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November 4, 2016, 15:56 
Interpolation on staggered grid

#1 
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Hi,
I'm a beginner trying to implement a basic/trivial 2D PIC simulation. I have a question regarding bilinear interpolation of a quantity (e.g. velocities) on staggered grid (the grid is on the picture below); I use two 2D arrays to store the velocities  one array for xcomponent and the other for ycomponent. One of the steps in the PIC algorithm is transfering velocities from particles to grid(s). Let's consider transfering the xcomponent of velocity from particles to the xgrid: for each particle I need to find 4 surrounding staggered points (green bars in the image above) to perform the (inverse) bilinear interpolation. Here's my question: how do I interpolate the xvelocity on the top and bottom boundary cells where there are no staggered points (green bars) below my particle (blue dot in the image)? How would you interpolate the blue particle's xvelocity component to the grid in the image above? My attempts to solution:


November 4, 2016, 16:19 

#2  
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Filippo Maria Denaro
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Quote:
Consider that you know the velocity on the boundary... for example, if the boundary is a wall and you work with viscous flows, you have zero xcomponent velocity so that you can add two values in the black nodes and using the bilinear interpolation taking into account that you have a rectangular grid the height being a half of the upper. 

November 4, 2016, 18:42 

#3  
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I'm sorry, but I think I don't understand the following part of your answer: Quote:
Another question: Should the 12 boundary gridcells be marked as a wall? Then there could not be a particle in the 12 boundary cells and thus there would always be some green bars below my particle to store the velocity on (that would mean that particles are allowed only in the 4 cells inside the grid from the image). Is such an approach correct? 

November 5, 2016, 03:26 

#4 
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Filippo Maria Denaro
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Yes, you can use the two lower black nodes (lower boundary) to complete the interpolation region. Setting wall or not as BC depends on your physical problem, is not a matter of interpolation.


November 5, 2016, 05:28 

#5  
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Quote:
given a 4x4 array of cells (like in the image) I create:
The 2 added rows of black nodes at the top and bottom rows for xcomponent grid (in the case of the ycomponentgrid it'd be the leftmost and the rightmost columns of black nodes) are used ONLY for interpolation when transfering velocities from particles to grid and in the opposite direction. When solving the Poisson equation to get divergencefree field, I'd ignore the 2 rows of black nodes and I'd use just the green bars. Is the above right? 

November 5, 2016, 06:02 

#6 
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Filippo Maria Denaro
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The pressure equation requires only red and green values for computing the source term but it is solved on the centre of the cells


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implementation, pic, staggered grid 
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