Hi Kaustubh,

Mesh convergence study generally ensures that the computed results converge to the actual solution whether the problem in hand in linear or nonlinear. Simple governing equations have close-form (analytical) solutions. But practical problems generally don’t have analytical solution. Or let us say even if they have, it is next to impossible to have them on paper. That’s why we need to solve them numerically.

All the problems have solution (even though they may not have analytical solution). In CFD or generally in simulation if you use very fine mesh you will have a solution that will be close to its true analytical solution (which only God can derive). But such a computation will generally (especially for large practical 3D problems) be prohibitively expensive computationally. We therefore need a solution which is close to ‘true solution’ but affordable.

That’s where the concept of mesh convergence has a role to play. Start from a relatively coarse mesh and solve your problem on it. You will get a solution which won’t be wrong but will be highly inaccurate. Next, solve the problem on a refined mesh initialized by coarse-mesh-solution (not necessary but it will reduce the amount of computation). Now you get a solution which is more accurate. Compute the %age variation of the current solution from the previous coarse-mesh-solution. Keep repeating this process until your current-mesh-solution varies by less than say 5% from previous coarse-mesh-solution. If that happens, you achieve ‘Mesh Convergence’ and you can be confident about your results.

FE solution gives the approximate results. If solution is useful if and only if it is close to exact solution. So try to built the FE model (mesh) which is independent of mesh size and number. For this it should required mesh convergence study. If you see any problem on simulation mesh convergence study is done before making final results.

In a nonlinear analysis when we get a converged solution for certain mesh size, if we reduce the size of mesh it affects solution convergence. on the other hand in a linear analysis the solution will converge regardless of mesh size. So how can we perform a mesh convergence study in a nonlinear analysis?

In my view, senstivity study on mesh size should be done no matter what type of simulation is. Richardson extrapolation method is recommended to estimate the discretization error. For a nonlinear analysis, maybe very fine mesh is needed to obtain mesh-independent convergent result.  At least, we know where we are and balance the efficiency and accuracy.

Hi Kaustubh,

Mesh convergence study generally ensures that the computed results converge to the actual solution whether the problem in hand in linear or nonlinear. Simple governing equations have close-form (analytical) solutions. But practical problems generally don’t have analytical solution. Or let us say even if they have, it is next to impossible to have them on paper. That’s why we need to solve them numerically.

All the problems have solution (even though they may not have analytical solution). In CFD or generally in simulation if you use very fine mesh you will have a solution that will be close to its true analytical solution (which only God can derive). But such a computation will generally (especially for large practical 3D problems) be prohibitively expensive computationally. We therefore need a solution which is close to ‘true solution’ but affordable.

That’s where the concept of mesh convergence has a role to play. Start from a relatively coarse mesh and solve your problem on it. You will get a solution which won’t be wrong but will be highly inaccurate. Next, solve the problem on a refined mesh initialized by coarse-mesh-solution (not necessary but it will reduce the amount of computation). Now you get a solution which is more accurate. Compute the %age variation of the current solution from the previous coarse-mesh-solution. Keep repeating this process until your current-mesh-solution varies by less than say 5% from previous coarse-mesh-solution. If that happens, you achieve ‘Mesh Convergence’ and you can be confident about your results.

Hello,

I went through various tests on meshing a rotor blade in my work published in Aerospace Science and Technology ( 2013, Truong et al.) There are two tests to be done:

(i) Choice of finite elements: depending upon your problem, if it is a 3-D model, there are linear

cubic elements ( 8 nodes) and nonlinear cubic elements. MSC Software suggests the use of linear cubic

elements if the model has rapid geometrical variation.

(ii) Convergence mesh: at least 3 models with 2 orders of magnitude for the number of elements. As it

is difficult to treat an important model, I suggest to use roughly 1:10:50 the ratio of number of

elements for the model. In practice, start from refined model and increase the coarsity of the model.

From the various models established, it is possible to look at the results convergence: if the

model with mid-range meshing definition doesn't converge to a few percent to the most refined model,

then the mesh is still too coarse.

Hope that these suggestions help.

Regards,

Rajeev KUMAR, Kaustubh Kulkarni

"Mesh convergence study has nothing to do with whether the problem in hand in linear or nonlinear. " This is not correct!

There are cases that "mesh convergence study" results depends on [time dependent and/or] non-linearity of the problem. Examples here:

Turbulence LES modeling (C. Roy- W. Oberkamps have nice discussion on that), Problems of Structural Vibration, Shallow Water Wave on wet and dry boundary, Internal waves in solids etc. See this paper for more examples:

QUANTIFICATION OF UNCERTAINTY IN COMPUTATIONAL FLUID DYNAMICS, By Patrick Roache, 1997, Annual Reviews of Fluid Mechanics.

Also this exact discussion is explicitly mentioned in the new AIAA standard of code verification in CFD (Not Published Yet, will be out by the end of this year, I was a external reviewer).

In all of those cases you can NOT perform mesh convergence study as nonlinearities, or mesh dependency of state equations, or small excitements by mesh dependence will change the problem path of solution. You still can do kind of mesh convergence study but you have to do it on the integrated values, instead of main state variables. For example: you cannot do mesh convergence study of LES turbulence solver in unsteady flow, but you can find the turbulence drag on a given airfoil in three different meshes and see if the drag force calculation is converging or diverging. This is mathematically means you perform couple of smoothing on your solution and use more forgiving norm of the result matrix.

Hope this helps you,

Kaveh

Kaveh Zamani, Kaustubh Kulkarni

I would like to thank Kaveh Zamani for raising this important issue. I certainly overlooked very specialized and highly non-linear cases you pointed out. While the main point of my writing was to drive home the point that irrespective of linear or nonlinear case, in the absence of analytical solutions, mesh convergence lends a kind of credibility or legitimacy to the computed results, however, I agree that my statement " Mesh convergence study has nothing to do with whether the problem in hand in linear or nonlinear." was too sweeping a statement. I will edit that part of my answer. Kaveh, thanks also for sharing some pretty useful references.

(Maybe a bit late to respond but I think an important point should also be added to the other comments.)

Answer. No, implementing a mesh convergence study is not always practically possible. A famous example is Locally dependent damage models. Indeed, in these cases, by changing the mesh density, you might get a different result, and the nonlinearities would even compound the situation. Generally speaking, such methods should be avoided, if possible, to reduce the mesh dependent errors.