Junchao,
Like David mentioned, the practical difficulty that the
penalty method poses is in matrix-conditioning, but this problem has been
largely alleviated by pre-conditioners/solvers today. Another problem is
that the penalty method can have a disproportionate impact on the
evaluations of certain functionals (quantities of interest) of the
solution, such as the boundary flux. But this can also be overcome.
The penalty method can be viewed in many ways. One of the simplest way is
to think of a penalty method, is to think of solving a constrained
optimization problem. So. instead of solving,
Lu = f in the interior,
u = g on the boundary.
One can think of solving the optimization problem,
Find u which minimizes: J(u) = 1/2 * ||Lu - f|_{interior}| + penalty * ||u
- g||_{boundary}
Such formulations are very common in optimization.
Another interpretation is to think of applying a Robin/Mixed boundary
condition instead of a Dirichlet one, so we solve:
Lu = f in the interior,
u = g + (1\penalty) * grad_n(u) on the boundary.
I believe by smoothing function, you are either implying using a Lagrange
multiplier or the Nietsche method. Perhaps you need to do this because your
Dirichlet boundary data (g) is not smooth ? The Lagrange multiplier should
be possible to do in libMesh, not sure about Nietsche. But I would start
with penalty even in this case.
If your boundary data is smooth enough, you can just use the Dirichlet
enforcement as the others have pointed out.
Thanks.
On Tue, Jan 12, 2016 at 2:36 PM, Junchao Zhang <junchao.zh...@gmail.com>
wrote:
> I took some notes when my boss told me the smoothing function months ago.
> He thought we should treat interior points equally with boundary points and
> thus penalty method is garbage in his words:). For a domain [0, 1] x [0,
> 1], we can multiply unknown u with a function such as x(1-x)y(1-y), making
> the boundary all zeros.
>
> I am a guy from computer science. Understanding FE is hard for me. Hope I
> can get some hints here.
> Thank you.
>
> --Junchao Zhang
>
> On Mon, Jan 11, 2016 at 5:46 PM, Geordie McBain <gdmcb...@freeshell.org>
> wrote:
>
> > 2016-01-12 8:44 GMT+11:00 Junchao Zhang <junchao.zh...@gmail.com>:
> > > Hello,
> > > I am learning FEM by reading the Libmesh example adaptivity_ex2.C
> and
> > > have two questions.
> >
> > > 2) The code uses the penalty method to impose boundary conditions. I
> > was
> > > told the penalty method is bad and a smoothing function with zero on
> > > boundary is preferred. Could someone point me to literatures or
> examples
> > on
> > > smoothing?
> >
> > Hello. I'm not sure that 'the penalty method is bad'. There can be
> > some technical issues with it in specialized problems, but if you're
> > only just learning FEM, I wouldn't worry about them for the moment. A
> > good introductory reference for the penalty method is the section
> > 'Essential boundary conditions' on pp. 121-122 of
> >
> > Becker, E. B., G. F. Carey, & J. T. Oden (1981). Finite Elements,
> > An Introduction,
> > Volume 1 of The Texas Finite Element Series. Englewood Cliffs, New
> > Jersey: Prentice-Hall
> >
> > which includes: 'The penalty method is not only easy to implement and
> > mathematically sound but it is also easy to motivate by physical
> > arguments.'
> >
> > I haven't heard of the 'smoothing function' idea, but if you do need
> > or would prefer to avoid the penalty method, libMesh does have the
> > DirichletBoundary, as in introduction_ex4.C.
> >
>
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--
Vikram Garg
Postdoctoral Associate
Predictive Engineering and Computational Science (PECOS)
The University of Texas at Austin
http://web.mit.edu/vikramvg/www/
http://vikramvgarg.wordpress.com/
http://www.runforindia.org/runners/vikramg
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