How to apply flux on a curved surface?

variational formulation
1

Hi everyone,

I’ve have implemented flux boundary conditions as follows:

#Define Problem
u = TrialFunction(V)
v = TestFunction(V)

#Variables
D = Constant(1.71*pow(10,-5)) # mm^2/s
sim_time = 61 # seconds
time_steps = 122
dt = sim_time/time_steps
tol = 1E-14

#Define Boundary Conditions
k1 = Constant(0.009221) # per second
A1 = Constant(0.24) # micro-gram per mm^2
Flux = Expression('t<30+1e-9 ? k1*A1*exp(-1*k1*t) : 0', degree = 1, k1=k1, A1=A1, t=0)
u_D = Constant(0)

boundaryconditions = { 
    1 : {"Neumann" : 0},
    2 : {"Neumann" : 0},
    3 : {"Neumann" : 0},
    4 : {"Neumann" : 0},
    5 : {"Neumann" : 0},
    6 : {"Neumann" : 0},
    7 : {"Neumann" : Flux},
    8 : {"Neumann" : 0},
    9 : {"Neumann" : 0},
    10 : {"Neumann" : 0}
}

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

integrals_N = []
for i in boundaryconditions:
    if "Neumann" in boundaryconditions[i]:
        if boundaryconditions[i]["Neumann"] != 0: #if not zero flux
            g = boundaryconditions[i]["Neumann"]
            integrals_N.append(D*dt*v*g*ds(i))

F = u*v*dx + D*dt*dot(grad(u),grad(v))*dx -sum(integrals_N) - v*u_n*dx
a,L = lhs(F),rhs(F)

#Solving Problem
u = Function(V)
t = -1*dt

for i in range(time_steps):
    t+=dt
    Flux.t = t
    solve(a==L,u,bcs)
    vtkfile<<(u,t)
    u_n.assign(u)

The problems I’m having are:

  1. The flux does not seem to be implemented uniformly? Does this have to do with a curved surface?
    Flux
    Flux.png1460×808 466 KB
  2. How do I minimize the negative values? To make it more obvious I’ve scaled the same image to exclude the negative values.
    Flux_scaled
    Flux_scaled.png1460×808 425 KB

Any help will be greatly appreciated.

2

For anyone who made a similar mistake: The error lies in the flux function which should not include the diffusivity term. So it should be:

integrals_N.append(dt*v*g*ds(i))
1 Like
3

Hi, could you please explain how to apply flux boundary condition for me?Does anybody has a reference example?Thank you.

4

Hi, you can refer to the example above. Simply use an expression to to define your flux and subsequently use that expression in your numerical integration (see “integrals_N.append(…)” and “F = u * v * dx + … - sum(integrals_N)” ).
The only mistake I had made was multiplying the diffusivity term again when its value was already captured in my expression. Hope that helps :slight_smile:

5

Thank you.I will have a try.Is this flux a scalar or a vector?

6

Its applied normal to the surface specified by ‘ds’ in the code

7

OK.Thank you very much.