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Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Geometric cell alignment on geodesic grids
Pedro S. PeixotoSaulo R. M. Barros
Applied Mathematics DepartmentUniversity of São Paulo
Brazil
PDEs 2014 Talk
CAPES Funding is acknowledged
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Summary
Summary
1 Geometric Alignment Theory
2 Application on Vector Reconstructions
3 Conclusions
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Motivation
Icosahedral Grid
Icosahedral grid (Voronoi grid)
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Motivation
Finite-Volume Discretization
div(~v)(P0) ≈ 1|Ω|
n∑i=1
~v(Qi ) · ~ni li .
−1.1e−04
−8.8e−05
−6.6e−05
−4.4e−05
−2.2e−05
0.0e+00
2.2e−05
4.4e−05
6.6e−05
8.8e−05
1.1e−04
Rotation vector field - Glevel 6 - 40962 nodes
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Theoretical results
Divergence Operator Discretization
Assuming the vector field given at the midpoints of the edges of thepolygon, or precisely calculated:
div(~v)(P0) ≈ 1|Ω|
∫Ω
div(~v) dΩ =1|Ω|
∫∂Ω
~v · ~n d∂Ω ≈ 1|Ω|
n∑i=1
~v(Qi ) · ~ni li .
On a plane:Average approximation: 2nd order if P0 is the centroidDivergence approximation: 1st order only in generalRectangle: 2nd orderOdd number of edges (triangle, pentagon): 1st order only, even ifregular.General quadrilaterals and hexagons?
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Theoretical results
Geometric Alignment
Definition (Planar aligned polygon)
A polygon on a plane with an even number of vertices, given byPin
i=1, is aligned if for each edge ei = PiPi+1 the correspondingopposite edge ei+n/2 = Pi+n/2Pi+n/2+1 is parallel and has the samelength as ei .
Definition (Spherical aligned polygon)
A spherical polygon with an even number of edges is aligned if itsradial projection onto the plane tangent to the sphere at its centroid isa planar aligned polygon.
OBS: Geodesics are straight lines in the projected plane
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Theoretical results
Alignment Index
Proposition (Alignment Index)
A polygonal cell Ω is aligned if, and only if, the nondimensional Ξ(Ω)is zero
Ξ(Ω) =1
nd
n/2∑i=1
|di+1+n/2,i − di+n/2,i+1|+ |di+1,i − di+n/2+1,i+n/2|
d = 1n
∑ni=1 di,i+1
di,j is distance metric betweenpolygon vertices Pi and Pj ,i , j = 1, ...,nThe greater Ξ, the greatermiss-alignment
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Theoretical results
Main result on the sphere
Theorem
Let ~v be a C4 vector field on the sphere and Ω an aligned sphericalpolygon with n geodesic edges, diameter d and area |Ω| satisfyingαd2 ≤ |Ω| ≤ d2, for some positive constant α. Then there is aconstant C, which independs on the diameter d, such that∣∣∣∣∣div(~v)(P0)− 1
|Ω|
n∑i=1
~v(Qi ) · ~ni li
∣∣∣∣∣ ≤ Cd2,
where P0 is the mass centroid of Ω.
See Peixoto and Barros (2013)
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Numerical Results
Alignment Index
Alignment index Div. of Rotation
0.0e+00
1.0e−01
2.0e−01
3.0e−01
4.0e−01
5.0e−01
6.0e−01
7.0e−01
8.0e−01
9.0e−01
1.0e+00
−1.1e−04
−8.8e−05
−6.6e−05
−4.4e−05
−2.2e−05
0.0e+00
2.2e−05
4.4e−05
6.6e−05
8.8e−05
1.1e−04
Glevel 6 - 40962 nodes
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Numerical Results
Grid imprinting
For Ξ < 1/100:glevel % Align Cells
4 30.99%5 49.22%6 70.12%7 84.24%8 91.85%
- Differences in cell geometry results indifferences in the convergence orders ofthe discretization
- Aligned cells have faster convergencethan non-aligned cells
- Badly aligned cells will have largererrors, and if these are related to thegrid structure, we will have gridimprinting
Analogous theorems for Rotational (curl) and Laplacian operatorTheory constructed generally for any geodesic grid
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Numerical Results
Locally refined SCVT grids
Grid - 40962 cells Diameters:
9.6e−03
1.2e−02
1.4e−02
1.7e−02
1.9e−02
2.2e−02
2.4e−02
2.6e−02
2.9e−02
3.1e−02
−2.2e−04
−1.8e−04
−1.3e−04
−9.0e−05
−4.5e−05
0.0e+00
4.5e−05
9.0e−05
1.3e−04
1.8e−04
2.2e−04
0.0e+00
1.6e−01
3.2e−01
4.8e−01
6.4e−01
8.0e−01
9.5e−01
1.1e+00
1.3e+00
1.4e+00
1.6e+00
Div Rotation Align. Ind.
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Summary
1 Geometric Alignment Theory
2 Application on Vector Reconstructions
3 Conclusions
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Vector Reconstructions
Vector reconstruction
Analysed Methods:- Perot’s method- Klausen et al method (RT0
generalized to polygons)- Polynomial (Least Sqrs.)- RBF? Hybrid (Perot and Linear LSQ)
See Peixoto and Barros (2014) - under review - JCP Preprint
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Vector Reconstructions
Perot’s Method
~u0 =1|Ω|
n∑i=1
~ri ui li ,
- Divergence Theorem based- Low cost- Exact for constant fields- 1st order only in general- 2nd order on aligned cells
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Vector Reconstructions
Hybrid scheme
0.0e+00
1.0e−02
2.0e−02
3.0e−02
4.0e−02
5.0e−02
6.0e−02
7.0e−02
8.0e−02
9.0e−02
1.0e−01
PEROT - Max Error = 9.7E-2 RMS=1.3E-2
HYBRIB - Max Error = 4.3E-2 RMS=1.2E-2
Vector recon. to Voronoi cell nodesRossby-Haurwitz wave 8 - Icosahedral grid level 7HYBRID: 84% Perot’s method and 16% Linear LSQ method
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Vector Reconstructions
Hybrid scheme
Perot’s method on well aligned cells (majority) and linear LSQ methodon ill aligned cells (minority)
2nd order accurateLow cost on fine grids (cost dominated by Perot’s method)No pre-computing needed (less memory usage compared toRBF and LSQ)Applicable to any geodesic grid (tested in locally refined SCVT)
Application
2nd order semi-Lagrangian transport model for staggered Voronoigrids
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Summary
1 Geometric Alignment Theory
2 Application on Vector Reconstructions
3 Conclusions
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Conclusions
Conclusions
Geometrical alignment:Better understanding of grid imprinting
General Mathematical proofs:Plane and sphere for arbitrary polygons
Alignment index:Tool for development of numerical methodsTool for grid development
Analysis maybe be extended to other operators anddicretizations
Where are we going with this?
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Conclusions
Analysis of shallow water model
Thuburn et al (2009) tangent vector reconstruction
0.0e+00
7.3e−01
1.5e+00
2.2e+00
2.9e+00
3.6e+00
4.4e+00
5.1e+00
5.8e+00
6.6e+00
7.3e+00
Error of Rossby-Haurwitz wave 8 - Icos glevel 6
Geometric Alignment Theory Application on Vector Reconstructions Conclusions
Conclusions
References
Peixoto, PS and Barros, SRM. 2013. Analysis of grid imprintingon geodesic spherical icosahedral grids, J. Comput. Phys. 237(March 2013), 61-78.
Peixoto, PS and Barros, SRM. 2014. On vector fieldreconstructions for semi-Lagrangian transport methods ongeodesic staggered grids, J. Comput. Phys. (under review)
Preprints available at www.ime.usp.br/∼pedrosp
Thank you very much!
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