# Nonlinear finite elements/Quiz 1

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## Quiz[edit]

*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

- Derive a symmetric weak form for the Laplace equation in 1-D (an insulated rod).
- What are the expressions for the components of the finite element stiffness matrix () and the load vector () for this 1-D problem?
- Assume that the one of ends of the rod is maintained at a temperature of (which is nonzero) and the other end has a prescribed heat flux of . 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.)
- List the steps needed to solve the modified problem using finite elements.