Crossed gradient field

1

Hello everybody,

i have the following problem as you can see in the attached picture.
My gradient field is crossed. I would like to implement homogenous Neumann conditions on the left and all over the right side. But since the flux would be zero and therefore this part would fall out of the solution I don’t know how to handle it.

Could somebody please give me a hint?
I have tried to make the discretization finer since somebody was giving me this advice. But unfortunately that doesn’t helped.

Thank you!

solution1|640x480

2

Ho,
It would be beneficial to have a minimal example (I.e. a code) or mathematical description of your problem. For Poisson problems and linear elasticity, a homogeneous Neumann condition is enforced by simply not including the boundary integrals after integration by parts:
https://bitbucket.org/fenics-project/dolfin/src/master/python/demo/documented/neumann-poisson/demo_neumann-poisson.py.rst
(Replace g with 0).

Note that as long as your problem is not a pure Neumann condition problem, you do not have to think about Lagrange-multipliers for uniqueness

3

Thank you again for the great hint dokken.

This is a part of my code:

W = VectorFunctionSpace(mesh, "Lagrange", 1)
V = FunctionSpace(mesh, "Lagrange", 1)
u = TrialFunction(V)  #  For linear case
test_u = TrialFunction(W)
#u = Function(V)      # For nonlinear case
v = TestFunction(V)
uh = Function(V)
g = Constant(0.0)


c = Constant(1.0e0)
x = SpatialCoordinate(mesh)
f = sin(2.0*pi*x[0])*cos(2.0*pi*x[1])
n = FacetNormal(mesh)

# Tangential gradient operator:
def grad_t(u):
    return grad(u) - dot(grad(u),n)*n

# Randbedingungen-----------------------------------------------------------
class Left(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and (between(x[1], (-1.0, 1.0)) and between(x[0], (4.0, -4.0)))

class Right(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and ((between(x[1], (2.0, 2.0)) and between(x[0], (1.0, 3.0))) and (between(x[1], (0.1, 0.0)) and between(x[0], (2.0, 2.0))) and (between(x[1], (2.0, 0.0)) and between(x[0], (0.1, 0.0))) and (between(x[1], (5.0, -3.0)) and between(x[0], (2.0, 0.0))))

##Facetfunction
#facetfunc = MeshFunction("size_t", mesh, mesh.geometry().dim()-1)
#bnd = Right()
#bnd.mark(facetfunc, 3)
#File("facetfunc.pvd").write(facetfunc)


class Bottom(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and (between(x[1], (5.0, -3.0)) and between(x[0], (4.0, -4.0)))

class Top(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and (between(x[1], (1.0, 3.0)) and between(x[0], (-1.0, 1.0)))


# Initialize sub-domain instances
left = Left()
top = Top()
right = Right()
bottom = Bottom()

# Initialize mesh function for boundary domains
boundaries = MeshFunction("size_t", mesh, mesh.topology().dim()-1, 0)
boundaries.set_all(4)
left.mark(boundaries, 1)
top.mark(boundaries, 2)
right.mark(boundaries, 3)
bottom.mark(boundaries, 4)


# Define Dirichlet boundary condition at top boundary
bc2 = DirichletBC(V, 0.0, boundaries, 2)

#Take a look at the solution:
ds=Measure('ds', domain=mesh, subdomain_data=boundaries)
area_gamma1 = assemble(1*ds(4))
leitfaehigkeit=8.6e6

a = inner(grad(u), grad(v))*dx
L = g*v*ds(1)+g*v*ds(3)+leitfaehigkeit*(1/area_gamma1)*(-zz[1, 3])*v*ds(4)
solve(a==L,uh, bc2)

p = plot(uh)
plot(grad(uh))
cbar = plt.colorbar(p)
plt.show()
plt.savefig('solution' + str(1) + '.png')

zz is a numerical solution from an ODE-System.

4

One can take for testing just anything.