1. 3D spherical gridding based on equidistant, constant volume cells for FV/FD methods A new method using natural neighbor Voronoi cells distributed by spiral functions German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf

2. Introduction to common 3D spherical grids -Most grids base on triangulated platonic solids (convex polyhedra such as the cube, dodecahedron, tetrahedron, icosahedron,...) -Domain decomposition through subdivisions of the platonic solids areas -Grids extend radial through a projection of the grid from the center to shells -Only axisymmetric alignment; could lead to increased numerical instabilities (oscillation) -Non-uniform cell size requires additional expensive compensation computations and leads to higher inner shell resolution, which is not desired in most cases (surface resolution matters!) -Only fixed resolution steps (TERRA) German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf TERRA grid setup and shell extension based on icosaeder subdivisions (Baumgardner, 1988) Solve these problems through new ditribution method?

3. Basic Equations: Archimede‘s Spiral: German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf In 2D cartesian coordinates: Spherical representation in 3D cartesian coordinates:

4. Second, incomplete elliptic integral: The arc length equations Archimede‘s spiral (polar) arc length: German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf - Non analytically inversible already! General arc length definition for 3D curves: Arc length for spherical spiral:

5. Equidistant point distribution over the arc length German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf Arc length for spherical spiral -We are interested in α for specific lengths s (s[i] = Resolution * i), which leads to an inversion of a non-analytically solvable integral ↷ Computational expensive calculations But: Easy parallel distribution possible Equiangular > Equidistant

6. Radial extension of the spiral sphere German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Shell generation through radial re-computation (not projection!) of the new shell for the desired resolution -Shell count and overall point count is a result of inner radius, outer radius and desired resolution: -Boundary shells added before inner and after outer shell -Results in equidistant point distribution within a spherical region -„Overturning“ of the spherical spiral function leads to better distribution Comparison of the TERRA grid to the spiral grid (Surface resolution = 130km, Earth mantle): TERRA:1,4M PointsSpiral:923.000 Points32 Shells

7. The dampening factor German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Required for an optimal equidistant distribution -Used as factor for the resolution to calculate the radial shell distance and α max -Dampening factor is optimal if the mean length of all connections of a Delauney triangulation equals the desired resolution Spiral sphere sideview d * res

8. The influence of the dampening factor on edges German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf

9. The influence of the dampening factor on volumes German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf

10. The influence of the dampening factor on distance German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf

11. Cell generation German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Two methods: -Projection of a 2D spherical Voronoi tessellation of every generated shell from the sphere center; leads to a non-uniform but axisymmetric grid! -Complete 3D Voronoi tessellation -Natural neighbor Voronoi cells lead to increased accuracy of the model 2D spherical Voronoi diagramOne shell of a complete Voronoi d.

12. Cell generation – complete 3D Voronoi diagram German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Outer shell points remain as open cells and inner shell points would connect throughout the center, but both can be used as boundary zones Cut through the two-sphere in positive domain; Inner radius = 1 Outer radius = 2 Resolution = 0.1 Shells = 12 (+ 2 boundary) Points (complete): 62529

13. Cell generation option – Centroidal Shift (CVD) German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Generator points are not necessarily the center of the cell -Optional shift of generator points (from spiral) towards the center of mass of the cell -Lloyd’s algorithm iterates until the generator points reach the center point within a given criteria -Requires recomputation of Voronoi diagram on each iteration -Smoothes cell properties, but not volumes Example of Lloyd’s algorithm in 2D, random generator- point distribution CVDs do not necessarily tend to equally sized cells!

14. Statistical analysis – Distance histogram German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf min = 0.0733mean = 0.0999max = 0.1435 σ = 0.01476skew = 0.104 min = 0.0716mean = 0.0977max = 0.1383 σ = 0.01148skew = 0.779

15. Statistical analysis – Face histogram German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf min = 10mean = 14.513max = 20 σ = 0.93145skew = 0.381 min = 9mean = 14.126max = 19 σ = 0.85666skew = 0.156

16. Statistical analysis – Volume histogram German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf min = 5.0747e-4mean = 5.6241e-4max = 6.14050e-4 σ = 6.7013e-6skew = 0.111 min = 4.4102e-4mean = 5.6271e-4max = 6.58129e-4 σ = 1.8974e-5skew = -0.436

17. Statistical analysis – Volume distribution German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf

18. Statistical analysis – Volume distribution German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf

19. Possible domain decomposition for parallelization German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Cones used to split the sphere into N even regions with an equivalent amount of cells -Halo zone is defined by all cells that get cut through the cone plus their natural neighbors for interpolation -Works with any even CPU counts -Zone cutting and grid information can be cached -Numbering system makes parallelization easy: One dimensional count from north-pole to south-pole, halo zones could be defined by only two numbers; complete sphere fits into 2D array: [Shell_Index, Point_Index]

20. -A scalar quantity diffuses through space with a rate of The diffusion equation discretized German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Area between cells act as energy distribution ratio to complete cell area Cell surrounded by its 13 of 14 neighbors

21. Summary German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Reliable, almost constant resolution throughout the sphere -Free choice of resolution (and therefore grid points) -Efficient parallelization through cone subdivisions -Cell volume is almost constant -Accurate diffusion through natural neighbors -No oscillation effects

22. Outlook German Aerospace Center BerlinThermodynamics of Planetary Interiors, www.dlr.de/pf -Wide range of applications: global modeling of seismology, geodynamics, electromagnetism, atmospherics… -Implementation of advection; complex but not computational expensive geometric algorithm -Implementation of grid algorithms in MPI (C or Fortran)