Multigrid Solvers in CFD. David Emerson. Scientific Computing Department STFC Daresbury Laboratory Daresbury, Warrington, WA4 4AD, UK
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1 Multigrid Solvers in CFD David Emerson Scientific Computing Department STFC Daresbury Laboratory Daresbury, Warrington, WA4 4AD, UK 1
2 Outline Multigrid: general comments Incompressible vs Compressible Flows Structured vs Unstructured Grids Results from both approaches 2
3 Multigrid: General Comments Originally proposed(?) by A. Brandt (Multi-level adaptive solutions to boundary-value problems, Mathematics of Computation, V31, pp , 1977) Basic strategy is to solve a discrete set of equations on multiple grids (levels) to obtain an accurate solution in less time (than on fine grid) Basic premise: on the fine grid, iterative solutions for Ax=b work well and can deal with high frequency errors, but are poor at removing low frequency errors. MG solves problem on coarser mesh to start removing these low frequency errors (now appearing as high frequency) 3
4 Multigrid: Early Issues Multigrid makes use of multiple grids/levels to smooth/correct the errors arising from solvers For sequential codes, each level has to be stored; when multigrid was developed c1977 the available memory was very small (storage would roughly double using MG) For parallel codes, lots of small messages requires low latency network. Again, initially perceived as a problem. We wrote a paper (published 1996) comparing PCCG and MG on the T3D for Poisson equation. We used shmem (latency ~6 us) and wrote our own global sum routine (there was no intrinsic routine available) and demonstrated MG was competitive with CG methods. 4
5 Multigrid: Later Issues Scalability as parallel MG became established, the main question was if it would efficiently scale to 1000s processors. This was a particular issue/concern for unstructured grids We need to understand whether multigrid scales to many processors. Again, assume a d-dimensional grid. h d On the fine grid we have N d If we have P processors, we have N / P d Communication is related to the surface e.g. N / P 2/3 d d N 2 N 1 d d d nd Computation P P 1 2 2/3 2/3 d d 1 2 d n d N 2 d 1 d N Comms P P 1 2 d 1/3 6 N In 3-D, we see that comp/comm 7 P 5
6 CFD: Incompressible vs Compressible Vast majority of flows in CFD are solved by discretising the PDEs for continuity, momentum, and energy i.e. we solve the Navier-Stokes equations on a discrete set of grid points. The equations are nonlinear and have elliptic, parabolic, and/or hyperbolic aspects. In CFD we generally classify flows as either incompressible (the density of the fluid doesn t change with time e.g. water and other liquids) or compressible (density changes need to be accounted for through an equation of state e.g. gas flow * ). For incompressible fluids, we have Div(U)=0 by definition. Linking the pressure-velocity field results in having to solve a Poisson equation. 6
7 CFD: Structured vs Unstructured Fluids engineering is about solving problems with complex geometries (if symmetry or periodicity exists we will exploit it). We can do this using different grid types, broadly referred to as structured or unstructured grids. A key feature of fluid modelling is resolving flow features near a solid boundary (where boundary layers form). This is critical for turbulent flows because we want to predict viscous drag/heating. To capture these effects we use grid stretching (to predict gradients at the wall). However, MG not so good for anisotropic grids so we transform to a computational domain. For unstructured meshes, we usually use mixed grid types (hexahedra, tetrahedra, prisms ) 7
8 Example (1) Consider a very simple aerofoil: NACA0012 C-grid O-grid 8
9 Example (2) Consider a very simple aerofoil: NACA0012 Unstructured-grid Unstructured-grid 9
10 Example (3) Note different grid types Overlapping grid system on the space shuttle Slotnick, Kandula and Buning (AIAA ) 10
11 Unstructured: another issue The use of unstructured grids has a number of issues relating to how the coarser mesh is generated. One possibility is to employ a sequence of independently generated coarse meshes. In this approach, the meshes are generally nonnested. An alternative strategy is to form the coarse mesh by agglomerating neighbouring nodes of a fine mesh. In this approach, seed nodes are merged with neighbouring nodes to form coarser mesh nodes. Looking at it from another perspective, control volumes of nodes to be merged are agglomerated to form polyhedra. The resulting coarse mesh is not a conventional shape. 11
12 MG: FD and FV methods Finite difference (upper) and finite volume (lower) schemes; grid layout is similar. Data can be stored at the node or the cell centre. 12
13 MG: agglomeration Data structure more random 13
14 Results: FLITE3D Sequence of meshes for a M6 wing showing fine mesh, dual mesh, and agglomerated meshes. 14
15 Results: FLITE3D YF Hu, DR Emerson, M Ashworth, KCF Maguire and RJ Blake Parallelizing FLITE3D a multigrid finite element Euler Solver Int. J. Numer. Meth. Fluids 2001; 35:
16 Results: THOR Resi level 2 level 3 level 4 level Driven Cavity Re=200 Processors: 4 Grid size 300 x Cyc (Iter) Number of cycles to converge using THOR, an in-house structured grid CFD code for incompressible low speed flow 16
17 Results: THOR Resi level 2 level 3 level 4 level Driven Cavity Re=200 Processors: 4 Grid size 300 x CPU time (s) Saving in CPU time using MG to accelerate convergence in THOR 17
18 MG: final comments MG used extensively in CFD, serial & parallel MG still has challenges in parallel (not all cycles efficient V best?) More recent developments are based on Algebraic MG (many variants, particularly for efficient parallelism/scalability) AMG used in other disciplines (Electromagnetics, QCD mentioned) Future hardware: MG has been implemented on GPUs MG on MIC? Some exploratory work done 18
19 Multigrid Solvers in CFD David Emerson Scientific Computing Department STFC Daresbury Laboratory Daresbury, Warrington, WA4 4AD, UK 19
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