Approximation Technique
5.2 In the Case of Grid-Shape Asymmetry
In general, the spacecraft we intend to place an ion engine on is not cylindrically symmetric. Since there will be asymmetries in the shape of the craft, we can expect there will be certain regions around the engine that are more sensitive to sputtering than others. As a result, an optimum engine grid shape could, in all likelihood, be asymmetric also. Unlike the axisymmetric NSTAR grid shape, which has a corresponding axisymmetric plume potential (Figure 4.7), the plume obtained using the procedure outlined in the previous chapter will be asymmetric for any arbitrary grid shape. While the procedure for integrating the equations of motion change very little between a 2D-axisymmetric and 3D-asymmetric plume, obtaining a fully three-dimensional plume of sufficient resolution could be impractical; to complete an optimization calculation, a new plume is required to be computed at each new grid configuration from each iteration. It would be preferable if an approximating method could be found such that the need for finding the complete three dimensional potential is eliminated, but that still gives sufficiently accurate trajectories and results.
It was hypothesized that approximating the potential as axisymmetric, despite the asymmetry of the grid shape, could yield sufficiently correct results. A sensible candidate for the two-dimensional potential assumed to be axisymmetric would be the potential in the radial plane that the target point falls on. This hypothesis was tested using a highly asymmetric grid shape considered to be at the engineering limits of the asymmetry possible to construct. The shape of the grid was that of a
“barrel vault” and is shown in Figure 5.7. The radius of curvature in the second principal axis is that of the NSTAR — 50.8 cm. The two-dimensional potentials occuring in the two radial planes along each of the principal axes are shown in Figure 5.8.
The contribution to the CEX-ion energy distribution at the target point (60,0,3) was computed for this grid shape using both a full asymmetric-3D plume potential as well as the axisymmetric
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Figure 5.7: Barrel-vault grid shape. Shape chosen to represent an asymmetric grid. The radius of curvature in the direction of the second axis is the same as for NSTAR — 50.8 cm. For this example, Axis 1 corresponds to thex-axis. The target point is again located atS(˜x) = (60,0,3).
extension of the 2D potential that lies in the same plane as the target point and the grid axis (x−z plane). For this comparison, thex-axis corresponded to the first principal axis and they-axis corre- sponded to the second principal axis. For both cases, both the full-trajectory as well as the energy- and angle-corrected LOS methods were used. The resulting contributions from three different beam- lets are shown in Figures 5.9 through 5.11. The estimated contributions from the 2D approximation is shown for both the full-trajectory analysis assuming a 2D potential, as well as for the LOS energy- and angle-corrected approximation. The contributions predicted using all three methods are similar for the first two beamlets. The approximations for the first beamlet under-estimate the sputtering rate contribution predicted using a full asymmetric plume by approximately 20% and 27%, for the 2D full-trajectory analysis and the LOS energy- and angle-corrected approximations, respectively.
The maximum energy predicted by both the 2D and 3D plumes are approximately the same, at 72 eV and 75 eV, respectively.
For the second beamlet, the 2D full-trajectory approximation over-estimates the sputter rate contribution by approximately 11%, and the 2D energy- and angle-corrected approximation under- estimates the contribution by approximately 9%. The maximum energy predicted by both the 2D
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Figure 5.8: Barrel-vault plume potentials. Two examples of the aymmetry of the plume due to the shape of the barrel-vault grid. Top figure: Plume potential of Axis 1 from Figure 5.7. Bottom figure: Plume potential of Axis 2 from Figure 5.7.
and 3D plumes were identical at 69 eV. Note that though the hole from which this beamlet originates is located on the plane from which the 2D plume was obtained, the contributions are not the same.
This is a result of the fact that the scattering centers located within the beamlet itself are not restricted to this same plane.
The 2D approximation was inadequate for estimating the contribution from the third beamlet shown in Figure 5.11. Though the 2D energy- and angle-corrected approximation yielded a result similar to the full-trajectory analysis using the 2D plume, both of these methods under-estimated the contribution by more than 75%. The maximum energy estimated by the 2D approximation was 45 eV, 10 eV less than the maximum energy predicted using the full 3D plume.
Though this analysis indicates that approximating an asymmetric plume by the symmetric ex- tension of the 2D plume corresponding to the plane in which the target point is located is inaccurate, it must be considered that the total predicted sputtering rate contribution from the third beamlet is two orders of magnitude less than that from the first beamlet. Though the 2D approximation yields inaccurate values for the third beamlet, the error contribution is small. Though the 2D approxima- tion is not an ideal approximation to make, we proceeded with using this approximation for ease of implementing the model into a grid shape optimization which we discuss further in Chapter 7.
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109 Figure5.9:Full-beamlet3Dapproximationcomparison:Hole(x,y)k=(-0.111,5.96).Leftfigureisthespatiallyintegratedenergydistribution (leftaxis)andenergy-differentiatedsputteringcontribution(rightaxis).Rightfigureistheenergy-integratedsputteringcontribution(leftaxis)and cumulativeintegraloverbeamlet-shellradiusofthesputteringcontribution(rightaxis).Thesolidlineindicatesthecorrectcomputationusingthe full3Dpotentialtocomputetrajectories.Thedashedlineisthecorrectcomputationusingthe2D-potentialassumption.Thedash-dotlineisthe energy-andangle-correctedLOSassumptionusinga2Dpotential.
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109 Figure5.10:Full-beamlet3Dapproximationcomparison:Hole(x,y)k=(5.994,0).Leftfigureisthespatiallyintegratedenergydistribution(left axis)andenergy-differentiatedsputteringcontribution(rightaxis).Rightfigureistheenergy-integratedsputteringcontribution(leftaxis)and cumulativeintegraloverbeamlet-shellradiusofthesputteringcontribution(rightaxis).Thesolidlineindicatesthecorrectcomputationusingthe full3Dpotentialtocomputetrajectories.Thedashedlineisthecorrectcomputationusingthe2D-potentialassumption.Thedash-dotlineisthe energy-andangle-correctedLOSassumptionusinga2Dpotential.
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109 Figure5.11:Full-beamlet3Dapproximationcomparison:Hole(x,y)k=(0,11.92).Leftfigureisthespatiallyintegratedenergydistribution(left axis)andenergy-differentiatedsputteringcontribution(rightaxis).Rightfigureistheenergy-integratedsputteringcontribution(leftaxis)and cumulativeintegraloverbeamlet-shellradiusofthesputteringcontribution(rightaxis).Thesolidlineindicatesthecorrectcomputationusingthe full3Dpotentialtocomputetrajectories.Thedashedlineisthecorrectcomputationusingthe2D-potentialassumption.Thedash-dotlineisthe energy-andangle-correctedLOSassumptionusinga2Dpotential.