Due to the accuracy of recent tools for Schwarz-Christoffel (SC) numerical mapping and numerical inversion (SCNI) [1, 2], two-stage SC+FD procedures have been already proposed to map the analysis domain to a rectangle, avoiding field singularities and providing a close boundary, and then to cope with the inhomogeneous dielectric in the rectangular domain: see for instance [3, 4, 5] for both plane and axially symmetrical geometry, where comparison is made to finite element (FEM) direct calculations in the original domain. Nowadays, the standard tool to solve plane field problems are FEM codes, though SC maps maintain some utility, looking for reference and comparison data, mainly in a homogeneous medium. Nevertheless, it appears that compound SC+FEM procedures can be worth of attention, taking again advantage of the SC stage to avoid singularity problems, but using a FEM stage to solve the Laplace or Poisson equations with inhomogeneous mediums in the SC-transformed domain. In the paper, after providing accurate SC+FD calculations, we compare the results with those obtained from direct FEM calculations in the original domain, using the half cross-section of a shielded dielectric-supported stripline as case study, both for the thin and for the thick strip cases. Then, we introduce a significant comparison with SC+FEM results, showing possible benefits. Due to the severe field singularity at the π-angle corner at the end of the half thin strip, this geometry is representative of many boundary shapes, and represent a simple but significant test of accuracy, even though it is normally linked to the old thin film technology, golden copper on alumina substrate, when it allowed to calculate capacitances and characteristic impedances in the quasi-transverse electromagnetic propagation mode.
2.1 SC+FD and FEM thin strip results
The geometry is shown in Figure 1, and its SC conformal map is shown in Figure 2, left hand. In air, the aspect ratio base/height of the rectangle provides by inspection the capacitance of the line, C0sc= 1.92249066·10–11 F/m. In Figures 1 and 2, solid lines represent conductor sides, the strip at the upper side and the metal case at the bottom, thick dashed lines Neumann sides, and thin dashed lines the two interfaces between different dielectrics (air, ϵr = 1, and alumina, ϵr = 9.5): plot1 maps the interface at the bottom plane of the dielectric, and plot2 maps that at the upper plane of it. To solve the Laplace equation in inhomogeneous dielectric, FD successive over-relaxation (SOR) procedures can be easily written using five-point equations.
Square meshes are used here for the sake of simplicity and speed, but the final results are corrected according to the aspect ratios of SC map and of the FD mesh outer rectangle.
The accuracy of direct FEM calculations on the original strip geometry and in vacuum can be preliminarily appreciated making comparison of the results in Table 1 with the SCNI capacitance C0sc = 1.92249066·10–11 F/m, to be considered virtually exact. Direct FEM capacitances (from the energy) agree with C0sc within a 0.5% error with 6,308 nodes, and a 0.1% error with a locally dense mesh of only 1,856 nodes but geometrical progression, with rate 6 and 20 elements along the half stripline. The saving of degrees of freedom between case A and case B is impressive but it can be justified by the severe singularity at the end of the stripline. The capacitance value for case A computed in vacuum from the dielectric flux on the outer conductor was C0FEM = 1.9227·10–11 F/m, and this seems to be the best result, with a percent error of about 0.01%.
Therefore, we assume the value CFEMA = 3.29654·10–11 F/m obtained from a locally dense mesh as a reliable FEM reference in inhomogeneous dielectric. A very dense mesh of 23,800 nodes leads to CFEMB = 3.301·10–11 F/m, different from CFEMA of about 0.13%, and a mesh of 6,308 nodes leads to CFEMC = 3.30852·10–11 F/m. We note that following one of the first proposed variational calculations , the capacitances in air, 1.9177·10–11 F/m, and in inhomogeneous dielectric, 3.2828610–11 F/m, agree within some 0.25% with C0sc and with CFEMA.
Considering Table 1 and Table 2, we see that probably an uncertainty of about 0.1% can be considered the accuracy limit for the FEM results, both in the homogeneous and in the inhomogeneous dielectric cases. As for the dielectric interfaces in SC+FD calculations, experience shows that, using predictor-corrector (P-C) routines with suitable numbers of steps, the related errors can be made almost negligible. In the present case, they differ for less than some 0.02% using 4,000 instead of the 8,000 steps, the last leading to Figure 3.
To apply the suitable five-point equation to any mesh node neighboring a dielectric interface, they are placed on suitable staircases obtained from the P-C traces, with minimum steps of one mesh cell both in the horizontal and the vertical directions. Table 3 and 4 show SC+FD results for various numbers of mesh cells and for 8,000 and 4,000 P-C steps, with various percent errors and differences.
Considering Figure 2, Table 2 and Table 3 shows that 4,000 or 8,000 P-C steps are well enough. Moreover, the differences between the corresponding values with large numbers of cells exhibit magnitudes analogous to CFEMA-CFEMB. However, and this is perhaps the most important evidence, a small but persistent difference from CFEMA is found in the SC+FD values for large numbers of mesh cells, with a somewhat oscillating behavior.
In particular, with a large FD mesh, exhibiting 2,200·1,013 = 2286 cells, and with 8,000 P-C steps dielectric interfaces, the SC+FD capacitance was CSCFD = 3.29106·10–11 F/m, with differences from CFEMA and CFEMB of about –0.17% and –0.31%. This suggests that, despite of refined mesh discretization, the major cause of these errors remain discretization effects related, more than to the absolute dimensions of the meshes, to ratios of discretized lengths derived from the original geometries. In the current case, these appear unsuitable in particular to deal with the severe concentration of the dielectric flux lines in the short region of the bottom conductor facing the high permittivity dielectric. It was precisely this to suggest considering the SC+FEM procedures proposed in the following.
2.2 SC+FD and FEM thick strip results
When the stripline model exhibits a finite thickness, the geometry of the problem is slightly different, although hardly perceived at the scale of Figure 1, being the ratio between the total height and the thickness equal to 130. The field singularities at the left end of the conductor are now two, but the effect on the overall capacitance is milder because of the different exponent at the π/2 corners. The SC+FD procedure leads (Figure 2, right hand) to a rectangle with a slightly smaller aspect ratio than the thin strip, i.e., to a slightly large capacitance in vacuum, and the maps of the dielectric interfaces are something more complex.
The SC capacitance in vacuum is now C0scT = 1.97509295·10–11 F/m and two FEM capacitance values have been provided, both in vacuum and in inhomogeneous dielectric, with two meshes having both triangle densities matched to the needs in proximity of the end of the stripline. Accordingly, Table 5 and Table 6 report the same evaluations as Tables 1 and 2 did for the thin stripline case, and we note immediately that the FEM accuracy seems to be somewhat better than with the more severe singularity of Figure 1. Due to the matched mesh densities, we can immediately appreciate the good uniformity and accuracy of these results, despite of the modest number of triangle for the case TB.
Table 7 shows the comparison of the SC+FD calculations in inhomogeneous dielectric with the capacitance of the FEM case TA.
It is analogous to the previous Table 3, but the last column reports the differences in the SC+FD capacitances obtained with 1000 instead of 4000 steps in the P-C mappings of the dielectric interfaces, instead of the comparison of a second FEM values. As in vacuum, the accuracies are perhaps somewhat better than in Table 3, but show in Figure 3 (dashed line) a similar behavior, leading again to the proposal of SC+FEM calculations, although limited to smaller numbers of FD cells and perhaps less worrying.
2.3 The SC+FEM procedure and its accuracy
To make it easier to represent the details of Figure 1 at the input of FEM codes, we will continue the discussion using a second stage map analogous to Figure 2, but in which plot1 was replaced with a perfect circle, and plot2 by a vertical straight segment, being unchanged the boundary vertices and the end points of the interfaces. Of course, the new geometry is no longer the map of a strip, but this is unimportant when discussing the possible accuracy of second stage calculations. This trial geometry is shown in Figure 4.
The FEM second stage results, with number of nodes ranging from 276 to 6351, ranged from 3.32221·10–11 to 3.32187·10–11 F/m, showing immediately differences of the order of only 0.01%, provide good reference values. In contrast, the behavior of the FD capacitances was again unsatisfactory. As for the thin strip, oscillatory differences are found for large numbers of cells ranging from zero to some –0.2%. Therefore, we find the same conditions we had when comparing direct FEM and SC + FD results.
This suggested a new step of analysis. Starting from the same geometry, the lengths of the various segments of the boundary have been replaced by numbers with integer ratios among them, to secure the possibility of positioning any vertex or interface ending point on the nodes of the FD mesh. This was possible with special care of the segments of the bottom conductor and with modest errors in regions of negligible influence.
This time, a reliable FEM reference was CFEMstage2 = 3.3224·10–11 F/m, from the energy, with 25,884 nodes, locally dense mesh with nonlinear rule of the second order (and, with a very similar value, C = 3.32270·10–11 F/m with a linear rule). The equipotential lines and field magnitudes are shown in Figure 5. The FD results have been very satisfactory too, as the results were, for instance, CFDstage2 = 3.32176·10–11 F/m with the quarter circle modeled with 8,000 P-C segments, and 3.32200·10–11 F/m with 4000. The maximum difference among these second stage FEM or FD values is of about 0.03%.
This allows us to draw important conclusions. Assuming FEM results on the original geometry as the reference, we can expect that the results obtained from a FEM second stage geometry, i.e., a virtually exact rectangular boundary and very accurate maps of the dielectric interfaces, will be very similar to the reference. At the same time, it was shown that the FD results for the same second stage geometry can be good in general and in addition that they are very good, and very similar to the FEM results, if the details of the maps of the interfaces are well represented by the FD mesh.
A well-known case study has been used to enlighten the different behaviors of FEM and two step SC+FD procedures in presence of strong field singularities both in vacuum and in inhomogeneous dielectric structures. Of course, the dominant role of FEMs in engineering calculation is confirmed, but care must be taken to secure suitable mesh densities in critical regions, near to the same source of field singularities to which the SC+FD calculations are instead insensitive by nature. The major sources of errors are in this case the discretization processes of the conductor segments bordering different dielectrics and of the dielectric interfaces. However, the discrepancy between the results with different number of cells in the FD second calculation stage can be easily reduced to some 0.1%, of the same order of the difference with the best FEM results with reasonable number of nodes. Therefore, two-stage calculations can be recommended to confirm or compare FEM results, to exploit the capabilities of the SC maps in case of geometry with severe corner singularities, or when in presence of open domain boundaries.
The behavior of the differences between SC+FD and FEM results suggested the advantage of performing the second stage calculations by means of FEMs instead of FDs, to better cope to discretization problems when significant length ratios observed in the SC maps cannot be accurately represented in the second stage FD mesh. Probably, this would allow a faster computation too. Defining a second stage FD map analogous to that obtained for the stripline geometry, we have shown that the differences among SC+FEM and direct FEM results can be reduced to some 0.01%, i.e., about an order of magnitude better than with SC+FD calculation. Unfortunately, representing the second stage by means of general purpose FEM codes are not always easy or even possible.
Driscoll T.A., Trefethen L.N., Schwarz-Christoffel Mapping, Cambridge University Press, Cambridge 2002, 9-30, 70-74. Google Scholar
Costamagna E., Di Barba P., Mognaschi M.E., Savini A., Fast algorithms for the design of complex-shape devices in electromechanics, in Wiak S., Napieralska-Juszczak E. (Eds.), Computational Methods for the Innovative Design of Electrical Devices, Springer, Berlin, 2010, 59-86. Google Scholar
Costamagna E., Fanni A., Computing capacitances via the Schwarz-Christoffel transformation in structure with rotational symmetry, IEEE Trans. on Magnetics, 1998, 34, 5, 2497-2500. CrossrefGoogle Scholar
Gish D.L., Graham O., Characteristic impedance and phase velocity of a dielectric-supported air strip transmission line with side walls, IEEE Trans. on MTT, 1970, 3, 131-148. Google Scholar
About the article
Published Online: 2017-12-29
Citation Information: Open Physics, Volume 15, Issue 1, Pages 839–844, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0099.
© 2017 E. Costamagna and P. Di Barba. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0