Chapter 1, section 4 of User's Guide for CPO2D and CPO3D

(or proceed to section 5)

1.4 The technique used to obtain the surface charges

The surface charge density s on each segment is taken to be uniformly distributed over its surface, on the assumption that the segments have been chosen to be sufficiently small. This represents the only significant approximation made in the present version of the Boundary Element Method, apart from any numerical approximations in the calculation of the potentials and fields. (Using a charge density that varies linearly over the surface of each segment would seem to be better, but extensive tests have shown that the computing time is significantly longer and that for the same total computing time higher accuracy is obtained with more segments and a uniform distribution on each of them.)

The technique that is used in the Boundary Element Method to determine the segment charges in non-space-charge simulations is also simple. The question that has to be answered is:

What are the charges qi on the segments i (where i = 1 to n and n is the total number of segments) when the potentials Vi are applied to the segments (or more precisely, to the parent electrodes)?

In the Boundary Element Method the problem is looked at from the reverse direction, and the question is asked:

What are the potentials Vi that would result from a given set of charges qi?

To answer this we note that the potentials and charges are related through the linear equation where rj is the mid-point of segment j, and Pi(rj) is the potential at rj due to a unit charge on segment i. In other words, the potential Vj of segment j is due to the charges qi on all the segments i, including the self-potential due to the charge qj on the segment j itself. Clearly the coefficients Pi(rj) can be calculated, essentially by using Coulomb's Law and averaging over the areas of the segments. There are n of these equations, for j = 1 to n. There are also n unknowns, qi, for i = 1 to n. Nothing else is unknown, because all the segment parameters and hence all the coefficients Pi are known (in principle).

The n equations in n unknowns can therefore be solved for a given set of Vj's to give the segment charges qi. (Technical note: this is achieved by inverting a matrix using the well-known LU decomposition routine).

The electrostatic problem is then completely solved. The system of electrodes has become a set of segments each of which carries a known charge uniformly distributed over its surface. Potentials and fields can now be calculated anywhere in space, again essentially by using Coulomb's Law. Laplace's equation, the equation for the electrostatic field, has therefore been solved.

Note that in the present form of the BEM the potentials at the centres of the segments are exact but the potentials calculated on other parts of the segments are not exact. Similarly in the FDM and FEM the potentials at the mesh points are exact but the potentials between those points are not exact. In potential plots in the FDM and FEM the mesh points are usually used for the plotting and so the potential distribution appears to be exact over the whole boundary, which it is not! On the other hand in potential plots in the BEM the plotting points usually do not coincide with the centres of the segments and so the potential distribution looks less exact than in the in the FDM and FEM, which it is not! And note also that in the BEM the potential distribution is much more accurate in the interior region, where the particle fly, than in the boundary region.

First technical note, on ‘unit charges’:

The number N of different voltages Wk applied to the electrodes is usually smaller (in fact usually much smaller) than the number n of segments. The values of Wk are given initially in the data file but can be changed at run-time. The program therefore calculates N sets of charges qi, where each set corresponds to one of the Wk being unity while all the other values are zero. These N sets are called the 'unit charges'. Appropriate linear combinations of these unit charges are used later to determine the actual set of charges qi for a particular set of voltages Wk.

Second technical note, on singularities:

Many authors emphasise the difficulties in calculating the ‘self’ matrix element Pi(ri), which is the potential at rj due to a unit charge on the same segment i. The supposed difficulties are caused by the fact that the potential at a point due to a charge at that point is singular. In CPO3D there is no difficulty because an analytical formula is used for the potential at the centre of a uniformly charged triangular or rectangular segment due to the charge on that segment. In CPO2D there is also no difficulty because although an analytical formula does not exist it is easy to integrate over a narrow uniformly charged segment in such a way that the singularity is taken fully into account. The details will be given in a future publication.

Third technical note, on the nature of the matrix:

Every segment couples to every other segment and so all the matrix elements are non-zero and the matrix is ‘dense’ For example if three electrodes are in a line, held at potentials V1, V2 and V3, then electrodes 1 and 3 are not hidden from each other because if V1 is changed then the surfaces charges on electrode 1 are changed but so are the charges induced on electrode 2, which then causes the charges on electrode 3 to change as well (assuming that we want the potentials of electrodes 2 and 3 to be unchanged). In the case of two isolated cavities, in principle if the voltages of electrodes inside one of them are changed then there should be no effect on the charges on any electrodes inside the 2nd cavity. So some of the matrix elements would be zero. In the CPO programs we would expect that these elements would be very small. In fact in this case it would be better to simulate the 2 cavities separately.

(proceed to section 1.5)