Reynolds Module: Iterating Elmer solvers and integral calculation

[josie]

Hello Elmer users,
I am fairly new to Elmer and am working with the Reynolds module for thin films. A relatively common approach taken to avoid coupling the Reynolds equation with 3D elastic deformation is the half-space approximation, which involves an integral formula for the film height ("Gap Height") in terms of the pressure:

filmheight.png (15.38 KiB) Viewed 2215 times

This leads to an iterative scheme where a gap height is computed from the film pressure and vice versa until some form of convergence is acheived.

In order to use this method with the Elmer Reynolds module, I need to compute the integral (with a singular integrand) for each relevant point (i.e. each integration point and/or vertex?) and also iterate these computations.
For the iterations, I am considering external scripting like is advised in this forum post: viewtopic.php?t=3305 Is this still best practice?
For the integral, I am planning to write a solver module that adapts the calculation of the force integral in the Reynolds Postprocessing solver. Are you aware of a simpler way or other modules with similar integral calculations?
I am also not sure of the best format to input the computed, variable gap height -- the most direct way would probably be if it's possible to pass an array with the values at integration points, which could then be used to set up the matrix for the pressure calculation. Do you have any recommendations here?

Thanks in advance!

[raback]

Hi

Sorry, could you ellaborate what is your integral a little bit. What is the area for (x',y') and is (x,y) a node on the elastic object? And where do you get R_x and R_y from. Relevant reference might help.

Personally I seldom use external scripting for the reason that I know Elmer fairly well. Coupling via memory gives superior speed. Sometimes coupling via scripting gives flexibility. I see this more an advantage in pre- and post-processing than in strongly coupled multiphysics cases.

-Peter

[josie]

Hi,
Sorry for the lacking clarity -- I have attached a reference that explains the method (see eqns 3 and 20).
The missing background is that we have a 2D, rectangular computational domain with an approximately elliptical region of higher pressure centered around (0,0) with half-axes of the ellipse along the coordinate directions.
The integral in the equation is over the whole 2D computational domain, i.e. the whole rectangle.
The integral formula for the film height H can be evaluated for any (x,y) in the computational domain (but should be evaluated only for the points needed to set up the finite element matrix for the next iteration).
R_x and R_y are constants that come from the problem geometry and do not depend on x or y.

I agree that external scripting is not the nicest method -- I'm new to Elmer and am not sure how an iteration of elmer solvers can be done differently, however. The link I posted was all I could find on the forum (though it's possible I missed some results due to many posts about iterative matrix equation solvers). How could this (iterating elmer solvers until convergence) be done within Elmer?

Thanks for the fast response!

[raback]

Hi Josie,

I have some concerns for the approach due to the huge effort needed for the integration: if I understood correctly the expression for the gap height including integral over the whole domain, needed to computed in every Gaussian quadrature where the height is needed.

Let's take bilinear quads with NxN domain. You have 4N integration points, and for each need a vector of size N. So the projection storage grows like 4N^2 while the sparse matrix grows like 9N. So you will hit a wall at some stage. The integral could basically be evaluated also using multipole expansion but that machine is even heavier than standard FEM.

With those reservations I would personally make a small subroutine that would compute the gap heights and store them as a field living on IP points. For each such field there would be a multiplier of the nodal pressure values created. So you would have a 4NxN dense matrix.

Elmer has basic gaussian quadratures. What you also need here is some singular quadratures that are at least needed when r and r' are on the same element.

For higher order methods you would need even more points and your scaling laws could look a little different. I would perhaps still use rather low order elements and perhaps eventually make the mesh finer with large pressure gradients.

-Peter