[CIG-SHORT] on the viscosity coefficient

Thu Apr 22 14:54:38 PDT 2010

Ikuo Cho,

Thanks very much for your detailed explanation.  After going through your explanation, it appears that the difficulty arises through a difference in how we define the power-law exponent.  Using the following definitions:

A_ET = A_E * exp(-Q/(R*T))
sigma_d = sigma_1 - sigma_3

PyLith 1.4.2 defines the flow law as:

epsilondot_11 = A_ET * sigma_d^n

In the derivation of Yuta Abe, the flow law used is:

epsilondot_11 = A * S_1^n

The differences arise because in the case of PyLith, the flow law is a function of the differential stress, while in the case of the derivation of Yuta Abe, the flow law is a function of the maximum deviatoric stress (S_1).  The relationship between the two is:

S_1 = 2 * sigma_d/3

Combining the two equations for epsilondot_11, we get

A_ET = (2/3)^n * A,

which I believe is what was used by Yuta Abe.  Does this sound correct?

Thanks,
Charles

p.s.  The value of A_ET is what is used as input for PyLith 1.4.2 (and an upcoming version 1.4.3).  In the next major release of PyLith, a different set of parameters will be used to define power-law flow.

On 20/04/2010, at 5:23 PM, Ikuo Cho wrote:

> Charales,
> 
>> This can be derived from the equations in the manual and I have
> verified that this is what is used in the code.
> 
> Thank you for the verification.
> 
>> As far as I know, the value that Yuta Abe used in the analytical
> solution probably corresponded to A_M * exp(-Q/RT)... Does that sound
> correct? 
> 
> No. A correct value that he had to use in the analytical solution was
> A=A_M * exp(-Q/RT)*(sqrt(3)/2)**(n-1). 
> He did mistakenly use a value of A_T, which equals to (2/3)**n A.
> See the pdf file for the details.
> 
> Regards,
> Ikuo Cho
> 
> On Wed, 14 Apr 2010 14:42:55 +1200
> Charles Williams <willic3 at gmail.com> wrote:
> 
>> Dear Ikuo,
>> 
>> I apologize for being so slow to respond to this.  I just returned from a trip and have started looking at things.  I'm still somewhat puzzled.  Internally, PyLith 1.4.2 uses a viscosity-coefficient, which can be defined as:
>> 
>> viscosity-coefficient = (1/(A_T * sqrt(3)**(n+t)))**(1/n)
>> 
>> This can be derived from the equations in the manual and I have verified that this is what is used in the code.  As far as I know, the value that Yuta Abe used in the analytical solution probably corresponded to A_M * exp(-Q/RT), which should be A_T * sqrt(3)**(n+1)/2.  Does that sound correct?  It's also possible that the power-law-coefficient used in his analytical solution is defined differently.  Let me know what you think about this, because it would be good to get this resolved.
>> 
>> Thanks,
>> Charles
>> 
>> 
>> On 17/03/2010, at 11:03 PM, Ikuo Cho wrote:
>> 
>>> Charles,
>>> 
>>> I discussed with my colleague, Yuta Abe, on the fit between the final
>>> output from PyLith 1.4.2 and an analytical solution.
>>> 
>>> Yuta Abe Wrote:
>>>> By the way, the final output of the PyLith program coincided with
>>> the analytical solution.
>>> 
>>> We found that an input parameter for the analytical calculation was
>>> incorrect when he wrote the above report, and consequently analytical
>>> solution does not fit the  numerical results. He did use a value of A_T
>>> as both the "powerlaw-coefficient" for PyLith and a parameter value for
>>> the analytical calculation, although he had to use values of A_T and
>>> (3/2)**n A_T for the "powerlaw-coefficient" and the analytical
>>> calculation, respectively.
>>> (As the result, he observed a good fit between the numerical and
>>> analytical solutions.)
>>> 
>>> I wanted to ask from this fact the possibility that the
>>> "powerlaw-coefficient" is actually defined by A_T'=(3/2)**n A_T,
>>> instead of A_T defined in (5.75) in PyLith 1.4.2.
>>> Is it difficult to check?
>>> 
>>> By the way, I recently installed PyLith from the repository. I also made
>>> comparison between the numerical and the analytic solutions for the same
>>> problem. (I noticed a small difference in the definition of A_T between
>>> the released and repository versions.)
>>> They showed a good fit in this case. 
>>> 
>>> Ikuo Cho
>>> 
>>> On Fri, 5 Mar 2010 07:14:30 +1300
>>> Charles Williams <willic3 at gmail.com> wrote:
>>> 
>>>> Dear Yuta Abe,
>>>> 
>>>> I will look at the code and the manual to see if there is a problem.  As I mentioned in a previous e-mail to cig-short, we are changing the input parameters for power-law materials, so 'eta' will no longer be a parameter in upcoming versions.  I will let you know what I find out about version 1.4.2, though.
>>>> 
>>>> I'm glad the final output matches the analytical solution.  Would it be possible for you to describe the problem?  It may be useful as an example problem or benchmark.
>>>> 
>>>> Cheers,
>>>> Charles
>>>> 
>>>> On 4/03/2010, at 9:07 PM, Yuta Abe wrote:
>>>> 
>>>>> Dear PyLith Developers:
>>>>> 
>>>>> I have a question about the viscosity_coefficient "eta", which is one of the physical properties that you get in the output "cel_info_fields" when you carry out analysis of a power-law viscoelastic material using PyLith 1.4.2.
>>>>> 
>>>>> I thought the values of "eta" could be calculated using the equations (5.74), (5.75) and (5.76) by substituting the power-law coefficient "At" and the power-law exponent n. I substituted "At"=1.99e-41 and n=3 into those equations, and obtained " eta"=1.77e+13 as a result. However, the value of "eta" in info.vtk file which was obtained as an automatic output of the PyLith program for the same values of "At"=1.99e-41 and n=3 was "eta"=1.77e+18, 100,000 times as large as the above value. By the way, the final output of the PyLith program coincided with the analytical solution.
>>>>> 
>>>>> I would like to find out the origin of this difference, so please kindly tell me how the value of "eta" is calculated within the PyLith program.
>>>>> 
>>>>> 
>>>>> ------------------------------------------------------
>>>>> Yuta ABE
>>>>> Active Fault and Earthquake Reserch Center
>>>>> National Institute of Advanced Industrial Science and Technology,Japan
>>>>> tel; +81-29-861-3686
>>>>> email; yuta-abe at aist.go.jp
>>>>> -------------------------------------------------------  
>>>>> _______________________________________________
>>>>> CIG-SHORT mailing list
>>>>> CIG-SHORT at geodynamics.org
>>>>> http://geodynamics.org/cgi-bin/mailman/listinfo/cig-short
>>>> 
>>>> Charles A. Williams
>>>> Scientist
>>>> GNS Science
>>>> 1 Fairway Drive, Avalon
>>>> PO Box 30368
>>>> Lower Hutt  5040
>>>> New Zealand
>>>> ph (office): 0064-4570-4566
>>>> fax (office): 0064-4570-4600
>>>> C.Williams at gns.cri.nz
>>>> NOTE NEW E-MAIL ADDRESS
>>>> 
>>> 
>>> ----------------------------------------------------------
>>> Ikuo Cho  ( ikuo-chou at aist.go.jp )
>>> Geological Survey of Japan,
>>> National Institute of Advanced Industrial Science and Technology
>>> Tsukuba Central 7, Tsukuba 305-8567 Japan
>>> Tel +81-29-861-3891, Fax +81-29-861-3682
>>> ----------------------------------------------------------
>>> 
>> 
>> Charles A. Williams
>> Scientist
>> GNS Science
>> 1 Fairway Drive, Avalon
>> PO Box 30368
>> Lower Hutt  5040
>> New Zealand
>> ph (office): 0064-4570-4566
>> fax (office): 0064-4570-4600
>> C.Williams at gns.cri.nz
>> NOTE NEW E-MAIL ADDRESS
>> 
> 
> -- 
> Cho Ikuo <ikuo-chou at aist.go.jp>
> <equation.pdf>

Charles A. Williams
Scientist
GNS Science
1 Fairway Drive, Avalon
PO Box 30368
Lower Hutt  5040
New Zealand
ph (office): 0064-4570-4566
fax (office): 0064-4570-4600
C.Williams at gns.cri.nz
NOTE NEW E-MAIL ADDRESS

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