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From: KIRK, BENJAMIN (JSCEG) (NASA) <benjamin.kirk1@na...>  20050107 14:43:34

It's not clear that you need to use the constraint matrix for this = case... For example, consider Laplace's problem in 2D: =20  div(grad(u)) =3D 0 with u=3Dg on some part of the boundary, du/dn=3Dh on the remainder. In the weak statement you perform integrationbyparts, and the term = du/dn appears in a boundary integral and you can use this to impose a Neumann = BC in the RHS of the system. See, for example, http://cfdlab.ae.utexas.edu/~benkirk/seminar/talk.pdf, Especially page = 4. In a similar way, Robin BCs can be implemented by letting h =3D alpha u = + beta du/dn. In this case you have two boundary terms that contribute to the system matrix. For implementation, you simply need to define a finite element object = that lives on the boundary and can be used to integrate along the edge to = provide the required term. Clear? Unfortunately, none of the examples show this. If you have any more questions let me know, it would be straightforward to modify ex14 to = show this procedure since it has the exact solution's derivative available. Regardless, we should probably do that since it seems silly to have 14 examples, all with Dirichlet BCs! Ben =20 Original Message From: libmeshusersadmin@... [mailto:libmeshusersadmin@...] On Behalf Of Michael Schindler Sent: Friday, January 07, 2005 3:26 AM To: libmeshusers@... Subject: [Libmeshusers] Constraint matrix for boundary conditions Hello, I would like to enforce nonDirichlet boundary conditions via the = constraint matrix. Does someone have an example how to implement this for e.g. = Neumann or Robin BC? Thanks, Michael. =20 "A mathematician is a device for turning coffee into theorems" Paul Erd=F6s.  The SF.Net email is sponsored by: Beat the postholiday blues Get a = FREE limited edition SourceForge.net tshirt from ThinkGeek. It's fun and = FREE  well, almost....http://www.thinkgeek.com/sfshirt _______________________________________________ Libmeshusers mailing list Libmeshusers@... https://lists.sourceforge.net/lists/listinfo/libmeshusers 
From: Michael Schindler <mschindler@us...>  20050107 15:05:06

Hello Ben, On 07.01.05, KIRK, BENJAMIN (JSCEG) (NASA) wrote: > For example, consider Laplace's problem in 2D: > >  div(grad(u)) = 0 > > with u=g on some part of the boundary, du/dn=h on the remainder. > > In the weak statement you perform integrationbyparts, and the term du/dn > appears in a boundary integral and you can use this to impose a Neumann BC > in the RHS of the system. See, for example, > http://cfdlab.ae.utexas.edu/~benkirk/seminar/talk.pdf, Especially page 4. Do you mean just to replace du/dn in the boundary integral by h in order to "enforce" du/dn = h ?? This would be the FEM standard procedure, I guess. Exactly at this point I have some doubts. Consider a different setup, where I add the equation du/dn = h weighted with some penalty, to the linear system to be solved. This works quite well. The first prodecure is equivalent to this penaltyprocedure, if one sets the penalty to 1 which is not a really good value for a penalty. > It's not clear that you need to use the constraint matrix for this case... Therefore, I wanted to look for an alternative approach to enforce the boundary condition  and came across the constraint matrices. > For implementation, you simply need to define a finite element object that > lives on the boundary and can be used to integrate along the edge to provide > the required term. Clear? Yes, clear. For an element I would get a constraining equation (again for the case dn/du = h) h = \sum_i w_i u_i where u_i are the degrees of freedom and w_i are some weights determined by the geometry of the element. Now comes the next difficulty. I have an idea how the method DofMap::constrain_matrix_and_vector works. It creates a matrix C from the constraints and uses these constraints while inserting the correct values into the big system matrix. I would expect this _only_ to work for h = 0. Do I miss something here? > Unfortunately, none of the examples show this. If you have any more > questions let me know, it would be straightforward to modify ex14 to show > this procedure since it has the exact solution's derivative available. > Regardless, we should probably do that since it seems silly to have 14 > examples, all with Dirichlet BCs! This would be great! Thanks, Michael.  "A mathematician is a device for turning coffee into theorems" Paul Erdös. 
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