Ch. Hafner, Computational Optics Group, IFH, ETH, 8092 Zurich, Switzerland
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Antisymmetric case, circular wires
Assume that the center of the first wire is located at xc=+a in the xy plane, whereas the center of the second wire is at xc=-a. The radii of both wires are equal to r. The potential of the first wire is +1V, whereas the potential wire is –1V.
Analytic solution, Mirror Charge Method
The problem can be solved analytically using conformal mapping, an appropriate bipolar coordinate system (with foci at ) or with the mirror charge method with line charges located at the foci of the bipolar coordinate system.
Note: "Mirror charges" are the charges we would see when our eyes were able to see charges. In this case, we would see the electrodes as mirrors. If we would replace the right wire by a light source at the position of a mirror charge and the left wire by a perfect mirror, we could really see the image of the light source in the mirror exactly at the place where we put the mirror charge.
Since the mirror charge method is closely related to MMP, one can easily use it to obtain the analytic solution with the 2D MMP solver contained in MaX-1.
Figure 1: Mirror charge solution of the antisymmetric case of two circular wires. Red crosses indicate the locations of the mirror charges. The black cross is a dummy multipoles expansion with zero amplitude (used as "excitation"), the black lines perpendicular to the boundaries indicate the error distribution. Project: WIRES000.PRO
To be more specific, we select a=0.5m and r=0.3m. We immediately obtain x=± 0.4m, y=0m for the locations of the two mirror charges. Since an mirror charge is nothing else than a monopole or a zero order multipoles, we can place zero order multipoles at these locations, let MMP compute the strengths of them and obtain the following field with a very high accuracy.
Notes for the MMP modeling
The boundaries are two circles. Impose "specific" boundary conditions with only "V" turned on. Set the boundary values +1 and –1 respectively.
After placing the two mirror charges (2D multipoles with maximum orders 0, only 1 parameter per multipole, add a third, dummy expansion (for example, a 2D multipole) with only one parameter. This is used as "excitation". Make sure that its amplitude is set equal to 0.
Since only "electric" multipoles are required, you may turn the "E" filter in the "Expansion" dialog on or you may select "E wave" in the "Project" dialog.
In order to determine the strengths (amplitudes) of the mirror charges, you may insert the values of the analytic solution or, you use the MMP point matching technique. Since there are only one unknown per wire, one matching point per wire is sufficient. When you set the values in the "Matching point definition" group of the "MMP" dialog correctly, MMP will create only one matching point per conductor and it will compute the resulting 2 by 2 matrix system very quickly.
In order to evaluate the error distribution along the boundaries, it is reasonable to drastically increase the number of matching points per wire to, for example, 100. However, you will see that the resulting error is numerically zero (machine accuracy).
Taking the YZ symmetry into account
By taking the symmetry of the problem into account, you may even reduce the MMP matrix size. When you set the "YZ plane" symmetry number 2 in the "Project" dialog, you may delete the second boundary and the second expansion and you obtain a single boundary (with a single matching point on it) and a single monopole expansion (beside the dummy expansion that is not computed) and the MMP matrix becomes a 1 by 1 matrix.
Figure 2: Same as in Figure 1 when the symmetry with respect to the y axis is used. Project: WIRES001.PRO
Simple Method of Moments (MoM) solution
The standard MoM approximates the surface charge distribution on the surfaces of the wires by a series expansion. In general, the basis functions of this series expansions are arbitrary surface charge distributions. The corresponding field is obtained from Coulomb integrals. Since these integrals cannot be solved analytically in most cases, one usually prefers subdomain basis functions, i.e., surface charge distributions that are non-zero only on a small segment of the boundary. As a consequence, the Coulomb integrals extend over the segments only. When these segments are curved, one still does not obtain analytic solutions. Therefore, one usually approximates the boundary by a polygon. As a consequence, corners are introduced and it is well known that the electric field (gradient of the potential) will be singular at convex corners, i.e., the polygon approximation makes it impossible to obtain correct field values on the boundary. Although this effect is dramatic for near-field computations, one can obtain accurate field values a bit away from the boundaries, and one also can obtain accurate results for integral values such as the capacity.
Since Coulomb integrals over straight segments were not implemented in MaX-1, we try an approximation that is even more rough: We approximate the surface charge distribution by a discrete set of charges, i.e., 2D multipoles of order 0 placed on the surfaces of the wires. In order to compute the corresponding parameters, we can again use the point matching technique with as many matching points as unknowns. Since both the matching points and the monopoles are placed on the surfaces of the wires, we would encounter numerical problems when we place matching points and monopoles at identical positions. Therefore, we may prefer to move the monopoles slightly away from the boundary.
Note that neither the placement of the monopoles nor the placement of the matching points is unique. It seems to be reasonable to use an equal spacing with constant distances between neighbor matching points and neighbor monopoles. Obviously, we then have two special cases: 1) The matching points are at the same positions as the monopoles (which requires moving the monopoles or the matching points a bit away in order to avoid numerical problems) and 2) The matching points are in the centers between two neighbor monopoles.
Figure 3: MoM approximation with 10 monopoles at (almost) same positions as the matching points. Project: WIRES002.PRO
Figure 4: MoM approximation with 10 monopoles at centers between neighbor matching points. Project: WIRES003.PRO
As one can easily see from Figures 3 and 4, the matching points should be placed between the monopoles. However, the MMP matrix is much bigger than in the "mirror charge" solution and the errors are quite high on the boundaries.
Notes for the MMP modeling
Start with the symmetric "mirror charge" model.
Increase the number of matching points in the "MMP" dialog to the desired number. 10 points were used for the figures 3 and 4. As in the previous model, it is reasonable to evaluate the errors in more points in order to see the errors between the matching points.
For generating a set of monopoles along the boundary, you best take advantage of the automatic multipole setting feature. This requires the definition of a dummy monopole expansion at an arbitrary location with a negative object number. You may convert the first expansion of the "mirror charge" model into such a dummy expansion by setting its "Object #" equal to –1 in the "Expansion" dialog. After this, press the "Generate 2D…" button to open the "Generate expansions along 2D boundary" dialog. Define the number of monopoles (for example 10) you would like to create in the "Multipoles/domain, method" box. Select a unique color number (for example 2, i.e., red) for the multipoles to be generated. Specify the distance factor for the distance of the multipoles from the boundary in the "Dmin/D=" and "Dmax/D=" boxes. Since you wish to have the monopoles on the boundary, this factor should be 0, but MaX-1 inserts the default value – that is non-zero – when you do this. Therefore, you should insert a very small, but positive value, for example 1.0e-6. Press the "Generate!!" button to let MaX-1 generate a set of monopoles. Check if the desired expansions are there. If you would like to generate another set, reopen the "Generate expansions along 2D boundary" dialog and press the "Delete color" button to remove all expansions with the specified color number (for example 2, i.e., red). After this, you modify the parameters of the automatic pole setting routine and press the "Generate!!" button again.
Check the positions of the matching points by setting the "error scaling" factor 0.1 in the "MMP" dialog before you let MaX-1 draw the boundary and the expansions.
For obtaining a different position of the monopoles with respect to the matching points, you can rotate all expansions around the center of the wire (at x=0.5, y=0). You can do this with the movie directive "ROTate EXPansions". For example, when you want to rotate the expansions with the numbers 2 up to 11 around x=0.5, y=0 with an 18 degree angle, enter the directive "ROTate EXPansions 2-11 .5 0 19" in the "Movie" dialog and run it.
As soon as you have the monopoles at the desired locations, you can proceed as before in the "mirror charge" model.
Taking also the XZ symmetry into account
You can obtain a more efficient solution by taking not only the YZ symmetry but also the XZ symmetry into account. This requires the discretization of only one half of the right electrode, which is a bit more difficult than the definition of a full circle. However, it allows you to reduce the number of unknowns and the number of equations by a factor of 2, which reduces the matrix size by a factor of 4 and the computation time for solving the matrix by a factor of almost 8. Since we have a small problem with a matrix that is already small, we continue without taking the XZ symmetry into account. However for those who have not much experience, it would be a good exercise.
Method of Auxiliary Sources (MAS) solution
When monopoles, i.e., sources of the electromagnetic field are placed on the surfaces of the electrodes for approximating the "true" surface charge distribution, one obviously obtains numerical problems and inaccurate field values on the surfaces and near the surfaces. The accuracy of the electric field is even worse that the accuracy of the potential. Therefore, the method cannot be used for high voltage applications where it is important to know the maximum values of the electric field. The standard MoM way towards more accurate field computations consists in applying smoother basis functions for the approximation of the surface charge density. Although this allows one to improve the accuracy, it is important to know that quite sophisticated basis functions are required when the maximum values of the electric field shall be evaluated precisely. As mentioned before, the standard MoM subdomain basis functions (constant charge density on a segment of the surface or linear charge density distribution on a segment of the surface) are not sufficiently good for this purpose. Accurate computations require the approximation of the surface charges wit sufficiently smooth entire domain basis functions. The drawback of such basis functions is that one cannot solve the corresponding Coulomb integrals analytically in most cases and that the numerical cost for computing the Coulomb integrals is high. The most simple way out is moving the monopole charges (that were used before) away from the boundaries, to the interior of electrodes. This placement seems to be unphysical, but it is obvious that it removes the numerical problems caused by the singularity of the field at the position of the monopole charge. When one now computes the field of such an "auxiliary source" or "fictitious source" outside the electrode, one can see that it coincides with the field of a certain surface charge distribution. Moreover, this surface charge distribution is smooth and continuous, i.e., one has found an appropriate entire domain basis function in terms of MoM. The evaluation of the surface charge distribution that is simulated by a monopole charge inside the electrode is quite time consuming, but it is not necessary to perform this computation because the Coulomb integral of the surface charge distribution is already known: It is the field of the monopole charge that is known analytically! Therefore, using auxiliary sources means implicitly using smooth surface charge distributions - entire domain MoM basis functions – without the requirement of an explicit evaluation of the corresponding Coulomb integrals. As a consequence, this approach is superior and much cheaper at the same time. Note that there are several names that were introduced for this method: Method of Auxiliary Sources (MAS), Method of Fictitious Sources, Charge Simulation Method, Image Charge Method, etc. Some of these methods were also applied to dynamic problems. In fact, there are several differences in the philosophy behind these methods that also affect the way how the monopoles are placed. For example, MAS experts use concepts such as conformal mapping, continuation of the field, caustic surfaces, etc. for finding optimal or almost optimal locations of the auxiliary sources. In the following, we consider a simplified MAS, where we distribute the monopoles along some auxiliary line along the boundaries.
Although MAS is much superior to MoM form the point of view efficiency and accuracy of the field near the surfaces, it should be mentioned that its application is a bit more difficult because the monopole sources must be placed properly along the boundaries, together with the corresponding matching points on the boundaries. Although this is quite easy, it is a source of errors for careless users. Moreover, the placement of the monopoles can affect the accuracy of the solution quite much. For sophisticated users, this is an advantage because they can find good or even optimal positions of the auxiliary sources. This allows one to obtain highly accurate results with a minimal numerical effort. For our problem of two circular electrodes, we can find the optimal places of the auxiliary sources from conformal mapping. First, we map the two circular electrodes on two parallel, straight lines. The charge distributions on these lines are uniform. Such charge distributions are obviously best approximated by a set of auxiliary charges that are uniformly distributed along a line parallel to the (conformally mapped) electrodes. Moreover, one can easily see that the approximation is improved when the auxiliary charges are moved further away from the electrodes. The further away the auxiliary charges are, the bigger the distance between neighbor charges may be. Finally, the optimal solution is a single auxiliary charge for each electrode at an infinite distance form the surface. We now use the inverse conformal mapping to obtain the optimal positions of the auxiliary charges and find that these are at the positions where the "mirror charges" were, i.e., a sophisticated MAS user will obtain the mirror charge result that is extremely efficient (2 by 2 system matrix) and accurate (machine precision) at the same time.
For more complicated geometries, finding an appropriate conformal mapping may bee too time-consuming. Therefore, one will usually prefer to find sub-optimal places for the auxiliary sources. Sub-optimal placement is rather arbitrary. For example, we can use a uniform set of monopole sources at a constant distance from the surface of each electrode. This can be easily done with the automatic multipole setting procedure of MaX-1 that was also used for obtaining the simple MoM solution.
As in the MoM case, we can use one matching point for each auxiliary source (monopole) for computing the strengths of the sources. When we use uniform monopole distributions, it is reasonable to also use uniform matching point distributions. As before, we can still put the matching points at different locations with respect to the monopoles. Figure 5 shows the result for the matching points at positions with minimum distance from the monopoles (Remember that this was the bad case for the simple MoM solution!). Obviously, a drastic error reduction is obtained even when the distances of the auxiliary sources from the boundaries are relatively small.
Figure 5: MAS approximation with 10 monopoles and matching points at the boundary points with the shortest distances from the monopoles. Project: WIRES004.PRO
As one can see from the black lines perpendicular to the electrodes in Figure 5, the errors along the boundaries oscillate and are much bigger between the matching points than near the matching points. This unbalanced error distribution could be avoided by the generalized point matching technique that uses more matching points then necessary. With MaX-1, you can easily increase the number of matching points and see how the errors behave.
Instead of increasing the number of matching points, you can also rotate the locations of the monopoles as in the simple MoM case.
Of course, it is interesting to find the optimal distance of the monopoles from the surfaces. This can be done very easily with the automatic pole setting routine of MaX-1: After deleting the monopoles of the previous model (use "Delete color"), increase the values in the "Dmin/D=" and "Dmax/D=" boxes and press the "Generate!!" button again. As you can see from Figure 6, the errors are reduced when the monopoles are moved away from the boundaries.
Figure 6: Same as in Figure 5 with auxiliary sources further away from the boundary. Project: WIRES005.PRO
Finally, all monopoles are set in the centers of the circular wires. This obviously causes numerical problems. In fact, the condition number of the MAS matrix increases continuously with the distance of the auxiliary sources from the surface of the electrodes and it becomes infinite when all sources are placed exactly in the centers of the electrodes. Increasing condition numbers cause a reduced accuracy of the results obtained with the matrix solver. It is important to know that the loss of accuracy depends not only on the condition number, but also on the matrix solver. The standard matrix solvers of the MaX-1 code can handle quite ill-conditioned matrices. Therefore, you will observe the loss of numerical accuracy only when the auxiliary charges are very close to the each other (near the centers of the electrodes). Figure 7 shows that accurate results may be obtained even in almost extreme situations.
It should be mentioned that the centers of the auxiliary circles, where we placed the auxiliary sources coincide with the center of the (circular) electrode. These auxiliary circles are not optimal, as one can see from the conformal mapping mentioned above. When we map the lines parallel to the mapped (straight) electrodes back to the original planes, we obtain circles that are the coordinate lines of a bipolar coordinate system with foci at the locations of the mirror charges. When we use these "bipolar" circles as auxiliary lines for distributing the auxiliary sources, we obtain more accurate results when we correctly distribute the sources and the corresponding matching points. Note that the correct locations are obtained from conformal mapping and that the corresponding distributions are not uniform. Therefore, the "optimal" placement of the auxiliary sources is much more time-consuming than our sub-optimal, uniform placement along concentric circles. If you have enough time, don’t hesitate to find the optimal placement along a "bipolar" circle.
In the limit of a "bipolar" circle with zero radius, we have all monopoles at the same locations as the mirror charges. In this case, only one auxiliary source per electrode is required and we obtain the optimal result, i.e., the result of the mirror charge method.
Remark: when you want to compute the field of two circular wires with a total charge different from zero, the mirror charges are not sufficient. In this case, it is optimal to place the auxiliary charges along the cut line (from the focus to the center of the electrode, along the x axis).
Figure 7: Same as in Figure 5 with auxiliary sources very close to the center of the electrode. Project: WIRES006.PRO
The loss of accuracy due to the bad condition number of the MAS matrix (obtained when several auxiliary sources are very close to each other) can be avoided by replacing such clusters of monopoles by multipoles of appropriate orders. When we want to remove the cluster of 10 monopoles near the center of the electrode as shown in figure 7, we must know that each order of a multipole has 4 parameters (except order 0 that has only 2 parameters), 2 of them corresponding to the E wave (in electrodynamics) and two of them corresponding to the H wave. Both pairs contain a cosine and a sine part (indicating the angular dependence of the field around the multipole – for order 0, the sine terms are missing). For our simple electrostatic problem, we only require multipoles of the E type. Therefore, we obtain 2N+1 unknowns when we use an E multipole of the orders 0, 1, …, N. In order to replace the 10 monopoles, we therefor would require a multipole with maximum order N=4.5. Since the maximum order must be an integer, we set N=5 and obtain 11 unknowns, i.e., a slightly bigger MMP matrix. Note that you now need at least 11 matching points along the electrode, i.e., you should increase the "Min.points/segment" value in the MMP dialog from 10 to 11. This leads to the solution shown in Figure 8.
If you prefer, you can also set N=4, obtain a slightly smaller matrix and less accurate results.
Figure 8: Same as in Figure 7, auxiliary sources replaced by a multipole (orders 0 up to 5) in the center of the electrode. Project: WIRES007.PRO
When you replace the monopole cluster of the MAS with the "bipolar" circles by a multipole in the focus of the bipolar coordinate system, you place a multipole at the location where the mirror charge was placed. Since the zero order of this multipole is nothing else then a charge, you will obtain the optimal solution (machine precision) even when you reduce the maximum order of the multipole to N=0. Otherwise, you can observe that all higher orders obtain zero amplitudes (within machine precision).
Generalized Point Matching
In the previous computations, we always used as many equations as unknowns, i.e., a square MMP matrix. For obtaining such a matrix, we selected the appropriate number of matching points. Although it is quite easy to find this number, this simple point matching technique has several drawbacks. It has already been shown that the accuracy of the results depends very much on the location of the matching points in the simple MoM solution that was considered. Although this is less dramatic in the MAS case, it also holds for the MAS and for the multipole solution as well. Furthermore, you can observe in the Figures 4-8 that the error distributions "oscillate", i.e., the errors are zero in the matching point and that maximum errors are observed somewhere between the matching points. This means that the error distribution is quite unbalanced. In more complicated situations, this may cause severe problems. Finally, it can be difficult to find an appropriate set of matching points when the geometry is more complicated than in our simple example.
These problems can be overcome by using more matching points than required, for example, by doubling the number of matching points in the MMP dialog. To illustrate this, we take the simple MAS example of Figure 5 and double the number of matching points. As you can see in Figure 9, the errors are now both smaller and more balanced. Nevertheless, you can also see that the error distribution is still not very balanced, but this is due to the sub-optimal placement of the auxiliary sources. You can see, that the errors are higher to the left hand side of the right electrode. When you move the auxiliary circle (where the auxiliary sources are placed) slightly to the left, it will coincide with a "bipolar" circle!
Figure 9: Same as in Figure 7, auxiliary sources with generalized point matching, overdetermination factor 2. Project: WIRES008.PRO
Adaptive setting of auxiliary sources
Finding 1) the "bipolar" circles, 2) the correct non-uniform distribution of the auxiliary sources on such a circle, and 3) the appropriate matching points for them can be done with conformal mapping, but this is not easy and time-consuming. Moreover, this technique cannot be used in complicated situations. However, from the error distribution of a non-optimal distribution of auxiliary sources, we obtain hints how to modify the locations in order to obtain more accurate results: The auxiliary sources should be moved away from regions with relatively low errors and closer to areas with relatively high errors. Since the same holds for multipole distributions, MaX-1 contains some routines that adapt the locations of 2D multipoles (and auxiliary sources) using the error information of a previous run. Note that there are many different ways to such an adaptive setting of multipoles. Since the optimal method has not been found, several methods were implemented and can be tested. Using them requires some experience. At the moment, it is not clear whether it is more reasonable to use adaptive methods or the more simple overdiscretization. Maybe, the adaptive method is convenient for complicated situations, there the overdiscretization would be too time- and memory-consuming. Although it is much easier to increase the accuracy in our simple case using the mirror charge method, we now try the adaptive method. The following steps are required: After a first MAS computation, we save the error distribution by pressing the "Write errors…" button in the MMP dialog. Then, we read the error distribution in the function array by pressing the "Read…" button in the Function dialog. If you wish, you may let MaX-1 plot the error distribution as a function plot, but this is not necessary. Now, you open the Expansion dialog and press the "Modify 2D…" button. In the "Modify 2D multipoles" dialog that will pop up, you specify the numbers of the expansions to be modified and the "Weight function argument". Since the error distribution is stored in the first argument of the function array (see Function dialog), select the "Weight function argument" equal to 1. For a first attempt, you may leave all other parameters as they are. Press the "Adapt LOCation!!" button and let MaX-1 redraw the expansions for inspecting the new locations. When you rerun MMP, you should obtain the field and error distribution shown in Figure 10.
Figure 10: Same as in Figure 9, after adapting the locations of the auxiliary sources. Project: WIRES009.PRO
As you can see, the errors are smaller than before, but the error distribution is still not balanced. Note that more accurate results can also be obtained with a regular distribution of the auxiliary sources, simply by moving the sources away from the boundary.
Try to modify the parameters of the adaptive multipole routine in such a way that you obtain a more balanced error distribution.
Antisymmetric case, non-circular wires
In order to obtain more insight, we now consider a more complicated boundary that can still be defined analytically. Such boundaries can most easily be obtained with MaX-1 by defining a formula along a circular boundary. For example, Figure 11 shows a boundary defined as , where r0=0.3m and a=0.1m. Note that a local polar coordinate system with origin at x=0.5m, y=0 is used for defining this boundary. For defining such a boundary, define a circular boundary with center at x=0.5m, y=0 and radius r0=0.3m. After this, enter the formula sin(mul(6,c1,v)) in the Boundary dialog and specify amplitude 0.1 in the corresponding box. Finally, specify an appropriate number of spline points in the "Spline points" box. For this simple example, 30 spline points are enough. Note that MaX-1 will evaluate the specified formula only for computing the spline points. It will not use the formula for computing the locations of the matching points along the boundary. The matching points will be located along the (cubic) spline approximation of the boundary, i.e., the code will work on the approximated boundary that is only piecewise analytically defined (by cubic splines). This reduces the accuracy that might be obtained with the correct, analytic definition of the boundary. The main reasons for this approximation is that derivatives of the boundary are required in general. In general, the symbolic evaluation of the derivatives and the interpretation is complicated and time-consuming. Furthermore, one rarely has analytically defined boundaries in practical applications. Finally, spline approximations allow one to obtain faster convergence, when the derivatives of the boundary are not continuous in some points.
Figure 11: Simple MAS solution with 30 auxiliary sources. Project: WIRES010.PRO
Since the boundary is known analytically, MAS experts would be able to find an appropriate conformal mapping, optimal positions for the auxiliary sources, and so on. However, this requires much brain work and computation time is inexpensive. Therefore, we use a simple MAS version with a distribution of the auxiliary sources at a fixed distance from the boundary as in the Figures 5 and 6. Remember that MaX-1 can automatically generate such a set of monopoles. Figure 11 shows such a MAS solution with 30 monopoles in the left electrode. YZ plane symmetry has been used. Therefore, the left electrode is not modeled explicitly.
Incidentally, the same procedure might be used for obtaining a simple MoM solution with monopoles on the boundary. Since we already know that this solution is quite inaccurate, we omit it. Instead of this, we may obtain more accurate results by moving the auxiliary sources further away from the boundary. As you can see from Figure 12, many of the auxiliary sources are close to the three centers of curvature of the three boundary points with the maximum distance from the center of the wire. As in the circular case, this causes an ill-conditioned matrix. Despite of this, we obtain more accurate results than before.
Figure 12: Same as Figure 11 with auxiliary sources moved further away from the boundary. Project: WIRES011.PRO
Notes for the MMP modeling
Since we use 30 spline points for defining the boundary, MaX-1 splits the boundary in 30 segments of equal lengths. When we select "Min.points/segment" equal to 1, MaX-1 will generate at least 30 matching points. To make sure that you obtain not more than 30 matching points (you have 30 unknowns!), select a big "Max. distance" and set 0 in the "Min. points/wavelength" and "Min. Overdetermination" boxes of the MMP dialog.
When you prefer MAS with generalized point matching, you can try twice as many matching points by setting "Min.points/segment" equal to 2.
In order to avoid the numerical problems caused by the ill-conditioned matrix, we can use the same multipole expansion as in the circular case. Since we had 30 unknowns for the previous MAS solution, we now select an E type multipole with maximum order N=14 that has 2N+1=29 unknowns. In order to obtain a square MMP matrix, we should now reduce the number of spline points from 30 to 29. Since MaX-1 can handle overdetermined systems of equations, we also can leave the number of spline points unchanged and obtain Figure 13.
Figure 13: Multipole solution obtained with 29 unknowns and 30 equations. Project: WIRES012.PRO
As you can see, the simple multipole solution is highly inaccurate. This has two main reasons: 1) We have placed 30 matching points along the boundary with a constant distance between neighbor points. This is a non-optimal distribution for the multipole expansion. For this expansion, we should place the matching points in such a way that the angles between neighbor matching points seen from the origin of the multipole are equal. 2) The multipole expansion is not very well suited for the boundary we have.
Multipole solution with generalized point matching
You may overcome the problem of finding an appropriate matching point distribution by increasing the number of matching points, i.e., by using the generalized point matching technique. When you double the number of matching points, the computation time is increased by the factor 2, as illustrated in Figure 14.
Note that the error distribution is now much more balanced. From this, we can expect that a stronger overdetermination would not reduce the error considerably anymore.
However, although the error is reduced by a factor of almost 100, the accuracy is still worse than the MAS accuracy with auxiliary sources at a sufficient distance from the boundary. Instead of taking this as a good reason to prefer MAS, we introduce additional multipole expansions in the following.
Figure 14: Multipole solution obtained with 29 unknowns and 60 equations. Project: WIRES013.PRO
The distribution of auxiliary sources in Figure 12 shows 3 different clusters of sources. Since these clusters cause numerical problems, it is reasonable to replace them by three multipole expansions, i.e., by a multiple multipole expansion. This is a true MMP solution (both the simple multipole solution and the MAS solution may be considered as special cases, the first one uses M=1 multipoles and the second one uses multiple multipoles of order N=0).
For obtaining good MMP solutions, it is almost impossible to work with the simple point matching technique, i.e., the generalized point matching technique should always be used. For simple cases as here, twice as many matching points as necessary should be sufficient.
In order to obtain a computation time similar to the one of the previous examples, we first set three multipoles with maximum orders N=4, i.e., 9 unknowns per multipole or 27 unknowns in total. Since we had 30 unknowns before, we can add one more multipole with maximum order N=1 in the center of the electrode. This leads us to the solution illustrated in Figure 15.
Note that the 4 multipoles in Figure 15 were placed manually, rather than in the centers of curvature of the three matching points with the biggest distances from the center of the electrodes. These centers are important for MAS: The auxiliary line where the auxiliary sources are placed, should be constructed in such a way that these centers of curvature are surrounded. The auxiliary line should never be further away from the boundary than the radius of curvature in the closest matching point. Theoretically, the multipoles can be placed anywhere inside the electrode, but placing them near the centers of curvature is reasonable and leads to accurate results.
We have inserted a fourth multipole in the center of the electrode for obtaining 30 unknowns as in the MAS solution. However, when you delete this multipole, you can see that it is not important.
Note that the MMP accuracy that was obtained here is almost the same as the MAS accuracy with the auxiliary sources far away from the boundary. Since the multipoles replace clusters of auxiliary sources, this is not surprising.
Figure 15: MMP solution obtained with 4 multipoles, 30 unknowns, 60 matching points. The maximum orders of the three multipoles indicated by red crosses are equal. Project: WIRES014.PRO
MMP solution with adapted orders
Since we observe relatively high errors on the part of the boundary that is close to the second (left hand side) electrode that is not explicitly modeled, we can try to obtain more accurate solutions by adapting the locations of the multipoles with those routines we already used in the circular case for adapting the locations of the auxiliary sources. However, this requires quite much experience and is not easy. At the same time, we only have four multipoles and we have placed them at reasonable locations. When you remember that each multipole replaces a cluster of auxiliary sources, you will understand that moving sources to a certain position is essentially the same as increasing the multipole order at this position, i.e., instead of adapting the position, we also can adapt the maximum orders of the multipoles, which is much easier.
In the case considered here, we can increase the maximum order of the lower left multipole by 2 and decrease the maximum orders of the upper and the lower right multipoles by one. When we do this, we still have 30 unknowns and obtain the result illustrated in Figure 16.
As you can see the error distribution is now more balanced than before. The maximum error has been reduced by a factor of 3. At the same time, the average error has also been reduced by a factor of 2.
Compared with the MAS, we have more freedom in setting the multipoles, which means that we must specify not only the locations, but also the orders. Therefore, MMP is more complicated. At the same time, we need much less multipoles than auxiliary sources and adapting the orders is much easier than adapting the locations. This makes the MMP modeling easier. However, the MMP feature and the automatic multipole setting routines of MaX-1 allow you to try both MMP and MAS. In both cases, finding accurate solution with a minimal number of unknowns is a delicate topic for advanced users, whereas finding quick and dirty solutions is quite easy.
Figure 16: MMP solution obtained with 4 multipoles, 30 unknowns, 60 matching points. The maximum orders of the three multipoles indicated by red crosses are equal . Project: WIRES015.PRO