1 Introduction
In fluid flow simulations, the incompressibility constraint is often used to bypass fast acoustic waves and thereby allow the solution to evolve on a convective timescale that is most relevant for many engineering problems. This model leads to a Poisson problem for the pressure that is invariably the stiffest substep in timeadvancement of the NavierStokes equations. For large 3D problems, which mandate the use of iterative methods, the pressure solve thus typically encompasses the majority of the solution time.
For the spectral element (SE) discretization, the Poisson system matrix contains nonzeros for elements with a polynomial degree of (i.e., approximately
unknowns.) Through the use of tensorproductsum factorization, however, the SE matrixvector product can be effected in only
reads and operations, even in the case of complex geometries [parter_spectral_1979, deville_highorder_2002]. Consequently, the key to fast SEbased flow simulations is to find effective preconditioners tailored to this discretization. Many methods, including geometric multigrid approaches with pointwise Jacobi and Chebyshevaccelerated Jacobi smoothers [sundar_comparison_2015, adams_parallel_2003, Kronbichler2019], geometric multigrid with overlapping Schwarz smoothers [lottes_hybrid_2005, stiller_nonuniformly_2017, loisel_hybrid_2008], and preconditioning via loworder discretizations [parter_spectral_1979, olson_algebraic_2007, bellomaldonado_scalable_2019, canuto_finiteelement_2010], have been considered.In this work, we explore the parallel performance of these methods and extend the Schwarzsmoothing based multigrid of [lottes_hybrid_2005] to support restrictiveadditive Schwarz ideas of Cai and Sarkis [ras99] as well as Chebyshevaccelerated smoothing. In addition, we extend the loworder preconditioning strategy of [bellomaldonado_scalable_2019] to run on GPU architectures through the use of AmgX [naumov_amgx_2015]
as an algebraric multigrid (AMG) solver for the sparse system. Numerical results are shown for the SEbased pressure Poisson problem as well as for the NavierStokes equation. All methods considered are implemented by the authors in the scalable opensource CFD code, nekRS
[fischer_nekrs_2021]. Special focus is given to performance and scalability on largescale GPUbased platforms such as OLCF’s Summit. A significant contribution is the introduction of a polyalgorithmic autotuning strategy that is designed to ensure that users realize optimized performance over a broad range of problem formulations, archictectures, and processor counts.The structure of this paper is as follows. Section 2 outlines the spectral element formulation for the Poisson problem in , as well as describe the various preconditioners considered. A brief description of several model probems of interest are presented in section 3. Numerical results are highlighted in section 4. Finally, a brief summary of the survey of solver techniques considered is provided in section 5.
2 Background and Implementation
We introduce here basic aspects of the SE Poisson discretization and associated preconditioners along with an important autotuning strategy.
2.1 SE Poisson Discretization
Consider the Poisson equation in ,
(1) 
Boundary conditions for the pressure Poisson equation are either periodic, Neumann, or Dirichlet, with the latter typically applicable only on a small subset of the domain boundary, , correspondong to an outflow condition. Consequently, Neumann conditions apply over the majority (or all) of the domain boundary, which makes the pressure problem more challenging than the standard allDirichlet case.
The SE discretization of (1) is based on the weak form, Find such that,
(2) 
where is a finitedimensional approximation comprising the basis functions used in the SE discretization, , , that vanish on and represents the discrete inner product based on GaussLobattoLegendre quadrature in the reference element, . The basis functions allow us to represent the solution, , as , leading to a linear system of unknown basis coefficients,
(3) 
with respective mass and stiffnessmatrix entries, , and .
is tesellated into nonoverlapping hexahdedral elements, , for , with isoparametric mappings from to provided by , for . Each is a thorder Lagrange cardinal polynomial on the GaussLobattoLegendre (GLL) qudrature points, . Similarly, the test and trial functions are written in local form as . Continuity is ensured across the interface between adjacent elements by enforcing when
. From this, a globaltolocal degreeoffreedom mapping,
, can be represented by a Boolean matrix , such that . The assembled stiffness matrix is then , where comprises the local stiffness matrices, . Similarly, . The SE formulation uses coincident GLL quadrature and nodal points, such that is diagonal. Moreover, is never formed, as it would contain nonzeros in the general case. Rather, the tensorproductsum factorization [parter_spectral_1979] allows for to be evaluated in time with storage, as described in detail in [deville_highorder_2002].2.2 Preconditioners
In the current study, all preconditioners are applied in the context of restarted GMRES. Although is symmetric positive definite (SPD), many of the preconditioners are asymmetric. Further, a recent study [fischer_highly_2021] has shown the benefits of projectionbased GMRES over flexible conjugate gradients because the effectiveness of the overall pressure solution strategy (which includes projection onto prior solutions [fischer_projection_1998]) generally ensures a low enough iteration count, , such that the costs in GMRES are not overly onerous.
2.2.1 Semfem
In [parter_spectral_1979], Orszag suggested that constructing a sparse preconditioner based on the loworder discretizations with nodes coinciding with those of the highorder discretization would yield bounded condition numbers and, under certain constraints, can yield for secondorder Dirichlet problems. This observation has led to the development of preconditioning techniques based on solving the resulting low order system [olson_algebraic_2007, bellomaldonado_scalable_2019, canuto_finiteelement_2010].
In the current work, we employ the same loworder discretization
considered in [bellomaldonado_scalable_2019]. Each of the vertices of
the hexahedral element is used to form one loworder, tetrahedral element,
resulting in a total of eight loworder elements for each GLL subvolume
in each of the highorder hexahedral elements.
This loworder discretization is then used to form the sparse operator, .
The socalled weak preconditioner, , is used to precondition the system.
Algebraric multigrid (AMG), implemented in CUDA in AmgX [naumov_amgx_2015],
is used with the following setup to solve the low order system:

PMIS coarsening

0.25 strength threshold

Damped Jacobi relaxation ()

One Vcycle for preconditioning

Smoothing on the coarsest level
We denote this preconditioning strategy as SEMFEM.
2.2.2 multigrid, Schwarz Smoothers
Another preconditioning strategy for the SEbased Poisson problem is to use geometric multigrid (pMG). The classical single pass Vcycle is summarized in Algorithm 1. In our application, we limit pMG preconditioning to a single Vcycle pass. Alternative strategies, such as F or Wcycle multigrid are not considered.
The SEbased additive Schwarz method (ASM) presented in [lottes_hybrid_2005, loisel_hybrid_2008] solves local Poisson problems on subdomains that are extensions of the spectral elements. The formal definition of the ASM preconditioner (or pMG smoother) is
(4) 
where is the restriction matrix that extracts nodal values of the residual vector that correspond to each overlapping domain, as indicated in Fig. 1b. To improve the smoothing properties of the ASM, we introduce the diagonal weight matrix, , which scales each nodal value by the inverse of the number of subdomains that share that node. Although it compromises symmetry, postmultiplication by was found to yield superior results to pre and postmultiplication by [stiller_nonuniformly_2017, lottes_hybrid_2005].
In a standard Galerkin ASM formulation, one would use , but such an approach would compromise the storage complexity of the SE method. To construct fast inverses for , we approximate each deformed element as a simple boxlike geometry, as demonstrated in Fig. 1a. These boxes are then extended by a single degreeoffreedom in each spatial dimension to form overlapping subdomains with interior degreesoffreedom in each domain. (See Fig. 1). The approximate box domain enables the use of the fast diagonalization method (FDM) to solve for each of the overlapping subdomains, which can be applied in time in [lottes_hybrid_2005]. The extendedbox Poisson operator is separable, with the 3D form
(5) 
where each represents the extended 1D massstiffness matrix pairs along the given dimension [lottes_hybrid_2005]. The FDM begins with a preprocessing step of solving a series of small,
, generalized eigenvalue problems,
(6) 
and defining and , to yield the similarity transforms
(7) 
From these, the inverse of the local Schwarz operator is
(8) 
where
(9) 
is a diagonal matrix. This process is repeated for each element, at each multigrid level save for coarsest one. Note that the perelement storage is only for the matrices and for . At each multigrid level, the local subdomain solves are used as a smoother. On the coarsest level (), however, BoomerAMG [henson_boomeramg_2002] is used to solve the system with the same parameters as the AmgX solver in section 2.2.1, except using Chebyshev smoothing. Unless otherwise noted, all pMG preconditioners use a single BoomerAMG Vcycle iteration in the coarsegrid solve.
Presently, we also consider a restrictive additive Schwarz (RAS) version of (4), wherein overlapping values are not added after the action of the local FDM solve, following [ras99]. RAS has the added benefit of reducing the amount of communication required in the smoother. (Formally, RAS can be implemented by simply changing .) Note that, in the case of ASM and RAS smoothers without Chebyshevacceleration, an additive Vcycle with no postsmoothing is used to avoid residual reevaluation in Algorithm 1 [fischer04].
2.2.3 Chebyshev Acceleration
A notable improvement over standard Jacobismoothed multigrid is to use Chebyshevacceleration [sundar_comparison_2015, adams_parallel_2003], as described in Algorithm 2 for a given surrogate smoother, . While is typically based on Jacobi smoothing (e.g.,[Kronbichler2019]), it is also possible to consider the use of overlappingSchwarz (4) as the smoother, which is a new approach that we explore here.
Algorithm 2 requires approximate spectral bounds, , of the smoothed operator, . Using 10 rounds of Arnoldi iteration, we generate as an initial proxy for . The bounds employed in Algorithm 2 are then determined through sensitivity analysis similar to that performed by Adams and coworkers [adams_parallel_2003]. The analysis is conducted using a challenging model problem, namely the Kershaw case of subsection 3.2, with , , and relative (norm) residual tolerance of . A secondorder Chebyshevaccelerated ASM smoother is used in the pMG Vcycle preconditioner, which we denote as the current default preconditioner in nekRS.
The results of the sensitivity study are shown in Table 1. The required iteration counts are seen to be sensitive to underestimation of , as also found by Adams and coworkers [adams_parallel_2003]
. On the other hand, results are less sensitive to the minimum eigenvalue estimate, provided
, as this delimits between the low frequencies that are handled by the coarse grid correction and the high frequencies that are eliminated by the action of the smoother. The choice of the bounds (relative to ) is not universal: (1/30,1.1) [adams_parallel_2003], (0.3,1) [baker_multigrid_2011], (0.25,1) [sundar_comparison_2015], and (1/6,1) [zhukov_multigrid_2015] have all been considered. Based on the results of Table 1, we set as a conservative choice. Lastly, we denote a pMG preconditioner using order Chebyshevaccelerated smoother with a multigrid schedule of as Cheb(), for the remainder of this work.1000  1000  1000  1000  1000  
1000  1000  1000  110  64  
1000  1000  1000  50  40  
1000  1000  121  40  38  
159  45  43  42  43 
2.3 Online Preconditioner Tuner
For any given preconditioner, iteration counts can vary dramatically across the range of fluid dynamics applications and computational meshes used with nekRS. Moreover, even for a fixed problem, the optimal choice of parameters for minimumtimetosolution can vary with the number of processors, , because the coarsegrid solve costs can dominate the solution for large . In such cases, additional smoothing, which reduces the number of cycles and thus the number of coarsegrid solves, can be beneficial. Identifying an optimal pMG strategy can be a daunting and expensive task even for sophisticated users, particularly if it requires queuing multiple large jobs, where such optimization is most warranted. Further, as shown in Table 2, the space of solver parameters is large.
Consequently, it makes sense to automate the tuning process, so that the pMG parameters are selected in context, at the correct processor count. We have developed an autotuner where the user can prescribe when the tuning should take place (e.g., on step 1000 or every 500 steps)^{1}^{1}1In turbulent flow simulations, it may take several hundred or thousand timesteps before the solution is rich in high wavenumber content; tuning too early might lead to suboptimal parameters.. Two tuning strategies are currently available: either an exhaustive search at the prescribed step(s) or a set of exhaustive searches over multiple timesteps. The former ensures that all trials solve the same problem, collecting several trials of each pMG preconditioner for the same step to mitigate the effects of system noise. The latter performs the exhaustive search over multiple timesteps, minimizing the cost,
(10) 
where and denote the iterations and time per iteration for preconditioner at sample . This approach seeks to minimize the effects of system noise by considering the minimum time per iteration for a candidate method, while also taking into account the convergence rate of the method. The use of projection during the exhaustive search, moreover, is also considered. Once determined, the optimal parameters are used through the rest of the simulation and are saved to a file so that subsequent restarts or simulations in the same class can leverage the results of the initial autotuning.
In the sequel, we consider the online tuner with Jacobi, ASM, and RAS as candidate smoothers for Chebyshevaccelerated pMGbased preconditioning. The Chebyshev order is varied between one and three. An exhaustive search is performed over this parameter space in order to minimize the pressuresolve timetosolution. To mitigate the effects of system noise on the results, three trials of each pMG preconditioner are performed over the search space. We note that our principal objective is not to squeeze out a few percent over a raft of good choices, but rather to ensure that a case does not run with an unfortunate set of parameters for which the performance is significantly substandard.
We further remark that the overall solver space is vast—we list many possible considerations in Table 2—and that the subset chosen here, while based on significant practical experience, is just a first step towards comprehensive coverage for any problem on any architecture.
Parameters  

Krylov Solver  flexible CG, GMRES 
GMRES  restart size, classical/modified GramSchmidt 
Preconditioners  pMG, SEMFEM 
pMG  cycle type, schedule, smoother, coarse grid solver, coarse grid discretization 
Cycle Type  V,W,Fcycle 
Schedule  all levels, by halfs, twolevel, 
Smoother  GaussSeidel, Jacobi, ASM, RAS, Chebyshevacceleration 
ASM/RAS  amount of overlap, local subdomain solve techniques (FDM), weights 
Chebyshevacceleration  Order, spectral bounds 
Coarse grid  direct solve, AMG, smooth 
Coarse grid discretization  Galerkin, nonGalerkin 
SEMFEM  loworder discretization, AMG solver settings 
AMG  cycle type, smoothers, interpolators, coarsening type, strength threshold 
3 Test Cases
We describe four model problems that are used to test the SE preconditioners. The first is a standalone Poisson solve, using variations of the Kershaw mesh. The others are modestscale NavierStokes problems, where the pressure Poisson problem is solved over multiple timesteps. The problem sizes are listed in Table 3 and range from relatively small (=16M points) to moderately large (=645M).^{2}^{2}2Larger cases for recent fullscale runs on Summit with =51B are reported in [fischer_highly_2021].
Case Name  

Kershaw  47K  7  16M 
146 pebble  62K  7  21M 
1568 pebble  524K  7  180M 
Boeing Speed Bump  885K  9  645M 
3.1 Poisson
The Kershaw family of meshes [kolev_ceed_2021, kershaw_differencing_1981]. has been proposed as the basis for a highorder Poissonsolver benchmark by Center for Efficient Exascale Discretization (CEED) within the DOE Exascale Computing Project (ECP). This family is parametrized by an anisotropy measure, , that determines the degree of deformation in the and directions. As decreases, the mesh deformation and aspect ratio increase along with it. The Kershaw mesh is shown in Fig. 2 for several values of . The domain with Dirichlet boundary conditions on . The right hand side for (1) is set to
(11) 
where is a random, continous vector vanishing on . The linear solver terminates after reaching a relative residual reduction of . Since this test case solves the Poisson equation, there is no timestepper needed for the model problem.
3.2 NavierStokes
For the pressurePoisson tests, three flow cases are considered, as depicted in Fig. 3. The first case corresponds to turbulent flow through a cylindrical packedbed with 146 spherical pebbles. The second is a similar configuration with 1568 pebbles. Both bed flows are at Reynolds number , based on sphere diameter, . Time advancement is based on a twostage 2ndorder characteristics timestepper with CFL=4 ( for the 146pebble case, for the 1568pebble case). An absolute pressure solver tolerance of is used. A restart at and convective time units is used for the 146 and 1568pebble cases, respectively, to provide an initially turbulent flow.
The third case, shown in Fig. 3c, is direct numerical simulation (DNS) of seperated turbulent flow over a speed bump at . This test case was designed by Boeing to provide a flow that exhibits separation. A DNS of the full 3D geometry, however, remains difficult [shur_direct_2021]. Therefore, this smaller example proves a useful application for benchmarking solver performance. This case uses a 2ndorder timestepper with CFL=0.8 () and an absolute pressuresolve tolerance of . A restart at convective time units is used for the initial condition.
In all cases, solver results are collected over 2000 timesteps. At each step, the solution is projected onto a space of up to 10 prior solution vectors to generate a highquality initial guess, . Projection is standard practice in nekRS as it can reduce the initial residual by orders of magnitude at the cost of just one or two matrixvector products in per step [fischer_projection_1998]. The perturbation solution, , is typically devoid of slowly evolving low wavenumber content. Moreover, the initial residual is oftentimes sufficiently small that the solution converges in iterations, such that the overhead of GMRES is small. Testing the preconditioners under these conditions ensures that the conclusions drawn are relevant to the application space.
4 Results
Here we consider the solver performance results for the test cases of Section 3. We assign a single MPI rank to each GPU and denote the number of ranks as . All runs are on Summit. Each node on Summit consists of 42 IBM Power9 CPUs and 6 NVIDIA V100 GPUs. We use 6 GPUs per node unless .
4.1 Kershaw Mesh
The Kershaw study comprises four tests. For each of two studies, we consider the regular box case (
) and a highly skewed case (
). The first study is a standard weakscale test, where and are increased, while the polynomial order is fixed at and the number of gridpoints per GPU is set to . The range of processors is =6 to 384. The second study is a test of the influence of polynomial order on conditioning, with and fixed, while ranges from 3 to 10. Both cases use GMRES(20).The results of the weakscaling study are shown in Fig. 5. For both =1.0 and , the iteration count exhibits a dependence on problem size, as seen in Fig. 5a,d. The timepersolve also increases with , in part due to the increase in iteration count, but also due to increased communication overhead as increases beyond unity. Table 4 indicates the maximum number of of neighboring processors for the assembly () graph of , which increases with . In addition, the number of grid points per GPU () is relatively low. These two factors cause an increased sensitivity of the problem to the additional communication overhead as the number of GPUs is initially increased. The number of neighbors, however, will saturate at larger (i.e., productionlevel) processor counts.
Nodes  1  2  4  8  16  32  64 
Max Neighbors  5  11  16  24  20  24  29 
The influence of polynomial order is illustrated in Fig. 4. For , iteration counts are essentially independent, as seen in Fig. 4a. For , however, a slight upward trend in the iteration count is observed for SEMFEM and for the pMG preconditioners with ASM, RAS, and ChebyshevJacobi smoothing (Fig. 4b). Figures 4c and d demonstrate that the time per pressure solve is strongly dependent polynomial order. For SEMFEM, there is an increase in time as a result of increased overhead for the underlying AMG solver. For the pMGbased methods, higher orders are generally faster. This performance gain can be attributed to surfacetovolume effects in the evaluation of the operators and smoothers. Application of and (smoother) is highly vectorizable, whereas application of (assembly) involves a significant amount of indirect addressing. The discrete surface to volume ratio for a spectral element of order is 296/512 60%. For lower this value is larger. This situation is exacerbated in the case of Schwarzbased smoothers, where the overlap contributes substantially to the work and communication for the smallelement (i.e., low) cases. In addition, nekRS [fischer_nekrs_2021] uses a single element per thread block, which limits the amount of work available for a streaming multiprocessor for relatively small polynomial orders.
4.2 NavierStokes Results
We consider scalability of nekRS for the cases of Fig. 3. All simulations except one use GMRES(15) with an initial guess generated by conjugate projection onto 10 prior solutions [fischer_projection_1998]. Due to memory constraints, the 1568pebble case with uses GMRES(10) with only 5 solutionprojection vectors. For each case, two pMG schedules are considered: and for ; and and for . Other parameters, such as the Chebyshev order and the number of coarse grid BoomerAMG Vcycles are also varied. Results are shown in Fig. 6.
In all the performance tests conducted, the pMG preconditioner with ChebyshevJacobi smoothing is outperformed by the other preconditioners, whether using one or two Vcycle iterations in the AMG coarsegrid solve. For each case, the fastest preconditioner scheme varies. In the 146pebble case, using ChebRAS(2),(7,5,3,1) yields the smallest time per pressure solve. However, in the 1568pebble case, SEMFEM is a moderate improvement over the second best preconditioner, ChebASM(2),(7,5,3,1). Lastly, pMG with a schedule and ChebyshevRAS (of any order) yield the best scalability and lowest time per pressure solve for the Boeing speed bump case.
Across all cases, pMG with a good coarsening schedule, Chebyshev order, and smoother choice, such ASM or RAS, is better or at least comparable to SEMFEM. The optimal coarsening schedule and smoother choice, however, is not always obvious, especially when the relative ranking depends on the number of nodes used for the simulation. For example, ChebASM(2),(7,5,3,1) is faster than ChebRAS(2),(7,5,3,1) for larger node counts. For smaller node counts, however, this order is reversed. These results make apparent the benefit of developing an autotuner, however rudimentary.
Lastly, solver performance degrades whenever is sufficiently small, regardless of the solver considered. For the various preconditioners considered, at 2 nodes (), the strongscale limit of 80% efficiency is far surpassed in the 146pebble case. This leaves the effective strong scale limit in the neighborhood of 12 nodes ( to ). For the 1568pebble case, running on 12 nodes () yields a parallel efficiency around 6070%, depending on the specific solver. Similarly, the parallel efficiency for the Boeing speed bump case for the fastest timetosolution preconditioners drops below 70% when using more than 48 nodes, placing the strong scale limit around .
4.3 Online Tuner
Autotuning results are presented in Table 5. Autotuning trials are performed on Step 1000 and timings are based on the last 1000 timesteps of the simulation, therefore excluding the cost of the tuning step.^{3}^{3}3The overhead of the parametric search for production simulations will be amortized over the remainder of the run, typically comprising – steps, and over the full simulation campaign, which might require mutiple runs. The table reports , the minimum time per solve from the autotuner. Also shown are , the autotuning speedup relative to the default multigrid smoother; , the speedup relative to the fastest preconditioner in Fig. 6; and , the speedup relative to the worst preconditioner in Fig. 6. Here, the default multigrid smoother is ChebASM(2),(7,3,1) for and ChebASM(2),(9,5,1) for .
In all cases, ChebJac(2),(7,5,3,1) () ChebJac(2),(9,7,5,1) () with two coarse grid iterations was the worst performer.^{4}^{4}4Presently, the tuner always uses a single coarse grid iteration. In many cases, the default preconditioner performance is close to the optimal found by the autotuner. In the 146pebble case, switching from the default to ChebRAS(2),(7,5,3,1) yields speedups of 15%, 20%, and 10% for 1,2, and 3 nodes, respectively. Moreover, the autotuner typically chose a preconditioner comparable to the default. However, in the 1568pebble case with solution projection turned on, the autotuner selected the rather poor options of ChebyJac(2),(7,3,1) or ChebyJac(3),(7,3,1). This choice was due to the initial residual being close enough to the specified tolerance that these suboptimal preconditioner happened to provide the fastest solution for that particular timestep. This projectiondependence is evident in Tables 4(a) and 4(b). This sensitivity suggests that one should sample over multiple timesteps, rather than taking several samples from the same timestep. However, as seen in the differences between Tables 4(c) and 4(d), this strategy alone is not always sufficient to avoid erroneous preconditioner selection.




5 Conclusions and Future Work
In this work, we introduce Chebyshevaccelerated ASM and RAS smoothers for use in multigrid preconditioners for the spectral element Poisson problem. Further, we compare the performance of Schwarzbased smoothers, Chebyshevaccelerated Jacobi smoothing, and SEMFEMbased preconditioning for a suite of challenging test problems. We conclude that the Chebyshevaccelerated Schwarz smoothers with multigrid, as well as the loworder SEMFEM preconditioner solved with AmgX, are performant on GPU machines such as OLCF’s Summit. We also introduce an online autotuner for the purposes of preconditioner selection and smoother optimization. Additional tuning options, such as using the SEMFEM discretization as the coarse grid solver, varying the AMG solver settings for the coarse grid solve as well as the solver location (CPU/GPU), varying the Chebyshev order and number of sweeps on each level of the multigrid heirarchy, experimenting with F and Wcycle multigrid, and varying the Chebyshev eigenvalue bounds, are avenues for future performance optimization.
6 Acknowledgements
This research is supported by the Exascale Computing Project (17SC20SC), a collaborative effort of two U.S. Department of Energy organizations (Office of Science and the National Nuclear Security Administration) responsible for the planning and preparation of a capable exascale ecosystem, including software, applications, hardware, advanced system engineering and early testbed platforms, in support of the nation’s exascale computing imperative. This research also used resources of the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract DEAC0500OR22725. The authors thank YuHsiang Lan for providing visualizations and mesh files for the pebble cases. The Ramesh Balakrishnan for providing mesh files for the Boeing speed bump.
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