Thanks to David and Vikram, your explanation helps me a lot. I'll do
further exploration from your points.
My current work is just to repeat what adaptivity_ex2.C does, with aim of
showing we can solve PDEs with another programming model.
--Junchao Zhang
On Tue, Jan 12, 2016 at 3:06 PM, Vikram Garg <vikram.v.g...@gmail.com>
wrote:
> 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.
>> >
>>
>> ------------------------------------------------------------------------------
>> Site24x7 APM Insight: Get Deep Visibility into Application Performance
>> APM + Mobile APM + RUM: Monitor 3 App instances at just $35/Month
>> Monitor end-to-end web transactions and take corrective actions now
>> Troubleshoot faster and improve end-user experience. Signup Now!
>> http://pubads.g.doubleclick.net/gampad/clk?id=267308311&iu=/4140
>> _______________________________________________
>> Libmesh-users mailing list
>> Libmesh-users@lists.sourceforge.net
>> https://lists.sourceforge.net/lists/listinfo/libmesh-users
>>
>
>
>
> --
> 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
>
------------------------------------------------------------------------------
Site24x7 APM Insight: Get Deep Visibility into Application Performance
APM + Mobile APM + RUM: Monitor 3 App instances at just $35/Month
Monitor end-to-end web transactions and take corrective actions now
Troubleshoot faster and improve end-user experience. Signup Now!
http://pubads.g.doubleclick.net/gampad/clk?id=267308311&iu=/4140
_______________________________________________
Libmesh-users mailing list
Libmesh-users@lists.sourceforge.net
https://lists.sourceforge.net/lists/listinfo/libmesh-users
