Integration over circle/meshing a circle

I am supposed to make a double integral over a circle to calculate the radiated field from a circular aperture (with a given distribution of currents). My issue is how to define the integration domain. A solution, tested but not good is to define two vectors: rho and phi vectors to select points. Here the issue is the farther I go, the less dense my points are, since the number of points per quadrant remains constant. The other solution is to build a square grid/domain and to "cut the angles". Actually I don't know how to do that.

Any solution?

[Mohammad Abouali]

In both cases that you mentioned(i.e. polar coordinates and cube sphere mesh), if you keep the number of cells constant, then the farther you go your cell need to cover more surface, hence, less dense, or lower resolution. So, both would suffer from the same problem.

If you want the cells to have a constant area (regardless of how far you are from the center), then you need to increase the number of cells, which in polar means reducing your dRho and dPhi.

Note that the number of cells increase with second power of R (or distance from the center). So if you are twice away from the center, then you need at least 4 times more cells to have the same density of cells.

You could get some other methods, but then you need to provide more info on how the data is generated or calculates. Generally some more details. But the above general approach would work, but not necessarily the optimum for certain cases.