What is a Singularity?
A singularity is a function’s divergence into infinity. Simulation occasionally produces stress (or heat flux) singularities.
How do they occur? Mathematically, the solver uses matrices to represent the elastic field (displacements of the elements). When a highly localized load is applied, the gradients of the displacement vectors begin to diverge, causing the roots of the matrices to go to infinity. For a simplified explanation, see the stress equation below. Stress goes to infinity due to force applied in a very small area.
Where do Singularities occur?
Singularities are usually seen at points, edges, or reentrant corners. Reentrant corners are interior corners, with angles pointing into the part. The high stress concentrations are usually seen near 90 degree corners, but can potentially occur for any angle less than 180 degrees.
Why don’t they occur in real life?
Think of the common case of singularities created on interior corners. In software, that corner is perfectly sharp. In real life, there will always be a slight bend. Also, the part may deform slightly, or “slip”, and allow the faces of the corner to slide against each other. The slight bend and additional friction allow for a converging stress.
“Adjust your legend’s color settings to grey out above the material’s yield point. This prevents singularities from overshadowing other important stress results!” – Joe Engineer, Know It All, GoEngineer
Is it a Singularity?
There are two methods to determine whether a stress concentration is a singularity.
Method for SOLIDWORKS 2016 and earlier:
First, apply sensors in high stress locations. Secondly, activate trend tracker to monitor those sensors. Third, use increasingly fine mesh controls in the high stress regions. As mesh fineness increases in the high stress fields, the stress sensors will either converge or diverge, viewable in your trend tracker graph.
- Have you not used Trend Tracker before? It is explained at the end of this webinar.
Method for SOLIDWORKS 2017 and later:
Create a Stress Hot Spot Diagnostics graph. This plot shows the variation of von Mises stresses. Areas with high, rapidly changing von Mises stresses, indicative of a singularity, will be highlighted in grey.
Want to see how to create these hot spot plots? Watch our What’s New in 2017 video here.
How to Avoid?
You now understand singularities and have determined you have a singularity in your part. If you want to fix it, you must adjust your part or adjust the settings in your study.
- Adjust Geometry
Locate reentrant corners where singularities most commonly occur. Since the force transfer trying to go through that edge is causing the singularity, provide a larger surface area in order to distribute the loads. Fillets or chamfers are commonly used. The pictures below demonstrate this concept.
Figure 1. Re-entrant corner vs. Fillet
Don’t let the fear of singularities prevent you from defeaturing your models. Remember, sometimes fillets are useful, but they always cause a finer mesh and longer calculation times.
If non-filleted re-entrant corners are unavoidable in your part, design nearby support to prevent crack growth. This support must dissipate the load over a larger surface area before allowing the load to affect the reentrant corner.
- Adjust Fixtures
A fixture applied at a point or edge can cause a singularity. To understand why, consider a simple cantilever beam problem. The force being applied on one end of the beam must be counteracted by the fixed end. If the element along the edge of the beam is forced to remain rigid while simultaneously counteracting the force, a singularity is created. In this situation, applying the fixture to the face rather than the edge will prevent divergence.
If fixing the face is not applicable, include the component from the assembly that is creating the fixture. A fraction of the component may also be used as long as it is large enough to allow for complete load transfer. Only replace components with fixtures if the component is much stiffer than the study’s part, or if the fixtures are far removed from important results.
Realize that some connectors act as fixtures too. Check if your connector is assuming infinite stiffness by accessing the Solidworks Help menu. If so, adjust your connectors or include the modeled part in the study. The assembly shown below is connected by a tight-fit bolt connector. The tight fit allows the shank to deform, but the nut is still infinitely stiff. The actual bolt needs to be modeled.
Figure 3. Singularity due to Connector.
- Adjust Loads
Don’t apply force or a heat source at points or small edges. This causes a sharp inflection point in the data input into the deflection equations, which leads to the singularity. If a thin loading area is required, create split lines and apply force to the area between them, as in the picture below. This will allow convergence.
- Adjust Mesh
Weak singularities may form at the interface of two bonded materials in a thermal study. This is due to the more rigid nature of a bonded element. If necessary, provide a thermal resistance along the interface—this will detract from the implied “perfect” bond.
In nonlinear studies where parts are yielding, a single element’s calculations may collapse if it is too different relative to its neighbors. Fix this by either reducing the aspect ratio in the standard global mesh or applying a mesh control.
How can I ignore the singularity?
If you have confirmed the singularity, you know it is due to idealization, or the stress is far removed from the important areas of the model, it is allowable to ignore it.
If you are concerned about the stress near the singularity, use an Energy Norm Error plot. Singularities and high stress change areas will produce high energy norm errors.
- Refine your mesh in these areas until you have reduced the error region.
- Create sensors very near the error region.
- Confirm those sensors are converging on a stress result.
More practically, the stress results near the error region will likely realistically represent the stress in the singularity regions.
Singularities are idealization errors that you can avoid by understanding the assumptions made within the software. First confirm singularities using a hot spots plot, or a trend tracker graph with increasingly finer mesh. Then adjust your geometry, connections, fixtures, loads, or mesh depending on the location of the singularity. If that does not help, use the Energy Norm Error plot and refined mesh to determine what results can be trusted.