3.5 Application to the Frequency Domain Method
3.5.2 Multimode Cavity Problems
Multimode cavity problems by their very nature possess many resonant modes. This has the effect of significantly increasing the condition number of the system, even when the mesh is relatively coarse. In practice this manifests itself by producing very slow convergence of the conjugate gradient method. The performance of both the Bi-CG method and QMR methods were tested for the linear system arising from the frequency domain discretisation of a cavity loaded with a tray of mashed potato. One quarter of the problem was modelled. The field results for this problem are discussed in§6.4 along with a more detailed description of the geometry. Figure 3.5 shows the comparative rates of convergence for the two methods with and without preconditioning using shifted incomplete Cholesky factorization. The convergence rates are very similar for both the QMR and Bi-CG algorithms, with the QMR giving smoother convergence as expected.
0 5,000 10,000 15,000 20,000 25,000 Iteration, i
10-10 10-8 10-6 10-4 10-2 100 102
r r0i
SIC Bi-CG SIC Bi-CG
SIC QMR SIC QMR
Bi-CG Bi-CG
QMR QMR
Figure 3.5: Convergence of the QMR and Bi-CG algorithms with and without precon- ditioning for a multimode cavity containing a mashed potato load. (Mesh 1 of Table 3.3)
The preconditioner is effective at reducing the number of iterations, however, a large number of iterations were still needed to produce a satisfactorily small residual.
The mesh used for Figure 3.5 was very coarse so further tests were carried out using a total of three different meshes with differing degrees of refinement. Table 3.3 gives the number of degrees of freedom and the number of non-zeros in the coefficient matrix for the three meshes. The first mesh was a coarse mesh with a small amount of refinement near the waveguide aperture and in the potato. The second mesh had more refinement inside and around the dielectric while the third mesh was refined throughout the whole domain. It can be seen from Figure 3.6 that the first two meshes produce systems that converge in a number of iterations approximately equal to half the number of degrees of freedom whereas the third shows much slower convergence. Since the rate of convergence is dependent upon the condition number this would suggest meshes 1 and 2 produce a system with a moderate condition number giving fairly slow convergence. Mesh 3 would appear to have a much higher condition number that causes the very slow convergence
3.5 Application to the Frequency Domain Method 77
0 20,000 40,000 60,000 80,000
Iteration, i 10-10
10-8 10-6 10-4 10-2
Mesh 2 Mesh 2 Mesh 1 Mesh 1
r r0i
Mesh 3 Mesh 3
Figure 3.6: Reduction in the residual using the SIC-QMR algorithm with three different meshes for the mashed potato problem.
of the SIC-QMR algorithm.
The major difference between meshes 2 and 3 is the discretisation of the air in the cavity. The former has a very coarse discretisation with the largest elements being ap- proximately one fifth of a wavelength in size while mesh 3 has elements of approximately one tenth of a wavelength. Meshes 1 and 2 cannot be expected to give good accuracy because of this coarse discretisation: a coarse mesh being able to only represent a small number of mode patterns accurately so artificially restricting the solution, and therefore the accuracy. The finer mesh allows considerably more modes to be represented by the discretisation, some of these modes may have high Q-factors, and indeed as suggested by Bossavit [1995] may be “air modes”, that is, ones with an infinite Q-factor. The presence of these modes near or at the excitation frequency may cause a loss of unique- ness of the solution in the discrete problem and result in a very ill-conditioned system even though the continuous problem is well posed. This very slow convergence seems to be characteristic of multimode cavities when loaded with a dielectric having a high permittivity.
Mesh
Degrees of
Freedom Non Zeros Final Residual Iterations Time 1 22,600 365,148 4.9×10−10 11,270 4 hours 2 39,278 649,246 5.0×10−10 20,316 10 hours 3 51,388 832,002 3.7×10−7 67,000 43 hours
Table 3.3: Comparison of performance of SIC-QMR for three meshes used for the mashed potato problem.
To determine the effect of the permittivity of the load on the convergence, the prob- lem was re-run using mesh 3 but with different values of the permittivity and the results are shown in Figure 3.7. As the permittivity of the load increases the convergence of the SIC-QMR method becomes progressively slower. The increase in condition number with increasing variation in material properties has been noted previously [Vavasis, 1993]. In multimode cavities this may be due to the higher number of modes that are capable of being supported by the more heavily loaded cavity. The higher permittivity will also cause more energy to be reflected from the surface of the load, in some cases this will reduce the coupling of a particular mode to the load. This is a serious computational problem since for microwave heating it is common to have a large cavity loaded with a large block of dielectric which has both a high permittivity and high loss factor, the example of mashed potato being typical.
At each iteration of the SIC-QMR it is necessary to perform two substitutions (one forward, one backward) plus a matrix-vector multiplication which will require a total of ≈2τc complex floating point operations. The use of ICCG(0) preconditioning in conjunction with Eisenstat’s procedure may allow this to be reduced to ≈ τc, halving the work per iteration. Even so when the number of iterations required for realistic loads of food like materials is> nthe solutions will take a considerable amount of time as shown in Table 3.3 unless a supercomputer is used.
The problems examined here are relatively small, with the largest having 51,000 unknowns. Many problems will be considerably larger making the slow convergence even more problematic since the work per iteration will increase. The larger problems are also