# Grid refinement or Higher order scheme

 Register Blogs Members List Search Today's Posts Mark Forums Read

 December 13, 2021, 14:13 Grid refinement or Higher order scheme #1 Member     Yusuf Elbadry Join Date: Sep 2018 Posts: 65 Rep Power: 7 Hello, I hope you are all fine, I am solving the heaving airfoil problem while using the Eulerian grid (Level Set Method). The 2D Incompressible N-S equations are solved using GLS FEM. I am simulating the flow at Re=800, Angle of Attack =20 Deg. I am using bilinear shape function for geometry and solution and I am using an explicit scheme. I have a problem with the accuracy of the result, I am comparing my result (Cl, CD) with the literature and the error is not small, and I was thinking to use a second-order shape function for both the geometry and solution to get a better result and better shape for the airfoil instead of stairs. Would this be better or should I use a more refined grid with a very small time step? MohamedAdawy likes this.

December 13, 2021, 15:49
#2
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,777
Rep Power: 71
Quote:
 Originally Posted by AeroSonic Hello, I hope you are all fine, I am solving the heaving airfoil problem while using the Eulerian grid (Level Set Method). The 2D Incompressible N-S equations are solved using GLS FEM. I am simulating the flow at Re=800, Angle of Attack =20 Deg. I am using bilinear shape function for geometry and solution and I am using an explicit scheme. I have a problem with the accuracy of the result, I am comparing my result (Cl, CD) with the literature and the error is not small, and I was thinking to use a second-order shape function for both the geometry and solution to get a better result and better shape for the airfoil instead of stairs. Would this be better or should I use a more refined grid with a very small time step?

Before introducing a higher order scheme, you must verify that your code works well for via via refined grids. If your solution does not improve under grid refinement, your problem is not the low accuracy order but some different issues.

December 13, 2021, 16:04
#3
Member

Join Date: Sep 2018
Posts: 65
Rep Power: 7
Quote:
 Originally Posted by FMDenaro Before introducing a higher order scheme, you must verify that your code works well for via via refined grids. If your solution does not improve under grid refinement, your problem is not the low accuracy order but some different issues.
If the solution accuracy got better while using a fine grid, would it be better to use a higher-order scheme to reduce the number of elements and get a better airfoil shape, or should I continue with the refined grid? I know that this question is related to the computation time but I would like to reduce the computation time as possible as I can, as I am doing these simulations on my personal computer.

December 13, 2021, 16:19
#4
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,777
Rep Power: 71
Quote:
 Originally Posted by AeroSonic If the solution accuracy got better while using a fine grid, would it be better to use a higher-order scheme to reduce the number of elements and get a better airfoil shape, or should I continue with the refined grid? I know that this question is related to the computation time but I would like to reduce the computation time as possible as I can, as I am doing these simulations on my personal computer. Thank you for your reply.

For an arbitrary fixed grid of size h, you cannot be sure that the solution obtained with a second order scheme is better than a first order scheme. You need to have a sufficient grid resolution to get a real improving. That is a classical topic in CFD.

December 13, 2021, 16:58
#5
Member

Join Date: Sep 2018
Posts: 65
Rep Power: 7
Quote:
 Originally Posted by FMDenaro For an arbitrarily fixed grid of size h, you cannot be sure that the solution obtained with a second-order scheme is better than a first-order scheme. You need to have a sufficient grid resolution to get a real improving. That is a classical topic in CFD.
Yeah, I got that. I just thought that if I used a 2nd order shape function or higher, I can get a better shape of the airfoil while using a small number of elements and nodes compared to the case while using a bilinear shape function in which I have to use a very very fine grid near the airfoil.

For example, if I have an airfoil with a chord of 1 (nondimensional length), I will need elements with a side length (in y-dir) of 3e-3 or less to have a smooth airfoil shape, If the airfoil will move in the y-direction with an amplitude of 0.5c (from -0.5c to 0.5c), then I need at least 333 divisions in this direction, and let us say I will have one-third of this number in the x-direction, this may result in around 350,000 elements near the airfoil where it moves, not mentioning the rest of the domain. This may result in a grid of more than 600,000 elements.

That is why I thought of using high order shape function, to reduce the number of elements and nodes.

I really appreciate your help and guidance and thank you for your patience.

December 14, 2021, 04:24
#6
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,777
Rep Power: 71
Quote:
 Originally Posted by AeroSonic Yeah, I got that. I just thought that if I used a 2nd order shape function or higher, I can get a better shape of the airfoil while using a small number of elements and nodes compared to the case while using a bilinear shape function in which I have to use a very very fine grid near the airfoil. For example, if I have an airfoil with a chord of 1 (nondimensional length), I will need elements with a side length (in y-dir) of 3e-3 or less to have a smooth airfoil shape, If the airfoil will move in the y-direction with an amplitude of 0.5c (from -0.5c to 0.5c), then I need at least 333 divisions in this direction, and let us say I will have one-third of this number in the x-direction, this may result in around 350,000 elements near the airfoil where it moves, not mentioning the rest of the domain. This may result in a grid of more than 600,000 elements. That is why I thought of using high order shape function, to reduce the number of elements and nodes. I really appreciate your help and guidance and thank you for your patience.

I don't understand, the shape functions defines a certain functional relation for the unknown variables, what do you mean now for the highlighted statement? The better description of the airfoil geometry is a different issue.
However, grid size and accuracy order defines different topics and different limits.The former fix the range of resolved wavenumber in the solution, the latter is the theoretical and asymptotical velocity rate of the error decreasing. You could have also a spectral accuracy but if you have fixed the grid size you cannot resolve nothing greater than the Nyquist frequency.

December 14, 2021, 13:59
#7
Member

Join Date: Sep 2018
Posts: 65
Rep Power: 7
Quote:
 Originally Posted by FMDenaro I don't understand, the shape functions defines a certain functional relation for the unknown variables, what do you mean now for the highlighted statement? The better description of the airfoil geometry is a different issue. However, grid size and accuracy order defines different topics and different limits.The former fix the range of resolved wavenumber in the solution, the latter is the theoretical and asymptotical velocity rate of the error decreasing. You could have also a spectral accuracy but if you have fixed the grid size you cannot resolve nothing greater than the Nyquist frequency.
If I understand correctly, in the Finite Element Method, you have shape functions for each variable or known and you have a shape function for the geometry. For the geometric shape function, if it is a bilinear, it means you have a rectangular element with 4 nodes, if it is bi-quadratic, you can have 8 or 9 nodes per element, this means, if you used a bilinear shape function, you will connect every two nodes by a line, and if you used a bi-quadratic shape function, you will connect every two nodes by a quadratic curve, this will lead to a better representation to the body (if you are using a fixed grid and the body is moving and you are using an algorithm to define which point is inside or outside the body).

Please correct me if I am wrong or if there is something I am missing.

December 14, 2021, 15:35
#8
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,777
Rep Power: 71
Quote:
 Originally Posted by AeroSonic If I understand correctly, in the Finite Element Method, you have shape functions for each variable or known and you have a shape function for the geometry. For the geometric shape function, if it is a bilinear, it means you have a rectangular element with 4 nodes, if it is bi-quadratic, you can have 8 or 9 nodes per element, this means, if you used a bilinear shape function, you will connect every two nodes by a line, and if you used a bi-quadratic shape function, you will connect every two nodes by a quadratic curve, this will lead to a better representation to the body (if you are using a fixed grid and the body is moving and you are using an algorithm to define which point is inside or outside the body). Please correct me if I am wrong or if there is something I am missing.

But this is a further topic. Think about a BEM formulation (panel method), you can describe the airfoilf by using straight segment or curved step. Then you prescribe a functional relation for the distribution of singularity.

In you case you want to use elements that are not linear triangles but curved triangles and then use high order shape functions.

Again, on the same and arbitrary mesh size h you cannot say a-priori that the error is for sure lesser for the quadratic shape function. You need to assess that by a grid refinement study. Only at a certain value h you get a monotonically expected rate for the error.

December 15, 2021, 14:02
#9
Member

Join Date: Sep 2018
Posts: 65
Rep Power: 7
Quote:
 Originally Posted by FMDenaro But this is a further topic. Think about a BEM formulation (panel method), you can describe the airfoilf by using straight segment or curved step. Then you prescribe a functional relation for the distribution of singularity. In you case you want to use elements that are not linear triangles but curved triangles and then use high order shape functions. Again, on the same and arbitrary mesh size h you cannot say a-priori that the error is for sure lesser for the quadratic shape function. You need to assess that by a grid refinement study. Only at a certain value h you get a monotonically expected rate for the error.
I am sorry, but I didn't understand the part related to the BEM and the panel method. I drew this comparison between a grid with bilinear and bi-quadratic shape function for the geometry, I meant that increasing the order of the geometrical shape function will allow me to get a better shape for the body.

Attached Images
 Capture.jpg (48.2 KB, 6 views)

December 15, 2021, 14:59
#10
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,777
Rep Power: 71
Quote:
 Originally Posted by AeroSonic I am sorry, but I didn't understand the part related to the BEM and the panel method. I drew this comparison between a grid with bilinear and bi-quadratic shape function for the geometry, I meant that increasing the order of the geometrical shape function will allow me to get a better shape for the body. I really appreciate your patience and your reply.
You are confusing the things in the problem. From your figure, the airfoil is still described by a step-by-step approximation. What you need is to adopt curved elements. On the other side, quadratic shape functions on linear-edge element can be effective in improving the fluid solution.

December 16, 2021, 13:26
#11
Member

Join Date: Sep 2018
Posts: 65
Rep Power: 7
Quote:
 Originally Posted by FMDenaro You are confusing the things in the problem. From your figure, the airfoil is still described by a step-by-step approximation. What you need is to adopt curved elements. On the other side, quadratic shape functions on linear-edge element can be effective in improving the fluid solution.
Yes, it is defined step by step exactly as in LBM for example.
I will try using a higher-order shape function for both and solution and geometry.
Thank you

December 16, 2021, 13:33
#12
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,777
Rep Power: 71
Quote:
 Originally Posted by AeroSonic Yes, it is defined step by step exactly as in LBM for example. I will try using a higher-order shape function for both and solution and geometry. Thank you

You could search for suitable isoparametric elements. On the other hand, the airfoil is better described by a triangular tesselation.

December 16, 2021, 14:15
#13
Member

Join Date: Sep 2018
Posts: 65
Rep Power: 7
Quote:
 Originally Posted by FMDenaro You could search for suitable isoparametric elements. On the other hand, the airfoil is better described by a triangular tesselation.
I know that there is a Lagrangian Element(9 nodes in case of bi-quadratic shape function) and Serendipity Element(8 nodes in case of bi-quadratic shape function), can I find a different element than these two types of the element?

December 16, 2021, 14:25
#14
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,777
Rep Power: 71
Quote:
 Originally Posted by AeroSonic I know that there is a Lagrangian Element(9 nodes in case of bi-quadratic shape function) and Serendipity Element(8 nodes in case of bi-quadratic shape function), can I find a different element than these two types of the element?

Lagrangian simplex, that is triangular elements (or tetrahedron in 3D), having 3, 6, 10, ... nodes

 Tags fem, high order scheme., lsm