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

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

Solution

Part 1

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,

$\underset{\mathrm{\Omega }}{\int }w\frac{{\partial }^{2}T}{\partial x^2}dx=0$

or

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

Part 2

The stiffness matrix terms are

$K_{ij}^e=\underset{{\mathrm{\Omega }}^e}{\int }\frac{\partial N_i^e}{\partial x}\frac{\partial N_j^e}{\partial x}dx$

The load vector terms are

$f_i^e={N_i^e\left(\frac{\partial T}{\partial x}\right)|}_{{\mathrm{\Gamma }}^e}$

Part 3

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

$\left[\begin{array}{ccc}{K}_{11}& K_{12}& 0\\ K_{12}& K_{22}& K_{23}\\ 0& K_{23}& K_{33}\end{array}\right]\left[\begin{array}{c}{T}_1\\ T_2\\ T_3\end{array}\right]=\left[\begin{array}{c}{f}_1\\ 0\\ Q_2\end{array}\right]$

If $T_1$ is not zero, the reduced system of equations will be

$\left[\begin{array}{cc}{K}_{22}& K_{23}\\ K_{23}& K_{33}\end{array}\right]\left[\begin{array}{c}{T}_2\\ T_3\end{array}\right]=\left[\begin{array}{c}-K_{12}{T}_1\\ Q_2\end{array}\right].$

Part 4

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

$\frac{\partial }{\partial x}(\kappa (T)\frac{\partial T}{\partial x})=0.$

Since $\kappa$ is a function only of temperature, we can take it outside the derivative to get

$\kappa (T)\frac{{\partial }^{2}T}{\partial x^2}=0⟹\frac{{\partial }^{2}T}{\partial x^2}=0.$

The equation does not change!

Part 5

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.