Chapter 1, section 5 of User's Guide for CPO2D and CPO3D


(or proceed to chapter 2)


1.5 Comparison with other methods


As we have seen the potentials and fields are obtained indirectly in the BEM, in two stages, firstly obtaining the surface charges and then using these charges to find the potentials and fields at any point. This gives a fast and accurate way of solving electrostatic problems, and is ideally suitable for space-charge and cathode simulations.


On the other hand the more widely used Finite Difference and Finite Element Methods (FDM and FEM) solve Laplace’s equation in one stage. A set of mesh points (also called grid points or nodes) is set up to cover the surfaces of the electrodes and also the volume of space that they occupy. In the FDM the mesh points make an array of square or cubic cells in 2D or 3D systems respectively. The known potentials of the points on the boundaries are used to deduce the approximate potentials of the inner mesh points, using a numerical approximation to Laplace’s equation. Potentials and fields at a general point in space are then obtained by interpolation between the mesh points, which can cause discontinuities in the field. The iterative method used to solve the numerical approximation to Laplace’s equation is accurate, but the mesh potentials themselves are not accurate -on the contrary these potentials and the resulting fields can be very inaccurate particularly near the edges or corners of electrodes. For more details see note on tests of accuracy and speed or further information on potential and field evaluations.


See also published comparisons of BEM with other methods, which show that typically the BEM is about two orders of magnitude more accurate than the other methods, for the same computing time, or equivalently is typically about 2 orders of magnitude faster for the same accuracy.


We have no doubt that those researchers who adopt a practical approach and use different program packages to compare the speed and accuracy obtained for a set of electrostatic simulations that are relevant to them and that have known solutions will finish by selecting CPO2D and CPO3D. As far as we know, only CPO2D and CPO3D are supplied with a wide-ranging set of ‘benchmark’ test files. As well as being fast and accurate, setting up a simulation is easy with the BEM because only the electrode surfaces have to be modelled.





  • In the CPO programs there is no need for an enclosing box.


  • The electrodes are subdivided into triangular or rectangular segments each of which carries an electrical charge on its surface.


  • The charges on the segments depend on the voltages applied to the parent electrodes.


  • The CPO programs calculate the charges on the segments without having to create a mesh of artificial points in the space inside the system of electrodes.


  • The CPO programs then use the charges to determine the potentials and fields anywhere in the system.


  • The Boundary Element Method is ideally suited to space-charge and cathode simulations.




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