A possible algorithm for the FDM is provided in the following pseudocode :

Algorithm $1$ Pseudocode for the FD method for the heat equation

INPUT: Thermal conductivities k

The

chip, k

The

board, the source schip and the boundary conditions Tbnd$.$

OUTPUT: The array $2$D of the temperature T$.$

procedure finite$\_$difference$\_$method

$$ Build the matrix A and the right$-$hand side b$,$

$$ Solve equation $(2\mathrm{.}1)$ for the $1$D array of temperatures x$.$

$$ Build and output the $2$D array of the temperature T$.$

end procedure

You may find it easier to use a function that allows to get the right indices to$-$and$-$fro k $($i$,$ j$)$ and

a functions that output the thermal conductivity k

The and the source term s given the indices $($i$,$ j$).$

$3$ Applications

Use the developed algorithm to solve the heat transfer for the following two cases :

$1.$ The case of a chip on the board.

$2.$ The case of a chip on the heat sink.

You will use the following parameters :

a$.$ Geometric dimensions : length of the board$/$heat sink L $=80$cm and centered around $(0,0),$ the

length of the squared chip l $=20$cm and centered around $($cx$,$ cy$)=(20$cm$,20$cm$).$

b$.$ Material properties : k

The

chip $=0\mathrm{.}023$W m$1$K$1$

$,$ k

The

board $=0\mathrm{.}15$W m$1$K$1$ and k

The

heatsink $=237$W

m$1$K$1$

$.$

You will also consider time$-$independent densities of heat source with increasing values