Taking the Robin boundary conditions into account
Computation of the right-hand side :
The vector which must be assembled is defined by :
with :
- i
are
shape functions defined on the reference interval
- is the number
of nodes per edge
- n1 and n2 are the start and end nodes
of the edge
-
are uA are the values of the boundary condition on the edge
Computation of the element matrix :
The matrix which must be assembled is defined by :
Taking the boundary conditions of first type into account :
To take these boundary conditions into acount, we modify the right-hand size using the following method :
with :
- g1(xj) : the value of the Dirichlet boundary
condition at the point xj
- : containing
the numbers of the nodes of the edge E(e3) which do
not have Dirichlet boundary conditions.
- : containing
the numbers of the nodes of the edge E(e3) which have
Dirichlet boundary conditions.
We also modify the stiffness matrix with :
with
: containing the numbers of the nodes of the edge E(e3)
Assembling process :
For each border domain of type 3
For each edge of the domain
Computation of f(e3) und K(e3)
Modification of f(e3) and K(e3) to take the boundary conditions of first type into account
For each node of the edge, which has i as global number and k as local number
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For each node of the edge, which has j as global number and l as local number