Update Wake integration to account for non uniform grid

[elenafuengar]

To have the Smart grid fully functional for Wakefield simulations, we need to adapt the wake potential integration in wakeSolver.py to account for non uniform integration path. Hence, dz is no longer a float but an array of values:

# ... inside WakeSolver ...
# both in calc_long_WP_3d(self, **kwargs)
# and in calc_long_WP(self, Ezt=None,**kwargs):

                        for n in range(len(s)):
                            for k in range(nz):
                                ts = (z[k]+s[n])/self.v-zmin/self.v-self.t[0]+ti
                                it = int(ts / dt)  # find index for t
                                if it < nt:
                                    WP[n] = WP[n] + (Ezt[k, it]) * dz  # compute integral

                            pbar.update(1)

                    WP = WP/(self.q*1e12)     # [V/pC]
                    WP_3d[i0+i,j0+j,:] = WP

[Antoniahuber]

At the moment there are problems with charge accumulation, if the arrays are non-uniform:

The origin of the charge accumulation being the non-uniform grid can be seen well in the LHC bell, where the boundaries between the regions cause numerical errors:

[Antoniahuber]

I found the source of the charge accumulation to be in the matrix multiplication of the discretized Maxwell equations. In a uniform grid the distances of the primal and the dual grid are always the same. In a non-uniform grid there is a difference between the distances and therefore the charge probably came from that difference:

New Matrix Multiplication:

Old Matrix Multiplication:

Same fields with new Matrix implementation: