Using solution on from one FE problem as a boundary condition for another

[Ernesto Ismail]

Hi Everyone,

I am currently working with a staggered coupled system where I am solving several finite element problems in sequence. Up until now my boundary conditions have been fairly straightforward and I've been able to get meaningful results from my work. 

The finite element schemes I use for each step are fairly different, with some being discontinuous and of varying orders. 

I am now in a position where I need to use the solution of one of my steps as the boundary condition for the next (I actually need to combine two of them with a simple arithmetic function, but baby steps for now). I presume I need to use some sort of projection to achieve this, but I'm not exactly sure how to go about it. I thought to try and use the project_boundary_values() function to do what I need, but I am unsure how I would structure the boundary_functions argument.

Any advice or a pointer to an appropriate resource would be much appreciated.

Thanks,

Ernesto

[Ernesto Ismail]

Hi Again,

On digging a bit deeper I came across Functions::FEFieldFunction. 

Would using this be a useful approach? It would mean that in every time-step this function would be searching for the appropriate cell and evaluating the field as the boundary values are updated. The triangulation is the same for each sub-problem I solve, so perhaps there is some way to update the inhomogeneity from within the assembly process when I know which cell I'm working with?

Thanks,

Ernesto

[Wolfgang Bangerth]

Ernesto,

> I am currently working with a staggered coupled system where I am solving
> several finite element problems in sequence. Up until now my boundary
> conditions have been fairly straightforward and I've been able to get
> meaningful results from my work.
>
> The finite element schemes I use for each step are fairly different, with some
> being discontinuous and of varying orders.
>
> I am now in a position where I need to use the solution of one of my steps as
> the boundary condition for the next (I actually need to combine two of them
> with a simple arithmetic function, but baby steps for now). I presume I need
> to use some sort of projection to achieve this, but I'm not exactly sure how
> to go about it. I thought to try and use the project_boundary_values()
> function to do what I need, but I am unsure how I would structure
> the boundary_functions argument.

Do you need to enforce these boundary conditions strongly? If you can enforce
them weakly, then the solution of one problem only enters the weak form of
another problem (in that case through a boundary integral) and that is really
not very different from the way we do this in other problems where we have
multiple DoFHandlers -- e.g., in step-31. (Though my usual advice still holds
true that I think everything becomes simpler if you pack everything into one
DoFHandler).

It gets more complicated if you want to impose strong boundary conditions.
You've already found the FEFieldFunction function, but it's complicated and
slow. A simpler way would be if you looked at the implementation of the
interpolate_boundary_values() function and identified where it is asking the
Function object for its values. That's the place where you'd need to somehow
combine your existing solutions into something else and turn that into a
constraint.

Best
W.

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

[Ernesto Ismail]

Dear Prof. Bangerth,

Thank you for your reply. 

I would ideally have liked to impose the boundary conditions strongly but it looks like doing so will not be straight-forward. I will do some investigations into applying the boundary conditions weakly and attempt that first.

One approach I had tried was to use Lobatto quadrature and use a quadrature point history data structure to transfer the solution at the boundary. The problem with this approach was that the differing element types lead to some strange behaviours along the boundary.

Thanks again,

Ernesto

[Daniel Arndt]

Ernesto,

Imposing strong boundary conditions should not be overly difficult in this case.

Just as Wolgang said you should have a look at interpolate_boundary_values() and replace the Function object evaluations on each cell by the function values given through

a FEValues object initialized for your first solution. Assuming that you are using interpolatory ansatz functions, this interpolation has to happen at the support points of your target finite element.

In particular, these support points should be the quadrature points for the FEValues object.

Best,

Daniel

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[Jean-Paul Pelteret]

Hey Ernesto,

I’ve used the approach outlined by Wolfgang and Daniel in my own work (multi physics problems, single DoFHandler with BC’s being transferred between a primary and secondary problem solved on subsets of the domain). Overall, it works really well so I’d back up their suggestion and recommend that you adopt it. Its mostly really just a question of finding common support points on the two surfaces, and seeing which DoF indices link up and transferring data from one solution vector to the other. I was looking though my code to see if I could share a snippet with you, but its so close in structure to the interpolate_boundary_values() (and still retains all of the original comments in the deal.II source :-) that its not really worth it. 

Cheers,

J-P

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[Ernesto Ismail]

Hi Daniel and J-P,

Thanks for your advice. I will try what you suggest.

I feel like a bit of an idiot here, but I can't seem to find the source code for the interpolate_boundary_values() function. I've rooted around various source folders on Git and most of them don't contain much e.g. https://github.com/dealii/dealii/blob/master/source/numerics/vector_tools_project.cc I think I'm just not familiar enough with how the project is laid out. Can anyone point me in the right direction?

Thanks,

Ernesto

On Wednesday, 15 January 2020 23:08:58 UTC+2, Jean-Paul Pelteret wrote:

Hey Ernesto,

I’ve used the approach outlined by Wolfgang and Daniel in my own work (multi physics problems, single DoFHandler with BC’s being transferred between a primary and secondary problem solved on subsets of the domain). Overall, it works really well so I’d back up their suggestion and recommend that you adopt it. Its mostly really just a question of finding common support points on the two surfaces, and seeing which DoF indices link up and transferring data from one solution vector to the other. I was looking though my code to see if I could share a snippet with you, but its so close in structure to the interpolate_boundary_values() (and still retains all of the original comments in the deal.II source :-) that its not really worth it. 

Cheers,

J-P

On 15 Jan 2020, at 07:48, Daniel Arndt <d.arn...@gmail.com> wrote:

Ernesto,

Imposing strong boundary conditions should not be overly difficult in this case.

Just as Wolgang said you should have a look at interpolate_boundary_values() and replace the Function object evaluations on each cell by the function values given through

a FEValues object initialized for your first solution. Assuming that you are using interpolatory ansatz functions, this interpolation has to happen at the support points of your target finite element.

In particular, these support points should be the quadrature points for the FEValues object.

Best,

Daniel

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[Ernesto Ismail]

Hi Everyone,

In case anyone ever follows this thread and needs to find the code associated with interpolate_boundary_values(), I found https://github.com/dealii/dealii/blob/master/include/deal.II/numerics/vector_tools.templates.h

I will report back on my implementation.

Thanks,

Ernesto