Hi Nazaruddin Sir,

You can non-dimensionalize the equations to find non-dimensional groups. Now these groups are used for scaling the geometry and boundary conditions. This is the standard technique to solve large domains. For instance, flow-field calculations over very large skyscappers. Designing of combustors, flow over airplane, waterships and submarine etc.

It is clear that, by making nodes smaller and smaller, we can obtain more exact results, but it has high cost,so in my point of view, firstly we should check grid in-dependency,it means by changing cell smaller and smaller the results do not change a lot,another way would be like,mesh concentration,we know our problem,some parts are more significant,thus we can concentrate small mesh there and in other part meshes could be a bit larger.

The best suggestion for problems involving partial differential equations is to use the separation of variables to reduce the dimensionality of the problem, for details please consult: "Notes on Separation of Variables" available at

https://math.mit.edu/~stevenj/18.303/separation.pdf

In all problems related to data analysis, the dimensionality reduction technique is very useful

https://members.loria.fr/moberger/Enseignement/AVR/Exposes/TR_Dimensiereductie.pdf

Thanks dear Syed Mughees Ali for your reply,

Yes I know, there is a rule between a model and prototype. But many of the problems could not treat as in Dimensional Analysis. For instance, the simulation of combustion in internal combustion engines, the virus movement in a room, etc. So , we need another method to reduce the cell number....

Dear Keivan Tavakoli ,

Thank you for your suggestion. We can do that, but we still have much number of cells, if the geometry is complicated and large....

Thank you very much dear Janusz Pudykiewicz for your share.

I will learn it. But I am not sure it can work when we use an application software such as Ansys Fluent or Comsol....

Dear Nazaruddin Sinaga,

Effectively, it is often impossible to do numerical calculations when we have a very large number of cells. To reduce the number of cells in a numerical simulation without losing the precision of the numerical results, you have two solutions:

· The first one consists to use adaptive mesh refinement procedures. In the other word, the mesh is fine and the number of cells is high only in the parts where the geometry is complex or the objects are thin. Otherwise, the cell size may be larger.

· The second one consists to use the domain decomposition methods. It consists in replacing the resolution of the problem posed on a potentially large domain Ω and complicated by a succession of resolutions of the same problem on subdomains of Ω, denoted by Ωi, supposedly simpler, smaller, etc ...

I hope I was useful for you.

With best regards

Thank you so much dear Ouaïl Ouchetto for your reply.

Could you share me some references for the domain decomposition methods, please?

Domain decomposition methods have been applied to several problems. We can find several references. This book is one of the most cited references:

https://books.google.co.ma/books?hl=fr&lr=&id=h7EVoI2g1nkC&oi=fnd&pg=PA1&dq=domain+decomposition&ots=uZKIqP73N8&sig=nY3JnwU5v-MaGThpg9S619Iv-Qw&redir_esc=y#v=onepage&q=domain%20decomposition&f=false

Dear Ouaïl Ouchetto ,

Thank you very much for the share.

you welcome

Thanks Nazaruddin Sinaga Sir

In addition to the excellent answers posted in this thread, I have some practical observations from my work with simulations based on the Euler, Navier-Stokes and réaction-diffusion equations performed in meteorological context.

1.We should remember always the principles of approximate computing

https://www.osti.gov/pages/servlets/purl/1286958

The approach advocated in the above article can lead to huge savings of computer resources.

2.Many problems could be solved very effectively with the use of hybrid Lagrangian-Eulerian methods. I consider the hybrid approach to be one of the most effective techniques for the simulations of continuous systems. One of the best examples is provided by the semi-Lagrangian schemes.

3.And last but not least, the time integration schemes should be reconsidered,; particularly for the multi scale problems, one of the best candidates for the time stepping is a class of the Exponential integration schemes.

By observing these simple principles, it is possible to perform simulations with a million nodes on a current laptop computer. The recent desktops can increase the number of degrees of freedom to ten millions as mentioned in the original question.

You may also apply for computer time from computing centres or university resources. Check within your university for references, typically you need to write a research proposal. For that, it will help a lot if you can show you use some of the resource-saving techinques already suggested here.

The other advice provided here is very good. I hope are reading between the lines and seeing that this is a hard problem. The simple answer is that you may not be able to achieve the ends you seek, unless the current setup is quite bad.

Most models in my field (ocean physics) are the result of person-decades of serious work by experts with backgrounds extending to person-centuries. And, since they tend to be difficult to use, they are seldom deployed by a novice working in isolation, which means that they are usually efficiently-configured from the start.

So, if you are working with a system you developed by yourself, you may be able to improve the efficiency, but, if not, then you are likely already at the limit where increasing speed means decreasing resolution to an unacceptable degree.

I no not see any citations in your question. You ought to start by reading the literature. There are many books. Since this topic normally covers many terms at university, you have much to learn. I think you will enjoy it.

Dear Janusz Pudykiewicz and M. Mesrizadeh

Thank you for your reply and suggestions.

Yes, I was thinking to use hybrid Lagrangian-Eulerian methods. From some literature I've read, it is one candidate to solve my problem. But I have not tried this method to may cases.

I hope you are always fine, safe, and well protected.

Kind regards.

Thanks dear Salvo Cielo for your reply.

I have not such resources in my university. Maybe I have to propose to purchase an HPC to facilitate my researches.

Best regards.

Sorry, dear M. Mesrizadeh

Do you mean the SPH method?

Thank you.

Ok dear M. Mesrizadeh

Thank you.

Good topic

Good discussion. Follow up.

Nazaruddin Sinaga, With all due respect,

If you ask me, I would say the best method is drawing your body and domain on a paper, consider the minimum geometrical element size, specify required sizes such as domain or volume control, and calculate roughly your desired mesh on paper before joining into the software like what we are doing in Gambit. As a result, you will have a clear landscape of what are you heading to and it will help you not to be drawn in software automatic meshing.

The second step after meshing is the conduction of the mesh metric on your simulation. All metrics must be near to one except for the skewness angle. The most vital metrics enumerated as follows based on their priority.

Face validity and cell quality: the value of less than 0.5 will result in negative cells.

cell skewness angle which with a high value near to 1 could result in diffusion.

chevron, warpage = negative cell vol.

volume ratio should be near to 1 which means size of grids should not be increased with 1/2 ration.

volume change= must be near to 1 otherwise it may result in instability of residuals,

Considering all the above is imperative, before any further action for reduction of total grid size, since your results may out wrong.

Further, technically two options are available for valid grid size reduction; simplification of geometry, particularly in the minimum-sized edges. Decreasing domain size if it would not affect the rest of the simulation.

this is a general prescription for every simulation, though for specific cases another advice may work.

regards,

If you want to obtain accurate results using fewer DOFs, then you have to generate the meshes by paying a lot of attention to the regions of interest and those that are not of much interest. This is painstaking. You can achieve only so much with such an approach. We have to accept the fact the size of the problem we can solve on personal computers is limited. However, there are some techniques one can employ for solving large-scale problems using fewer resources.

For solid mechanics problems, one can employ sub-structuring and sub-modelling techniques to overcome the difficulties associated with modelling large-scale problems.

In sub-structuring, we create macro-elements of components or zones and use them in subsequent analyses of large models. These macro-elements represent equivalent mass and stiffness of the components/zones.

In sub-modelling, we analyse the large-scale model using a coarse mesh, by either not caring much about small-scale features such as fillets and holes or by completely ignoring them. Later, we focus on these small features for potential stress concentration regions, by creating refined meshes around such features. For the sub-model, we map the boundary conditions from the coarse model.

I have not seen such techniques being employed for problems in fluid mechanics. I don't know how effective and accurate they can be because the size of the domain affects the solution in CFD. Nevertheless, I believe that one can still employ sub-modelling technique with reasonable accuracy for extremely large-scale models, for examples, cities -> buildings -> rooms; or ocean/river -> a particular shore; or complete aeroplane -> inside of an engine.

Dear Ahmed Ramadhan Al-Obaidi , Danu Sulistiawan , Hamed Majidian , and Chennakesava Kadapa ,

Thank you very much for your reply and valuable contributions...🙏👍

Dear all respective researchers,

If you have any update knowledge or information about the topic of discussion, please share here.

Best regards,

Nazar

https://www.quora.com/How-do-we-reduce-the-number-of-cells-in-a-numerical-simulation-but-give-good-results