Homework 1.

IVÁN SALVADOR VILLANUEVA ROSAS

Consider a slab of thickness L and constant conductivity k (W m-1 °C-1). Suppose that energy is generated at a uniform rate of f0 (W m^3) inside the wall.

We wish to determine the temperature distribution in the wall when the boundary surfaces of the wall are subject to the following three different sets of boundary conditions:

We have the next second order differential equation.

0=∫0

L

w∗[−ddx ( kA∗dTdx )−A f 0]After applying the integration by parts in order to reduce the grade of the differential equation

0=∫0

L

(kA dwdx dTdx −A f 0w)dx−wkA dTdx ∨L0

Using the Galerking Method for expressing as a finite element equation:

K ije=∫

0

L

kAdψ i

e

dxdψ j

e

dxdx

f ie=∫

0

L

A f 0ψ iedx

And

[K e ] {Te }={f e}+{Qe}

First the Integration interval, Gaussian interpolation and Gaussian points for integration were defined:

In the next step, the K matrix and F matrix for element 1 were calculated:

Equally for element 2

The assembly of the global matrices was made:

With the following results:

Applying boundary conditions:

Solving the problem:

T 1={185.62500000000003 }

Finally plotting the result:

10 20 30 40 50

50

100

150

200

250

300

Where the blue function is the FE approximation and the orange plot is the analytic solution to the problem.