3d spherical gridding based on equidistant , constant volume cells for fv/fd methods

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 Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf

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 resolutionsteps (TERRA)


TERRA grid setup and shell extension based on icosaeder subdivisions(Baumgardner, 1988)

Solve these problems through new ditribution method?

Basic Equations:

Archimede‘s Spiral:


In 2D cartesian coordinates:

Spherical representation in 3D cartesian coordinates:

x f rcoscos2

max

y f rsincos2

max

z f r sin2

max

r f ax f siny f cos

-10

-5

0

5

10-10

-5

0

5

10

-20

0

20

-10

-5

0

5

10

Second, incomplete elliptic integral:

The arc length equations

Archimede‘s spiral (polar) arc length:


- Non analytically inversible already!

General arc length definition for 3D curves:

s12a12 Log1 2

sx 2 y 2 z 2

Arc length for spherical spiral:

s r E

max,

max2

2

E, m0

1 mSin2

Equidistant point distribution over the arc length


Arc length for spherical spiral

s r E

max,

max2

2

-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


-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:

ShellCountOuterR InnerR

Resolution Dampening 2

PointCount i0

ShellCountE, 2 InnerR

Resolution Dampening 2 i 1

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 Points Spiral: 923.000 Points

32 Shells 32 Shells

The dampening factor


-Required for an optimal equidistant distribution

-Used as factor for the resolution to calculate the radial shell distance and αmax

-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 distance


Cell generation


-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 diagram One shell of a complete Voronoi d.

Cell generation – complete 3D Voronoi diagram


-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)


-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


min = 0.0733 mean = 0.0999 max = 0.1435 σ = 0.01476 skew = 0.104min = 0.0716 mean = 0.0977 max = 0.1383 σ = 0.01148 skew = 0.779

Statistical analysis – Face histogram


min = 10 mean = 14.513 max = 20 σ = 0.93145 skew = 0.381min = 9 mean = 14.126 max = 19 σ = 0.85666 skew = 0.156

Statistical analysis – Volume histogram


min = 5.0747e-4 mean = 5.6241e-4 max = 6.14050e-4 σ = 6.7013e-6 skew = 0.111min = 4.4102e-4 mean = 5.6271e-4 max = 6.58129e-4 σ = 1.8974e-5 skew = -0.436

Statistical analysis – Volume distribution


Possible domain decomposition for parallelization


-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 dimensionalcount 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


-Area between cells act as energy distribution ratio to complete cell area

Cell surrounded by its 13 of 14 neighbors

2

t

tn,iN

ne

1n i

nit

ii

ti

dtti

i

S

A

S

dt

step Time :dt

functionindex cellNeighbor :n)N(i,

i cell of neighbors ofNumber :ne

i cell of n wall of Area:A

i cell of Surface :S

i

ni

i

Summary


-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

3d spherical gridding based on equidistant , constant volume cells for fv/fd methods

Documents

spiral grid surface

spherical spiral function

desired resolutionshell

d spherical gridding

higher inner shell resolution

spherical representation

spherical regionoverturning

new shell