I started using FreeFEM. I find it very easy to understand and use. I started liking it and considering to integrate it into my research workflow. I am looking for some clarification regarding the application of boundary conditions.

I read in the FreeFEM documentation that Dirichlet boundary conditions are imposed using the penalty method using a very high value (10^30) for the penalty constant.

My experience indicates that such an approach results in ill-conditioned matrices and poses many issues in the modal analysis (eigenvalue problems) and buckling analysis due to the sensitivity of results to the penalty constant.

I am wondering you have any plans for implementing the elimination approach for imposing Dirichlet BCs in FreeFEM in the future.

Hello Chenna,
Please have a look at these slides. You can do unsymmetric elimination (tgv = -1) or symmetric elimination (tgv = -2) for homogenous Dirichlet boundary conditions.

when you compute generalize eigen value problem
the problem this find smallest eigen value (not all) of
A u = lambda B u
if you use generalize inverse mode you compute eigen value
of compute eigen value of
A^-1 B u = lambda^-1 u

to remove this problem you just put
tgv trick on A and no boundary condition on B then you get correct results with non problem.

if you add tgv=-2 on both matrix A, B then you get a lot of eigen values 1 form boundary part,
and so you have to remove value.

Form me, the tgv technique is very simple and in fact they do not change the condition number a level of
floating point number.

That is not correct to put tgv = -2 on B, I think. If you put tgv = -20 on B, then you won’t have spurious eigenvalues.
Reminder: tgv = -20 => put the complete row and column to 0 (dangerous, singular symmetric system), tgv = -10 => put the complete row to 0 (dangerous, singular nonsymmetric system).

Also, if you solve block systems iteratively, e.g., [[A, B]; [C, D]], and don’t impose Dirichlet boundary conditions exactly, it’s a nightmare with penalization. If you put tgv = -1 on A and D, and tgv = -10 on B and C, then no problem.

I must say that this increased my confusion. I will go through the presentation to understand how these different options work.

I think a cleaner implementation in which only the unknown DOFs are solved would be good idea. Having said that, I can understand that you might have your reasons for following the approach you have implemented now.

Don’t worry, if you want to do elimination, just use tgv = -1. The issue raised by Frédéric can be bypassed by some other tgv values, but this is really for power-user.

Just a small remark, Dirichlet unknowns are actual unknowns. If you are referring to an implementation where we’d actually remove the associated rows/columns from the coefficient matrix, then this is usually a bad idea. It requires a lot of bookkeeping (because you need to modify the sparsity pattern of the systems, which is sparse, and operations on sparse matrices are usually slow), for almost zero advantages.