Integral and boundary condition

Question

Hello all,

I try to implement the solution of the boundary integral of u^2.
So integral_(Neumann boundary) u^2 ds
This is equivalent to || u^2 || ^2_(Neumann boundary)

I want to solve it with the following norm
dolfin.fem.norms.norm(v, norm_type='L2', mesh=None)

So i started

n = 50
mesh = UnitSquareMesh(n,n)
V = FunctionSpace(mesh, "Lagrange, 1")

p = Expression('5.0')
u = Function(V)
u.interpolate(p)

nrm = norm(p, 'L2', mesh)

but this will solve it on the whole mesh.
Is it possible to solve in only on the boundary.

Thanks

Comments

wrong comment.

Answer 1

You can code this norm yourself, instead of using the dolfin.norm function. Staying close to your example, this can be done as follows:

from dolfin import *

n = 50
mesh = UnitSquareMesh(n,n)
V = FunctionSpace(mesh, "Lagrange", 1)

p = Expression('5.0')
u = Function(V)
u.interpolate(p)
# Compute L2 norm of u over boundary
bnrm = sqrt(assemble(u*u*ds))
print bnrm

Comments

Thank you.
I want to have 3 out of 4 boundaries to be Dirichlet and the 4th one Neumann.
Is it possible to do it like this ?

from dolfin import *

n = 50
mesh = UnitSquareMesh(n,n)
V = FunctionSpace(mesh, "Lagrange", 1)

p = Expression('5.0')
u = Function(V)
u.interpolate(p)

# Dirichlet boundary
def boundary(x):
    return x[0] < DOLFIN_EPS or x[1] < DOLFIN_EPS or x[1] > 1 - DOLFIN_EPS
u0  = Constant(0.0)
bc  = DirichletBC(V, u0, boundary)

# Compute L2 norm of u over boundary
temp = assemble(u*u*ds)
bc.apply(temp)
bnrm = sqrt(temp)
print bnrm

---

You want to integrate over the Neumann boundary, right? Then I would use the following approach:

from dolfin import *

class Right(SubDomain):
    def inside(self,x,on_boundary):
        return on_boundary and near(x[0],1) 

n = 50
mesh = UnitSquareMesh(n,n)
V = FunctionSpace(mesh, "Lagrange", 1)

p = Expression('5.0')
u = Function(V)
u.interpolate(p)

boundaries = FacetFunction("size_t", mesh) 
boundaries.set_all(0)

# Now mark your Neumann boundary
rightbound = Right()
rightbound.mark(boundaries, 1)
ds=Measure('ds')[boundaries]

# And integrate over Neumann boundary
bnrm = sqrt(assemble(u*u*ds(1)))
print bnrm

Hope this solves your issue?

---

double reply sorry

---

That's right. I want to integrate over the Neumann boundary.
But this is just a short extract of the whole programm in which I already used
the Dirichlet boundary

# Dirichlet boundary
def boundary(x):
    return x[0] < DOLFIN_EPS or x[1] < DOLFIN_EPS or x[1] > 1 - DOLFIN_EPS
u0  = Constant(0.0)
bc  = DirichletBC(V, u0, boundary)

I may try your approach. But is it also possible to solve the problem as I mentioned?

from dolfin import *

n = 50
mesh = UnitSquareMesh(n,n)
V = FunctionSpace(mesh, "Lagrange", 1)

p = Expression('5.0')
u = Function(V)
u.interpolate(p)

# Dirichlet boundary
def boundary(x):
    return x[0] < DOLFIN_EPS or x[1] < DOLFIN_EPS or x[1] > 1 - DOLFIN_EPS
u0  = Constant(0.0)
bc  = DirichletBC(V, u0, boundary)

# Compute L2 norm of u over boundary
temp = assemble(u*u*ds)
bc.apply(temp)
bnrm = sqrt(temp)
print bnrm

Otherwise I need to change the whole programm.
Does the bc.apply() command solve the problem?
(I want to have 3 Dirichlet and 1 Neumann boundary.)

I tried it like this, but i received an error do to the bc.apply(temp)..
Can someone help me with this?

---

Hi,

I tried to use your solution, but I received the following error error.
"Notation dx[meshfunction] is deprecated. Please use dx(subdomain_data=meshfunction) instead."
Sorry but perhaps it is a stupid question: How do I fix this?
I did it like this:

ds=Measure('ds', domain=mesh, subdomain_data=boundaries)

from dolfin import *

class Right(SubDomain):
    def inside(self,x,on_boundary):
        return on_boundary and near(x[0],1) 

n = 50
mesh = UnitSquareMesh(n,n)
V = FunctionSpace(mesh, "Lagrange", 1)

p = Expression('5.0')
u = Function(V)
u.interpolate(p)

boundaries = FacetFunction("size_t", mesh) 
boundaries.set_all(0)

# Now mark your Neumann boundary
rightbound = Right()
rightbound.mark(boundaries, 1)
ds=Measure('ds', domain=mesh, subdomain_data=boundaries)

# And integrate over Neumann boundary
bnrm = sqrt(assemble(u*u*ds(1)))
print bnrm

Am I right?