How to access Dof values of solution vector according to their geometric location?

[Tim hyvärinen]

Hi, deal.ii developers and users

Recently, I get this question as been mentioned in subject.

I have a saddle point problem on hand. The system of vector-valued PDEs is non-linear. I implement the solver with Newton iteration without line search. And I also need to set up a specific initial configuration for iteration. 

Here comes the question, I don't have clear idea how to access and write down a specific value on the 0-th solution vector according to every Dof's geometric location on grid.  

Let's say we have a 3-components vector-valued unknown. Geometry domain is  2 * 2 * 2 size cube.  I want to set Dofs, which are inside ball x^2+y^2+z^2=1,  to be  (1, 1, 1) at every vertex. While the rest of Dofs of the cube to be (1,0,0) at very vertex. 

I went through docs of DoFHandler,  DoFTools, ComponentMask and CellAccessor. Unfortunately, I only figured out that I can access certain DoFs in solution vector by loop over IndexSet, which is generated by DoFTools::extract_dofs(DofHandler., ComponentMask). I still no idea how to control the access through geometric information i.g., vertex's coordinates.

Is there any clue or idea how this could be done?

Best,

Timo

[Wolfgang Bangerth]

On 3/25/24 09:13, Tim hyvärinen wrote:
>
> Let's say we have a 3-components vector-valued unknown. Geometry domain
> is  2 * 2 * 2 size cube.  I want to set Dofs, which are inside ball
> x^2+y^2+z^2=1,  to be  (1, 1, 1) at every vertex. While the rest of Dofs
> of the cube to be (1,0,0) at very vertex.
>
> I went through docs of DoFHandler,  DoFTools, ComponentMask and
> CellAccessor. Unfortunately, I only figured out that I can access
> certain DoFs in solution vector by loop over IndexSet, which is
> generated by DoFTools::extract_dofs(DofHandler., ComponentMask). I still
> no idea how to control the access through geometric information i.g.,
> vertex's coordinates.

Tim:
you need to turn the problem around. You're *first* trying to partition
your DoFs into ones inside/outside the sphere and in a *second* step for
each element of the set assign them a value. Instead, combine the steps:
think of it as a loop over all DoFs and for each you determine whether
it is inside or outside the sphere, and assign a value.

In practice, the way to do this is to call
VectorTools::interpolate()
with a function object that will look like this:

class StartingValues : public Function<dim>
{
double vector_value (const Point<dim> &p) {
if (p inside sphere)
return 1,1,1
else
return 1,0,0
}
};

Best
W.