Re: [deal.II] Error in the second derivative of solution...?

[Jean-Paul Pelteret]

Thats great! I'm glad that you've managed to work it out :-)

Best,
Jean-Paul

On Wednesday, May 17, 2017 at 10:06:18 AM UTC+2, hanks0...@gmail.com wrote:
>
>  Thank you very much!!
>
> It seems that I can solve my problem.
>
> Thank you!
>
> Kyusik. 
>

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Re: [deal.II] Error in the second derivative of solution...?

[hanks0227]

 Thank you very much!!

It seems that I can solve my problem.

Thank you!

Kyusik. 

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Re: [deal.II] Error in the second derivative of solution...?

[Jean-Paul Pelteret]

Hi Kyusik,

What makes me confused here is that, according to your explanation, two 
> different DoFHandler could be used simultaneously.
>

Yes, that it correct. You can convert between cell iterators. So I do the 
following:

  typename Triangulation::active_cell_iterator
  cell = triangulation.begin_active(),
  endc = triangulation.end();
  for (; cell!=endc; ++cell)
  {
const typename DoFHandler::active_cell_iterator cell_1 
(, cell->level(), cell->index(), _handler_1);
typename DoFHandler::active_cell_iterator cell_2 
(, cell->level(), cell->index(), _handler_2);
  ...}

and then follow the usual process of FEValues reinitialisation
 

> Also, It seems that I have to use two different FEValues one for 
> solution the other for the vector...
>

Yes you would need to. The one relates to the scalar FE system and the 
other to the new vector FE system.
 

> In the assemble_system() for M and f, can I use these two things 
> simultaneously...?
>

Sure. You use one for assembly and one for the extraction of gradient data 
from the scalar solution field.

Regards,
Jean-Paul
 

>
> Thank you
>
> Kyusik.
>
>  
>

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Re: [deal.II] Error in the second derivative of solution...?

[hanks0227]

Dear Jean-Paul,

Thank you again for a detailed explanation.

However, please excuse my ignorance... 

>
> 1. Create a vector-valued finite element system (maybe an 
> FESystem(FE_Q(1),dim) ) and use it to distribute DoFs onto a new 
> DoFHandler.
> 2. Create a mass matrix "M" with the new DoFHandler.
> 3. Create a RHS vector "f" of the form f_i = \int_\Omega f(x) \phi_i(x) dx 
> . Here f(x) = \grad \psi (x) is the gradient of the solution evaluated at 
> point x and \phi_i is a vector-valued shape function. So you need to 
> obviously compute the gradient of the solution at all quadrature points 
> when doing integration - you would do this with the function I mentioned 
> previously.
>
> What makes me confused here is that, according to your explanation, two 
different DoFHandler could be used simultaneously.

I mean let's say dof_handler1 is for solution and dof_handler2 is for M

To get \grad\psi, I have to use dof_handler1, however, to assemble matrix M 
and vector f I have to use dof_handler2...

So, in the assemble_system() I don't know whether I have to use 

cell= dof_handler1.begin_active(), endc = dof_handler1.end();
or 
cell= dof_handler2.begin_active(), endc = dof_handler2.end()

Also, It seems that I have to use two different FEValues one for 
solution the other for the vector...

In the assemble_system() for M and f, can I use these two things 
simultaneously...?

Thank you

Kyusik.

 

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Re: [deal.II] Error in the second derivative of solution...?

[Jean-Paul Pelteret]

Dear Kyusik,
 

> I'm sorry to tell you that actually I don't know how to project 
> grad_solution into the vector-valued FE space
>
> Do you happen to know some example tutorial code that includes it?
>

Off the top of my head, no I don't know of an example tutorial that 
demonstrates it. But you're effectively solving a least-squares problem for 
the solution gradient so its relatively straight-forward to do:
1. Create a vector-valued finite element system (maybe an 
FESystem(FE_Q(1),dim) ) and use it to distribute DoFs onto a new 
DoFHandler.
2. Create a mass matrix "M" with the new DoFHandler.
3. Create a RHS vector "f" of the form f_i = \int_\Omega f(x) \phi_i(x) dx 
. Here f(x) = \grad \psi (x) is the gradient of the solution evaluated at 
point x and \phi_i is a vector-valued shape function. So you need to 
obviously compute the gradient of the solution at all quadrature points 
when doing integration - you would do this with the function I mentioned 
previously.
4. Solve the linear system M.d = f, where "d" represents the (continuous) 
gradient of the solution to the scalar problem.

There's some useful information in the introduction to the VectorTools 
namespace 
 
(namely 
on projection and the form of the RHS vector), so you can look there as 
well for more information. 

As I said before, you could avoid these complexities by expanding your 
original equation into the two separate terms. However, the benefit to the 
projection approach is that you can evaluate this term for one order lower 
polynomial since you don't need to directly compute second derivatives. 
With the projection one might incur some loss of accuracy of the gradient 
field, but I'm not quite sure to what extent this would affect your 
calculation. 

Regards,
Jean-Paul

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Re: [deal.II] Error in the second derivative of solution...?

[hanks0227]

Dear Jean-Paul Pelteret,

Thank you very much for your help. It is very helpful.


> However, you'll note that this function only makes sense (and its 
> therefore only defined) for a vector field. Your solution field is scalar 
> though, which presents a bit of an issue. On the face of it you'd have to 
> first project the gradient of the solution (scaled by the function "f") 
> onto an axillary vector-valued FE space and then compute the divergence of 
> that projected field. 
>

I'm sorry to tell you that actually I don't know how to project 
grad_solution into the vector-valued FE space

Do you happen to know some example tutorial code that includes it?

Thank you.

Kyusik.   

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Re: [deal.II] Error in the second derivative of solution...?

[Jean-Paul Pelteret]

Dear Kyusik,

So my new question is , in deal.II, could I get local divergence of certain 
> function to get surface integral?
>
>>
Yes it is. You can extract the divergence of a function using, for example, 
the FEValuesView function get_function_divergences 

. 

However, you'll note that this function only makes sense (and its therefore 
only defined) for a vector field. Your solution field is scalar though, 
which presents a bit of an issue. On the face of it you'd have to first 
project the gradient of the solution (scaled by the function "f") onto an 
axillary vector-valued FE space and then compute the divergence of that 
projected field. 

I think that what you could alternatively do is to expand your integrand 
into a gradient term and a Laplacian term. You can quite easily point-wise 
compute these individual quantities using the scalar FEValueViews functions 
get_function_gradients 

 
and get_function_laplacians 
,
 
or those equivalently 

 
defined 

 
for FEValuesBase.

Does this help?

Regards,
Jean-Paul

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Re: [deal.II] Error in the second derivative of solution...?

[hanks0227]

Dr. Wolfgang Bangerth,

Thank you very much for your kind reply.

But, could I have one more question?

>
> I don't know whether the problem really comes from that, but you can't 
> take 
> the second derivative of a finite element solution without problem. The 
> issue 
> is that it exists, of course, in the interior of elements.


I just had a good idea for the calculation q without second derivative 
yesterday.

But I need the surface integral of divergence of f(R,solution)*grad_solution

where f is function of position and solution.


So my new question is , in deal.II, could I get local divergence of certain 
function to get surface integral?

Thank you.

Kyusik.


>
>
>
>

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Re: [deal.II] Error in the second derivative of solution...?

[Wolfgang Bangerth]

On 05/14/2017 01:57 AM, hanks0...@gmail.com wrote:


I guess this problem comes from the second derivatives of solution.


I don't know whether the problem really comes from that, but you can't take 
the second derivative of a finite element solution without problem. The issue 
is that it exists, of course, in the interior of elements. But it is a 
delta-function like thing at the interfaces between cells because the finite 
element solution is continuous but not continuously differentiable across the 
interface (it has a "kink"), and consequently the second derivative consists 
of delta functions. I suppose you are not taking them into account in your 
formulation.


If you do need second derivatives, you need to either use a C^1 element (quite 
difficult to do), or use a mixed formulation in which you rewrite the equation 
 -Delta q = f  into

  u + nabla q = 0
  -div u  = -f
and then use a continuous element for the variable u. This way u=-grad q,
and Delta q = -div u, which you can compute because u is continuous.

Best
 W.

--

Wolfgang Bangerth  email: bange...@colostate.edu
   www: http://www.math.colostate.edu/~bangerth/

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