${V}^{e}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{j=1}^{2}{N}_{j}^{e}\text{\hspace{0.17em}}{V}_{j}^{e}}.$ |
(12.94) |

${N}_{j}^{e}$ denotes the interpolation or shape functions which are given by

${N}_{1}^{e}=\frac{{r}_{2}^{e}-r}{{l}^{e}}\text{\hspace{1em}}\mathrm{and}\text{\hspace{1em}}{N}_{2}^{e}=\frac{r-{r}_{1}^{e}}{{l}^{e}}$ |
(12.95) |

where ${l}^{e}={r}_{2}^{e}-{r}_{1}^{e}\text{\hspace{0.17em}}.$ The functions ${N}_{j}^{e},$ which depend on the independent variable *r*, are shown in Figure 12.10. ${N}_{1}^{e}\text{\hspace{0.17em}}({N}_{2}^{e})$ equals 1 at node 1 (2) and zero at node 2 (1); at any intermediate point its value is a fraction between 0 and 1. This is an obvious result given the expression, as in Equation 12.94, for voltage at any point within the element, ${V}^{e}={N}_{1}^{e}\text{\hspace{0.17em}}{\nabla}_{1}^{e}+{N}_{2}^{e}\text{\hspace{0.17em}}{\nabla}_{2}^{e}\text{\hspace{0.17em}}.$ For example, at node 1, ${V}^{e}={V}_{1}^{e}$ leading to ${N}_{1}^{e}\text{\hspace{0.17em}}=1$ and ${N}_{2}^{e}\text{\hspace{0.17em}}=0$. Thus, for the nodes, ${\left({N}_{i}^{e}\right)}_{j}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\delta}_{ij},$ where *δ _{ij}* = 1 ...

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