Chapter 14, Problem 28P

To determine

Calculate the global coefficient matrix of the two element region.

Answer to Problem 28P

The global coefficient matrix is $\underset{\_}{\left[\begin{array}{cccc}1& 0& -0.5& -0.5\\ 0& 0.5& -0.5& 0\\ -0.5& -0.5& 1.5& -0.5\\ -0.5& 0& -0.5& 1\end{array}\right]}$.

Explanation of Solution

Calculation:

Refer to Figure 14-61 in the textbook for the two-element region.

Modify Figure 14-61 by mentioning global number. The modified figure as shown in Figure 1.

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

From Figure 14.61, the given values are,

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

Calculate the value of $P_1$.

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

Calculate the value of $P_2$.

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

Calculate the value of $P_3$.

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

Calculate the value of $Q_1$.

$\begin{array}{c}{Q}_1=x_3-x_2\\ =0-2\\ =-2\end{array}$

Calculate the value of $Q_2$.

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

Calculate the value of $Q_3$.

$\begin{array}{c}{Q}_3=x_2-x_1\\ =2-0\\ =2\end{array}$

Consider the expression for the area A.

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

Substitute 2 for $P_2$, $-2$ for $P_3$, 0 for $Q_2$, and $2$ for $Q_3$ in Equation (1).

$\begin{array}{c}A=\frac{1}{2}\left[\left(\left(2\right)\left(2\right)-\left(-2\right)\left(0\right)\right)\right]\\ =2\end{array}$

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 0.5, 0, $-0.5$, 0, 0.5, $-0.5$, $-0.5$, $-0.5$, and 1 respectively.

Write the expression of the element coefficient matrix C.

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

Substitute 0.5 for $C_{11}$, 0 for $C_{12}$, $-0.5$ for $C_{13}$, 0 for $C_{21}$, 0.5 for $C_{22}$, $-0.5$ for $C_{23}$, $-0.5$ for $C_{31}$, $-0.5$ for $C_{32}$, and 1 for $C_{33}$.

$C^{\left(1\right)}=\left[\begin{array}{ccc}0.5& 0& -0.5\\ 0& 0.5& -0.5\\ -0.5& -0.5& 1\end{array}\right]$

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

From Figure 14.61, the given values are,

$x_1=0,x_2=4,x_3=2,y_1=0,y_2=0,\text{and}{y}_3=2$.

Calculate the value of $P_1$.

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

Calculate the value of $P_2$.

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

Calculate the value of $P_3$.

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

Calculate the value of $Q_1$.

$\begin{array}{c}{Q}_1=x_3-x_2\\ =2-4\\ =-2\end{array}$

Calculate the value of $Q_2$.

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

Calculate the value of $Q_3$.

$\begin{array}{c}{Q}_3=x_2-x_1\\ =4-0\\ =4\end{array}$

Substitute 2 for $P_2$, 0 for $P_3$, $-2$ for $Q_2$, and 4 for $Q_3$ in Equation (1) to find area A.

$\begin{array}{c}A=\frac{1}{2}\left[\left(\left(2\right)\left(4\right)-\left(0\right)\left(-2\right)\right)\right]\\ =4\end{array}$

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 0.5, 0, $-0.5$, 0, 0.5, $-0.5$, $-0.5$, $-0.5$, and 1 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 0.5 for $C_{11}$, 0 for $C_{12}$, $-0.5$ for $C_{13}$, 0 for $C_{21}$, 0.5 for $C_{22}$, $-0.5$ for $C_{23}$, $-0.5$ for $C_{31}$, $-0.5$ for $C_{32}$, and 1 for $C_{33}$.

$C^{\left(2\right)}=\left[\begin{array}{ccc}0.5& 0& -0.5\\ 0& 0.5& -0.5\\ -0.5& -0.5& 1\end{array}\right]$

Calculate the global coefficient matrix.

$\begin{array}{c}C=\left[\begin{array}{cccc}{C}_{11}^{\left(1\right)}+C_{11}^{\left(2\right)}& C_{12}^{\left(1\right)}& C_{12}^{\left(1\right)}+C_{13}^{\left(2\right)}& C_{13}^{\left(1\right)}\\ C_{21}^{\left(2\right)}& C_{22}^{\left(2\right)}& C_{23}^{\left(2\right)}& 0\\ C_{21}^{\left(1\right)}+C_{31}^{\left(2\right)}& C_{32}^{\left(2\right)}& C_{22}^{\left(1\right)}+C_{33}^{\left(2\right)}& C_{23}^{\left(1\right)}\\ C_{31}^{\left(1\right)}& 0& C_{32}^{\left(1\right)}& C_{33}^{\left(1\right)}\end{array}\right]\\ =\left[\begin{array}{cccc}0.5+0.5& 0& 0-0.5& -0.5\\ 0& 0.5& -0.5& 0\\ 0-0.5& -0.5& 0.5+1& -0.5\\ -0.5& 0& -0.5& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& -0.5& -0.5\\ 0& 0.5& -0.5& 0\\ -0.5& -0.5& 1.5& -0.5\\ -0.5& 0& -0.5& 1\end{array}\right]\end{array}$

Conclusion:

Thus, the global coefficient matrix is $\underset{\_}{\left[\begin{array}{cccc}1& 0& -0.5& -0.5\\ 0& 0.5& -0.5& 0\\ -0.5& -0.5& 1.5& -0.5\\ -0.5& 0& -0.5& 1\end{array}\right]}$.