Boundary Condition of Heat Transfer Problem

General
1

Dear all, I want to solve heat transfer problem on a rectangular plate, where the boundary conditions are:
(i) Initial temperature of the model is 0 degree Celsius.
(ii) The right boundary is suddenly heats up to 400 degree Celsius.
(iii) all other boundaries are insulated.
I defined the boundary conditions as follows:

#Initial temperature
u_0 = Constant(0.0)  
u0=interpolate(u_0, V)

#Right Boundary
def right_boundary( x, on_boundary):
    return on_boundary and ((abs(x[0])-1)< DOLFIN_EPS)

bc1= DirichletBC(V, Constant(400.0),right_boundary)

# Insulated Boundary
class InsulatedBoundary(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and (near(x[0], 0) or near(x[1], 1) or (near(x[1],0)))
bc2 = DirichletBC(V, Constant(0.0), InsulatedBoundary())
bc=[bc1, bc2]

However, i am not getting my desired result. i guess, my defined insulated boundary is not correct. Please anyone help me to figure out my mistake.

2

When you say the boundary is insulated, I guess that shouldn’t mean that the boundary value should be 0? Doesn’t insulation yield something closer to a Neumann condition, i.e.
\frac{\partial u}{\partial n} = 0?

You can check your boundary conditions with the following code:

from dolfin import *

mesh = UnitSquareMesh(10, 10)
V = FunctionSpace(mesh, "Lagrange", 1)


# Initial temperature
u_0 = Constant(0.0)
u0 = interpolate(u_0, V)

# Right Boundary


def right_boundary(x, on_boundary):
    return on_boundary and ((abs(x[0])-1) < DOLFIN_EPS)


bc1 = DirichletBC(V, Constant(400.0), right_boundary)

# Insulated Boundary


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


bc2 = DirichletBC(V, Constant(0.0), InsulatedBoundary())
bc = [bc1, bc2]

u = Function(V)
u.vector()[:] = -400 # This will be the value of all interior nodes.
[b.apply(u.vector()) for b in bc]
with XDMFFile("test.xdmf") as xdmf:
    xdmf.write(u)