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4.1 Wider prospects for the algorithm

4.1.1 The need to adapt the meshing

The 3d space charge calculation implemented in Astra uses rectangular mesh, each mesh having the same dimension. Eventhough this can be satisfying for cylinderical or spherical distribution which keeps a trajectory close to one of the laboratory axis (e.g. z), a lack of efficiency will be found for distorded distributions or trajectories which are not parallel to one of the frame axis. This is illustrated in figure 4.1 where the rectangular meshing along the frame axis results in a lot of empty mesh. Since the number of meshes in each direction strongly influence the computational cost, an adapted meshing should be applied. Reducing the number of empty mesh and increasing the resolution of the mesh on the distribution by keeping the implemented method of rectangular meshing will be performed by rotating the frame axis. As illustrated in figure 4.1, this will result in a better adapted meshing, increasing the accuracy of the calcul for a same computational cost (number of mesh)

**Figure 4.1:** Problem of unadapted mesh. On the left, one can see that a lot of cells are empty, so that computational time is lost, and excessive number of particles are present in other cells, resulting in a bad numerical solution. An adapted mesh as presented on the right figure, would solve these problems.

4.1.2 No momentum direction favorised

Astra was implemented to study particles travelling along the z axis, the main compoment of momentum being $\beta_z$ . Thus several approximations were done knowing that few deviations from the z axis would occured, mainly $\v{\beta} \approx \beta_z \v{e}_z$ . As soon as elements such as dipoles are introduced in the beam lattice, the main direction of the momentum is changed and these approximations are not valid any more. Moreover, the introduction of a dipole will introduce situations of undaptated meshing as described in the previous section 4.1.1. The algorithm we implemented does not favorised any direction of momentum and designs an adapted grid around the distribution that is not parallel to the frame axis. A similar approach is actually followed by Klaus Floettman for the next version of Astra, but has not been released at the time this paper is written.

Emmanuel Branlard
http://emmanuel.branlard.free.fr/