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

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

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                       ARE THERE QBC, NON-HOMOGENEOUS
                 BC, ESSENTIAL BC BY MODIFYING EQUATIONS?
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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

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  [ 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.

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  [ 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>