicdesign of new launch vehicles.
1 Introduction
NASA has renewed interest in traditional rocket designs such as the Saturn V, whichsuccessfully launched many orbital and lunar missions during the Apollo programof the 1960s and 1970s. The ‘vertical stack’ design places the payload at the top ofthe rocket where a significant fraction of total drag force is generated. The currentpaper seeks to improve the aerodynamic performance of a ‘vertical stack’ launchvehicle by identifying the best of six shroud shapes for a Saturn V geometry andascent trajectory.
This study is performed with the NASA-developed software Cart3D, a parallelCFD package that includes an embedded boundary automatic Cartesian mesh gen-erator and an inviscid flow solver, see Aftosmis et al. (2000) as well as an adjoint-
Marshall Gusman*, e-mail: [email protected] Housman*, e-mail: [email protected] Corp., 465 S Mathilda Ave., Sunnyvale, CA 94086 USA
Cetin Kiris, e-mail: [email protected] Ames Research Center, Moffett Field, CA 94035 USA
*This work was performed under NASA contract NNA06BC19C to ELORET Corp.
1
2 Marshall Gusman, Jeffrey Housman, and Cetin Kiris
based mesh refinement capability, see Nemec and Aftosmis (2007), and Nemec etal. (2008). Solutions are marched in time to steady-state using a five-stage Runge-Kutta scheme with local time stepping and a multigrid W-cycle. The inviscid flowassumption is valid provided boundary layer effects are negligible over the regionof interest. The combination of large Reynolds numbers and relatively short shroudlengths results in minimal boundary layer development and negligible skin frictioneffects. The software’s parallel efficiency and solution-adaptive meshing providesan excellent utility for shape trade analyses in the early stages of rocket design.
2 Adaptive Meshing
Adaptive mesh refinement is an intelligent alternative to a global refinement ap-proach because it requires fewer resources to achieve the same level of solutionaccuracy. Conversely, better accuracy can be achieved with the same resources byadaptively refining the mesh in only the important regions. The adjoint-based adap-tive mesh refinement method involves choosing a functional of interest, J(Q), whichis a scalar quantity that depends on the flow solution, Q. In this study, the functionalis the drag force on the shroud.
The accuracy of the flow solution and functional approximation are highly de-pendent on the discretization error associated with the numerical scheme and com-putational domain. The discretization error can be reduced by globally refining themesh, but we soon discover that global refinement is inefficient and unaffordable. Amore cost-effective approach is to restrict refinement to the most critical regions ofthe flow domain.
2.1 Adjoint Method
The critical regions are identified by solving the adjoint (or dual) problem. Thissolution can be used to produce error estimates representing each cell’s contribu-tion to the global error in the functional. A user-specified error tolerance criterion isthen used to identify which cells should be refined to improve accuracy. After ob-taining the newly refined mesh, the flow solution is recomputed and the adaptationprocedure is repeated until the functional error tolerance is met. The following pro-cedure identifies the critical regions and refines the mesh to improve the functionalestimate.
Functional Approximation: From a given solution, the functional on a uniformlyrefined mesh can be approximated with a Taylor series expansion about a solutionQH
h , which is interpolated from the current mesh (H) to the uniformly refined mesh(h).
J(Qh)≈ J(QHh )+
(∂J
∂Qh
)(Qh−QH
h). (1)
Adjoint-Based Adaptive Meshing in a Shape Trade Study for Rocket Ascent 3
Similarly, the residual on the finer mesh is approximated as,
R(Qh)≈ R(QHh )+
(∂R
∂Qh
)(Qh−QH
h). (2)
Setting R(Qh) to zero and rearraging Equation 2 gives the following expression
(Qh−QH
h)
=−(
∂R∂Qh
)−1
R(QHh ). (3)
Substituting Equation 3 into Equation 1 gives the approximation of the functionalon the fine mesh,
J(Qh) ≈ J(QHh )−
(∂J
∂Qh
)(∂R
∂Qh
)−1
R(QHh )
.= J(QHh )−ψ
Th R(QH
h ), (4)
where ψTh satisfies the adjoint equation(
∂R∂Qh
)T
ψh =(
∂J∂Qh
)T
. (5)
Since the solution of Equation 5 on the fine mesh is too costly, the adjoint equationsare solved on the coarse mesh for ψH and then interpolated to the finer mesh. Thisleads to the following estimate,
J(Qh)≈ J(QHh )− (ψH
h )T R(QHh )︸ ︷︷ ︸
computable correction
−(ψh−ψHh )T R(QH
h )︸ ︷︷ ︸remaining error
. (6)
Adaptation Criteria: The functional approximation described above contains a cor-rection term and a term representing the remaining error. Since the correction termis computable, it can be used to correct the functional on the given mesh. The re-maining error term cannot be affordably computed and must be driven to zero byrefining the cells which contain the largest remaining local errors. Since the remain-ing error is not known, the local error estimates are approximated with trilinear andtriquadratic interpolations of ψH onto the embedded fine mesh, denoted ψT L andψT Q respectively. This leads to the following local error estimate for each cell,
e = |(ψT Q−ψT L)T R(QHh )|. (7)
Next, a user-defined tolerance εTOL is used to define a threshold representing therequired accuracy of the computed functional. Since the remaining global error issimply the sum of the remaining local errors over all cells, then εLOCAL = εTOL/Nwhere N is the total number of cells in the current mesh. Thus a cell is flagged forrefinement if e > εLOCAL.
4 Marshall Gusman, Jeffrey Housman, and Cetin Kiris
(a) Level 0 (22k cells) (b) Level 1 (28k cells) (c) Level 2 (62k cells)
(d) Level 3 (229k cells) (e) Level 4 (868k cells) (f) Level 5 (3.8M cells)
Fig. 1 Mesh adaptation on baseline Saturn V shape, Mach 1.70.
Solve and Repeat: A newly refined mesh is generated based on the flags fromthe adaptation criteria. The flow solution is recomputed on the fine mesh as before,and the procedure is repeated until the estimated remaining global error is less thanεTOL.
2.2 Adapted Meshes
The adapted meshes show that refinement is limited to the regions affecting thefunctional of interest (drag on the shroud). Figure 1 shows a cutting plane of theadapted mesh at successive levels of refinement for the baseline shape in supersonicflow. Mesh points cluster where the largest errors are estimated; at the leading shockstructure and near the body, while regions downstream of the shroud are ignored.Localized refinement keeps the computational cost to a minimum while capturing ahigh level of accuracy. A slice of the mesh for subsonic flow in Figure 2(a) showsrefinement upstream and downstream of the shroud. The adjoint-based adaptationdetects flow influence from both directions and refines appropriately. A slice of thetransonic mesh (b) reveals refinement upstream of the shroud and an abrupt halt torefinement after the shroud. This is caused by a supersonic bubble that exists justdownstream of the shroud and prevents information and error from traveling up-stream and affecting the shroud. With supersonic flow, downstream refinement isunnecessary. When the freestream flow is supersonic, even less refinement is neces-sary. The mesh is only refined in regions of high error (like shocks) that influencethe functional, and all other regions are left coarse. The critical diamond-shapedregion of influence and dependence is apparent in Figure 1(f). Upstream of the bow-shock the flow is uniform and there is no error in the solution, and downstream of
Adjoint-Based Adaptive Meshing in a Shape Trade Study for Rocket Ascent 5
(a) Mach 0.60 (b) Mach 0.95
Fig. 2 Slice through the computational domain showing mesh refinement at subsonic and transonicconditions.
(a) (b)
Fig. 3 Solution improvement with mesh adaptation: (a) Convergence of drag coefficient, and (b)decrease in estimated functional error.
the shroud there is zero functional sensitivity to errors in the solution. Refinementonly occurs where the product of residual- and adjoint-sensitivity is high. A briefverification of the adaptive meshing procedure is presented for the baseline shape atMach 1.7. Figure 3(a) shows the drag coefficient (CD) as a function of the total num-ber of cells as the mesh is refined. Initially, the 22,000-cell mesh under-predicts CDby approximately 7% in comparison to the drag estimate on the final 3,800,000-cellmesh. The corrected functional is also shown, which is calculated using the com-putable correction term from Equation 6. On a given mesh, the corrected functionalis an estimate of the functional value that would be obtained on a mesh uniformlyrefined by one extra level. Both values converge towards CD ≈ 0.805. Progressiveimprovements to the functional accuracy are observed in Figure 3(b), where thefunctional error estimate is plotted versus mesh size. The log of error falls linearlywith the log of the number of cells. Extrapolating this linear behavior to the finalmesh, the error is approximately 0.003, or less than 0.5% of the total value. The ad-
6 Marshall Gusman, Jeffrey Housman, and Cetin Kiris
joint solution is not computed on the final mesh because no further mesh refinementis needed to meet the error tolerance, however, it could be computed to obtain a finalcorrection term and a conservative estimate of the remaining error.
3 Shape Trade Study
This study compares the drag performance of six shroud shapes at transonic tosupersonic conditions and zero angle of attack. Specifically, the baseline shapeclosely represents the command module flown on the Apollo missions, and consistsof a conical shape with rounded tip. The dimensions of the baseline and alterna-tive shapes are defined by the original Apollo command module, with total lengthL = 2.8788 meters, measured from the tip to the cylindrical stack, and maximumradius R = 1.9558 meters. The alternative shapes consist of a family of power-lawshapes defined by Equation 8 with exponents N = 0.4,0.5, and 0.6, and two Sears-Haack shapes defined by Equation 9 with coefficients C = 0.0 and 0.33. The Sears-Haack shapes are blended with a sphere of radius 25.4cm to form a blunt nose thatcoincides with the blunt tip of the baseline conical shape, and is necessary for man-ufacturing feasibility and improved subsonic performance. Combined, these shapescomprise a set of realistic rocket fairing designs that are likely to exhibit improvedaerodynamic performance over the baseline. Each shroud is mounted on a simpli-fied Saturn V vertical stack and simulated at twelve points in the ascent trajectory,with Mach numbers ranging from 0.6 to 5.0.
r = R(x/L)N (8)
r = R
√θ − 1
2 sin2θ +C sin3θ
πwhere θ = arccos(1− x/L) (9)
Figure 4 shows dimensional pressure contours on the shroud surface and a slicethrough the pressure field at the Mach 2.0 trajectory point. The baseline conicalshape exhibits a constant surface pressure in supersonic flight because the flow doesnot accelerate after the initial deflection at the tip of the shroud. The constant flowvelocity corresponds to a constant high pressure along the shroud. In contrast tothe conical shape, the high curvature at the nose of the power-law and Sears-Haackshapes accelerates the flow and lowers pressure over most of the shroud, reducingthe integrated axial pressure force. Figure 5 shows surface pressure distributionsalong the length of each shroud at Mach 2.0. Peak pressures exist at the leadingpoint, and then drop as the flow approaches the base of the shroud. In contrast tothe baseline, the Sears-Haack shapes have roughly linear descents from peak tominimum pressure, and the power-law shapes drop quickly then level out towardsthe base.
Adjoint-Based Adaptive Meshing in a Shape Trade Study for Rocket Ascent 7
(a) Baseline (b) Sears-Haack C = 0.00 (c) Sears-Haack C = 0.33
(d) Power-law N = 0.4 (e) Power-law N = 0.5 (f) Power-law N = 0.6
(g) Legend
Fig. 4 Pressure contours on a slice through the domain at Mach 2.0.
Fig. 5 Pressure distribution from tip to base shrouds (vertical dashed line) at Mach 2.0.
3.1 Drag Profiles
Every Sears-Haack and power-law shape is superior to the conical shape at everyMach number tested in the study. The drag coefficients plotted in Figure 6 revealthe improvements of each alternative shroud shape over the baseline. At subsonic
8 Marshall Gusman, Jeffrey Housman, and Cetin Kiris
Fig. 6 Drag coefficient versus Mach over ascent trajectory.
conditions, the Sears-Haack C=0.33 has the lowest drag coefficients, but loses thisadvantage in the supersonic regime. Characteristics that make a shape efficient atlow speeds do not necessarily apply in supersonic flow, as shown by the trade-offsinherent in each of the shapes. There is a notable performer in the study, however.The power-law N=0.4 shape is nearly as good as the Sears-Haack in subsonic flowand keeps this advantage in the transition through transonic and then supersonicflow. It’s maximum drag coefficient is reached at Mach 5.0, but the atmosphere hasbecome thin at this point, and dynamic pressure has decreased dramatically.
3.2 Drag Loss
A metric known as ‘drag loss’ is introduced to quantify shroud performance overthe entire ascent trajectory. Drag loss represents the total velocity decrease due todeceleration from aerodynamic drag in reference to a zero-drag body, see Martinez-Sanchez (2005). This evaluation method is advantageous because it considers ve-hicle weight, time, and dynamic pressure in addition to the drag coefficient. Theshroud with the lowest drag loss will enable larger payloads and greater flexibilityin launch and orbital trajectories.
To find drag loss, the dimensional drag force on the body is converted to anacceleration by dividing by the vehicle’s instantaneous mass. This is then integratedover the flight time to obtain a velocity term. The computed drag loss is then ameasure of the shroud’s overall effect on the final velocity of the rocket. For thediscrete evaluation of drag loss, a trapezoidal integral approximation is used, hencevalues for dimensional drag D and mass m are averaged between trajectory points i
Adjoint-Based Adaptive Meshing in a Shape Trade Study for Rocket Ascent 9
Fig. 7 Drag loss comparison of all shroud shapes.
at time ti and ti+1.
Drag Loss =∫
∞
0
Dm
dt
≈n−1
∑i=1
Di+1/2
mi+1/2(ti+1− ti). (10)
The six shroud drag losses are compared in Figure 7. Of the six, the baseline isclearly the worst performer. The Sears-Haack shapes provide approximately a 20%improvement over the baseline, and the power-law shapes are between 20− 33%better. A trend in the power-law shapes indicates that the blunter (i.e. N = 0.4)shapes have better overall performance over the launch trajectory.
4 Conclusion
Benefits of the adjoint-based adaptive meshing algorithm have been illustrated inflow conditions from subsonic to supersonic. The algorithm detects functional sen-sitivity to the mesh and generates an optimal mesh for the given geometry and flowconditions. Localized refinement improves the functional accuracy while maintain-ing the computational affordability of a large database of simulations. This tech-nique was applied to a set of axis symmetric shroud shapes for the Saturn V launchvehicle, which are evaluated for their drag performance across the Saturn V launchtrajectory. Then, a quantitative comparison of each shape’s drag performance wasmade with a benchmark quantity called drag loss. The most blunt power-law shape(N = 0.4) has the lowest drag loss, with a 33% improvement over the baselineshroud. This shape trade study has demonstrated how significant improvements canbe achieved for both fluid dynamic computations and future rocket designs.
Acknowledgements The authors would like to thank Drs. Mike Aftosmis, Marian Nemec, andShishir Pandya for their support.
10 Marshall Gusman, Jeffrey Housman, and Cetin Kiris
References
1. Aftosmis, M., Berger, M., & Adomavicius, G. 2000. “A Parallel Mulitlevel Method for Adap-tively Refined Cartesian Grids with Embedded Boundaries,” AIAA-2000-0808.
2. Martinez-Sanchez, M. 2005. “Orbital Mechanics: Review Staging,” From MIT OpenCourse-Ware lecture notes, 16.512 Rocket Propulsion, Lecture 32, Fall 2005.
3. Nemec, M., & Aftosmis, M. 2007. “Adjoint-Based Adaptive Mesh Refinement for EmbeddedBoundary Cartesian Meshes,” AIAA-2007-4187.
4. Nemec, M., Aftosmis, M., & Wintzer, M. 2008. “Adjoint-Based Adaptive Mesh Refinementfor Complex Geometries,” AIAA-2008-0725.
5. Saturn V Flight Evaluation Working Group, 1969. “Saturn V Launch Vehicle Flight Evalua-tion Report,” AS-503 Apollo 8 mission, February 1969.
Mesh Quality Effects on the Accuracy of Eulerand Navier-Stokes Solutions on UnstructuredMeshes
Aaron Katz and Venkateswaran Sankaran
Abstract We examine discretization error for standard node- and cell-centeredschemes using the Method of Manufactured Solutions. We find that for isotropicgrids, node-centered approaches produce less error than cell-centered approachesfor comparable cell size. In contrast, cell-centered schemes produce less discretiza-tion error on stretched meshes. In 3D, careful treatment of non-planar faces is nec-essary to avoid first-order errors. We introduce a new corrected scheme which en-hances the accuracy of the node-centered scheme to third-order.
1 Introduction
The need to resolve high Reynolds number viscous flows around complex geometryplaces high demands on flow solvers. Unstructured schemes have acheived main-stream use for such problems due to their ability to automatically discretize complexdomains (Mavriplis, 2007). As the complexity of CFD simulations increases, theneed for rigorous verification becomes paramount. The difficulty of obtaining exactsolutions for verification has led to the use of the Method of Manufactured Solu-tions (MMS) (Roache, 2002), which has acheived widespread use to evaluate orderof accuracy (Pautz, 2001; Roy, 2005) and grid quality effects (Luke et al, 2008; Sunet al, 2010). In this work we evaluate the performance of node- and cell-centeredschemes on a variety of cell-types. We confirm many previous findings as well ashighlight additional findings, including a new third-order scheme and implicationsfor 3D prismatic meshes.
Aaron KatzUS Army Aeroflightdynamics Directorate (AMRDEC), Moffett Field, CA 94035, e-mail:[email protected]
Venkateswaran SankaranUS Army Aeroflightdynamics Directorate (AMRDEC), Moffett Field, CA 94035, e-mail:[email protected]
1
2 Aaron Katz and Venkateswaran Sankaran
Fig. 1 MMS solutionsused for a) Euler, b) scalarconvection-diffusion, c)boundary layer, and d) curvedboundary layer tests
(a) (b) (c) (d)
2 Description of Schemes Tested
Discretization errors are evaluated for 2D node- and cell-centered schemes as wellas a 3D cell-centered prismatic scheme applied to a general convection-diffusionequation. The 2D node-centered scheme is derived from a Galerkin finite elementmethod, with control volume face areas computed as the sum of the dual facetstouching each edge. Upwinding is introduced via artificial diffusion using linearleast squares reconstruction. The viscous discretization is also based on a Galerkinapproximation, as described by Barth (Barth, 1991).
In this work, we present a new “corrected” scheme, which obtains formal third-order accuracy for inviscid terms only on arbitrary triangular meshes with the addi-tion of a correction term at each node,
C0 = ∑i
14(∆r0i · (∇Fi −∇F0)) ·A0i, (1)
where ∇F is the gradient of the flux, computed to second-order accuracy with aquadratic least squares method. Niether second derivatives nor higher-order quadra-ture are needed as for cell-centered quadratic schemes (Delanaye and Liu, 1999).While the corrected scheme appears to work well for isotropic grids, detailed anal-ysis of the method for stretched grids is reserved for future work.
In the cell-centered formulation, the state variables are located at the cell-centers,and the faces are the boundaries of the primary cells. Reconstruction is performedby first obtaining nodal values of the variables with a linear least squares procedure,followed by a Green-Gauss gradient integration around the perimeter of each con-trol volume. The viscous terms are discretized with a combination of reconstructednodal and cell-center values to obtain an estimate of the gradient at the face quadra-ture points.
The 3D prismatic solver is based on the same approach. Since general polyhedralvolumes in 3D are composed of non-planar faces, we implement a flux integrationscheme in which each non-planar face is triangulated with a quadrature point placedat the center of each resulting triangular facet. We also implement a simplified fluxquadrature method that uses only a single quadrature point at the center of eachnon-planar face.
Mesh Quality Effects on Unstructured Meshes 3
3 Accuracy Tests on Isotropic and Stretched Grids
Using MMS solutions shown in Figure 1, we verifiy inviscid and viscous orders ofaccuracy using uniform and randomly perturbed grids of quadrilaterals, equilateraltriangles, and right triangles. The cell size for a given mesh and scheme is definedas ds = (Vtotal/ndo f )
1/d , where Vtotal is the total volume of the domain, ndo f is thenumber of degrees of freedom in the mesh (nodes or cells), and d is the number ofspatial dimensions. Inviscid and visous discretizations are tested in turn to isolatethe resulting discretization errors.
ds -1
L2
erro
r
500 1000 1500
10-6
10-5
10-4
10-3
10-2
Node, RT, reg.Node(cor.), RT, reg.Node, RT, pert.Node (cor.) , RT, pert.Node, ET, reg.Node (cor.), ET, reg.Node, ET, pert.Node (cor.), ET, pert.Node, Q, reg.Node, Q, pert.
11
1
2
1
3
(a) node-centered
ds -1
L2
erro
r
500 1000 1500
10-6
10-5
10-4
10-3
10-2
Cell, RT, reg.Cell, RT, pert.Cell, ET, reg.Cell, ET, pert.Cell, Q, reg.Cell, Q, pert.
12
(b) cell-centered
Fig. 2 Error convergence of inviscid terms, regular and perturbed isotropic grids.
Table 1 Order of accuracy of inviscid and viscous terms as predicted by the isotropic scalar MMStest. Perturbed and regular grid results given.
scheme quadrilateral equilateral triangle right trianglepert./reg. pert./reg. pert./reg.
Node-centered linear, inviscid 1/2 2/3 2/3Node-centered corrected, inviscid - 3/3 3/3Cell-centered, inviscid 2/2 2/2 2/2Node-centered linear, viscous 2/2 2/2 2/2Cell-centered, viscous 2/2 2/2 2/2
The results of the isotropic grid order of accuracy test indicate that triangles pro-duce less discretization error than quads for a given cell size, shown in Figure 2for the inviscid discretization, and Table 1 for both inviscid and viscous discretiza-tions. In the node-centered case, the median-dual approximation actually producesfirst-order errors for quads, consistent with the findings of Diskin et al. (Diskin andThomas, 2007). For the cell-centered case, second-order accuracy is maintained forquads, but at a higher level of error. On the other hand, triangles always produce
4 Aaron Katz and Venkateswaran Sankaran
third-order errors for the corrected scheme, with second-order accuracy observedon arbitrary meshes for both the node- and cell-centered schemes. With the properGalerkin-weighted MMS source term, the linear node-centered scheme show third-order accuracy, contrary to the findings of Diskin. The proper discretization of theMMS source term appears to be critical for correct prediction of orders greater thantwo (Pautz, 2001). Due to the third-order accuracy on regular triangles, along withthe unconditional third-order accuracy of the corrected scheme, nodal schemes ap-pear best suited for isotropic grids, as long as quads with median-dual approxima-tions are avoided. All viscous schemes tested result in second-order accuracy forwith comparable levels of error.
Along with isotropic grids, the performance of the schemes on stretched meshesof various types is tested. A total of four mesh types are used, including flat andcurved surface boundary layer type meshes, shown in Figure 3, along with the cor-responding manufactured solutions shown in Figures 1(c) and 1(d). Three levels ofmesh stretching are tested, including wall cell aspect ratios of 102, 104, and 106. Asin the isotropic tests, inviscid and viscous discretizations are tested independentlyto isolate the effects of mesh stretching on each discretization type.
Fig. 3 Flat and curved quadand triangular stretched gridsused to assess grid qualityeffects.
ds -1
L2
erro
r
500 1000 1500
10-4
10-3
10-2
10-1
100
Node, RT, A=10 2
Node, RT, A=10 4
Node, RT, A=10 6
Cell, RT, A=10 2
Cell, RT, A=10 4
Cell, RT, A=10 6
1
2
(a) node- and cell-centered rt. triangles
ds -1
L2
erro
r
500 1000 1500
10-4
10-3
10-2
10-1
100
Cell, RT, A=10 6
Cell, Q, A=10 6
1
2
(b) quad and rt. triangular cell-centered
Fig. 4 Comparison of error convergence of viscous terms for the curved boundary layer case.
The curved surface error convergence for right triangles is shown in Figure 4.Similar trends are seen for the inviscid terms and for the flat plate case. The slopesin Figure 4(a) indicate the cell-centered scheme produces less discretization errorthan the node-centered scheme independent of aspect ratio. Furthermore, Figure4(b) shows that quads produce significantly less error for a given cell size than
Mesh Quality Effects on Unstructured Meshes 5
Fig. 5 Error convergenceshowing first-order effectsof simple flux integrationon non-planar faces for 3Dprismatic grids.
ds -1
L2
erro
r
20 40 60 80
10-10
10-9
10-8
10-7
10-6
single pt. quad., regsingle pt. quad., pert.tringulated face, reg.triangulated face, pert.
1
2
11
triangles for stretched meshes. Cell-centered quads of aspect ratio 106 produce lesserror than node-centered triangles of aspect ratio 102. These results are in contrastto the isotropic results, which favor node-centered schemes on triangular grids.
4 Implications for Non-Planar Faces in 3D
The low errors produced by the cell-centered scheme on quads motivate the useof cell-centered prismatic meshes in 3D. Here, we present an order of accuracystudy regarding the non-planar control volume faces characteristic of 3D prismaticgrids. Regular and perturbed prismatic grids are studied, exhibiting planar and non-planar faces, respectively. Single point and triangulated face quadrature methodsare evaluated. With single point quadrature, the convective flux is evaluated once atthe face center of each quadrilateral face. With triangulated face quadrature, eachquadrilateral face is first triangulated, with a quadrature point placed at the center ofeach triangular facet for a total of two quadrature points per quadrilateral face.
The results of the study are shown in Figure 5 and summarized in Table 2. Thesingle-point quadrature is unable to retain second order accuracy in general. In con-trast, the triangulated face quadrature maintains second-order accuracy, even on per-turbed grids with non-planar faces. This result is either largely unknown in the lit-erature, or largely ignored, as pointed out by Delanaye and Liu (Delanaye and Liu,1999). However, face triangulation was also advocated in the work of Liu and Vi-nokur (Liu and Vinokur, 1998).
Table 2 Order of accuracy for planar and non-planar faces for cell-centered prismatic grids.
quadrature scheme planar faces non-planar faces
single point 2 1triangulated face 2 2
6 Aaron Katz and Venkateswaran Sankaran
5 Conclusions
The MMS procedure is used to predict order of accuracy precisely and evaluatemesh quality effects for a variety of cell types and schemes. Nodal approaches,including a new corrected scheme, produce less error than cell approaches onisotropic grids, showing third-order accuracy on triangles in many cases. However,the median-dual approach on quads produces first-order errors. In contrast, cell-centered schemes produce less error on stretched meshes, especially for quads. Thestretched mesh results encourage the use of cell-centered prismatic meshes in 3D forhigh Reynolds number flows. In 3D, first-order errors arising from non-planar facesare observed. Second-order accuracy is recovered by triangulation of non-planarfaces. Future work will focus on the effects of curvature and high aspect ratio in 3D.
Acknowledgements Development was performed at the HPC Institute for Advanced RotorcraftModeling and Simulation (HIARMS), supported by the DoD High Performance Computing Mod-ernization Office (HPCMO). Material in this paper is a product of the CREATE-AV Element ofthe Computational Research and Engineering for Acquisition Tools and Environments (CREATE)Program sponsored by the U.S. Department of Defense HPC Modernization Program Office.
References
Barth TJ (1991) Numerical aspects of computing viscous high reynolds numberflows on unstructured meshes. AIAA paper 1991-0721, AIAA 29th ASM, Reno
Delanaye M, Liu Y (1999) Quadratic reconstruction finite volume schemes on 3darbitrary unstructured polyhedral grids. AIAA paper 1995-3259, AIAA 14th CFDConference, Norfolk
Diskin B, Thomas J (2007) Accuracy analysis for mixed-element finite-volume dis-cretization schemes. NIA Report 2007-08, National Institute of Aerospace
Liu Y, Vinokur M (1998) Exact integrations of polynomials and symmetric quadra-ture formulas over arbitrary polyhedral grids. J Comp Phys 140:122–147
Luke E, Hebert S, Thompson D (2008) Theoretical and practical evaluation ofsolver-specific mesh quality. AIAA paper 2008-0934, AIAA 46th ASM, Reno
Mavriplis DJ (2007) Unstructured mesh discretizations and solvers for computa-tional aerodynamics. AIAA paper 2007-3955, AIAA 18th CFD Conf., Miami
Pautz S (2001) Verification of transport codes by the method of manufactured solu-tions: The attila experience. Tech. Rep. LA-UR-01-1487, Los Alamos
Roache P (2002) Code verification by the method of manufactured solutions. Trans-actions of the ASME 124:4–10
Roy C (2005) Review of code and solution verification procedures for computationalsimulation. Journal of Computational Physics 205:131–156
Sun H, Darmofal D, Haimes R (2010) On the impact of triangle shapes for boundarylayer problems using high-order finite element discretization. AIAA paper 2010-0542, AIAA 48th ASM, Orlando
Analysis of a RK/Implicit Smoother for MultigridR. C. Swanson,∗ E. Turkel † and S. Yaniv ‡
Abstract
The steady-state compressible Navier-Stokes equations are solved with a finite-volume, second-orderaccurate scheme. The equations are solved with a multigrid algorithm that uses a 3-stage Runge-Kuttascheme with an implicit preconditioner as a smoother. We analyze this smoother in which the implicitsystem is approximately inverted by a few symmetric Gauss-Seidel relaxation sweeps. The analysis forthe linear system determines the Fourier spectrum of the multigrid smoother. Improved performance ofthe algorithm based on the analysis is demonstrated by computing laminar flow in a rocket motor andturbulent flow over a wing.
1 Introduction
Multigrid algorithms with an explicit Runge-Kutta (RK) scheme and implicit residual smoothing (IRS) arethe foundation of many existing aerodynamic prediction codes. This class of methods has recently beensignificantly improved by replacing the scalar form of IRS with a matrix form. The matrix form allows alarge CFL number, resulting in faster propagation of information (enhancing multigrid efficiency since theNavier-Stokes equations contain a hyperbolic part) and reduced discrete stiffness. In this paper we analyzethis multigrid smoother (RK/implicit scheme), introduce improvements in efficiency, and demonstrate therobustness of the modified algorithm.
2 RKI Algorithm
To discretize the governing fluid dynamic equations we apply a cell-centered, finite-volume approach. Theadvection terms are approximated with three point central differencing plus numerical dissipation. A matrix-valued or Roe-type dissipation is applied such that the scheme is second order in smooth regions of the flowfield and first order in the neighborhood of shock waves. The viscous terms are discretized with a second-ordercentral difference approximation. See [1, 2] for details.
For solving the discrete equations we use the full approximation storage (FAS) multigrid algorithmdescribed in [2]. This algorithm uses full coarsening and either a V-type or W-type cycle. The operatorsfor restricting residuals and flow variables to the coarser grids are determined by a conservative residualsummation and volume weighting, respectively. Coarse grid corrections are transferred to finer grids bylinear interpolation operators. The RK scheme of the multigrid smoother has three stages with coefficients[α1, α2, α3] = [0.15, 0.4, 1.0]. The solution vector W on the q-th stage of the RK scheme is given by
W(q) = W(0) + δW(q) = W(0) − αq∆t
VLW(q−1) = W(0) − αq∆tR(q−1), (1)
where L is the complete difference operator for the system of equations, ∆t is the time step, V is the volumeof the mesh cell being considered, and R(q−1) represents the residual function for the (q − 1)-th stage. Ageneral form for the residual function can be written as
R(k) =1
VLW(k) =
1
V
[LcW(k) −
k∑r=0
γkr LvW(r) +k∑
r=0
γkr LdW(r)
], (2)
with Lc, Lv, and Ld denoting convective, viscous, and dissipative operators. For consistency∑
γkr = 1.The coefficients γkr are the weights of the viscous and dissipative terms on each stage, and for the 3-stagescheme,
γ00 = γ1; γ10 = 1− γ2, γ11 = γ2; γ20 = 1− γ3, γ21 = 0, γ22 = γ3. (3)
∗NASA Langley Research Center, Hampton, VA 23681, USA, email: [email protected]†School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel, email: [email protected]‡Israel Military Industries Ltd. (IMI), Ramat Hasharon, Israel, email: [email protected]
When the weights [γ1, γ2, γ3] are [1.0, 1.0, 1.0], this is called standard weighting. Based upon analysis andnumerical testing we have determined that the modified weights [1.0, 0.5, 0.5] lead to improved robustnessof the smoother. In particular, the Mach number used in calculating the preconditioner needs to be cutoffto prevent a zero Mach number. The modified weights allow a lower level for these cutoffs which leads to afaster rate of convergence.
Letting Li be an implicit operator, we define the following replacement for the explicit update in Eq. (1):
δW(q)
= −αq∆t
VP LW(q−1) = −αq
∆t
VP
∑all faces
F(q−1)n S, (4)
where P is the implicit preconditioner defined by the approximate inverse L−1i , Fn is the normal flux density
vector at the cell face, and S is the area of the cell face. A first-order upwind approximation based on theRoe scheme is used for the convective derivatives in the implicit operator, which is defined by[
I + ε∆t
V∑
all faces
An S
]δW
(q)= −αq
∆t
V∑
all faces
F(q−1)n S. (5)
The matrix An is the flux Jacobian associated with Fn at a cell face, and ε is an implicit parameter.Using one-dimensional Fourier analysis and numerical testing we have found that a good choice for ε is 0.5.An approximate inverse of the implicit operator for the linear system is obtained with Gauss-Seidel (GS)relaxation. Complete discussion of the scheme is given in [3] and [4].
3 Fourier Analysis
To analyze the RK/implicit scheme we consider a finite domain with periodic boundary conditions. Wediscretize the linearized, time-dependent Navier-Stokes equations on a Cartesian grid with mf ×nf cells cov-ering the domain. For convenience, the Navier-Stokes equations are transformed from conservative variablesto primitive variables. We let Uj1,j2 denote the discrete solution vector of primitive variables that resides atthe mesh point (j1hx, j2hy). Then, the preconditioned form of Eq. 1 is Fourier transformed to obtain
U(q)k1,k2
= U(0)k1,k2
− αq∆t
VP LhU
(q−1)k1,k2
. (6)
Lh is the linearized discrete residual operator for the primitive variables. The transformed discrete vectorfunction is given by
U(q)k1,k2
=1
mfnf
mf−1∑j1=0
nf−1∑j2=0
U(q)j1,j2
e−i(j1θx+j2θy), (7)
where the phase angles θx and θy, along with the corresponding wave numbers are given by
θx = 2πk1mf
, θy = 2πk2nf
, k1 = −(1
2mf − 1), · · · , 1
2mf , k2 = −(
1
2nf − 1), · · · , 1
2nf . (8)
The transformed residual operator Lh is a function of the transformed shift operators, which are defined by
Ex ≡ eiθx , Ey ≡ eiθy , −π < θx ≤ π, −π < θy ≤ π. (9)
After applying the 3-stage RK scheme with standard weighting of dissipation, we have
Un+1k1,k2
= GrkiUnk1,k2
, (10)
where the amplification matrix
Grki = I− α1PLh + α2α1(PLh)2 − α3α2α1(PLh)
3,
I is the identity matrix and αq represents the product of the RK coefficient and the ratio of ∆t to V.Pierce and Giles [6] introduced a matrix preconditioner determined by the diagonal elements of the
residual function. This is equivalent to a block Jacobi preconditioner. In the RKI scheme we introduce off-diagonal terms. An approximate inverse of the resulting implicit system is obtained using a small number(usually two) of symmetric Gauss-Seidel sweeps. This method can be considered a block Gauss-Seidelpreconditioner.
In Fig. 1 we show the Fourier footprints for the RKI(3,3) scheme (3 stages, 3 dissipation evaluations) withblock Jacobi and block GS (i.e., RKI) preconditioners. The Fourier footprints are eigenvalue distributionscorresponding to the Fourier symbol of the operator V−1PL. For these plots the Mach number is M =0.5,the Reynolds number is Re=∞, and there is flow alignment (flow angle α=0◦). There is a moderate meshcell aspect ratio (AR) of five. The dashed and solid lines in the figures are the absolute stability curves withstandard and modified dissipation weighting. The eigenvalue distributions are associated with all modeshaving a high-frequency component in at least one direction. There are four eigenvalue footprints, whichcorrespond to the two convective (entropy and vorticity) modes and the two acoustic modes. The entropyfootprint is either above or below the real axis. The two acoustic footprints are above and below the realaxis, and the vorticity footprint lies between them. With block Jacobi there is poor damping of certain high-frequency modes corresponding to acoustic and convective modes. First-order upwind differencing is usedfor the residual function, but the character of the clustering does not change with second order. The blockGS preconditioner provides good damping except for certain convective modes, and this is a consequenceof the flow alignment that results in a vanishing eigenvalue. Most eigenvalues are well clustered away fromthe origin of the complex plane, which is necessary for effective damping. By introducing an appropriateentropy fix, all modes are effectively damped with block GS.
Fig. 2 displays contours of the spectral radius of the amplification matrix for RKI(3,3) when the Fourierangles −π ≤ θx, θy ≤ π with a flow angle 45◦. Two symmetric Gauss-Seidel (SGS) sweeps are used toapproximate the inverse of the implicit system. High-frequency components lie outside the dashed line square.In this comparison of damping behavior of the RKI(3,3) scheme with standard and modified dissipationweighting, the improved damping, especially of the highest frequencies, with modified weighting is evident.The modified weighting provides the additional advantage of extending the absolute stability curve of theRKI(3,3) scheme by a factor of two along the negative real axis in the complex plane, as seen in Fig. 1. Thisbenefit can be quite important when more dissipation is introduced into the scheme, which can occur, forexample, when there is a significant increase in the eddy viscosity of a turbulent flow. Moreover, it resultsin a higher level of robustness for the flow solver.
Fig. 3 shows the influence of two different values of Re, and representative corresponding values of AR,on the damping behavior of the RKI(3,3) scheme. We observe a change in the character of the dampingcurves due to the AR. There is still good smoothing of all high-frequency modes, and the smoothing factorsfor Re= 102 and Re= 106 are approximately 0.27 and 0.43, respectively. The good smoothing factors forthe various flow parameters indicate that the RKI(3,3) scheme is suitable for full-coarsening multigrid.
4 Numerical Results
To demonstrate the robustness of the present scheme we show a result for an axisymmetric solid rocket motor.The solid rocket motor flowfield is characterized by a high temperature and a very slow speed (M < 0.1)inside the chamber. The flow accelerates towards the nozzle throat and a highly supersonic flow developsin the diverging portion of the nozzle. Fig. 4(a) shows the full range of Mach contours in the chamber andnozzle. In Fig. 4(b) we zoom on the Mach values inside the motor where the sound speed is around 1000m/s and the Mach values are low.
The flow is solved using the Navier-Stokes equations for laminar steady flow. This computation was doneusing a three-dimensional (3-D) code with a single cell in the circumferential direction. An upwind schemewith a Van Albada limiter was used to discretize the flow equations. The RKI(3,3) scheme with three levelsof multigrid was used to solve the flow equations. One boundary condition was injection of flow on thepropellent surface according to Vielle’s law
m = ρpa
(p
pref
)n
where m is the injected mass flux, ρp is the propellent density, a is the burning rate, and n is the burningrate exponent. The other boundary conditions were as follows: no slip on the chamber and nozzle walls,extrapolation on the supersonic motor exit plane.
In Fig. 5 we compare the convergence behavior of the multigrid scheme using RKI(3,3) with that usingthe original (scalar) implicit residual smoothing [2]. A well converged solution is obtained with the matrixsmoother (CFL = 1000), whereas the computation with the scalar smoother (CFL = 2.5) exhibits a muchslower convergence.
To verify the effect of the new dissipation weights [1.0, 0.5, 0.5] in the RKI(3,3) scheme, we computedtransonic flow over an ONERA M6 wing with a free-stream M∞ = 0.83 and Re = 2.1158 × 107 using aBaldwin-Lomax turbulence model. A matrix-valued dissipation was used [5]. Entropy fixes were appliedto the eigenvalues of the preconditioning matrix in Eq. 5. For the entropy fix of the convective eigenvalueswe use a two-parameter function of the mesh aspect ratio. The two parameters multiply the bounds onthese eigenvalues. If these parameters are chosen too large, the convergence slows down. When they aretoo small, the scheme can become unstable. Hence, we want these parameters to be as small as possiblewithout destroying stability. Since this function is nonlinear, it cannot be modeled with the Fourier analysis.The two parameters are very dependent on the dissipation weighting used. In Table 1 we present theconvergence after 100 multigrid V-type cycles; NC means no convergence. We see that the convergencefor fixed cutoff parameters with the weights [1.0, 0.5, 0.5] is slightly better. What is more important isthat the new dissipation weights make the scheme more robust, allowing lower cutoffs which result in fasterconvergence. We have also done similar computations for two-dimensional transonic flow around the RAE2822 airfoil. The trends were the same, although the required cutoffs were slightly larger.
5 Concluding Remarks
Fourier analysis of the RKI(3,3) scheme has been considered to study the effect of various flow parameters.Using the analysis we have compared block Jacobi and block Gauss-Seidel preconditioners. The improvementof the current RK scheme, which uses block Gauss-Seidel, has been shown. The analysis has demonstratedthat this scheme has good smoothing and eigenvalue clustering properties for all high-frequency error com-ponents, making it a suitable smoother for full-coarsening multigrid. In addition, the advantages of a newweighting of dissipation, including an increased reliability of the scheme, have been discussed.
The implicit preconditioner introduced in [3], investigated and improved in [4], has been implementedin several three dimensional codes using either central-differencing with a matrix-valued dissipation or anupwind Roe scheme for the convective terms. Previously, acceleration of the convergence to a steady statehas been achieved for turbulent flows over airfoils and wings and for flows in turbomachinery. In this studywe have demonstrated that this scheme can be applied to flows inside rocket motors.
References
[1] A. Jameson, W. Schmidt, E. Turkel, Numerical solutions of the Euler equations by finite volume methodsusing Runge-Kutta time-stepping schemes, AIAA Paper 81-1259, 1981.
[2] R. C. Swanson, E. Turkel, Multistage schemes with multigrid for Euler and Navier-Stokes equations,NASA TP 3631, 1997.
[3] C.-C. Rossow, Efficient computation of compressible and incompressible flows. J. Comput. Phys. 220(2007) 879–899.
[4] R. C. Swanson, E. Turkel, C.-C. Rossow, Convergence acceleration of Runge-Kutta schemes for solvingthe Navier-Stokes equations, J. Comput. Phys. 224 (2007) 365–388.
[5] R. C. Swanson and E. Turkel, On Central Difference and Upwind Schemes, J. Comput. Phys. 101 (1992)292–306.
[6] N. A. Pierce, M. B. Giles, Preconditioned multigrid methods for compressible flow calculations onstretched meshes, J. Comput. Phys. 136 (1997) 425–445.
Re
Im
-10 -8 -6 -4 -2 0-3
-2
-1
0
1
2
3wgts: [1 1 1]
wgts: [1 .5 .5]
(a)
Re
Im
-10 -8 -6 -4 -2 0-3
-2
-1
0
1
2
3wgts: [1 1 1]
wgts: [1 .5 .5]
(b)
Figure 1: Fourier footprints of RK(3,3) scheme with two preconditioners for all modes with high-frequencycomponents (64 × 64, M = 0.5, α = 0◦, CFL = 103, AR = 5, Re = ∞). (a) Block Jacobi (residual: 1storder), (b) Block Gauss-Seidel (residual: 2nd order).
0.35 0.250.2
0.1
0.35
0.3
0.4
0.150.150.
1
0.1
0.35 0.25
0.1
0.2
0.2
0.35
0.2
0.4
0.6
0.8
0.8
θx
θ y
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
(a)
0.2
0.2 0.20.250.2
5
0.20.15
0.150.2
0.6
0.8
0.8
0.4
θx
θ y
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
(b)
Figure 2: Effect of dissipation weights on damping behavior of RKI(3, 3) scheme (64×64, M = 0.5, α = 45◦,CFL = 103, AR = 1, Re = ∞, 2 SGS). (a) wgts: [1, 1, 1], (b) wgts: [1, 0.5, 0.5].
0.250.2
0.15
0.25
0.20.25
0.90.15
0.4
θx
θ y
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
(a)
0.2
0.20.15
0.2
0.20.15
0.250.35 0.95
θx
θ y
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
(b)
Figure 3: Damping behavior of RKI(3, 3) scheme with variation in Re and AR (64× 64, M = 0.5, α = 45◦,2 SGS). (a) Re = 102, AR = 10, CFL = 103, (b) Re = 106, AR = 103, CFL = 104.
x
Y
0 0.1 0.2 0.3
-0.05
0
0.05
0.1
0.15M
2.11.71.30.90.50.1
(a)
x
Y
0 0.1 0.2 0.3
-0.05
0
0.05
0.1
0.15M
0.10.080.060.040.020
(b)
Figure 4: Rocket Motor
cycles
log
(err
)
0 200 400 600 800 1000 1200 1400
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
implicit smoothingRK/implicit smoothing
Figure 5: Convergence rate for rocket motor
Dissipation Weights Cutoff Parameters Residual Conv. Rate1.0, 0.5, 0.5 .10, .20 .327× 10−4 .87981.0, 0.5, 0.5 .10, .10 .445× 10−5 .86191.0, 0.5, 0.5 .08, .05 .472× 10−6 .84251.0, 0.5, 0.5 .08, .04 .230× 10−6 .83641.0, 0.5, 0.5 .06, .04 .182× 10−6 .83441.0, 1.0, 1.0 .10, .20 .348× 10−4 .89371.0, 1.0, 1.0 .10, .10 .500× 10−5 .86311.0, 1.0, 1.0 .08, .05 .604× 10−6 .84471.0, 1.0, 1.0 .08, .04 NC1.0, 1.0, 1.0 .06, .04 NC
Table 1: Convergence for ONERA M6 wing for various dissipation weights and cutoff parameters.
Anisotropic adaptive technique for simulationsof steady compressible flows on unstructuredgrids
O. Feodoritova1, D. Kamenetskii2, S. Kravchenko3, A. Martynov1, S. Medvedev1
and V. Zhukov1
Abstract This paper presents the anisotropic adaptive technique applied to solutionof the 2D compressible Navier-Stokes and Euler equations. The equations are dis-cretized by the higher order finite element scheme on unstructured triangular grids.Anisotropic grid refinement is applied for grid adaptation to provide accurate com-putations of multiscale solutions. A heuristic Hessian-based error indicator is usedto capture directional information. For finite elements of a polynomial order k > 1p−multigrid can serve as a linear solver, or preconditioner for GMRES. For the loworder k = 1 scheme the geometrical multigrid is employed based on hierarchicalstructures of the adaptive grid.
1 Introduction
We consider the governing equations (the Euler or Navier-Stokes equations supple-mented by Spalart-Allmaras turbulence model) written in the form of conservationlaws.
A framework of high resolution gridding technology is presented for automaticunstructured grid generation and viscous flow adaptation. The goal is to iterativelyresolve strongly anisotropic behavior of solution, i.e. grid cell aspect ratio andstretching direction must follow the numerical solution anisotropy at every pointin space (including boundary and shear layers, shocks etc.). At the same time theorder property for the discrete approximation must be maintained, i.e. the error ofnumerical solution must decrease in appropriate way on a sequence of adaptivelyrefined grids with increasing number of grid nodes.
1 Keldysh Institute of Applied Mathematics Moscow, Russia, e-mail: [email protected], e-mail:[email protected], e-mail: [email protected], e-mail: [email protected] Boeing Commercial Airplanes, Seattle, WA, USA, e-mail: [email protected] The Boeing Company, Chicago, IL, USA, e-mail: [email protected]
2 O. Feodoritova, D. Kamenetskii, S. Kravchenko et al.
a) b)
Fig. 1 a) Three-element airfoil. Left: macro-grid. Right: upper – fragment of macro-grid, lower– computational adaptive grid in this fragment. b) Slotted airfoil solution. Level lines of Machnumber and a fragment of the adaptive grid are shown.
In practice, higher order finite difference schemes are favorable due to lowercomputational cost. However, accuracy might be lost due to highly anisotropic grids.To overcome this difficulty, we use higher order finite element (FE) scheme [7],based on the SUPG scheme [5]. In combination with the presented solution-adaptiverefinement technique capable also for the dynamic grid adaptation the stabilized FEscheme provides a powerful tool both for steady and time-dependent flow modeling.
Higher order accurate schemes offer potentially large savings in computationalresources but might have difficulty in computing steady flow, when convergence ofthe discrete nonlinear steady state equations is desired. This is especially true forrather irregular and high cell aspect ratio adaptive grids, and special care must betaken to ensure the ”rock-solid” convergence of the nonlinear solver. The implicittime integration is used to march the solution to steady state.
On each time step we solve systems of linear equations arising from linearizationof the discrete nonlinear scheme. For finite elements of a polynomial order k > 1p−multigrid can serve as a linear solver, or preconditioner for GMRES. For the loworder k = 1 scheme the geometrical multigrid is employed based on hierarchicalstructures of the adaptive grid.
2 Anisotropic adaptation
Adaptive generation of computational grids is a necessary requirement for accuratemodeling of complex flows. It is a challenging problem that involves substantialmultidisciplinary effort including approximation theory, error estimates for Navier-Stokes equations and computational geometry. The problem becomes especiallycomplicated when the curved boundaries describing realistic geometries are to betreated.
Optimal computational grids for viscous flows feature a mixture of stretched gridcells with high aspect ratio needed to resolve thin boundary and shear layers and
Anisotropic adaptive technique for compressible flows on unstructured grids 3
isotropic cells for description of essentially isotropic flow features. As follows fromthe approximation theory, a Hessian based metric can be employed to describe opti-mal grid cell size and stretching direction distribution in space. The Hessian shouldbe estimated based on numerical solution and some regularization is needed. Theregularization can be directly applied to the Hessian matrices but the differencesin scales between boundary layers and far-field is so large that usual smoothingprocedures would significantly distort the grid optimality. Even if the problem ofisotropic grid generation had been satisfactory solved for infinite space with a givenmetric - and it is not the case in general 3D situation when the grid quality con-trol is essential - the presence of curved boundaries considerably aggravates theperspectives of the optimal grid generation based on the ”first principles” Hessianmetric. Another approach is to combine the regularization of the Hessian metric andthe grid elements in real space. The concept of macro-grid introduced in [6] is apractical implementation of such an approach at the expense of deviation from thestrict grid optimality but with a possibility to directly control the grid quality andoptimize boundary conforming grid. An initial grid is needed to begin the adapta-tion procedure. Usually a coarse isotropic grid is enough for that. The combinationof stretched and isotropic grid cells in the course of the adaptive grid generation isthen obtained by subsequent refinement/de-refinement operations on the macro-gridelement hierarchy: edges and cells (also faces in 3D case) followed by macro-celltriangulation (tessellation). The resulting computational grid is used by a solver.
The edge error indicator for an edge vector e
Et =
(w1
|∂u/∂e||∇u|1−qn
+w2|∂ 2u/∂e2|1/2
|H|(1−qn)/2
)|e|
is based on the optimal anisotropic estimates for linear interpolation error in Lp norm[1, 2]. The value of qn is determined by the value of p and the dimension of space n:qn = 1−1/(2p+n). The L∞ norm corresponds to qn = 1. Lower values of qn < 1,corresponding to weaker norms, provide a possibility to redistribute the adaptivegrid from sharp solution features (boundary layers and shocks) to other flow fieldregions. A reasonable minimal value of qn is determined by the condition p≥ 1. Themajorant function |H| for the Hessian, entering the error indicator, can be replacedby its maximal eigenvalue. A usual choice of the scalar function to adapt is theMach number. The main purpose of the first derivative term is a regularization. Theweights in the edge error indicator are adjusted in order to assure that a prescribedfraction of edges to be refined are flagged due to the second derivative term.
The goal function for the mechanics of the macro-grid adaptation is to providethe equidistribution of the error indicator Et over the edges of the computationalgrid with a reasonable accuracy. The macro-grid and the computational grid aredemonstrated in Fig.1a.
4 O. Feodoritova, D. Kamenetskii, S. Kravchenko et al.
3 Discretization
Three broad classes of difference schemes for achieving higher order accuracy werereviewed and unified under a common framework. These are finite volume, stabi-lized finite element and Discontinuous Galerkin (DG) methods [7]. The higher orderschemes are exceptionally efficient for smooth inviscid and viscous flows. For flowswith discontinuities in the solution or its derivatives, the accuracy of all higher orderschemes is degraded (asymptotically) to an order dependent on the solution smooth-ness. This suggests the need for local solution-based grid adaptation (h-refinement)in the vicinity of these discontinuities. The grid refinement test called the torture testwas proposed. It adds grid in the worst possible location as determined by solutionerror. Unbounded error growth suggests potential difficulties for incorporating a par-ticular discretization into a practical h-p refinement strategy. By applying a limiterto the gradient, the second-order finite volume scheme successfully passes the test.Other schemes that pass the torture test with bounded solution errors include themonotone DG(0) scheme and the higher order SUPG(1-2) and DG(1-2) schemes.Robust damped Newton method augmented by a time term was used for nonlinearsolution converged to double precision machine-zero.
The combination of the Streamwise Upwind Petrov-Galerkin/Galerkin LeastSquares methods SUPG(1)/GLS(1) with the node based DG(0) was shown to re-liably work with the anisotropic refinement framework described above.
In Fig.1b the steady state solution for the slotted wing is presented.
4 Matrix solver
On each time step we solve systems of linear equations arising from linearizationof the discrete nonlinear scheme. Two cases are considered: higher order accuratescheme with polynomial degree k > 1 and the lower order scheme with k = 1 thatcorresponds to linear finite elements. For model problems with smooth solutionsa polynomial degree k corresponds to the local error O(hk+1) in L∞ as well as inL1 and L2 norms. Polynomial degree k of the FE basis functions can be specifiedeither for each cell (providing a possibility of local p-refinement) or globally overthe whole triangulation.
In the case of k > 1 the choice of the FE basis affects numerical solution of linearsystems. We compared standard FE Lagrange (L) and hierarchical (H) bases. Inthe L–basis each local element function is a polynomial of the same degree k ≥ 1.The H–basis uses the standard linear element modes which are supplemented byhierarchical polynomials of higher degrees up to k. Each expansion coefficient ofgrid function is associated with a reference grid node (a vertex or an edge point); fork > 2 there are also the ”interior” unknowns eliminated by the static condensationprocedure (exact Schur complement).
We developed an iterative multilevel algorithm [3]. It is based on approximateelimination of higher order degrees of freedom and can be considered as an ap-
Anisotropic adaptive technique for compressible flows on unstructured grids 5
proximate Schur complement approach. Multigrid interpretation describes such atechnique much better, and this method is now known as p−multigrid [4]. In our al-gorithm the fine grid contains vertices of triangulation (c−nodes) and nodes relatedto unknowns on edges (e−nodes). The coarse grid contains only c−nodes.
The results of numerical experiments for a set of test problems (diffusion,convection-diffusion, Euler, Navier-Stokes) show capability of the proposed variantof p−multigrid. In most considered problems p−multigrid achieves convergencerate independent of k and h in the case of hierarchical finite elements.
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♦
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♦
♦
♦
♦
♦
♦ ♦
♦♦♦
♦♦
- c-points - r-points - e-points♦
a) b) c) d)Fig. 2 Agglomerations based on isotropic and anisotropic macro-grids.
For low order scheme with k = 1 the main difficulties, associated with the useof multigrid algorithm on unstructured grids, lie in the generation of coarser levels.Most of the known approaches are not suitable for anisotropic grids.
For unstructured anisotropic grids, we propose a specially designed method togenerate the set of coarse levels and corresponding multigrid operators, i.e. prolon-gation, restriction and coarse grid operators. This algorithm essentially uses the hi-erarchical macro-cell and macro-edge structures arising from the proposed adaptiveprocess. The coarse grid is not geometrically triangular anymore – it is just a markedsubset of fine grid nodes. In Fig.2a,b,c some possible agglomerated grid cells (coarsenodes are shown by gray circles) arising from the hybrid triangle/quadrangle macro-grid are shown. In case of highly anisotropic grids with relatively small number ofhighly populated macro-cells another type of agglomeration is more appropriate(Fig.2d).
To derive the coarse grid equations we use operator-dependent interpolation,operator-based elimination of internal degrees of freedom and Galerkin coarsegrid operators. The algorithm employs mainly the additive Schwartz method as asmoother based on decomposition of the computational domain into a set of subdo-mains. Such decomposition is usually produced in computations on multiprocessorcomputers.
The proposed multigrid algorithm is used as a preconditioner to accelerateKrylov subspace iterations, in particular the GMRES method. For the proposedApproximate Schur Algorithm (ASA) the number of GMRES iterations is almostindependent on the number of processors on middle stage of nonlinear iterationscompared to the growth for the baseline Alternating Schwarz Method (ASM).
6 O. Feodoritova, D. Kamenetskii, S. Kravchenko et al.
5 Conclusions
The right ingredients for working adaptive Navier-Stokes solver are: anisotropicadaptive grids, stabilized FE schemes ensuring convergence on irregular grids, andadvanced parallel matrix solvers.
An optimal combination of the heuristic Hessian based error indicator, as provid-ing most reliable directional information for anisotropic h-refinement, with theoret-ically backed error estimates (e.g. based on adjoint solution for functional outputsadaptation) has to be found.
Concerning the higher order schemes the following considerations should betaken into account. For nonlinear governing equations overshoots in the solutionare unacceptable in practice, as they may adversely impact solver convergence oreven preclude the existence of a discrete solution on a given grid. Therefore, higherorder schemes that allow nonphysical solution overshoots and oscillations must stillbe augmented with limiters before they can be employed routinely for computingsteady-state Navier-Stokes solutions. As for the h-p refinement, wherein higher or-der schemes are combined with h-refinement, the difficulty with this approach lies inestimating the local smoothness of numerical solution, in particular, for a hyperbolicproblem, where local criteria are not always sufficient.
The coarse grid operators based on the hierarchical grid topology seems to bevery useful for speed-up of convergence in massively parallel computations.
Real gains of the adaptive solvers are expected in real life 3D applications withrealistic geometries. There is still a way to go in this direction but the success ofthe 2D version is rather encouraging. Taking into account realistic 3D geometriesand generation of adaptive anisotropic surface grids are the next big challenges foradaptive grid mechanics in 3D.
References
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3. Feodoritova, O., Young, D., Zhukov, V. Iterative algorithms for higher order finite elementschemes. Russian J. of Math. Modelling, 16, (2004), N 7, 117–128. In Russian.
4. Helenbrook, B., Mavriplis,D., Atkins, H. Analysis of p-multigrid for continuous and discon-tinuous finite element discretizations, AIAA Paper 2003-3989, 2003.
5. Hughes, T. Recent progress in the development and understanding of SUPG methods withspecial reference to the compressible Euler and Navier-Stokes equations, Int. Journ. for Nu-mer. Met. in Fluids, 7 (1987), 1261-1275.
6. Martynov, A., Medvedev, S. A robust method of anisotropic grid generation. Grid Genera-tion: theory and applications. Computing Centre RAS, Moscow, 2002, p. 266–275.
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