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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
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
[ 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>
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