# Nonlinear finite elements/Quiz 1

${\displaystyle \nabla ^{2}T=0~\qquad ~{\text{or,}}\qquad {\frac {\partial ^{2}T}{\partial x^{2}}}+{\frac {\partial ^{2}T}{\partial y^{2}}}+{\frac {\partial ^{2}T}{\partial z^{2}}}=0~.}$
2. What are the expressions for the components of the finite element stiffness matrix (${\displaystyle K_{ij}}$) and the load vector (${\displaystyle f_{i}}$) for this 1-D problem?
3. Assume that the one of ends of the rod is maintained at a temperature of ${\displaystyle T_{1}}$ (which is nonzero) and the other end has a prescribed heat flux of ${\displaystyle Q_{2}}$. 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.