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How do I know the convergence of mesh size in my model?
Is there a method to know if the mesh has converged?
Hi Hemant,
Mesh density and solution convergence are closely related and the factors which determine that relationship can be controversial. In an effort to meet specific time and accuracy requirements, trade-offs involving modeling time, accuracy, computation time, and cost must be made.
The correct mesh, from a numerical accuracy standpoint, is one that yields no significant differences in the results when a mesh refinement is introduced. Mesh refinements must accurately represent the problem in question if they are to be used in the analysis. Mesh refinements by simple splitting of elements can be misleading unless the newly created nodes conform closely to the original geometry. As refinement progresses, the original element selection must retain its significance. For example, a shell model can be refined to the point that it loses its validity in the area of interest, creating a need for a solid element model.
Please go through our free ebook on Practical Aspects of Finite Element Simulation (a Study Guide) which can be downloaded form http://www.altairuniversity.com/free-ebooks-2/free-ebook-practical-aspects-of-finite-element-simulation-a-study-guide/. This contains all the basic theoretical concepts of Finite Element Analysis.
For evaluating 'Convergence' you have to perform several analysis by refining element size.
Warning: not all refinement could give you 'convergence' goal.
Hi there,
I've seen this question few times here; it shows that this is an important topic that should be reinforced across the community.
It’s important to distinguish between convergence and accuracy, two concepts that are often confused in FEA modeling.
Convergence refers to the numerical stability of your solution. It is governed by the truncation error, which measures how closely your discrete model approximates the underlying mathematical equations. Long story short, when running an FEA simulation, the solver iteratively refines the solution until the difference between successive iterations falls below a specified (by the analyst) numerical tolerance, this is where truncation error serves as a threshold.
However, converging a solution doesn't guarantee that the solution represents properly the reality and nature of the phenomena that it’s been analyzed. You may achieve convergence while still producing a model that misrepresents the natural behavior of what it is observed in the reality. This is where accuracy becomes critical.
To evaluate the accuracy of your FEA model, a mesh sensitivity study is essential. This involves running the same simulation with varying mesh densities, commonly categorized as coarse, fine, and very fine, while keeping all other modeling parameters constant.
If your key output variables (e.g., temperature, flow velocity, stress, displacement, voltage) exhibit little to no change as the mesh becomes finer, it’s a strong indication that your model has been well resolved. On the other hand, if significant differences remain between the fine and very fine meshes, this suggests that the existing mesh is still too coarse to capture the physical phenomena with accuracy.
In such cases, further mesh refinement or adjustments to the model setup (e.g., boundary conditions, element types, or solver settings) may be necessary to improve both resolution and confidence in your results.
For anyone beginning to formalize their understanding of FEM, I highly recommend the following two references. They are both accessible and grounded in the fundamental philosophy of the method, making it easier to connect the mathematics with the underlying physics:
Sadiku, M. N. O.A Simple Introduction to Finite Element Analysis of Electromagnetic Problems.IEEE Transactions on Education, 32(2), pp. 85–93, 1989.DOI: 10.1109/13.30767: This is a concise and intuitive introduction tailored for electromagnetic applications, but its clarity makes it valuable for understanding general FEM concepts as well.
Lewis, R. W., Nithiarasu, P., & Seetharamu, K. N.The Finite Element Method in Heat Transfer Analysis.John Wiley & Sons, 2004. ISBN: 978-0471908770: This book offers a solid mathematical and physical foundation, especially for problems involving heat transfer, but its structured approach can be generalized to other domains like fluid flow, mechanics, and more.
Both works do an excellent job of linking the mathematics of FEM to its physical meaning, which is essential when you're trying to understand fields like electromagnetics, heat transfer, mechanics, fluid dynamics, or even gravitational modeling.
Hope this comment helps!
