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

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

Solution

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

${K_{ij}^{e}=\int _{\Omega ^{e}}{\frac {\partial N_{i}^{e}}{\partial x}}{\frac {\partial N_{j}^{e}}{\partial x}}~dx}$

The load vector terms are

${f_{i}^{e}=\left.N_{i}^{e}\left({\frac {\partial T}{\partial x}}\right)\right|_{\Gamma ^{e}}~}$

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.

Nonlinear finite elements/Quiz 1