Chapter 14, Problem 29P

To determine

Calculate the global coefficient matrix of the two-element mesh.

Answer to Problem 29P

The global coefficient matrix is $\underset{\_}{\left[\begin{array}{cccc}0.8333& -0.667& 0& -0.1667\\ -0.667& 1.4583& -0.375& -0.4167\\ 0& -0.375& 0.625& -0.25\\ -0.1667& -0.4167& -0.25& 0.833\end{array}\right]}$.

Explanation of Solution

Calculation:

Refer to Figure 14-62 in the textbook for the two-element mesh.

For element 1, local numbering 1-2-3 corresponds to global numbering 1-2-4.

From Figure 14.62, the given values for element 1 are,

$x_1=0,x_2=1,x_3=2,y_1=1,y_2=0,\text{and}{y}_3=2$.

Calculate the value of $P_1$.

$\begin{array}{c}{P}_1=y_2-y_3\\ =0-2\\ =-2\end{array}$

Calculate the value of $P_2$.

$\begin{array}{c}{P}_2=y_3-y_1\\ =2-1\\ =1\end{array}$

Calculate the value of $P_3$.

$\begin{array}{c}{P}_3=y_1-y_2\\ =1-0\\ =1\end{array}$

Calculate the value of $Q_1$.

$\begin{array}{c}{Q}_1=x_3-x_2\\ =2-1\\ =1\end{array}$

Calculate the value of $Q_2$.

$\begin{array}{c}{Q}_2=x_1-x_3\\ =0-2\\ =-2\end{array}$

Calculate the value of $Q_3$.

$\begin{array}{c}{Q}_3=x_2-x_1\\ =1-0\\ =1\end{array}$

Consider the expression for the area A.

$A=\frac{1}{2}\left(P_2{Q}_3-P_3{Q}_2\right)$        (1)

Substitute 1 for $P_2$, 1 for $P_3$, $-2$ for $Q_2$, and 1 for $Q_3$ in Equation (1).

$\begin{array}{c}A=\frac{1}{2}\left[\left(\left(1\right)\left(1\right)-\left(1\right)\left(-2\right)\right)\right]\\ =\frac{3}{2}\end{array}$

Write the expression for co-efficient $C_{ij}$.

$C_{ij}=\frac{1}{4A}\left[P_i{P}_j+Q_i{Q}_j\right]$        (2)

By using Equation (2), the coefficient values $C_{11},C_{12},C_{13},C_{21},C_{22},C_{23},C_{31},C_{32},\text{and}{C}_{33}$ are calculated as $\frac{5}{6},-\frac{4}{6},-\frac{1}{6},-\frac{4}{6},\frac{5}{6},-\frac{1}{6},-\frac{1}{6},-\frac{1}{6},\text{and}\frac{2}{6}$ respectively.

Write the expression of the element coefficient matrix $C^{\left(1\right)}$.

$C^{\left(1\right)}=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]$

Substitute $\frac{5}{6}$ for $C_{11}$, $-\frac{4}{6}$ for $C_{12}$, $-\frac{1}{6}$ for $C_{13}$, $-\frac{4}{6}$ for $C_{21}$, $\frac{5}{6}$ for $C_{22}$, $-\frac{1}{6}$ for $C_{23}$, $-\frac{1}{6}$ for $C_{31}$, $-\frac{1}{6}$ for $C_{32}$, and $\frac{2}{6}$ for $C_{33}$.

$C^{\left(1\right)}=\left[\begin{array}{ccc}\frac{5}{6}& -\frac{4}{6}& -\frac{1}{6}\\ -\frac{4}{6}& \frac{5}{6}& -\frac{1}{6}\\ -\frac{1}{6}& -\frac{1}{6}& \frac{2}{6}\end{array}\right]$

For element 2, local numbering 1-2-3 corresponds to global numbering 4-2-3.

From Figure 14.62, the given values for element 2 are,

$x_1=2,x_2=1,x_3=3,y_1=2,y_2=0,\text{and}{y}_3=0$.

Calculate the value of $P_1$.

$\begin{array}{c}{P}_1=y_2-y_3\\ =0-0\\ =0\end{array}$

Calculate the value of $P_2$.

$\begin{array}{c}{P}_2=y_3-y_1\\ =0-2\\ =-2\end{array}$

Calculate the value of $P_3$.

$\begin{array}{c}{P}_3=y_1-y_2\\ =2-0\\ =2\end{array}$

Calculate the value of $Q_1$.

$\begin{array}{c}{Q}_1=x_3-x_2\\ =3-1\\ =2\end{array}$

Calculate the value of $Q_2$.

$\begin{array}{c}{Q}_2=x_1-x_3\\ =2-3\\ =-1\end{array}$

Calculate the value of $Q_3$.

$\begin{array}{c}{Q}_3=x_2-x_1\\ =1-2\\ =-1\end{array}$

Substitute $-2$ for $P_2$, 2 for $P_3$, $-1$ for $Q_2$, and $-1$ for $Q_3$ in Equation (1) to find area A.

$\begin{array}{c}A=\frac{1}{2}\left[\left(\left(-2\right)\left(-1\right)-\left(2\right)\left(-1\right)\right)\right]\\ =2\end{array}$

By using Equation (2), the coefficient values $C_{11},C_{12},C_{13},C_{21},C_{22},C_{23},C_{31},C_{32},\text{and}{C}_{33}$ are calculated as $\frac{4}{8},-\frac{2}{8},-\frac{2}{8},-\frac{2}{8},\frac{5}{8},-\frac{3}{8},-\frac{2}{8},-\frac{3}{8},\text{and}\frac{5}{8}$ respectively.

Write the expression of the element coefficient matrix $C^{\left(2\right)}$.

$C^{\left(2\right)}=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]$

Substitute $\frac{4}{8}$ for $C_{11}$, $-\frac{2}{8}$ for $C_{12}$, $-\frac{2}{8}$ for $C_{13}$, $-\frac{2}{8}$ for $C_{21}$, $\frac{5}{8}$ for $C_{22}$, $-\frac{3}{8}$ for $C_{23}$, $-\frac{2}{8}$ for $C_{31}$, $-\frac{3}{8}$ for $C_{32}$, and $\frac{5}{8}$ for $C_{33}$.

$C^{\left(2\right)}=\left[\begin{array}{ccc}\frac{4}{8}& -\frac{2}{8}& -\frac{2}{8}\\ -\frac{2}{8}& \frac{5}{8}& -\frac{3}{8}\\ -\frac{2}{8}& -\frac{3}{8}& \frac{5}{8}\end{array}\right]$

Calculate the global coefficient matrix.

$\begin{array}{c}C=\left[\begin{array}{cccc}{C}_{11}^{\left(1\right)}& C_{12}^{\left(1\right)}& 0& C_{13}^{\left(1\right)}\\ C_{12}^{\left(1\right)}& C_{22}^{\left(1\right)}+C_{22}^{\left(2\right)}& C_{23}^{\left(2\right)}& C_{23}^{\left(1\right)}+C_{21}^{\left(2\right)}\\ 0& C_{23}^{\left(2\right)}& C_{33}^{\left(2\right)}& C_{31}^{\left(2\right)}\\ C_{13}^{\left(1\right)}& C_{23}^{\left(1\right)}+C_{21}^{\left(2\right)}& C_{31}^{\left(2\right)}& C_{33}^{\left(1\right)}+C_{11}^{\left(2\right)}\end{array}\right]\\ =\left[\begin{array}{cccc}0.8333& -0.667& 0& -0.1667\\ -0.667& 1.4583& -0.375& -0.4167\\ 0& -0.375& 0.625& -0.25\\ -0.1667& -0.4167& -0.25& 0.833\end{array}\right]\end{array}$

Conclusion:

Thus, the global coefficient matrix is $\underset{\_}{\left[\begin{array}{cccc}0.8333& -0.667& 0& -0.1667\\ -0.667& 1.4583& -0.375& -0.4167\\ 0& -0.375& 0.625& -0.25\\ -0.1667& -0.4167& -0.25& 0.833\end{array}\right]}$.