3d spherical gridding based on equidistant , constant volume cells for fv/fd methods
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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)
German Aerospace Center Berlin Thermodynamics 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?
Basic Equations:
Archimede‘s Spiral:
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
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:
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
- 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
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
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
German Aerospace Center Berlin Thermodynamics 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:
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
German Aerospace Center Berlin Thermodynamics 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 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
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
Cell generation
German Aerospace Center Berlin Thermodynamics 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 diagram One shell of a complete Voronoi d.
Cell generation – complete 3D Voronoi diagram
German Aerospace Center Berlin Thermodynamics 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
Cell generation option – Centroidal Shift (CVD)
German Aerospace Center Berlin Thermodynamics 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!
Statistical analysis – Distance histogram
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
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
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
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
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
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
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
Statistical analysis – Volume distribution
German Aerospace Center Berlin Thermodynamics of Planetary Interiors, www.dlr.de/pf
Possible domain decomposition for parallelization
German Aerospace Center Berlin Thermodynamics 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 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
German Aerospace Center Berlin Thermodynamics 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
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
German Aerospace Center Berlin Thermodynamics 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
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