How to express temperature-dependent thermal conductivity

dolfinx
1

When the thermal conductivity is a constant, the general heat conduction problem is solved as follows. If the thermal conductivity is a function of temperature, for example, kappa = T**1.5, how should the corresponding variational equation be modified?

from mpi4py import MPI
from dolfinx.io import gmshio
from dolfinx import io
from ufl import (TestFunction, TrialFunction, dot, dx, grad)
from dolfinx.fem import (functionspace, dirichletbc, locate_dofs_topological)
from dolfinx.fem.petsc import LinearProblem

mesh, cell_markers, facet_markers = gmshio.read_from_msh("f.msh", MPI.COMM_WORLD, gdim=3)

q_v = 0.01    #power
kappa = 0.002  #thermal conductivity

VT = functionspace(mesh, ("CG", 1))
Tf = 1000.0
facets = facet_markers.find(6)
fdim = mesh.topology.dim - 1
dofs = locate_dofs_topological(VT, fdim, facets)
bcT = dirichletbc(Tf, dofs, VT)

T_ = TestFunction(VT)
dT = TrialFunction(VT)
aT = dot(grad(dT), grad(T_)) * dx
LT = q_v/kappa * T_ * dx

problem = LinearProblem(aT, LT, bcs=[bcT], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
Delta_Tf = problem.solve()
2

This would make the problem nonlinear. Essentially, you’re looking for this:
https://jsdokken.com/dolfinx-tutorial/chapter2/nonlinpoisson.html

3

Thank you for your reply. I’m currently working through this tutorial.