Short answer $(20$ points$)$

As shown in the left figure, a force of size $P$ in the $x-$direction is applied to node $2$ of a rod in which both ends are fixed, consisting of three elements and four nodes. The crosssectional area of elements $1$ and $2$ is $A,$ and element $3$ is $2A.$ Young's modulus of the aluminum is E$,$ and Young's modulus of copper is $\frac{E}{2}.$

$(1)$ Find the stiffness matrix of each element and the entire system for finite element analysis, and express the behavior of the entire system in the form of $\{F\}=[K]\{u\}$ determinant. $(10$ points$)$

$(2)$ Considering the boundary condition, represent the displacements at nodes $2$ and $3$ as A$,$ E$,$ L$,$ and P$.(10$ points$)$

Ref$)$ Displacement by simple tensile force P $=$ PL$/$AE $($L: Length, A: Cross$-$sectional area, E: Young's modulus$)$