So now that the importance of convergence has been discussed, how can it be measured? What is a quantitative measure for convergence? One way to measure it would be to compare with analytical solutions or experimental results.

Fig. 07: Definition of errors

As shown in Fig. 07, several errors can be defined for displacement, strains, and stresses. These errors could be used for comparison, and they would need to reduce with mesh refinement. However, in a FEM mesh, the quantities are calculated at various points (nodal and Gauss). In this case, where and at how many points should the error be calculated?

Fig. 08: Error norm and comparison with the element size

Alternatively, norms are defined such that the averaged errors over the entire structure or part of the structure can be calculated. As shown in Fig. 08, the error norms can also be compared to the size of the element. Here “c” is a constant of proportionality, while “h” is the element size, as defined in Fig. 08. Hence, several errors such as L2 and energy error norms can be defined as below:

However, in real practical applications, a non-dimensional version of the same is more useful to assess the actual degree of error. Therefore, in this instance, the root-mean-square value of the norms, as defined below, is used to plot the reduction in error.

The final topic is related to the rates at which these errors ideally decrease. If we are using linear or quadratic or cubic elements, how does one judge whether the error is decreasing at the right speed or the quality of the algorithms coded? As shown in Fig. 09, the L2-norm error decreases at the rate of p+1 and energy-norm at the rate of p.

Fig. 09: Convergence rates for different error norms in finite element analysis

Conclusion

I hope this article has given a comprehensive overview of convergence, convergence rates, and ways to accurately judge these aspects in a finite element analysis. Convergence plays an important role in the accuracy of solutions obtained using numerical techniques like FEA, and thus needs to be comprehensively analyzed in any given problem.

Learn how to get easy access to cloud-based tools for CAD and simulation by watching the recording of the webinar “How to Optimize Medical Devices with Cloud-Based Simulation” in partnership with Onshape. All you need to do is fill out this short form and it will play