Thermoelastic problem

1. PDE

The variational form for the thermoelastic problem is

\[\int_{\Omega} \kappa \nabla T \nabla \hat{T} d x + \int_{\Omega} \sigma:\nabla v d x -\int_{\partial \Omega}\kappa(T \cdot n) \hat{T} d \partial \Omega -\int_{\partial \Omega}(\sigma \cdot \eta) \cdot v d s=0,\]

where the \(\sigma\), \(v\) are the stress tenser and the test functions for the displacements; \(T\) and \(\hat{T}\) are the function and the test functions for the temperature field.

1. Code

import dolfin as df

def get_residual_form(u, v, rho_e, T, T_hat, KAPPA, k, alpha, mode='plane_stress', method='RAMP'):
    if method=='RAMP':
        C = rho_e/(1 + 8. * (1. - rho_e))
    else:
        C = rho_e**3

    E = k * C
    # C is the design variable, its values is from 0 to 1

    nu = 0.3 # Poisson's ratio


    lambda_ = E * nu/(1. + nu)/(1 - 2 * nu)
    mu = E / 2 / (1 + nu) #lame's parameters

    if mode == 'plane_stress':
        lambda_ = 2*mu*lambda_/(lambda_+2*mu)

    I = df.Identity(len(u))
    w_ij = 0.5 * (df.grad(u) + df.grad(u).T) - alpha * I * T
    v_ij = 0.5 * (df.grad(v) + df.grad(v).T)

    d = len(u)

    sigm = lambda_*df.div(u)*df.Identity(d) + 2*mu*w_ij

    a = df.inner(sigm, v_ij) * df.dx + \
        df.dot(C*KAPPA* df.grad(T),  df.grad(T_hat)) * df.dx

    return a