Constrains using null space in dolfinx

[francesco-ballarin]

@bagla0

Hi,
I am implementing nullspace to apply constrains in alternative to lagrange multiplier in dolfinx. The MWE is:

IMHO, @bagla0 you should have led with a bit more of context. By previous interactions, I guess that these questions are related to Mixed function space with dolfinx - how to implement an integral constraint - #6 by bagla0, but any other reader may not be aware of this.

@dokken

@bagla0

3b. I got error in c.interpolate. I don’t understand its significance of using x.shape. What should be for this example?

You have a function space V that is a vector function space.
The input to interpolate is a set of coordinates, x, of shape (3, number_of_points), where x[0] is the list of all x-coordinates, x[1] all y coordinates etc.
The input is designed this way so that you can used vectorized functions in numpy.
As your space is a vector space of dim=3, the output should be (3, number_of_points), not just number_of_points.

In addition to what @dokken is saying, keep in mind that should have two rigid body motions (in 2D) associated to a translation. A basis could be (1, 0) for displacement in the x-direction and (0, 1) for displacement in the y-direction. Therefore, rather than

c.interpolate(lambda x: np.ones(x.shape[1]))

you will have to come up with a lambda function that is non-zero only on one component of the displacement. To do that, follow the suggestions above.

@dokken

@bagla0

if I want a constraint like \int dv.dx(0) =0∫dv.dx(0)=0\int dv.dx(0) =0, how can I implement this ?

Not sure why you would want such a constraint, what is the physical interpretation of this.

Not sure either. Instead, @bagla0 you definitely need to add a third (in 2D) vector in null space to prevent rigid rotations. That’s not really my field, but I guess the third element in the null space could be something like (-y, x) in 2D. Maybe @bleyerj can confirm, or fix my suggestion if it is wrong. Depending on what the third vector looks like, you may need to carry out an orthonormalization with dolfinx.la.orthonormalize before passing the three vectors to PETSc.