Re: [Libmesh-users] Are there QBC, non-homogeneous BC, essential BC by modifying equations?

[John P. <jwp...@gm...>]

Hi Edgar,

Sorry for not responding, your email got kind of lost in my inbox.

LibMesh does support penalty BCs, some examples which use penalty BCs
are: adjoints_ex1,2,4; adaptivity_ex1,2,4. It sounds like you may be
interested in DirichletBoundary objects, which impose the BCs as
user-defined constraints. Examples using DirichletBoundary objects are:
adaptivity_ex3; adjoints_ex5,6,7; fem_system_ex1,3,4. The implementation of
DirichletBoundary objects corresponds to the description you gave in your
original email about modifying the matrix and rhs.

--
John


On Wed, May 5, 2021 at 2:50 PM edgar <edg...@cr...> wrote:

> Nevermind. I found this [1]:
>
>      - Cons of this approach :
>        - Works for an interpolary finite element basis but not in
> general.
>        - May be inefficient to change individual entries once the global
> matrix is assembled.
>      - Need to enforce boundary conditions for a generic finite element
> basis at the element stiffness matrix level.
>
> [1]
>
> https://users.oden.utexas.edu/~roystgnr/libmeshpdfs/roystgnr/sandia_libmesh.pdf
>
> On 2021-04-24 05:14, edgar wrote:
> > ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
> >                       ARE THERE QBC, NON-HOMOGENEOUS
> >                 BC, ESSENTIAL BC BY MODIFYING EQUATIONS?
> >            ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
> >
> >
> > Hello busy developers and scientists! I’ll try to make it short.
> >
> > [Logan12] calls them /nonhomogeneous boundary conditions/ (BC),
> > [Ashgar05] refers to them as /essential boundary conditions by
> > modifying
> > equations/ and I am sure that some people call them qualifying boundary
> > conditions (who?).
> >
> > The procedure consists, for example, from a 1D system like this (2
> > elements, 3 nodes), where nodes 1 (n1) and 3 (n3) have an imposed value
> > of f and node 2 (n2) has essential (Dirichlet) BC. The global matrix of
> > the system would be like this
> >
> > ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
> >  [ k_{1,1}  k_{1,2}  k_{1,3} ]     [ u_{1} ]     [ f_{1} ]
> >  [ k_{2,1}  k_{2,2}  k_{2,3} ]  ·  [ u_{2} ]  =  [ f_{2} ]
> >  [ k_{3,1}  k_{3,2}  k_{3,3} ]     [ u_{3} ]     [ f_{3} ]
> > ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
> >
> > The above mentioned procedure does this [Ashgar05]:
> >       (i) Insert 1 at the diagonal location and zero at all
> >       off-diagonal locations in the row corresponding to the
> >       specified degree of freedom.
> >
> >       (ii) Insert the known nodal value to the corresponding
> >       location in the right-hand side.
> >
> > ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
> >  [ k_{1,1}  0  k_{1,3} ]     [ u_{1} ]     [ f_{1} ]
> >  [    0     1     0    ]  ·  [ u_{2} ]  =  [  Δ{u} ]
> >  [ k_{3,1}  0  k_{3,3} ]     [ u_{3} ]     [ f_{3} ]
> > ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
> >
> > I read that libMesh solves equations with a penalty method [CMISS,
> > hooshi]. Does it also apply the method described above?
> >
> > That’s it. Thanks!
> >
> >
> >
> > [Logan05] Daryl L. Logan, A first course in the finite element method,
> > (2012), Cengage Learning, Stanford, CT, U.S.A.
> >
> > [Ashgar05] M. Ashgar Bhatti, Fundamental finite element analysis and
> > applications: with mathematica and matlab computations (2005), John
> > Wiley \& Sons.
> >
> > [CMISS]
> > <https://www.cmiss.org/openCMISS/wiki/LibmeshBoundaryConditions>
> >
> > [hooshi] <https://hooshi.gitlab.io/FILES/math521_libmesh.pdf>
>