Problem $5$: Isoparametric Element Mapping $(14$ Points$)$

Consider the quadratic triangular element shown below, with its coordinates given in physical xy coordinates. The local coordinates are defined with $(0,0)$ in the lower left $($node $1)$ and $N_1(r,s)=(1-r-s)(1-2r-2s)$

$N_2(r,s)=r(2r-1)$

$N_3(r,s)=s(2s-1)$

$N_4(r,s)=4r(1-r-s)$

$N_5(r,s)=4rs$

$N_6(r,s)=4s(1-r-s)Jdet(J)(r,s)=(\frac{2}{3},\frac{1}{2})(r,s)=(\frac{1}{4},\frac{1}{4})J=\left[\begin{array}{cc}\text{d}\text{e}\text{l}\frac{x}{d}\text{e}\text{l}\text{r}& \text{d}\text{e}\text{l}\frac{y}{d}\text{e}\text{l}\text{r}\\ \text{d}\text{e}\text{l}\frac{x}{d}\text{e}\text{l}\text{s}& \text{d}\text{e}\text{l}\frac{y}{d}\text{e}\text{l}\text{s}\end{array}\right]0.$

The shape functions for the $6-$node element are defined $in$ the local coordinate system:

$N_1(r,s)=(1-r-s)(1-2r-2s)$

$N_2(r,s)=r(2r-1)$

$N_3(r,s)=s(2s-1)$

$N_4(r,s)=4r(1-r-s)$

$N_5(r,s)=4rs$

$N_6(r,s)=4s(1-r-s)$

Compute the Jacobian matrix $J$ and determinant det$(J)$ for two locations inside the element:

$(a)At$ position $(r,s)=(\frac{2}{3},\frac{1}{2})$

$(b)At$ position $(r,s)=(\frac{1}{4},\frac{1}{4})$

Recall that $J=\left[\begin{array}{cc}\text{d}\text{e}\text{l}\frac{x}{d}\text{e}\text{l}\text{r}& \text{d}\text{e}\text{l}\frac{y}{d}\text{e}\text{l}\text{r}\\ \text{d}\text{e}\text{l}\frac{x}{d}\text{e}\text{l}\text{s}& \text{d}\text{e}\text{l}\frac{y}{d}\text{e}\text{l}\text{s}\end{array}\right]\mathrm{.}0$ and $0.$

The shape functions for the $6-$node element are defined $in$ the local coordinate system:

$N_1(r,s)=(1-r-s)(1-2r-2s)$

$N_2(r,s)=r(2r-1)$

$N_3(r,s)=s(2s-1)$

$N_4(r,s)=4r(1-r-s)$

$N_5(r,s)=4rs$

$N_6(r,s)=4s(1-r-s)$

Compute the Jacobian matrix $J$ and determinant det$(J)$ for two locations inside the element:

$(a)At$ position $(r,s)=(\frac{2}{3},\frac{1}{2})$

$(b)At$ position $(r,s)=(\frac{1}{4},\frac{1}{4})$

Recall that $J=\left[\begin{array}{cc}\text{d}\text{e}\text{l}\frac{x}{d}\text{e}\text{l}\text{r}& \text{d}\text{e}\text{l}\frac{y}{d}\text{e}\text{l}\text{r}\\ \text{d}\text{e}\text{l}\frac{x}{d}\text{e}\text{l}\text{s}& \text{d}\text{e}\text{l}\frac{y}{d}\text{e}\text{l}\text{s}\end{array}\right].$