Setting boundary conditions to faces, really simple question

lrunber · March 9, 2012, 09:25

Hi,
I am just wondering a very basic principle. If the linear system of equations is set up for each cell in the domain from the well known equation (where p is the cell center and n the neighbouring cell centres):

$a_p \phi_p + \sum a_n \phi_n = r$

Where are the boundary condition imposed, considering BC's are to be set at the appropriate faces and the values for $\phi$ are at the cell centers?

duri · March 9, 2012, 14:59

You can set up boundary conditions by two ways. In Ghost cell approach boundary values are stored in the ghost cells which is adjacent to boundary. Boundary fluxes are calculated from interior and ghost cell. Other way is to apply characteristics boundary condition where you can do it directly on face and estimate flux without additional cells.

lrunber · March 9, 2012, 15:12

Quote:

Originally Posted by duri

You can set up boundary conditions by two ways. In Ghost cell approach boundary values are stored in the ghost cells which is adjacent to boundary. Boundary fluxes are calculated from interior and ghost cell. Other way is to apply characteristics boundary condition where you can do it directly on face and estimate flux without additional cells.

Thank you very much duri, that is very informative.

I take then that the setting of a fixed value boundary ( e.g., $\phi_{boundary} = 1.0$ ) is done implicitly through either of the methods you mentioned above and there is no explicit imposing of $\phi_{boundary}$ within the algebraic system of equations?

sbaffini · March 9, 2012, 16:43

Besides the specific approach in use, in cell centered FV method you have variables stored in cell centers but you need their integrals on volume faces. When they are available they are used directly (velocity and/or mass flow inlets; wall normal derivative of the wall normal velocity component for incompressible flows; symmetry boundaries; outflows etc). In all the other cases, both on boundary and interior cells, some reconstruction/approximation is adopted by using known face or neighbor cell values.

lrunber · March 10, 2012, 02:49

Quote:

Originally Posted by sbaffini

Besides the specific approach in use, in cell centered FV method you have variables stored in cell centers but you need their integrals on volume faces. When they are available they are used directly (velocity and/or mass flow inlets; wall normal derivative of the wall normal velocity component for incompressible flows; symmetry boundaries; outflows etc). In all the other cases, both on boundary and interior cells, some reconstruction/approximation is adopted by using known face or neighbor cell values.

Thank you for your explanation Paolo, if you don't mind I want to write in my understanding and if you or anyone could please comment/correct?

In the cell centered FV method the variables are stored at the cell centers but the face values are required for the discretized forms of the convection and diffusion terms. Generally this is obtained by interpolation such as the popular upwind differencing. At a fixed value (Dirichlet) boundary this is set directly to the specified value (no interpolation is used). This results in the correct value being obtained for the boundary without direct modification of the algebraic system of equations. Subsequently, the setting of a fixed value boundary condition is more complex than directly imposing $\phi_{boundary}$ in $[A] [\phi] = [Q]$ , but is achieved through setting values at the faces where they are used in the discretized form of the equations which results in a modification of the coefficient matrix (

) and source vector (

).

sbaffini · March 10, 2012, 08:28

Dear lrunber,

you are essentially correct, however i'd like to stress some aspects.

1) When you put your equations under the matrix form, you have to remember that this implies that your variables are the cell centerd ones (in cell centered FV). As a consequence, using a face variable directly (a la Dirichlet) actually implies a modification in the corresponding cell equation (i assume there is only one neighbor cell as it is a boundary condition) with respect to other interior cell equations. That is, the corresponding neighbor cell coefficient is missing from the relative equation in the system.

2) In a general transport equation, face values are required for the variables and their face normal gradient. As you can only have one boundary condition (dirichlet or neumann, respectively), one of the two values required still need an approximation involving other cell values. Say, you have a dirichlet condition: the boundary normal gradient has to be approximated by values inside the domain (one-sided, in contrast to the ghost-cell approach). If you have a neumann condition then the face values will need an approximation involving interior cells (however, face values in finite volumes are only required for the convective term and this case has very few applications, mostly outflows)

Hope this helps more

lrunber · March 10, 2012, 09:45

Quote:

Originally Posted by sbaffini

Dear lrunber,

you are essentially correct, however i'd like to stress some aspects.

1) When you put your equations under the matrix form, you have to remember that this implies that your variables are the cell centerd ones (in cell centered FV). As a consequence, using a face variable directly (a la Dirichlet) actually implies a modification in the corresponding cell equation (i assume there is only one neighbor cell as it is a boundary condition) with respect to other interior cell equations. That is, the corresponding neighbor cell coefficient is missing from the relative equation in the system.

2) In a general transport equation, face values are required for the variables and their face normal gradient. As you can only have one boundary condition (dirichlet or neumann, respectively), one of the two values required still need an approximation involving other cell values. Say, you have a dirichlet condition: the boundary normal gradient has to be approximated by values inside the domain (one-sided, in contrast to the ghost-cell approach). If you have a neumann condition then the face values will need an approximation involving interior cells (however, face values in finite volumes are only required for the convective term and this case has very few applications, mostly outflows)

Hope this helps more

Thank you Paolo, this helps a lot. The implementation of BC's are a bit more complicated than I originally thought, but I now understand much better

March 9, 2012, 09:25	Setting boundary conditions to faces, really simple question	#1
lrunber New Member lazio Join Date: Apr 2010 Posts: 22 Rep Power: 16	Hi, I am just wondering a very basic principle. If the linear system of equations is set up for each cell in the domain from the well known equation (where p is the cell center and n the neighbouring cell centres): $a_p \phi_p + \sum a_n \phi_n = r$ Where are the boundary condition imposed, considering BC's are to be set at the appropriate faces and the values for $\phi$ are at the cell centers?

March 10, 2012, 08:28		#6
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,195 Blog Entries: 29 Rep Power: 39	Dear lrunber, you are essentially correct, however i'd like to stress some aspects. 1) When you put your equations under the matrix form, you have to remember that this implies that your variables are the cell centerd ones (in cell centered FV). As a consequence, using a face variable directly (a la Dirichlet) actually implies a modification in the corresponding cell equation (i assume there is only one neighbor cell as it is a boundary condition) with respect to other interior cell equations. That is, the corresponding neighbor cell coefficient is missing from the relative equation in the system. 2) In a general transport equation, face values are required for the variables and their face normal gradient. As you can only have one boundary condition (dirichlet or neumann, respectively), one of the two values required still need an approximation involving other cell values. Say, you have a dirichlet condition: the boundary normal gradient has to be approximated by values inside the domain (one-sided, in contrast to the ghost-cell approach). If you have a neumann condition then the face values will need an approximation involving interior cells (however, face values in finite volumes are only required for the convective term and this case has very few applications, mostly outflows) Hope this helps more lrunber likes this.

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March 9, 2012, 14:59		#2
duri Senior Member duri Join Date: May 2010 Posts: 245 Rep Power: 17	You can set up boundary conditions by two ways. In Ghost cell approach boundary values are stored in the ghost cells which is adjacent to boundary. Boundary fluxes are calculated from interior and ghost cell. Other way is to apply characteristics boundary condition where you can do it directly on face and estimate flux without additional cells.

March 9, 2012, 16:43		#4
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,195 Blog Entries: 29 Rep Power: 39	Besides the specific approach in use, in cell centered FV method you have variables stored in cell centers but you need their integrals on volume faces. When they are available they are used directly (velocity and/or mass flow inlets; wall normal derivative of the wall normal velocity component for incompressible flows; symmetry boundaries; outflows etc). In all the other cases, both on boundary and interior cells, some reconstruction/approximation is adopted by using known face or neighbor cell values.