Average over a sphere in uniform cartesian grid

zkwzju · December 16, 2014, 07:06

Hi,
Anyone has good idea how to calculate the average value of a scalar on a sphere (all points on a sphere surface) , the scalar field is discretized with a uniform Cartesian grid.
I have Googled all day long, but didn't get any good idea. 3D interpolation to get quadrature points on the sphere, and then integration is my last plan. Obviously, it is not convenient and expensive. I have huge data and large number of average to calculate.
Does anyone has a good idea to use only the index of the grid with sort of weight to get the final average value? Thanks.

Kun

mprinkey · December 16, 2014, 15:46

Intersection of a sphere with Cartesian grid lines will lead to many points onto which you can do 1D interpolation to the surface of the sphere. Those intersection points on the sphere can be assembled into a triangular tessellation of the sphere surface. With scalar values at each vertex of the triangular tessellation, you can integrate over each triangle and add them all up. This is very similar to how cut-cell Cartesian meshes are built. It is messy.

3D interpolation to spherical quadrature points will likely be easier to code because you won't have the tessellation book keeping to do, but may not be as accurate unless your scalar field is very smooth or you use many quadrature points. I'd make a general formulation for Lebedev quadrature and 3D interpolation for the sample, and then systematically increase the quadrature samples until the result converges.

Good luck.

zkwzju · December 16, 2014, 22:09

Thanks, Michael. I will follow the easier way as your suggestion. I didn't know Lebedev quadrature before, thanks for your info.

sbaffini · December 18, 2014, 16:55

I'm just guessing, but you could maybe assume your data to be a sum of sines and cosines in x, y and z (the result of a 3D FFT), make a coordinate substitution from cartesian to polar (center on the sphere center, of course) and try to figure out what the exact integral of a Fourier series over a sphere is. Then you simply apply the formulas to the result of your FFT.

In general, pick-up an interpolation kernel, choose your stencil, apply it to those stencils cut by the sphere, integrate manually, apply the result to the interpolation coefficients. The stencils should be non overlapping.

Second approach: use paraview.

Edit: ok, this was not so clever. There is no reason to say more than Lebedev. I didn't know it neither.

December 16, 2014, 07:06	Average over a sphere in uniform cartesian grid	#1
zkwzju New Member Kun Zhou Join Date: Dec 2014 Posts: 2 Rep Power: 0	Hi, Anyone has good idea how to calculate the average value of a scalar on a sphere (all points on a sphere surface) , the scalar field is discretized with a uniform Cartesian grid. I have Googled all day long, but didn't get any good idea. 3D interpolation to get quadrature points on the sphere, and then integration is my last plan. Obviously, it is not convenient and expensive. I have huge data and large number of average to calculate. Does anyone has a good idea to use only the index of the grid with sort of weight to get the final average value? Thanks. Kun

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December 16, 2014, 15:46		#2
mprinkey Senior Member Michael Prinkey Join Date: Mar 2009 Location: Pittsburgh PA Posts: 363 Rep Power: 25	Intersection of a sphere with Cartesian grid lines will lead to many points onto which you can do 1D interpolation to the surface of the sphere. Those intersection points on the sphere can be assembled into a triangular tessellation of the sphere surface. With scalar values at each vertex of the triangular tessellation, you can integrate over each triangle and add them all up. This is very similar to how cut-cell Cartesian meshes are built. It is messy. 3D interpolation to spherical quadrature points will likely be easier to code because you won't have the tessellation book keeping to do, but may not be as accurate unless your scalar field is very smooth or you use many quadrature points. I'd make a general formulation for Lebedev quadrature and 3D interpolation for the sample, and then systematically increase the quadrature samples until the result converges. Good luck.

December 16, 2014, 22:09		#3
zkwzju New Member Kun Zhou Join Date: Dec 2014 Posts: 2 Rep Power: 0	Thanks, Michael. I will follow the easier way as your suggestion. I didn't know Lebedev quadrature before, thanks for your info.

December 18, 2014, 16:55		#4
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,195 Blog Entries: 29 Rep Power: 39	I'm just guessing, but you could maybe assume your data to be a sum of sines and cosines in x, y and z (the result of a 3D FFT), make a coordinate substitution from cartesian to polar (center on the sphere center, of course) and try to figure out what the exact integral of a Fourier series over a sphere is. Then you simply apply the formulas to the result of your FFT. In general, pick-up an interpolation kernel, choose your stencil, apply it to those stencils cut by the sphere, integrate manually, apply the result to the interpolation coefficients. The stencils should be non overlapping. Second approach: use paraview. Edit: ok, this was not so clever. There is no reason to say more than Lebedev. I didn't know it neither.