Problem $1$: $2-$D Heat Conduction Element $(18$ Points$)$

The scalar two$-$dimensional heat conduction strong form can be stated as follows for a domain A and two$-$part boundary:

$k(\frac{de{l}^{2}T}{del{x}^2}+\frac{de{l}^{2}T}{del{y}^2})+Q=0inA$

$T\frac{?}{b}=ar(T)ondel{A}_T$

$k(\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}}{n}_x+\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{y}}{n}_y)=hondel{A}_q$

Show that the weak form for this boundary value problem can be expressed as below. Let $w$ be the weighting function. Specify the appropriate degree of continuity and any prescribed conditions for the trial temperature field $T$ and the weighting function $w.$ Hint: like some the problems in an open book exam, the solutions may be inferred from the textbook $($i$.$e$.$ check the headings$).$ Nonetheless, I ask you to verify it$,$ so show the steps.

Find TinS such that for all weighting functions winV :

${\displaystyle \underset{A}{\overset{\phantom{}}{}}}k(\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{x}}\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}}+\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{y}}\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{y}})dA-{\displaystyle \underset{A}{\overset{\phantom{}}{}}}$wQdA$-{\displaystyle \underset{del{A}_q}{\overset{\phantom{}}{}}}whds=0$

Let the finite element approximation for the temperature and weighting function be:

$T(x,y)$~~$T^h(x,y)={\displaystyle \underset{a=1}{\overset{3}{}}}{N}_a(x,y)T_a=NT,N_1=\frac{1}{2}x,N_2=\frac{1}{2}y,N_3=1-\frac{1}{2}x-\frac{1}{2}y$

$w(x,y)$~~$w^h(x,y)={\displaystyle \underset{b=1}{\overset{3}{}}}{N}_b(x,y)w_b=Nw$

for a triangular element with edge lengths of $2$ along the x and y axes and hypotenuse running between node $1$ and node $2,$ i$.$e$.$ the nodal $(x,y)$ coordinates are $(2,0),(0,2)$ and $(0,0).$ Let the conductivity $k=300.$ Develop the element stiffness matrix $K$ for this triangular element. Actual numerical values are desired for the final matrix. Does the matrix exhibit similar properties to the stiffness matrices we have found for mechanical elements $(2$ d solid, $1$ d bar, $1$ b beam$)?$ Name two example properties observed from the stiffness matrix.