Linear thermoelasticity in 2D domain

[Arijit-Majumdar]

Hi,
I am trying to solve linear thermoelasticity in a 2D beam in Firedrake. For the temperature I am solving $div(k\nabla T) = 0$. The variational equation is as follows
aT = Constant(k)*dot(grad(T_test),grad(T_trial))*dx
LT = Constant(hT)*T_test*ds(2)
solve(aT == LT, T_soln, bcs = bcT, solver_parameters={'ksp_type': 'gmres'})

The temperature profile looks fine to me. After solving for the temperature I am solving the displacement which follows these equations
$\nabla\cdot\sigma = f$
where $\sigma = \lambda tr(\epsilon) + 2\mu\epsilon$ and $\epsilon = 0.5(\nabla u + \nabla u^T) + \alpha T I_d$. Here $\lambda$ and $\mu$ are constants and $I_d$ is 2x2 identity matrix. I first made two functions for $\sigma$ and $\epsilon$
def epsilon(u, Temp):
return 0.5*(grad(u) + grad(u).T) + alpha*Temp*Id

def sigma(u, Temp):
return lambda_*div(u)*Id + 2*mu*epsilon(u, Temp)

Using the variational form I get is
a = inner(sigma(u, T_soln), epsilon(v,T_soln))*dx
L = dot(f, v)*dx
where u and v are the trial and test functions for displacement and T_soln is the temperature from solving the heat equation. But when I try to solve the above variational equation I get the following error
ArityMismatch: Adding expressions with non-matching form arguments ('v_0',) vs ().

I am providing link to my colab notebook. Any help is greatly appreciated.
Colab notebook : https://colab.research.google.com/drive/1JmDhteWrtnqQqy4Tq_OxndxmwrHbI2Us?usp=sharing

Thanks

[dham]

Your issue is in the the definition of epsilon. The first term contains u while the second term does not. When you pass in an Argument (a test function or trial function) in as u, this means the first summand has an argument and the second does not. That causes the error. Mathematically, what you have is invalid because a must be linear in both u and v, and that definition of epsilon is not linear.