Edit: The coefficient of the convective term, if it is correct to label it
that, is a 2x2 tensor. I realized I made this mistake as I thought about
this last night. In other words \sigma is 2x2.
Cheers,
Kyle
On Fri, Feb 13, 2015 at 6:05 PM, Kyle Lawlor <klawlor...@gmail.com> wrote:
> Hi, Dr. Guyer.
>
> Thanks for the response. For the reasons you mention I have started trying
> implement the two-dimensional case. I appreciate the comments and help.
>
> As a brief comment on the issues I am working out, the two-dimensional
> equation's for this problem are coupled. The two governing equations are
> for (\partial w/\partial t) and (\partial \vec{u}/\partial t). The equation
> for (\partial w/\partial t) involves a complicated convective term. The
> coefficient inside the convective term is a 3x3 tensor (\sigma) and the
> variable inside the convective term is grad(w). There is coupling that
> enters in 3x3 tensor involves fairly complicated combinations of spatial
> derivatives of w and the components of \vec{u}. I've written these out by
> hand, but I can type these up and send them at a later time if it's
> helpful.
>
> The paper that I'm referencing use spectral methods to solve the problem.
> I realize that for DiffusionTerms with a power greater than 3 the
> documentation recommends spectral methods. I'm thinking this problem may be
> one that needs to be solved through spectral methods.
>
> All of this is to say, in your experience does this problem look feasible
> for FiPy? I'll attach a snapshot of the equations for w and \vec{u}. A
> smaller concern is actually figuring out how to write the tensor (\sigma)
> in FiPy.
>
> Looking forward to hearing more.
>
> Cheers,
> Kyle.
>
>
> On Tue, Feb 10, 2015 at 1:48 PM, Guyer, Jonathan E. Dr. <
> jonathan.gu...@nist.gov> wrote:
>
>> I understand what you mean about it being simpler, but I tend not to work
>> in 1D, because it can be misleading. You've got combinations of even-order
>> and odd-order terms that don't make much sense to me if you think about
>> them in higher dimensions. Using \nabla instead of (\partial / \partial x)
>> makes it a bit clearer whether you're working with vectors or scalars.
>>
>> If u and w are scalars, then (\partial u/partial x) is a vector and it
>> doesn't make any sense to add (\partial^4 w\partial x^4), a scalar, to
>> (\partial u/\partial x)(\partial^2 w / \partial x^2), a vector.
>>
>>
>> On Feb 10, 2015, at 12:20 AM, Kyle Briton Lawlor <klawlor...@gmail.com>
>> wrote:
>>
>> > Hello, FiPy.
>> >
>> > I am presently working with a set of coupled 1D pde’s.
>> > Images with the equations are attached at the bottom of the email.
>> > The first image shows the equations.
>> > The second image shows the equations with a “sketch” of how I might
>> write the terms in FiPy and some questions I have.
>> >
>> > Roughly, the equations are solving for displacements of a line subject
>> to compressive stress.
>> > Eventually I would like to model the two-dimensional problem.
>> > However, I figure 1D is a good starting point as the 1D eq’s are quite
>> complicated themselves.
>> > Hopefully it is feasible to solve the 1D problem in FiPy.
>> > In the paper I’m referencing for these evolution equations, they solve
>> using a finite difference method.
>> >
>> > The two main variables in the problem as you can see in the pde’s are w
>> and u.
>> > w is the lateral displacement and u is the on-axis displacement.
>> > As you can see in the equations, there are quite complicated
>> coefficients on a few of the terms.
>> > Also there are two terms in particular that I am not sure how to write.
>> (See the second image; Term 1 and Term 2).
>> >
>> > I have not started coding up this problem, but I think some preliminary
>> guidance could be useful.
>> > This is a broad question but is solving these equations with FiPy
>> feasible?
>> >
>> > Look forward to hearing responses.
>> >
>> > Thanks,
>> > Kyle
>> > <Screen Shot 2015-02-09 at 11.59.39 PM.png>
>> > <Screen Shot 2015-02-09 at 11.59.58 PM.png>
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