A typical engineering design involves the prediction of deflections/displacements, stresses, natural frequencies, temperature distributions, etc. These parameters are used to iterate on material parameters and/or geometry to optimize their behavior. Traditional methods, like hand calculations, involved idealization of physical models using simple equations to obtain solutions. However, these approximations oversimplify the problem, and an analytical solution can only provide conservative estimates. Alternatively, FEM and other numerical methods are meant to provide an engineering analysis that takes into account much greater detail—something that would be impractical with hand calculations. FEM divides the body into smaller pieces, enforcing continuity of displacements along these element boundaries. More information on “how FEM works” and “how to learn FEM” can be found in the respective SimScale articles.
What is Convergence in Finite Element Analysis (FEA)?
For those using finite element analysis, the term “convergence” is often used. Most linear problems do not need an iterative solution procedure. Mesh convergence is an important issue that needs to be addressed. Additionally, in nonlinear problems, convergence in the iteration procedure also needs to be considered. So, what does this mean? In this article, we investigate and address issues related to this term.
Mesh Convergence: h- and p-refinement in Finite Element Analysis
One of the most overlooked issues in computational mechanics that affects accuracy is mesh convergence. This is related to how small the elements need to be to ensure that the results of the finite element analysis are not affected by changing the size of the mesh.
Fig. 01: Convergence of quantity with an increase in degrees of freedom
As shown in Fig. 01, it is critical to first identify the quantity of interest. At least three points need to be considered, and as the mesh density increases, the quantity of interest starts to converge to a particular value. If two subsequent mesh refinements do not change the result substantially, then one can assume that the result has converged.
Fig. 02: Mesh refinement of a structure
Going into the question of mesh refinement, it is not always necessary for the mesh in the entire model to be refined. Saint-Venant’s Principle enforces that the local stresses in one region do not affect the stresses elsewhere. Hence, from a physical point of view, the model can be refined only in particular regions of interest and further have a transition zone from a coarse to a fine mesh. There are two types of refinements (h- and p-refinement), as shown in Fig. 02. H-refinement relates to the reduction in the element sizes, while p-refinement relates to increasing the order of the element.
However, it is important to distinguish between the geometric effect and mesh convergence. Particularly when meshing a curved surface using straight (or linear) elements, which will require more elements (or otherwise mesh refinement) to capture the boundary exactly. As shown in Fig. 03, mesh refinement leads to a significant reduction in errors.
Fig. 03: Reduction in error with h-refinement of the curved surface
Such a refinement can allow an increase in the convergence of solutions without increasing the size of the overall problem being solved.
FEAConvergence in Presence of Singularities
After reading the above section, it feels safe to assume that once the stress converges in a particular part of the structure, using the same element size elsewhere should lead to converged solutions. However, this is not a valid assumption.
Most models have corners, both internal and external, where the radius is assumed to be zero. This is also the case in the presence of cracks. In these instances, the stresses are theoretically infinite. Now can you guess why airplane windows do not have corners but are rounded at the edges?
Fig. 04 Stress singularity
In the presence of the singularity, the mesh needs to be refined around it. However, as shown in Fig. 04, the more the mesh is refined, the more the stress continues to increase and tend towards infinity.
Hence, in the presence of fillets, it is generally more reasonable to assume an actual radius and then refine the region using a sufficient number of elements. For more details on mesh singularities, we recommend our recent article on the SimScale blog titled “Mesh Size Influence on Mechanical Stress Concentration”.
