in this section.
3.6.1 Diagonal Scaling
Condition Number
Problem n Diagonal Scaling No Scaling
S/C Waveguide 6,196 4.83 11.08
S/C Waveguide 80,334 3.73 9.00
S/C Waveguide (irregular mesh) 13,525 9.35 28.46 TM010 cavity (Prism elements) 22,174 241.78 3223.36 Mashed Potato loaded cavity 51,506 31.3 1439.24
Table 3.4: Condition number, κ1(A) of the matrices from the TDFE discretisation of various three dimensional problems (30 time steps/cycle).
Table 3.4 compares the condition numbers, calculated using Hager’s algorithm, for several problems with and without diagonal scaling. Since the condition numbers are very low the solution of the linear systems required by the algorithm can be carried out very quickly. It is interesting to note that for the waveguide problem the condition number actually reduced slightly as the mesh was refined, the complete reverse of the frequency domain case. The first two waveguide problems have meshes produced by dividing the domain into a regular hexahedral mesh and then splitting each hexahedron into five tetrahedra. This produces a mesh with only ten distinct shapes and orientations of the elements. The irregular mesh was produced using GEOMPACK which produces the tetrahedra directly using a Delaunay technique. The irregular mesh has a larger number of element sizes and orientations which produces a higher condition number. If nodal elements were used then the condition number of the mass matrix ( [T²]) becomes independent of both element size and shape and the size of the mesh [Wathen, 1987].
This is not the case for edge elements, and when an implicit time stepping scheme is used A contains contributions from [S] and it does not hold for nodal elements either.
The TM010 cavity mesh was constructed from prism elements the size of which varied
3.6 Application to the Time Domain Method 81
greatly throughout the mesh (see§5). This is the cause of the relatively large condition number.
The mesh used for the cavity loaded with mashed potato has localised refinement near the aperture and in the dielectric which produces elements with different sizes and orientations. This, coupled with the large variation in material properties causes the condition number, prior to scaling, to be significantly larger than for the waveguide problems which have uniform dielectric properties. Scaling has the effect of removing the dependency of the condition number on material properties. Since diagonal scaling is so effective at reducing the condition number it has been used for all the calculations.
The equations being scaled before the application of any further preconditioning.
3.6.2 Direct Methods
Direct methods suffer from exactly the same memory problems when applied to the time domain solution as for the frequency domain problem. This makes them applicable only to very small problems where they may be competitive with iterative methods. In§3.2 it was suggested that if the number of iterations, k, required for an iterative solution of (3.1) satisfied kτc < τf then the iterative method would be faster. For the smallest problem considered in Table 3.1 we see that τf ≈5τc, however, the very low condition number means that only 2–3 iterations of the SSOR-CG method is required, so even for this small problem the iterative method should be faster. For the larger problems the difference is much greater.
3.6.3 SOR Iteration
The time domain method permits the solution of equation (3.1) using SOR or Gauss- Seidel iteration. This method is very simple and can give good results when the initial guess is chosen according to (2.33). The SOR algorithm has a performance which is very dependent upon the choice of the over relaxation parameter ω. However, since we are solving many systems in succession, one at each time step, it is possible to continually vary ω and therefore dynamically optimise its value during the calculation.
Table 3.5 shows the performance of the SOR method for various problems along with the value of ω used. The systems were diagonally scaled prior to starting the SSOR
Problem n Iterations ω
S/C Waveguide 2,532 8 1.29
S/C Waveguide 22,812 8 1.27
S/C Waveguide 80,388 11 1.26
TM010 cavity 22,174 11 1.24
Mashed potato loaded cavity 51,506 11 1.21
Table 3.5: Performance of SOR iteration for different TDFE problems.
iteration.
3.6.4 Preconditioned Conjugate Gradients
The preconditioned conjugate gradient method was applied to the mashed potato loaded cavity problem. Table 3.6 shows the number of iterations required to reduce the residual to 1×10−6 both with and without SSOR preconditioning and for different initial guesses forx0. The system of equations was diagonally scaled prior to starting the solution for all problems. SSOR preconditioning was chosen since it allows Eisenstat’s procedure to be used which means the work per iteration for the preconditioned and unpreconditioned systems is about the same and no extra storage is required. Table 3.6 clearly shows the effectiveness of this preconditioner for this particular problem. The use of x0 = 0 has the advantage that, since r0 = b no matrix-vector multiplication is required prior to starting the iteration. This is not the case for the other two initial guesses where it is necessary to calculate the residual. The first and third options for choosing x0
appear equally good, however, it is found that option three is slightly better because it causes the number of iterations at each time step to be reduced slightly as the solution approaches the steady state condition. The results in Table 3.6 show that the SSOR preconditioned conjugate gradient method, when coupled with diagonal scaling, gives a very effective method of solving the system in the time domain.