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,

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, TERRA grid setup and shell extension based on icosaeder subdivisions (Baumgardner, 1988) Solve these problems through new ditribution method?

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

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

Equidistant point distribution over the arc length German Aerospace Center BerlinThermodynamics of Planetary Interiors, 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

Radial extension of the spiral sphere German Aerospace Center BerlinThermodynamics of Planetary Interiors, -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: Points32 Shells

The dampening factor German Aerospace Center BerlinThermodynamics of Planetary Interiors, -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

The influence of the dampening factor on edges German Aerospace Center BerlinThermodynamics of Planetary Interiors,

The influence of the dampening factor on volumes German Aerospace Center BerlinThermodynamics of Planetary Interiors,

The influence of the dampening factor on distance German Aerospace Center BerlinThermodynamics of Planetary Interiors,

Cell generation German Aerospace Center BerlinThermodynamics of Planetary Interiors, -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.

Cell generation – complete 3D Voronoi diagram German Aerospace Center BerlinThermodynamics of Planetary Interiors, -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

Cell generation option – Centroidal Shift (CVD) German Aerospace Center BerlinThermodynamics of Planetary Interiors, -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!

Statistical analysis – Distance histogram German Aerospace Center BerlinThermodynamics of Planetary Interiors, min = mean = max = σ = skew = min = mean = max = σ = skew = 0.779

Statistical analysis – Face histogram German Aerospace Center BerlinThermodynamics of Planetary Interiors, min = 10mean = max = 20 σ = skew = min = 9mean = max = 19 σ = skew = 0.156

Statistical analysis – Volume histogram German Aerospace Center BerlinThermodynamics of Planetary Interiors, min = e-4mean = e-4max = e-4 σ = e-6skew = min = e-4mean = e-4max = e-4 σ = e-5skew =

Statistical analysis – Volume distribution German Aerospace Center BerlinThermodynamics of Planetary Interiors,

Statistical analysis – Volume distribution German Aerospace Center BerlinThermodynamics of Planetary Interiors,

Possible domain decomposition for parallelization German Aerospace Center BerlinThermodynamics of Planetary Interiors, -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]

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

Summary German Aerospace Center BerlinThermodynamics of Planetary Interiors, -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

Outlook German Aerospace Center BerlinThermodynamics of Planetary Interiors, -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)