[CIG-CS] Discontinuous viscosity

[Sergio Zlotnik]

Hi Walter,
I'm very interested in what happen when the interface in not horizontal.
Using some similar techniques, I've found very different results when the
interface is slightly tilted.
Regards,
Sergio





Sergio Zlotnik
LaCaN, Laboratori de Calcul Numeric
Dept. Matematica Aplicada III
UPC-Barcelona Tech
Jordi Girona 1-3 E-08034
Barcelona, Spain

sergio.zlotnik at upc.edu
Ph: +34 93 401 7760
Fax: +34 93 401 1825
office: C2-203


2012/3/5 Walter Landry <walter at geodynamics.org>

> Hello Everyone,
>
> Back in May 2011, I ran a test for solving Stokes with discontinuous
> viscosity using deal.II and Gamr, my finite difference code.
>
>  http://www.geodynamics.org/pipermail/cig-cs/2011-May/000035.html
>
> That highlighted a problem that all codes encounter with discontinuous
> viscosities.  I think I now have a solution for this problem.  I have
> implemented a variation of the Immersed Interface Method (IIM) in a
> finite difference code.  The basic idea of IIM is to use normal finite
> differences and then add correction terms arising from the jumps.
>
> One interesting detail is that it is necessary to change the
> fundamental variables, because I do not have expressions for the jumps
> in velocity.  To that end, I use the augmented velocity, which is the
> ordinary velocity multiplied by the viscosity.
>
> In any case, to test the code I used SOLCX [1], a test used by other
> groups to gauge how well a code handles variable viscosity [2] [3].
> SOLCX is set in a unit box and has a sharp jump in viscosity at x=x_c.
> For these tests, I used x_c=0.4, which is not aligned with any grid
> boundary.  Moresi et al [3], with their finite element code, saw
> errors in pressure of 80% with a 32x32 grid when the elements were not
> aligned with the grid.  They do not report on convergence, though,
> based on the work I did in May 2011, I suspect that it diverges.
>
> In comparison, I am attaching a plot of the pressure solution and the
> error at y=0.25 for a viscosity ratio of 10^10.  This method should
> converge to first order on the boundary.  For this toy code I am using
> Gauss-Seidel relaxation.  The number of iterations is largely
> independent of the viscosity ratio.  For a 64x64 grid, using vx=vy=p=0
> as the initial guess:
>
>  viscosity ratio    # iterations
>  1                  18415
>  1e2                22685
>  1e4                22764
>  1e6                22765
>  1e8                22765
>  1e10               22765
>
> As resolution improves, the number of iterations increases, but the
> error in the pressure decreases.
>
>  viscosity ratio=10^10
>  h      # iterations  L_infinity(pressure)
>
>  1/32   5769          .0184
>  1/64   22765         .01562
>  1/128  92946         .00246
>
>
> As I mentioned, this is still a toy code.  It is hard coded for a
> horizontal viscosity jump.  The solver is just Gauss-Seidel
> relaxation, but it does use the same relaxation parameters as the full
> parallel variable viscosity Stokes solver in Gamr.  It does end up
> using a larger stencil than the straight Gamr code, but otherwise it
> should apply to the full code without too much trouble.  If you are
> interested, the code is available through mercurial with the command
>
>  hg clone http://geodynamics.org/hg/cs/AMR/Discontinuous_Stokes/
>
> One nice thing about this method is that it also works as a way of
> embedding boundaries.  So I can use it to model a spherical earth
> inside a rectilinear grid.
>
> So the next step is to get this method working for more general
> interfaces.  My plan is to embed SOLCX into a larger grid and then
> rotate it.  After that, I will use an analytic solution for a circular
> inclusion [4].  The last step is to put it into Gamr.
>
> Cheers,
> Walter Landry
>
>
> [1] Analytic solutions for Stokes’ flow with lateral variations in
> viscosity
>    Shijie Zhong
>    Geophys. J. Int.(1996) 124, 18-28
>    Section 2.3
>
> [2] Preconditioned iterative methods for Stokes flow problems arising in
>    computational geodynamics
>    Dave A. May, Louis Moresi
>    Physics of the Earth and Planetary Interiors 171 (2008) 33�#|47
>
> [3] The accuracy of finite element solutions of Stokes' flow with
>    strongly varying viscosity
>    Louis Moresi, Shijie Zhong, Michael Gurnis
>    Physics of the Earth and Planetary Interiors 97 (1996) 83-94
>
> [4] Analytical solutions for deformable elliptical inclusions
>    in general shear
>    Daniel W. Schmid and Yuri Yu. Podladchikov
>    Geophys. J. Int. (2003) 155, 269�#|288
>
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