Question 4.17: (a) Find the voltage distribution within the triangular elem...
(a) Find the voltage distribution within the triangular element if the voltages at the three nodes have the following values V_{1} = 8 @ (4, 0); V_{2} = 0 @ (0, 0); and V_{3} = 0 @ (4, 3).
(b) Determine the area of the triangle that is defined by the three nodes.
(a) The voltage distribution within the element is computed from (4.80)
V^{(e)} (x,y) = [1 x y]\left[\begin{matrix} 1 & x_{1} & y_{1} \\ 1 & x_{2} & y_{2} \\ 1 & x_{3} & y_{3} \end{matrix} \right] ^{-1} \left[\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \end{matrix} \right] (4.80)
V_{1,2,3}^{(e)} (x,y) = [1 x y] \left[\begin{matrix} 1 &4 &0 \\1& 0 & 0 \\ 1& 4 & 3\end{matrix} \right] ^{-1} \left[\begin{matrix}8 \\ 0 \\ 0 \end{matrix} \right]= [1 x y ]\left[\begin{matrix} 0 & 1 & 0 \\ 1/4& -1/4 & 0\\ -1/3& 0 &1/3\end{matrix} \right] \left[\begin{matrix}8 \\ 0 \\ 0 \end{matrix} \right] \\ = [1 x y ] \left[\begin{matrix} 0\\ 2 \\-8/3 \end{matrix} \right] = 2x -\frac{8}{3} y
The solution satisfies Laplace’s equation (4.76) and the boundary conditions (4.78).
∇^2V= 0 (4.76)
V^{(e) } (x_{j}, y_{j})= V_{j} ( j = 1,2,3) (4.78)
(b) The area of the triangle is determined from magnitude of the determinant of the square matrix given in (4.79).
\left[\begin{matrix} 1 & x_{1} & y_{1} \\ 1 & x_{2} & y_{2} \\ 1 & x_{3} & y_{3} \end{matrix} \right] ^{-1} \left[\begin{matrix} a \\ b \\ c \end{matrix} \right]= \left[\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \end{matrix} \right] (4.79)
A_{e} = \frac{1}{2} \left|\begin{matrix} 1 &4 & 0 \\ 1 & 0&0\\ 1& 4 & 3\end{matrix} \right| = 6