Consider a finite element with three nodes, as shown in the figure. When the solution is approximated

Question:

Consider a finite element with three nodes, as shown in the figure. When the solution is approximated using $u(x)=N_{1}(x) u_{1}+N_{2}(x) u_{2}+N_{3}(x) u_{3}$,

a. calculate the interpolation functions $N_{1}(x), N_{2}(x)$, and $N_{3}(x)$ if it is intended to obtain the displacement field in the following form: $u(x)=c_{0}+c_{1} \sqrt{x}+c_{2} x$; and

b. when the nodal displacements are given as: $u_{1}=0, u_{2}=0.5$, and $u_{3}=1$, sketch the function $u(x)$.

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