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Number of mesh elements and geometry size

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Dear all,

I was just thinking about the following...For the same number of mesh elements, does the accuracy of the solution depend on the size of the geometry you are solving on?

Let me give an example: If you are solving a PDE on a 100x100 squared geometry with 25 squared mesh elements, the size of each mesh elements is 20x20. If you now increase the size of the geometry to 1000x1000 and keep the number of mesh elements content to 25, then the size of the mesh elements is 200x200.

Does it make a difference?

Many thanks in advance

4 Replies Last Post 2014年7月28日 GMT-4 09:31
Ivar KJELBERG COMSOL Multiphysics(r) fan, retired, former "Senior Expert" at CSEM SA (CH)

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Posted: 1 decade ago 2014年7月24日 GMT-4 11:14
Hi

Think of the mesh elements as a discretisation: if you listen to music which sampling rate do you choose for best response ? and would you use the same for rock, classical, bass, treble ... type music ?

It's the same with FEM, the mesh is a discretisation (this is well identified in COMSOL as you define the physics on the geometry and then you decide how to mesh, while in classical "old 50 years or more) type FEM you define the mesh and get the physics already "squeezed" in the mesh, (this is numerically OK, but conceptually from the physics and FEM methodology not the correct way of using "physics - PDE's - mesh" concept)

To answer your question short, the accuracy depends on the optimum mesh density w.r.t. the dependent variable variations, not directly on the geometry size.

So be modern catch "the Concept", now finally possible also in FEM with COMSOL :)

--
Have fun COMSOLing
Ivar
Hi Think of the mesh elements as a discretisation: if you listen to music which sampling rate do you choose for best response ? and would you use the same for rock, classical, bass, treble ... type music ? It's the same with FEM, the mesh is a discretisation (this is well identified in COMSOL as you define the physics on the geometry and then you decide how to mesh, while in classical "old 50 years or more) type FEM you define the mesh and get the physics already "squeezed" in the mesh, (this is numerically OK, but conceptually from the physics and FEM methodology not the correct way of using "physics - PDE's - mesh" concept) To answer your question short, the accuracy depends on the optimum mesh density w.r.t. the dependent variable variations, not directly on the geometry size. So be modern catch "the Concept", now finally possible also in FEM with COMSOL :) -- Have fun COMSOLing Ivar

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Posted: 1 decade ago 2014年7月25日 GMT-4 05:31
Dear Ivar,

Thanks very much for your kind answer.

So the relevant parameter here is the mesh density. For example, for a transport problem in the radial direction, the relevant parameter to be optimised is the number of mesh elements in the radial direction rather the total number of elements in the geometry.

Kind regards!
Dear Ivar, Thanks very much for your kind answer. So the relevant parameter here is the mesh density. For example, for a transport problem in the radial direction, the relevant parameter to be optimised is the number of mesh elements in the radial direction rather the total number of elements in the geometry. Kind regards!

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Posted: 1 decade ago 2014年7月28日 GMT-4 08:27
Mesh elements assume a simplified situation, for example a linear distribution of temperature within the element. The complexity of the real heat distribution in the geometry is achieved by dividing it into a sufficiënt amount of mesh elements. The mesh elements should be small enough for the approximation to hold. If everything scales appropriately with the size of the geometry (for example if all dimensions of a geometry are multiplied by 10, and the temperature difference imposed is also multiplied by 10), then a meshing that is appropriate for the smaller geometry should also be appropriate for the bigger one (so the dimensions of the mesh elements can also be multiplied by 10 and the amount of mesh elements stays the same). However if not everything scales the same, you should re-check the appropriateness of your mesh size.
The 'easiest' check to see if your mesh is fine enough is repeating the calculation with a finer mesh. If the results are the same then your mesh was probably fine enough, otherwise you might need to make it even finer.
Mesh elements assume a simplified situation, for example a linear distribution of temperature within the element. The complexity of the real heat distribution in the geometry is achieved by dividing it into a sufficiënt amount of mesh elements. The mesh elements should be small enough for the approximation to hold. If everything scales appropriately with the size of the geometry (for example if all dimensions of a geometry are multiplied by 10, and the temperature difference imposed is also multiplied by 10), then a meshing that is appropriate for the smaller geometry should also be appropriate for the bigger one (so the dimensions of the mesh elements can also be multiplied by 10 and the amount of mesh elements stays the same). However if not everything scales the same, you should re-check the appropriateness of your mesh size. The 'easiest' check to see if your mesh is fine enough is repeating the calculation with a finer mesh. If the results are the same then your mesh was probably fine enough, otherwise you might need to make it even finer.

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Posted: 1 decade ago 2014年7月28日 GMT-4 09:31
Thanks for your answer.

The approach I have taken is to optimise the number of mesh elements in the transport direction. I am using a mapped geometry...

Thanks
Thanks for your answer. The approach I have taken is to optimise the number of mesh elements in the transport direction. I am using a mapped geometry... Thanks

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