Nonlinear finite elements/Quiz 1/Solutions

Quiz 1: Given
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 ~. $$

Part 1
Derive a symmetric weak form for the Laplace equation in 1-D (an insulated rod).

The 1-D version of Laplace's equation is

\frac{\partial^2 T}{\partial x^2} = 0 ~. $$ To derive the symmetric weak form we multiply the equation by a weighting function ($$w$$) and integrate by parts. Thus,

\int_{\Omega} w~\frac{\partial^2 T}{\partial x^2}~dx = 0 $$ or

{ \int_{\Omega} \frac{\partial w}{\partial x}~\frac{\partial T}{\partial x}~dx = \left.w~\left(\frac{\partial T}{\partial x}\right)\right|_{\Gamma} ~ \qquad \leftarrow \qquad \text{Symmetric Weak Form} ~. } $$

Part 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?

The stiffness matrix terms are

{ K^e_{ij} = \int_{\Omega^e} \frac{\partial N^e_i}{\partial x}\frac{\partial N^e_j}{\partial x}~dx } $$ The load vector terms are

{ f^e_{i} = \left.N^e_i\left(\frac{\partial T}{\partial x}\right)\right|_{\Gamma^e} ~ } $$

Part 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.

The finite element system of equations for a two element mesh (with linear shape functions) is

\begin{bmatrix} K_{11} & K_{12} & 0\\ K_{12} & K_{22} & K_{23} \\ 0 & K_{23} & K_{33} \end{bmatrix} \begin{bmatrix} T_1 \\ T_2 \\ T_3 \end{bmatrix} = \begin{bmatrix} f_1 \\ 0 \\ Q_2 \end{bmatrix} $$ If $$T_1$$ is not zero, the reduced system of equations will be

{ \begin{bmatrix} K_{22} & K_{23} \\ K_{23} & K_{33} \end{bmatrix} \begin{bmatrix} T_2 \\ T_3 \end{bmatrix} = \begin{bmatrix} -K_{12}T_1 \\ Q_2 \end{bmatrix} ~. } $$

Part 4
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.)

If the thermal conductivity ($$\kappa$$) is a function of temperature, the governing equation takes the form

\frac{\partial }{\partial x}\left(\kappa(T)\frac{\partial T}{\partial x}\right) = 0 ~. $$ Since $$\kappa$$ is a function only of temperature, we can take it outside the derivative to get

\kappa(T)\frac{\partial^2 T}{\partial x^2} = 0 \qquad \implies \qquad { \frac{\partial^2 T}{\partial x^2} = 0 ~.} $$ The equation does not change!

Part 5
List the steps needed to solve the modified problem using finite elements.

The standard steps for a linear ODE are applicable.


 * 1) Derive the symmetric weak form.
 * 2) Substitute the approximate solution into the weak form and find the symmetric element stiffness matrix and element load vector.
 * 3) Assemble global stiffness matrix and load vector.
 * 4) Apply boundary conditions.
 * 5) Solve.