Finite Volume Discretization on Irregular Voronoi Grids

Transcription

1 Finite Volume Discretization on Irregular Voronoi Grids C.Huettig 1, W. Moore 1 1 Hampton University / National Institute of Aerospace Folie 1

2 The earth and its terrestrial neighbors NASA Colin Rose, Dorling Kindersley What influence has a convecting mantle? Topography Plate Tectonics Volcanism Habitability History / Evolution What influences mantle convection? External: Impacts, long-term cyclic effects Internal: Composition, Size, Heat sources Folie 2

3 Why numerical modeling Approximate complex processes to simple models that supercomputers can handle Understand basic phenomena behind what we can observe today Folie 3

4 Simulation Setup Standard Boussinesq Free slip Finite V/D 3D / 2D Irregular Grid u 0 T u ( u) Ra QTer p 0 t T u T 2 T 1 0 BiCGStab solver Highly parallel DC IDL framework for evaluation / vis Independent C++ code, STL Ra Q g Hd k 2 5 ref ( T) exp( T z) ref P ( T ) S Folie 4

5 Numerical details The Voronoi Diagram The partitioning of a plane with points into convex polygons such that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other. Mathworld Folie 5

6 Numerical details The Voronoi Diagram Advantages Generator points can be arbitrary Face perpendicular to neighbors Related to Delaunay triangulation (dual) Cellular structure suited for Finite Volume Many regular grids are Voronoi by nature Cell surrounded by its 13 of 14 neighbors Folie 6

7 Numerical details The Grid We require a 3D spherical shell Cells should have equal volume and smooth interface to boundary Spirals to the rescue! Spiral can be used to distribute n points on a sphere equally (not perfect though) Equiangular > Equidistant Folie 7

8 Numerical details The Grid Two methods to generate a grid from spiral Project 2D spherical VD, cells get bigger on the outside Build spiral for each radial step Folie 8

9 Numerical details Grid Types One-eighth of an equidistant spiral grid and projected icosahedral grid Irregular Spiral grid Projected Icosahedra grid Folie 9

10 Numerical details Spiral properties Folie 10

11 Numerical details - discretization Based on Cartesian coordinates no advantage in spherical c. Co-located setup, 5 unknowns per node (u, v, w, p, T) Works on all Voronoi-grids (red: Voronoi cell for black nodes, blue: triangulation): Advantageous setup because Delaunay triangulation is Dual of Voronoi setup Folie 11

12 Numerical details - discretization Finite-Volume a good basis faces are orthogonal and half-way between nodes BUT: in case of irregular distribution skewed! (face center interpolation required) Correct linear interpolation term easy from triangulation Barycentric interpolation Folie 12

13 Numerical details - discretization Stress tensor requires derivatives of all velocity components at the face center, many weighting factors (Barycentric coordinates) required Finite-Difference approach with Cartesian cross at F-center Pre-calculated weights requires memory Exploiting range of weights to reduce memory demand (64-bit double 16-bit word) Sparse matrix class exploits recurring setup sequence Profiling (approx.): 71% MatMul, 9% MatSetup, 6% VecOps, 2% Other, (12% MPI) Folie 13

14 Pressure correction SIMPLE(R/C) iterative system to reduce divergence (ensure mass conservation) Each timestep: Solve Energy Equation Solve Momentum Equation Solve for delta-pressure from divergence: Folie 14

15 Under-relaxation for Velocity Required for pressure correction Allow the velocity in each inner iteration to only change by a fraction: Values typically bewteen Complementary of alpha appears in Laplacian for delta p Folie 15

16 Numerical details parallelization How to divide the spherical shell into equally (in terms of nodes) sized parts with a minimal slice area? Idea from quantum physics: the Thomson-problem In theory also best spherical discretization but only known for up to 381 electrons Minimizes boundary area for domain decomposition and equalizes volume Folie 16

17 Example Case Example case: Ra0=1000, γ=60 Temperature Velocity Folie 17

18 The problem: Spurious Solutions Net-Rotation The problem: Artificial momentum in lateral direction Also a solution to the Navier-Stokes Explicit dampening usually employed Source term could be identified: Momentum equation in the very first outer iterations Worst case scenario from initial condition: Pressure and velocity zero! Folie 18

19 Spurious Solutions Net-Rotation Momentum equation: Body force in radial direction from temperature gradient (Boussinesq): Folie 19

20 The problem: Spurious Solutions Net-Rotation Result Lateral part only Folie 20

21 The problem: Spurious Solutions Momentum eq. Newtonian fluid: Three assumptions were made by Stokes: The stress tensor is a linear function of the strain rates The fluid is isotropic For a fluid at rest, must be zero (so that hydrostatic pressure results) Applying these assumptions will lead to: Incompressible: λ=0, no vector Laplacian since we are curious about strongly varying viscosities Folie 21

22 Spurious Solutions Net-Rotation Momentum equation leaves trace in Stress tensor: Trace associated with divergence, gets damped during inner iterations BUT: Directional error such as lateral velocity stays Under-relaxation triggers the error too: α=1 leads to no error! Introducing divergence with compressible equation and λ=-1μ solves the problem: Folie 22

23 Future Adaptive mesh refinement Code exploits a lot of pre-computed factors Parallel efficiency Voronoi- refinement without complete rebuild possible? Free-surface Moving mesh requires rebuild Changing neighbors challenging for parallelization On/Off approach with simulated free-surface? THANK YOU! Folie 23