Charged particles motion inside magnetic field

variational formulation
1

Hello to everyone!
I am trying to model a velocity vector field of a stationary motion of charged particles inside magnetic field. The equation is very simple:

$$ \dot{\vec{u}}=A \vec{u}\cross\vec{B} $$
$$ A=const $$

but I have troubles composing variational formulation.
Taking this and this examples as guides, I came up with this .ufl file:

from ufl_legacy import (Coefficient, dx,
                        tetrahedron, Constant, VectorElement,
                        TestFunctions, TrialFunctions)

dt = 1
A = 1

V = VectorElement("Lagrange", tetrahedron, 2)

u1, u2, u3 = TrialFunctions(V)
un1, un2, un3 = TrialFunctions(V)
v1, v2, v3 = TestFunctions(V)
k = Constant(dt)
A = Constant(A)
B1 = Coefficient(P)
B2 = Coefficient(P)
B3 = Coefficient(P)

a = ((u1 - un1) / k) * v1 * dx - A * (u2 * B3 - u3 * B2) * v1 * dx \
  + ((u2 - un2) / k) * v2 * dx - A * (u3 * B1 - u1 * B3) * v2 * dx \
  + ((u3 - un3) / k) * v3 * dx - A * (u1 * B2 - u2 * B1) * v3 * dx
L = 0

I am planning to define non-constant magnetic field $$ \vec{B} $$ later in .cpp file.
But for now, running the ffc -l dolfin magnetic.ufl command yields ufl_legacy.log.UFLException: Invalid cell 1.
I believe that this error message is caused by incorrect variational formulation. Does anyone know how to fix it, or alternatively know of a different way of solving this equation?

Many thanks,
Michael

2

Hi Michael,

What is P (the argument to Coefficient) in this code?

Also, If I’ve interpreted your notation correctly, un1, un2 and un3 are the components of u at the previous timestep (please correct me if I’m wrong here). You have defined these un’s as trial functions, but I presume the values at the previous timestep are known and should therefore rather be defined as Function objects.

Cheers,
Halvor