Linear elastic problem
==========================

1. PDE
-------------
The variational form for the linear elastic problem is

.. math:: \int_{\Omega}\sigma:\nabla v d x -\int_{\partial \Omega}(\sigma \cdot \eta) \cdot v d s=0 ,

where the :math:`\sigma`, :math:`v` are the stress tenser and the test functions. 


2. code
-------------
.. code-block:: python

  import dolfin as df


  def get_residual_form(u, v, rho_e, E = 1, method='SIMP'):
      if method =='SIMP':
          C = rho_e**3
      else:
          C = rho_e/(1 + 8. * (1. - rho_e))
      

      E = 1. * 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


      w_ij = 0.5 * (df.grad(u) + df.grad(u).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 
      
      return a