[CIG-CS] Variable viscosity Stokes solver

[Shijie Zhong]

It would be interesting to benchmark the results with 10^n viscosity variations 
across an element against analytical solutions for Stokes' flow. If you are 
interested in the benchmarks, I would offer some help.

Cheers,

Shijie

Shijie Zhong, Professor 
Department of Physics
University of Colorado at Boulder
Boulder, CO 80309
Tel: 303-735-5095; Fax: 303-492-7935
Web: http://anquetil.colorado.edu/szhong

---- Original message ----
>Date: Mon, 11 Apr 2011 11:37:32 -0700 (PDT)
>From: cig-cs-bounces at geodynamics.org (on behalf of Walter Landry 
<walter at geodynamics.org>)
>Subject: Re: [CIG-CS] Variable viscosity Stokes solver  
>To: cig-cs at geodynamics.org
>
>Moving this discussing to the cig-cs list.
>
>Wolfgang Bangerth <bangerth at math.tamu.edu> wrote:
>> 
>>> Taras Gerya manages to get up to 10^6 viscosity jumps for this problem
>>> by using continuations.  However, when doing geophysical runs, he
>>> never needs it.
>>> 
>>> Rhea has some large viscosity variations in their simulations, but I
>>> do not think that their element-to-element variation is this large.
>> 
>> I don't think so either. In Rhea, the viscosity is determined by the 
>> temperature and flow field, and a variation in eta so large would require an 
>> element-to-element variation in temperature or strain rate that is also very 
>> large. I do not believe that you would be able to get such variations in 
>> solution fields of this magnitude from most reasonable finite element 
methods 
>> -- you need to stabilize methods to resolve variations this large, and this 
>> tends to spread the variation out over several cells, which would then also 
>> spread out the variation in viscosity. 
>> 
>> In other words, requiring a solver to be able to deal with such large 
>> viscosity variations is not something you'd likely encounter in a code in 
>> which the viscosity is not a parameter but a function of other variables.
>
>In problems with faulting, we can get arbitrarily small regions with
>arbitrarily large viscosity jumps.  We can apply a limiter in order to
>get a convergent algorithm, but the viscosity gradient will still be
>very sharp.  For example, in Gale's dike example, the viscosity
>variation is (I think) 10^5 element-to-element.  That is why I ended
>up using a direct solver for it.
>
>>> Dave May's solver handles 10^6, but he aligns the boundary of the
>>> falling box with the edges of the element.
>> 
>> That actually leads to another interesting question: if the interface is not 
>> aligned, what do you use to integrate the local matrices and vectors? If your 
>> coefficient, Gauss formula are no longer useful. Presumably an iterated 
Gauss 
>> formula, or something else of low order would be more appropriate.
>
>Gale partitions the element up with Voronoi cells centered around each
>particle.  The viscosity is assumed smooth on those cells.  So the
>integration is essentially done by using the particles as integration
>points.
>
>I have also seen good results for other problems by using a higher
>order field (e.g. fourth order) for material properties.  I do not
>know if that will work for these problems.
>
>Cheers,
>Walter Landry
>walter at geodynamics.org
>
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