When was the last time you actually tested your finite element analysis (FEA) results? Are you sure that every FEA analysis ever ordered is accurate? Will you trust your life to it? In truth, FEA results can be wrong. In most cases, the error originates with the wrong mesh size. Mesh sizes are the biggest source of error in an FEA simulation. Especially difficult because these errors only reveal as subtle differences in the results. It takes a trained engineer and formal review to confirm an accurate FEA mesh. Today I demonstrate that mistakes in FEA mesh sizing may be larger than you anticipated.
FEA owes its popularity to the robustness of its method and general applicability to any structural analysis. Gone are the days when we derive custom equations for every single structural analysis. To achieve this leap in efficiency, FEA gave up some accuracy. The process of discretization can potentially generate major errors.
The key to FEA’s generality is discretization: the process of generating the mesh. The supporting math behind discretization requires us to give up some accuracy when we model the structural equations. The complex curves of stress patterns get replaced with a series of straight lines. (Figure 2‑1) Figure 2‑1 overly simplifies the process of FEA, but the analogy aptly explains the problem of discretization. Notice that when the blue curve changes sharply, those straight red lines show a major difference from the true curve. On the bright side, with a small enough mesh those straight lines closely match the original curves and work just fine. But only if the cells of the mesh are small enough. That is why the engineer focuses on sizing the mesh correctly to match the conditions. Without the right size, we get errors and the wrong answer.
The discretization problem gets worse when we consider structural stresses. Stresses are the main output we need from an FEA structural analysis. They are the important part. But they are not the data output produced by the FEA model. At the core, FEA programs solve for deflections: the physical distance that each node moves. The important part is how stresses relate to the fundamental deflection data. Stresses derive from the gradient of the deflection. For accurate stresses, we need to accurately calculate how fast the deflection changes across our structure. Calculating the gradient can get tricky for our discretized mesh.
The errors from discretization get even larger when we consider the gradient. (Figure 2‑2) See how the discretized slope in Figure 2‑2 shows even larger error. The gradient (red lines) only match the blue curve at the single point locations. They deviate greatly the rest of the distance. The slopes don’t even match each other at the node locations. This can happen in actual FEA programs too. In all fairness to the FEA developers, they employ additional tricks to recover some of this lost accuracy. This analogy merely demonstrates that discretization errors may seem subtle, but they generate major differences in the final result.
We agree that discretization errors can be a major problem for FEA results. But many people want to believe that these errors would grossly distort the model and make any FEA results obviously suspect. WRONG! That is the curse behind the math of structural engineering. FEA often arrives at the correct stress pattern long before it gets the correct magnitude of those peak stresses.
As humans, we focus on the pattern when we view pictures of FEA stress results. Examine Figure 3‑1 through Figure 3‑3. These show stress results for a simple plate bending model. All three results had the same forces applied, but were analyzed with different mesh densities. Just from judging these pictures, when would you consider the results acceptable?
Now compare the peak stresses from those same models. (Table 3‑1) The larger element sizes showed greater error. Even worse, they under-reported the stress values. If you design a structure from such flawed FEA results, the steel will be too thin. This leads to earlier fatigue problems and potential structural failure in storm conditions.
Small Size (%)
|Large element size||200||1.62||17%|
|Medium element size||100||1.90||3%|
|Small element size||12||1.96||0%|
We need something that allows us to compare FEA results and search for patterns; build confidence that we have the right mesh settings. Enter the science of mesh independence analysis. Mesh independence analysis offers several techniques to compare how results vary with mesh size. By comparing different mesh sizes, we can easily identify good mesh settings.
Examine Figure 4‑1. This compares the same three mesh sizes, plus several more tested sizes. The results show a definite trend. Examining this graph, we know that around 50 mm, the results change very little with mesh size. They achieved a state of mesh independence; we now have confidence in the FEA analysis. Below that critical size, further changes to the mesh do not affect the accuracy of the answer.
Each FEA engineer has their own strategy for mesh independence analysis. The most important element is that you see some evidence to demonstrate that different mesh sizes were compared. If they are very good, the FEA engineer will also use the mesh independence analysis to estimate the margin of error in the results. All FEA analysis has some margin of error. That is the curse of discretization. But these should be very small levels.
Regardless of the margin of error, mesh independence analysis allows us to detect these errors and proceed with confidence that we understand the accuracy limits for each analysis. Approximations are unavoidable. It’s the price we pay for discretization. But with proper study and review by an FEA engineer, these errors are marginal. We can tame the dangers of discretization error and confidently deliver reliable FEA results.