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

[edgar <edg...@cr...>]

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>