Consider the problem of steady state heat conduction through a stepped rod of $6$ m length and circular cross$-$section with cross$-$sectional areas as shown in Figure $4.$ For the finite element modelling, you are required to use three $2-$node $($linear$)$ heat conduction elements of equal lengths. The left end of the rod is maintained at $\backslash (100^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ while the right is maintained at $\backslash (0^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The rod loses heat to the surrounding by convection through its lateral surfaces with a heat transfer coefficient $\backslash (100\backslash$mathrm$\{$~W$\}.\backslash$mathrm$\{$m$\}^\{-2\}.\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}^\{-1\}\backslash )$ while the temperature of the surrounding medium is maintained at $\backslash (0^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Take the thermal conductivity of the rod material as $\backslash (300\backslash$mathrm$\{$Wm$\}^\{-1\}\backslash$mathrm$\{$C$\}^\{-1\}\backslash ).$

Determine the nodal temperatures of the stepped rod by solving the global finite element equations. Determine how much heat enters$/$leaves the left$/$right ends of the rod and also how much heat leaves the lateral surface by convection. Take the necessary expressions for the heat conduction matrix and nodal heat flow vectors from the lecture slides.