Hi everyone,
From one of the other discussion posts, I saw that Simvascular uses the generalized-alpha method and the 'non-linear residual' as defined in equation (24) and (10) of [1]. Firstly, is this is the way Simvascular defines residual?
Qualitatively, I understand that the accuracy of the simulation improves as the residual decreases. Quantitatively, I want to write something along the lines of, "We required the scaled residual criteria defined in (citation 1) to be less than 0.0001 for convergence. This corresponds conceptually to a percentage error of less than 0.1 percent in the solution to the Navier-Stokes equations (citation 2)....."
Do you guys have any recommendations on what flow quantity or flow scale I can use to non-dimensionalize or scale the 'non-linear residual' I get from my simulations?
References
1.Jansen, K.E., Whiting, C.H. and Hulbert, G.M., 2000. A generalized-alpha method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Computer methods in applied mechanics and engineering, 190(3-4), pp.305-319.
Thanks a lot,
Sudharsan
How do we scale the 'non-linear residual'?
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Sudharsan Madhavan
- Posts: 5
- Joined: Mon May 25, 2015 2:22 pm
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Justin Tran
- Posts: 109
- Joined: Mon Sep 30, 2013 4:10 pm
Re: How do we scale the 'non-linear residual'?
Hi Sudharsan,
Thank you for the question! As you mentioned, the non-linear residual represents the error in the velocity and pressure fields. It is computed by plugging in the current solution for velocity and pressure into the Navier-Stokes equations (which should equal zero for a perfect solution) and getting the non-zero entry. What is probably not clear about SimVascular in particular is that the non-linear residual that is reported in the histor.dat is divided by the number of nodes in the mesh, i.e. it is effectively an "average" residual for the entire domain.
The units for the residual are thus the same as each of the terms in the Navier-Stokes equations. Because these systems tend to be heavily advection-dominated, you can try normalize by the advection term. Another option could be to normalize the residual at future timesteps by the residual at the initial state. I will say that I have not personally tried these, but this is what my intuition tells me.
For me personally though, I usually report the accuracy of my results as just the non-linear residual tolerance. The concept of non-linear residuals is very well known in the finite element fluids community so it should be sufficient to justify the accuracy of your simulations.
Thank you for the question! As you mentioned, the non-linear residual represents the error in the velocity and pressure fields. It is computed by plugging in the current solution for velocity and pressure into the Navier-Stokes equations (which should equal zero for a perfect solution) and getting the non-zero entry. What is probably not clear about SimVascular in particular is that the non-linear residual that is reported in the histor.dat is divided by the number of nodes in the mesh, i.e. it is effectively an "average" residual for the entire domain.
The units for the residual are thus the same as each of the terms in the Navier-Stokes equations. Because these systems tend to be heavily advection-dominated, you can try normalize by the advection term. Another option could be to normalize the residual at future timesteps by the residual at the initial state. I will say that I have not personally tried these, but this is what my intuition tells me.
For me personally though, I usually report the accuracy of my results as just the non-linear residual tolerance. The concept of non-linear residuals is very well known in the finite element fluids community so it should be sufficient to justify the accuracy of your simulations.