Mesh Sensitivity

Finite Element Mesh Variation as Quality Assurance in Space Charging

EMA3D® has tools which, when used in combination with Nascap-2K, provide the user a comprehensive and sophisticated space charging analysis framework.  One of the key capabilities that EMA3D provides is the ability to control the finite element mesh which serves as the basis for a Nascap-2K simulation object.

Control over the mesh within the EMA3D set of tools includes control over both the meshing algorithm and the mesh resolution.  Both of these aspects of the mesh are important to an analysis program.  Different mesh algorithms may represent different areas of the vehicle more accurately.  Some are stronger at maintaining the curvature of surfaces, while others are better at providing uniformity in the mesh gradient.

A finer mesh resolution generally provides more accurate results than a coarser mesh, however, a finer mesh takes longer to simulate.  In Nascap-2K the simulation time scales as the number of elements squared.  It is then desirable to find the coarsest mesh resolution which is still numerically accurate.

By using different simulation mesh algorithms and resolutions, the user can get a sense of the numerical accuracy and stability of their simulation results.  We present a flow chart that represents one possible strategy for using multiple meshes as the foundation of an accurate and stable simulation program:

In this post, we flesh out some of the details of our flow chart and show how using multiple meshes can provide insight and confidence in the numerical accuracy of simulation results.

We show our simulation model in the figure below, where we have chosen three different mesh algorithms with roughly the same resolution to begin our sensitivity analysis with.  In the EMA3D-Internal, the user starts with a geometrical model, assigns materials, and then meshes the geometry which gets directly exported as a Nascap-2K object.  The user can easily generate new meshes using a convenient mesh wizard.  New objects can be exported quickly, with the appropriate materials already assigned.

Our model shown in the Figure is a simple model with just four materials: solar panels (cyan), elimstat paint (magenta), windows (blue) and FRSI thermal material (red).  The three different meshes are generated by three different algorithms that provide different combinations of curvature sensitivity and uniformity.

We simulate the model out to 10,000 seconds in a geosynchronous ‘worst case’ plasma.  We consider both a shaded (eclipse) environment and one in which there is illumination on the back side of the vehicle (away from windows and FRSI).

We start by looking at the results for the shaded scenario across these three algorithms.  We will look at sensitivity to the mesh resolution further down.  Our results are shown in the Figure below, where we plot the absolute maximum voltage to plasma on the vehicle minus the absolute minimum voltage to plasma on the vehicle, all scaled relative to the DELM mesh result.  By plotting in this way, we have an easy comparison across meshes for a relatively intuitive physical quantity.

When looking the plot, a couple of observations are in order.  First, the green curve corresponding to the DELC mesh (curvature sensitive mesh) shows a discrepancy relative to the other two meshes.  Second the green curve also appears to have some numerical fluctuations early in the simulation.  Taken together, our initial impression is that the DELC mesh is less accurate than the DELM and DELT meshes.  Our initial impression is also that these two meshes, DELM and DELT, are likely accurate since they are stable and agree with each other. However, in order to confirm these initial impressions, we look at the 3D plot of the voltage to plasma at the end of the simulation, which we show in the following Figure.

The Figure shows a color representation of the induced voltages at the final moment of simulation.  We have included both the DELM and DELC meshes in the plot.  When looking at the 3D plot, we see that the DELC mesh has trouble capturing the spatial gradient on the solar arrays.  The results are choppy, especially compared to the smooth variation seen in the DELM mesh.  This confirms our initial impression that the DELC mesh results should be viewed with skepticism for this scenario.

We now consider a variation in the resolution of the mesh.  We have limited our analysis here to variations in the DELM mesh resolution.  The figure shows results in the same format as above for three different resolutions – labeled Fine, Medium and Coarse.  The number of mesh elements for each mesh is shown in the parenthesis next to the plot legend.

When looking at variations in the mesh algorithm there is always some uncertainty as to which mesh should represent ‘baseline’.  However, when looking at variations in the mesh resolution, we can usually safely assume that the finer mesh is more reliable.  Looking at the results in the Figure we see that the Coarse mesh (776 elements) shows clear discrepancy with the Fine mesh (3622 elements), but that the Medium mesh (2050 elements) agrees relatively well with the Fine mesh.  In this case, the user may make a judgment based on the need for simulation speed versus the importance of numerical precision whether the Fine or Medium mesh is more suitable for their needs.  It is also possible that another mesh, perhaps with 2500 elements, would be a good compromise.

We now consider the results across the three algorithms for the scenario with illumination, which are shown in the Figure.  It is interesting to note in these results how much the DELC mesh results fluctuate relative to the other two meshes early in the simulation, but then they all converge toward later times.

While it is reassuring to see consistency across the meshes in the ‘steady state’ of the results, the early fluctuations seen in the DELC mesh again indicate this mesh should be viewed with skepticism.  Although we have only plotted results relative to the DELM results as a ratio, the user can also plot the minimum and maximum voltages for each mesh separately.  Doing that would highlight clearly that the DELC mesh shows numerical instability at early times.

In this post, we have presented a strategy for using multiple meshes as the foundation of an accurate and stable space charging simulation program.  Our strategy takes advantage of the mesh and material capabilities present within the EMA3D/Nascap-2K interface.  Within this interface, the user can quickly generate simulation objects from different mesh algorithms and resolutions.  We have seen that numerical results can fluctuate between these different meshes and that by careful analysis the user can start to pinpoint which meshes are reliable and how fine a mesh resolution is required.  These capabilities help lay the groundwork for an accurate and reliable space charging analysis program.