Geometric cell alignment on geodesic grids › events › pdes-workshop › Presentations-Posters … · Finite-Volume Discretization div(~v)(P 0) ˇ 1 j j Xn i=1 ~v(Q i) ~n i l i:-1.1e-04-8.8e-05-6.6e-05-4.4e-05-2.2e-05

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


Summary

Summary

1 Geometric Alignment Theory

2 Application on Vector Reconstructions

3 Conclusions


Motivation

Icosahedral Grid

Icosahedral grid (Voronoi grid)


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


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?


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


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


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)


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


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


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.


Summary



3 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



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



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



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


Summary



3 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?


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


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!

http://www.ime.usp.br/~pedrosp

Documents

Geometric cell alignment on geodesic grids › events › pdes-workshop › Presentations-Posters … · Finite-Volume Discretization div(~v)(P 0) ˇ 1 j j Xn i=1 ~v(Q i) ~n i l i:-1.1e-04-8.8e-05-6.6e-05-4.4e-05-2.2e-05