Dielectrics and Neumann boundary conditions.

An option is available for treating systems that include dielectric materials (see the Price List).

Another similar ‘special’ option treats Neumann boundary conditions.

Available only in 3D.

We start here by considering dielectrics.

'The ‘dielectric' version of CPO3D and CPO3DS.

This version gives very accurate simulations of static non-space-charge systems which include dielectric materials that have a uniform dielectric constant.

Dielectric systems are very easy to set up. The interfaces that enclose each dielectric medium are represented by quasi-electrodes, called ‘dielectric electrodes’, that have the same choice of geometries as the normal ‘conducting electrodes’. After defining the dielectric electrodes all the user has to do is to specify the dielectric constant K1 of the medium and the dielectric constant K2 of the surrounding volume (usually 1). More than one dielectric medium can be included.

The dielectric electrodes can be extremely close to conducting electrodes, although not touching.

Important points (please print this page for reference):

1. Only uniform dielectric media are treated, so each medium has a constant dielectric constant.

2. The boundaries or interfaces that enclose each dielectric medium are represented by quasi-electrodes, called ‘dielectric electrodes’, that have the same choice of geometries as the normal ‘conducting electrodes’.

3. Each dielectric interface must be a closed surface -the program does not check for this. (The reason for this is that in the region at the free end of an open boundary the program cannot give the field ratio K2/K1 mentioned above.)

4. The user specifies 2 dielectric constants, K1 and K2, for each dielectric boundary between two media, and also specifies a ‘reference point’ and to say whether this reference point is inside or outside the first medium. For example if there is a cylindrical surface that enclosed a medium of dielectric constant 10 and the constant outside the cylinder is 1, then we would put K1 = 10, K2 = 1 and put the reference point anywhere inside the cylinder and state that it is ‘inside’. (On the other hand it might be more convenient to put K1 = 1, K2 = 10 and use the same reference point but state that it is ‘outside’.) More exactly, an ‘inside’ reference point does not have to be actually inside the first medium, but it must be on that side of all parts of the electrode. Similarly for an outside reference point.

5. A ‘dielectric electrode’ can be very close to a ‘conducting electrode’ but must not coincide with it nor cross over it. (The reason for this is that if the ‘free’ and ‘polarization’ charges mentioned above are not kept distinct from each other then there are not enough variables to be able to solve the equations mentioned above.) The usual checks for overlapping segments are disabled when dielectrics are present. It is therefore important to check manually for possible overlapping or closeness of the normal ‘conducting electrodes’ with each other -a good way of doing this is to run initially without the dielectric electrodes and with the overlapping check enabled. (Technical note: one test is carried out that gives an error message and refers to a minimum distance, which is the largest distance from the centre of a segment to a corner times the square root of the inaccuracy used for the dielectric calculation.)

6. When a ‘dielectric electrode’ and a ‘conducting electrode’ are very close they should have the same geometry with the same number of subdivisions, but with slightly different sizes or positions. The ‘transform’ option in CPO3D is very useful for doing this.

7. Do not use the dielectric version of CPO3D for non-dielectric systems, because several options are disabled in the dielectric version (for example, checks for overlapping segments, boundary potential improvements, solid electrodes, zero total charge).

8. The inaccuracy used for evaluation of the fields at the dielectric interfaces has a default value is 0.001. (Note that this inaccuracy is not the same as the inaccuracy used in calculating the surface charges of the conducting electrodes, which is always 1.E-7 and which cannot usually be changed by the user.) It might sometimes be better to reduce this inaccuracy (at the bottom of the 'tracing accuracy' sheet).

9. The calculation of the charges on the ‘dielectric electrodes’ is typically about 10 times slower than for the normal ‘conducting electrodes’.

Benchmark tests of dielectric systems are:

test3d22, spherical capacitor with dielectric in gap.

test3d23, parallel plate capacitor with dielectric in gap.

test3d24, field in a cavity inside a dielectric.

test3d25, field inside a dielectric sphere.

test3d26, field inside a dielectric cylinder.

test3d27, capacitance of coaxial cylinders with dielectric.

An ‘example’ file:

xmpl3d87, dielectric simulation of a magnetic yoke.

An example of a cathode in the presence of dielectric materials is given by xmpl3d90.

Technical notes:

When there are dielectric materials present it is convenient to consider the ‘electric displacement’ vector D, which is defined as D = KE, where K is the dielectric constant (also called the relative permittivity) and E is the electric field. When there is a boundary between two media of different dielectric constants K1 and K2 then the component of D perpendicular to the surface is the same on both sides of the boundary. This implies that E is discontinuous and that the ratio of the component of E perpendicular to the surface at the two sides of the boundary is K2/K1. ‘Polarization charges’ then exist at the boundary. The components of D and E parallel to the surface are the same on both sides. The electric field at any point is effectively due to the combined effects of all the charges -that is, all the ‘free surface charges’ on the conducting electrodes and all the ‘polarization charges’ at dielectric boundaries.

CPO3D adjusts all the charges, ‘free’ and ‘polarization’, to give the required electric field ratios K2/K1 at the dielectric interfaces, while at the same time reproducing all the applied potentials at the surfaces of the conducting electrodes. (The number of variables, the charges, is then equal to the number of conditions, the field ratios and applied potentials, so the relevant equations can be solved.).

Neumann boundary conditions

The CPO programs normally use ‘Dirichlet’ boundary conditions, in which the boundary potentials V (or more exactly, the electrode potentials) are specified. But there is a ‘special’ Neumann version in which the ‘Neumann’ boundary conditions are used. Here the values of dV/dn at the boundaries are specified, where dV/dn is the differential of V in the direction normal to the boundary. More exactly, in this version V can be specified for some electrodes and the field component En = -dV/dn can be specified for other electrodes.

The restrictions and advice are similar to those for dielectrics.

1. The interfaces that define each Neumann boundary condition are represented by quasi-electrodes, called ‘Neumann electrodes’, that have the same choice of geometries as the normal ‘conducting electrodes’.

2. The user specifies a field strength E for the interface. This field strength is the component of the field normal to the surface (it is not possible to specify the other components of the field because there are not enough variables). The user also specifies a ‘reference point’ and is asked to say whether this reference point is inside or outside the interface. For example if the interface is a cylindrical surface and if the field strength needs to be defined for the external surface of the cylinder then we can put the reference point anywhere inside the cylinder and state that it is ‘inside’. On the other hand it might be more convenient to use a reference point that is ‘outside’. An ‘inside’ reference point must be on that side of all parts of the electrode. Similarly for an outside reference point.

3. Do not use the Neumann version of CPO3D for systems that are defined by the more usual Dirichlet boundary conditions, because several options are disabled in the Neumann version (for example, checks for overlapping segments, boundary potential improvements, solid electrodes, zero total charge).

4. The inaccuracy used for evaluation of the fields at the Neumann interfaces has a default value is 0.001. (Note that this inaccuracy is not the same as the inaccuracy used in calculating the surface charges of the conducting electrodes, which is always 1.E-7 and which cannot usually be changed by the user.) It might sometimes be better to change this inaccuracy to 0.0001 (at the bottom of the 'tracing accuracy' sheet).

5. The calculation of the charges on the ‘Neumann electrodes’ is typically about 10 times slower than for the normal ‘conducting electrodes’.

A benchmark test of Neumann systems are:

28th benchmark test for CPO3D, Neumann boundary conditions.