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How to define a plane region with polygonal boundaries for 2D integration?

Posted 10 years ago

Hello Everyone,

I calculated the charge density over a plane with a finite element method , so I get a function like sigma=f[x,y].

Now I need to integrate the charge over a finite area (electrode) in order to calculate the collected charge by the electrode. The electrode has a closed polygonal contour like the example below:

In[267]:= fff

Out[267]= {{-6.17, -33.75}, {-9.35, -33.75}, {-9.35, -33.78}, {-9.35, \
-39.375}, {-9.35, -45.435}, {-9.35, -45.45}, {-9.22, -45.45}, {-9.22, \
-39.375}, {-6.17, -39.375}, {-6.17, -33.78}, {-6.17, -33.75}}

I think need to define a Boole function that is True when a point {x,y} belongs to the interior of the closed polygonal contour in order to Integrate f[x,y] as showed in the On line help:

Integrals over Regions

This does an integral over the interior of the unit circle. 

In[1]:= Integrate[If[x^2 + y^2 < 1, 1, 0], {x, -1, 1}, {y, -1, 1}]

Out[1]=  \[Pi]

Here is an equivalent form. 

In[2]:= Integrate[Boole[x^2 + y^2 < 1], {x, -1, 1}, {y, -1, 1}]

Out[2]=  \[Pi]

So my question is: what is the simplest way to express the fact that the current point {x,y} is inside my polygonal region (can be non-convex) ?

Thanks for your suggestions


POSTED BY: Christian Neel
7 Replies

Thanks again guys, got V10 now, it's a dream!


POSTED BY: Christian Neel

Also in V10,

Integrate[1, {x, y} \[Element] Polygon[points]]
NIntegrate[Cos[x] Sin[y], {x, y} \[Element] Polygon[points]]
POSTED BY: Ilian Gachevski

Hi Ilian,

Thanks also for this other solution.And I thought Polygon was only a graphic primitive...:-)

Definitively another good reason to switch to V10! *


*= incidentally i just got an upgrade proposal in my mailbox this morning :)

POSTED BY: Christian Neel

Hi, Christian,

If you have V10, then you can define more or less arbitrary regions, see documentation for ParametricRegion[] and ImplicitRegion[] as well.

points = {
   {-6.17, -33.75}, {-9.35, -33.75}, {-9.35, -33.78},
   {-9.35, -39.375}, {-9.35, -45.435}, {-9.35, -45.45},
   {-9.22, -45.45}, {-9.22, -39.375}, {-6.17, -39.375},
   {-6.17, -33.78}, {-6.17, -33.75}
R = BoundaryMeshRegion[points, Line[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1}]]
Integrate[1, {x, y} \[Element] R]
NIntegrate[Cos[x] Sin[y], {x, y} \[Element] R]


POSTED BY: Ivan Morozov

Hi Ivan,

Thanks for this tip! Great feature! I am currently working with a 8.04 but this is the opportunity to jump to V10!


POSTED BY: Christian Neel

In general I think you will have to break up the region into convex regions. I don't know if there is a routine that will triangulate it. Maybe if you gave a practical but more generic case?

However, in the case you present it is quite simple.

vertexPoints = {{-6.17, -33.75}, {-9.35, -33.75}, {-9.35, -33.78},  {-9.35, -39.375}, {-9.35, -45.435}, {-9.35, -45.45}, {-9.22, -45.45},  {-9.22, -39.375}, {-6.17, -39.375}, {-6.17, -33.78}, {-6.17, -33.75}};

which plots as:

  AbsolutePointSize[5], Point /@ vertexPoints},
 AspectRatio -> 1,
 PlotRange -> {{-10, -5}, {-50, -30}},
 Frame -> True,
 ImageSize -> 300]

enter image description here

The Boole function for the electrode region is given by:

  y_] := (-9.35 <= x <= -6.17) \[And] (-39.735 <= 
     y <= -33.78) \[Or] (-9.35 <= x <= -9.22) \[And] (-45.45 <= 
     y <= -39.735)

which plots as:

RegionPlot[inElectrode[x, y], {x, -10, -5}, {y, -50, -30},
 PlotPoints -> 30]

enter image description here

Thank you for the example David, I think you are right since at the present level of my design the electrodes are mostly built with a main rectangular collecting plate attached to a thin track, so every electrode can be partitioned exactly in two rectangles as you did. I think I'll try to build a routine to do the partitioning automatically because the rectangles are all parallel to axes.

For the future I'll need to look for a more generic method because the more advanced designs of electrodes (which I am studying) will be more corrugated and extracted from CAD files (DXF) so I will have no clue to automatize the process. So I guess you suggestion of triangularization will be one good option (at least will allow automatic processing).

Thanks again!


POSTED BY: Christian Neel
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