Nonlinear equation with space dependent term

variational formulation
1

Hello everyone, this is my first post, I am fairly new to fenics and I’m having an issue with the implementation of a non linear heat transport equation. I have this source term, dependent on the temperature and x y and z. How can I write this convolute term in my weak formulation to solve it? Thank you very much.Screenshot_28

2

Hi, what is \alpha(T) and R(T) ? T is a given function ? For the spatially dependent part you can use something like

a = ...
w = ...
l = ...
A = ...
R = ...
S = Expression("A*(1-R)*a*exp(a*x[2])*
                exp(-2*pow(x[0]/w,2)*exp(-2*pow(x[1]/l,2)", 
                A=A, R=R, a=a, w=w, l=l, degree=2)
3

Thanks for the response.
Alpha and R are functions depending on T, that is the main variable, it’s the temperature.

4

Could you give the implicit or explicit relationship for these variables, such that one can further help you.
One can use


x=SpatialCoordinate(mesh)
S = A*(1-R)*a*exp(a*x[2])*exp(-2*(x[0]/w)**2)*exp(-2*(x[1]/l)**2)

where you define your other variables prior in the script.

5

Thank you, these are the relationships for the variables, lam is a constant.
k = 1523.7*(T**(-1.226))
alpha = 4pik/lam
R = (7.188+k**2)/(21.912+k**2)

6

Thank you for the space dependent part, now I’m facing another issue, how can I use a numpy method as “exp” with a function?

7

Exp has its own dolfin function, which is loaded whenever you do either from dolfin import exp or from dolfin import *
MWE:

from dolfin import *
mesh = UnitSquareMesh(10,10)                                            
V = FunctionSpace(mesh, "CG", 1)                                        
x = SpatialCoordinate(mesh)                                             
u = Function(V)                                                         
a = exp(Constant(2)*u)                                                  
assemble(a*dx)