Nonlinear finite elements/Quiz 1

Quiz
''Answer the following questions. You have 15 minutes.''

Heat conduction in an isotropic material with a constant thermal conductivity and no internal heat sources is described by Laplace's equation

\nabla^2 T = 0 ~\qquad~ \text{or,} \qquad \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = 0 ~. $$


 * 1) Derive a symmetric weak form for the Laplace equation in 1-D (an insulated rod).
 * 2) What are the expressions for the components of the finite element stiffness matrix ($$K_{ij}$$) and the load vector ($$f_i$$) for this 1-D problem?
 * 3) Assume that the one of ends of the rod is maintained at a temperature of $$T_1$$ (which is nonzero) and the other end has a prescribed heat flux of $$Q_2$$. If we discretize the rod into two elements, what does the reduced finite element system of equations look like?  You do not have to work out the terms of the stiffness matrix - just use generic labels.
 * Now, assume that the thermal conductivity of the material varies with temperature. What form does the governing equation take? (We will call this the modified problem.)
 * 1) List the steps needed to solve the modified problem using finite elements.