To add on Demetris comments. When you are dealing with nonlinearities, the finer the mesh the larger your stiffness matrix gets therefore your solution might lead to divergence due to the derived unbalanced forces induced to your system during the analysis.

When you have a coarse mesh then you might get a singular matrix that is ill conditioned thus again face divergence issues given that your stiffness matrix gets deteriorated faster (you are unable to capture phenomena such as gradual plastification of your structural members).

Also share if anybody facing problems similar to the given question

What do you mean by "not working" ?

When refining your mesh the solution should converge (hopefully to the expected value), so if your "problem is not working", does it mean that it doesn't converge ? Or does it mean that it converges to a solution different from the one you expect ?

First of all your question is not clear but i guess you were willing to discuss about first cell height and its dependence on various mesh sizes and solutions but still i requested re-post your doubt in a specific way so i am able to discuss with you.

As said earlier in this conversation, you really need to be more specific about "not working" . (Convergence? Bad results compared some well known solutions? compared to experimental data? ). And also, give us the details of you application.

But indeed, there is some application where a finer mesh can bring some problems. I am working on a lagragian application on the simulation of diesel fuel spray. A finer mesh would be better for the Eulerian phase. However, an hypothesis of the Lagrangian method stipulates that the liquid phase can not be higher that 70% of the cell volume. So given the density of the spray, the mesh should not be finer...

It is still used and perform well and you can work with it. You just need to be aware of this and keep that in mind when analyzing your results.

It is a matter of the so called 'epsilon of the machine'. I repeat my answer of a relevant question about Matlab:

*****

The epsilon of the machine (short: eps) is the minimum distance that a floating point arithmetic program like Matlab can recognize between two numbers x and y.

Try this:

>> format long e

>> x=1;y=x+eps;

>> y-x

ans =

2.220446049250313e-016

>> x=1;y=x+eps/2;

>> y-x

ans =

0

You see that y-x=0 and Matlab cannot recognise a difference less than eps:

>> eps

ans =

2.220446049250313e-016

*******

So if you have a grid with distances smaller than or comparable to the 'epsilon', then the failure is guaranteed.

You need to specify what type of elements you are using and the type of problem you are solving so as to get an objective answer.

There can be many reasons. One is which type of method is applied and the method may not be convergent or stable. Second if it may be theoretically convergent, the machine used and size of the matrix , its solution procedure used may be creating problems. One has to analyse the problem nature, method sensitivity etc.

@Eric Cabrol - not working here means not converging - sorry for the confusion

Except from the hardware originated problem of 'epsilon' other possible reasons for a not converging algorithm are:

1)the constraints for convergence are not satisfied, even a slight departure will make great oscillations some times, if the problem is near an ill defined one

2)there exist local singularities at the solution, ie local extrema, where many descent methods will get trapped when you have a small mesh

Can you be more specific on your cases?

To add on Demetris comments. When you are dealing with nonlinearities, the finer the mesh the larger your stiffness matrix gets therefore your solution might lead to divergence due to the derived unbalanced forces induced to your system during the analysis.

When you have a coarse mesh then you might get a singular matrix that is ill conditioned thus again face divergence issues given that your stiffness matrix gets deteriorated faster (you are unable to capture phenomena such as gradual plastification of your structural members).

In finite element model updating (a kind of inverse analysis), if the updating parameters are wrongly chosen,it can ahppen that increasing FE mesh density does not lead to convergence.

Dear Keivan, let's say that we convert all values to [0,1] or [-1,1]. The problem of smallest distance between numbers is still present... Can you clarify more your argument, probably with a small example?

well one case of instability, un boundedness or non-convergance using fine mesh is the explicit finite difference method for say 1-D parabolic partial differential equations (PDEs). If you want to make the length domain finer than accordingly you have to reduce the time step size as well, otherwise solution can be unstable

You may also wish to review what sort of word precision you are using. Have you tried double precision with ILP64 (real*16 words)?

Dear All - Thanks for all of you for the inputs/comments

The question which I have posted is related to my friend's problem. The problem he was working was on a model in a hypersonic flow field. The model is in form of a cube and has a sphere cavity inside it. Hypersonic flow comes in from one face of the cube and goes outside from the opposite face meanwhile facing the sphere which is at the center of the cube.

I am not aware of the in-depth details of the problem.

By just hearing from him, It was a bit strange for me why his problem is not working for fine mesh. Because I was in a notion that fine mesh always gives better results.

Hence I have posted this question to make me aware of the concept behind it.

I understood a bit - hardware as well as other limitations exist when we go for fine meshes.

Thanks again to all

Fine meshes are not always better than coarse meshes. In fact it occurs in damage mechanics or when one has to solve shear band localisation and so on. The solution becomes mesh dependent and the error increase with finest mesh. It depends on equation stability when non linear effects occur. In damage mechanics non local models are used where some characteristic length of the problem is introduced, say there is a physical scale one cannot go below.

Our nimerical simulations should always be mesh-independent. For a coarse grid the "truncation error" is large. when we refine the mesh to a special size and beyond, the "round off error" becomes significant and may cause instability in the solution algorithm. I think this can be the real reason for convergence problem.

Depending on turbulence model used, for example, two most popular are k-epsilon vs. SST (which is combination of k-epsilon model for locations outside of the boundary layer, if the flow is within boundary layer - near surfaces, it is solved using k-omega model). If you use for example k-epsilon model, it is better to use slightly coarser grid near surfaces, because k-epsilon doesn't perform well in boundary layers, it performs well outside of those regions where it can assume logarithmic velocity profile. If the grid is too fine (low y+ values), it takes a lot more time for the solution to converge, it can even diverge, in this case it is advised to use scalable wall functions which use internal algorithm to evaluate potential areas where grid is too fine, and automatically "combines" elements together to get y+ value to be higher. If you on the other hand use SST model, it is advisable to use as finer mesh as possible, since SST model is consisted of k-epsilon (for higher y+ values) and k-omega for low y+ values, therefore to get better convergence, you need to use fine mesh near surfaces. To conclude, k-epsilon performs well in areas outside of the boundary layer, therefore you need to decrease number of elements in those areas (near surfaces), SST as combination of k-epsilon and k-omega turbulence models performs well in areas outside of the boundary layer and within boundary layer if the mesh is fine enough (y+ values as little as possible). Internal advice for using k-epsilon is to get y+ values above 20-30, and for SST to get them below the mentioned values on problematic surfaces.