Dear James
Why not simply impose the boundary condition directly on the degree of
freedom?
The degree of freedom need not be on the boundary of the domain. All
that is important is that you create a map between the degrees of
freedom you need to constrain and the amount you wish to constrain
them. The following example is used to fix the horizontal degree of
freedom at a specific node in the mesh:
{
std::map<unsigned int, double> boundary_values;
// points to constrain a dof
Point<deal_II_dimension> fix_point_A(-3.0, -7.7, 3.0); //
point to fix
DoFHandler<deal_II_dimension>::active_cell_iterator
cell = dof_handler.begin_active(), endc = dof_handler.end();
std::vector<bool> vertex_touched(triangulation.n_vertices(),
false);
for (unsigned int cellCount = 0; cell != endc; ++cell, +
+cellCount) {
for (unsigned int v = 0; v <
GeometryInfo<deal_II_dimension>::vertices_per_cell; ++v) {
if (vertex_touched[cell->vertex_index(v)] == false) {
vertex_touched[cell->vertex_index(v)] = true;
Point<deal_II_dimension> vertex_position = cell-
>vertex(v);
if (vertex_position(0) == fix_point_A(0) &&
vertex_position(1) == fix_point_A(1)) {
boundary_values[cell->vertex_dof_index(v, 0)]
= 0.0;
}
}
}
}
MatrixTools::apply_boundary_values(boundary_values, K,
delta_d, f_residual);
}
Regards
Andrew
Dear all,
I am doing calculations containing a number of metallic regions,
modeled by fixed solutions to Poisson's equation. That is, a good
number of internal DOFs should be fixed when solving Poisson's
equation. However, I cannot simply exclude the regions from the mesh
and use Dirichlet boundary conditions, since the cells are necessary
for subsequent calculations, and mesh refinement should extend into
the fixed regions.
I have so far not been able to solve this problem in a satisfying way
using deal.II. I have attempted the following schemes:
1. Using Dirichlet boundary conditions using
VectorTools::interpolate_boundary_values. This causes Deal.II to
abort with an exception complaining that the fixed DOFs are not on
a boundary surface - which is obviously true, but is that
necessary?
2. For each fixed degree of freedom i with value V_i, set the ith row
of the Laplace matrix to T_{ij} = \delta_{ij} and rhs_i = V_i.
This works, but destroys the symmetry of the Laplace matrix and
causes convergence to be very slow - in the examples I've tried,
solving Poisson's equation is slowed down by a factor of 10-15
compared to CG on the symmetric matrix.
3. My third idea was to use two meshes. I.e., I would build the full
mesh, doing all my refinement, but before solving Poisson's
equation, cut out the fixed regions and apply Dirichlet boundary
conditions to the surfaces of the "holes". After solving, embed
the solution back into the full mesh.
The problem with this is that I cannot seem to find any mechanism
in Deal.II to remove or deactivate cells from an existing mesh.
After pulling my hair for some time over this, I've concluded that
there really must be a better way of achieving this. After all, I'm
almost certain that I'm not the only one who needs to do FEM
calculations in which some of the internal DOFs are fixed. Needless to
say, any help will be greatly appreciated!
Thank you very much in advance.
With kind regards,
James Avery
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