Hello,
I am trying to implement plasticity into Elmer and I need some more information about the solution of the Navier-Stokes equations. I am especially interested in the method of stabilization. Do you use penalty formulation? Do you use some form of one-point integration of pressure when using bilinear elements? And what means artificial compressibility?
Thanks.
Regards,
Petra
Navier Stokes equations - stabilization
Re: Navier Stokes equations - stabilization
Hi,
Have a look at that paper: (from Wiki publication list)
volainen V., Järvinen J., Ruokolainen J. and Anttila O. A stabilized finite element analysis for three-dimensional Czochralski silicon melt flow. A paper presented in the FEM3D conference held in Jyväskylä, June 2000. pdf [~1.3MB].
Hope it may help,
LG,
Selim
Have a look at that paper: (from Wiki publication list)
volainen V., Järvinen J., Ruokolainen J. and Anttila O. A stabilized finite element analysis for three-dimensional Czochralski silicon melt flow. A paper presented in the FEM3D conference held in Jyväskylä, June 2000. pdf [~1.3MB].
Hope it may help,
LG,
Selim
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Re: Navier Stokes equations - stabilization
There are actually a number of techniques to solve the N-S equations in Elmer. The one referred in the paper is the default. Additionally, there is an own implementation of residual free bubbless that is used often when the standard stabilization fails.
The artificial compressibility (AC) in the N-S model of Elmer is mainly a strategy to deal with fluid-structure-interaction problems. There the idea is that the AC mimics the elastic response of the structure. The formulation of the continuity equation is almost similar to the classical iteration tricks of Chorin et al. from the '70s but I guess they were more related to improving convergence in segragated solvers. In Elmer the strong coupling between velocity and pressure usually eliminates such convergence problems.
The stabilized methods use the same basis functions for velocity and pressure. The same technique achieves the optimal upwinding and fulfills the inf-sup condition (Babushka-Brezzi). The other option to meet the condition is to use mixed finite elements i.e. different basis function for velocity and pressure. You will find in /elmerfem/fem/tests/StokesPFEM/Stokes.f90 an example with quadratic velocity/linear pressure. Such a compact solver could provide a more usefull starting point for your own work compared to the legacy N-S solver which carries a large amount of exotic features.
BR, Peter
The artificial compressibility (AC) in the N-S model of Elmer is mainly a strategy to deal with fluid-structure-interaction problems. There the idea is that the AC mimics the elastic response of the structure. The formulation of the continuity equation is almost similar to the classical iteration tricks of Chorin et al. from the '70s but I guess they were more related to improving convergence in segragated solvers. In Elmer the strong coupling between velocity and pressure usually eliminates such convergence problems.
The stabilized methods use the same basis functions for velocity and pressure. The same technique achieves the optimal upwinding and fulfills the inf-sup condition (Babushka-Brezzi). The other option to meet the condition is to use mixed finite elements i.e. different basis function for velocity and pressure. You will find in /elmerfem/fem/tests/StokesPFEM/Stokes.f90 an example with quadratic velocity/linear pressure. Such a compact solver could provide a more usefull starting point for your own work compared to the legacy N-S solver which carries a large amount of exotic features.
BR, Peter
Re: Navier Stokes equations - stabilization
Thanks much for your answers.
Is the StokesSolver the right solver for using mixed finite elements (e.g. linear in pressure and quadratic in velocities)? I was not able to understand the way to specify the mixed elements in the GRD file... But now i see there is a documentation to the StokesSolver, so I'll try to read it more carefully.
I am modeling a sort of sandbox experiment ( http://sci.gallaudet.edu/Ellsworth/SandboxExp1.html ), where I have prescribed motion of vertical boundaries. In fact I have quite good results when using artificial compressibility (and bilinear elements), much better than with stabilization. Anyway, it is not perfect, so I am trying to find the reason. I don't use the additional Artificial Compressibility solver - is that a mistake?
Regards,
Petra
Is the StokesSolver the right solver for using mixed finite elements (e.g. linear in pressure and quadratic in velocities)? I was not able to understand the way to specify the mixed elements in the GRD file... But now i see there is a documentation to the StokesSolver, so I'll try to read it more carefully.
I am modeling a sort of sandbox experiment ( http://sci.gallaudet.edu/Ellsworth/SandboxExp1.html ), where I have prescribed motion of vertical boundaries. In fact I have quite good results when using artificial compressibility (and bilinear elements), much better than with stabilization. Anyway, it is not perfect, so I am trying to find the reason. I don't use the additional Artificial Compressibility solver - is that a mistake?
Regards,
Petra
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Re: Navier Stokes equations - stabilization
The elementtypes are defined in the sif file for each solver. The initial mesh files (created in your case from .grd file) define the basic nodal elements which are used for geometry definition and as basis functions if not otherwise specified. When using mixed finite elements the element basis is selected by the highest degree and the redundant degrees of freedom are eliminated. Maybe somebody else could elaborate on what numerical technique you might use. Usually for novel equations we use the residual free bubble stabilization which is robust in that there is no need to specify and ad'hoc constants.
The fact that you obtain good results with AC comes maybe from the fact that you specify velocities on each boundary. Then during the iteration you may violate the conservation of mass on the BCs. Without the AC this results to spurious pressure pulses. Note that even if you use AC you're still using stabilization in the normal way. The AC solver is intended to scale the AC field optimally in the case of FSI. In your case its not applicable. Instead you should just experiment with the value of AC for optimal convergence.
BR, Peter
The fact that you obtain good results with AC comes maybe from the fact that you specify velocities on each boundary. Then during the iteration you may violate the conservation of mass on the BCs. Without the AC this results to spurious pressure pulses. Note that even if you use AC you're still using stabilization in the normal way. The AC solver is intended to scale the AC field optimally in the case of FSI. In your case its not applicable. Instead you should just experiment with the value of AC for optimal convergence.
BR, Peter
Re: Navier Stokes equations - stabilization
As far as I know, mixed elements are usually used when implementing visco-plastic rheology. Or some under-integration of stress terms or so.
When I use bubble stabilization, resulting pressure is oscilating and it has unrealistic values. Then, a "penalty function" is used to treat pressure (divergence of velocity is assumed "almost" zero) - this might be equivalent to the artificial compressibility(?).
About artificial compressibility: I have free surface on the top boundary, so volume should be conserved(?).
Regards,
Petra
When I use bubble stabilization, resulting pressure is oscilating and it has unrealistic values. Then, a "penalty function" is used to treat pressure (divergence of velocity is assumed "almost" zero) - this might be equivalent to the artificial compressibility(?).
About artificial compressibility: I have free surface on the top boundary, so volume should be conserved(?).
Regards,
Petra
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raback
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Re: Navier Stokes equations - stabilization
Stabilization and mixed elements are partially used to overcome the same problems. Maybe you could give some pointers to the equations you want to solve so the experts in numerical methods would have more to comment on.
The AC model is consistant when convergence is reached. Hence it should have only the effect of making the convergence path smoother. If you have a free surface problem I guess that you do not fix the normal velocity on the surface and hence there should be now problems in meeting the continuity equation. The AC method is most useful when during the iteration the suggested normal velocities do not result to zero integral. Such is the case in modeling blood flow in arteries, for example. This being the case, have you checked that you really have reached convergence without the AC method? If you have you might not have these oscillations. Also you could try to enhance convergence by setting the relaxation factor of N-S to 0.5-0.7 and performing just one iteration before updating the free surface, and use enough coupled system iterations to reach the convergence of the coupled system. I'm not saying that you should not use the AC method in this case, only that it might maybe not be needed if the problem setup is otherwise optimal.
BR, Peter
The AC model is consistant when convergence is reached. Hence it should have only the effect of making the convergence path smoother. If you have a free surface problem I guess that you do not fix the normal velocity on the surface and hence there should be now problems in meeting the continuity equation. The AC method is most useful when during the iteration the suggested normal velocities do not result to zero integral. Such is the case in modeling blood flow in arteries, for example. This being the case, have you checked that you really have reached convergence without the AC method? If you have you might not have these oscillations. Also you could try to enhance convergence by setting the relaxation factor of N-S to 0.5-0.7 and performing just one iteration before updating the free surface, and use enough coupled system iterations to reach the convergence of the coupled system. I'm not saying that you should not use the AC method in this case, only that it might maybe not be needed if the problem setup is otherwise optimal.
BR, Peter