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Important Mesh Theory to Accurately Run your Finite Element Analysis Simulations

Understanding fundamental theory of Finite Element Analysis (FEA) is critical to getting accurate results in your simulation. This article is going to focus on two primary aspects of this: Mesh Convergence and Stress Singularities. Deeply understanding these two theory topics can greatly increase your ability to get accurate results in simulation.   

Mesh Convergence

The accuracy of your simulation stress results is dependent on the density of the finite element meshA very coarse mesh has insufficient data points to accurately measure the maximum stress value and it will often under-represent the stress because of how the calculations are performed.  Mesh convergence means the mesh has been refined enough to make the results independent of the mesh, which stated simply means that the result is accurate to the real-world scenario. Compare the four models in the picture below. The only difference in these simulation setups is the mesh density in each setup is different. If you compare the max stress (primary value used to predict if the part will break) in each scenario, you will see that the max stress increases quite a bit and then levels off to the true stress value as you increase the mesh density. 

Important Mesh Theory to Accurately Run your Finite Element Analysis Simulations

With mesh density having a significant impact on the accuracy of the results, it raises the question why don’t analysts always just use a very fine mesh to get accurate results? The reason why this isn’t done is that the more mesh elements you have, the more equations the solver has to calculate and therefore increases your time waiting for results and you might even run out of computer resources! It could be very easy to create a mesh that takes several hours to solve that you could have gotten the same accuracy of results with a coarser mesh that takes a minute or two to solve. Refer to the graphic below for example data of mesh density (Number of elements) compared to stress results and time to solve. On the x axis is the number of elements and the blue line corresponds to the predicted stress in the model. As you increase the number of elements, you see that the stress converges to a value. The black line corresponds to comparing the number of elements to the time it takes to solve. You can see that having lots of elements can significantly increase the number of time without increasing the accuracy. Please note that the data is just used for illustration purposes. 

Important Mesh Theory to Accurately Run your Finite Element Analysis Simulations

There is a sweet spot that you want to have for your mesh. See the annotations below on the same graph to compare where you want to be at on the graph. For a last pass analysis, you want to be in the sweet spot of accurate results and low solve time but having a very coarse mesh can serve a purpose to verify your setup as well. 

Important Mesh Theory to Accurately Run your Finite Element Analysis Simulations

Stress Singularities

It is important to understand what a stress singularity is and what it is doingIn a simply stated way, there are certain conditions that the stress value will never converge to a single value as you increase the mesh density. This can happen most often on sharp internal corners. The main reasoning for this is that the virtual model has infinitely sharp corners but in reality nothing is infinitely sharp and has some curvature to it. The units for stress is a Force divided by an area (eg. Lbs/ in^2). If the area is going to zero (Due to an infinitely sharp corner) then as you increase the mesh density the stress will go to infinity. One thing to note is that your displacement values are not affected by singularities like this since there is no “area” in the units. When you experience this situation, there are two primary paths that you can take: 

  1. Choose to ignore the stress values in this stress singularity area if you can confirm that the loading will not cause a failure in that region before it fails in a location where the stresses are converged. 
  2. Add a small fillet to the area of concern which will make the stress area non-zero and should force the mesh to resolve the corner in a realistic way.  

Stress singularities are a mathematical issue that can apply to any finite element analysis study and do not accurately represent the realistic behavior of the modelUnderstanding this can help you identify why the maximum stress value in your study will not converge normally and what to do about it. 

Practicing and coming to understand these two aspects of finite element analysis will help you obtain accurate results for your simulation. If you want to learn more about being able to run accurate simulations check out our SOLIDWORKS Simulation Course. 

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