Please first review Finite Element Analysis $($FEA$)$ for nonlinear elasticity in Lecture $13.$ Now let's extend the method and code for three nonlinear spring finite elements. A constant force $P=5N$ is applied to the right of the third element. The goal is to solve the displacements $u_1,u_2,$ and $u_3$ by using force equilibrium. Since the equations will be nonlinear due to nonlinear springs, the problem can be solved using multivariate Newton$-$

Given: General formula for Jacobian of a system of $n$ equations:

$U$

Instructions: For questions $(1)$&$(2),$ some handwork is required, but NO need to submit the handwork.

$(1)$ By using force equilibrium at each node $($the circles above $u_1,u_2,$ and $u_3),$ write down the equations to be solved, one equation for each node. $($Hint: force generated by a spring is $F=k*u,$ where $k$ is stiffness, $u$ is the elongation. For the $2^{nd}$ and $3^{rd}$ springs, consider using $u_2-u_1$ and $u_3-u_2)$

$(2)$ Obtain function vector $\frac{?}{\text{b}\text{a}\text{r}}(f)$ and Jacobian Matrix J$.($Hint: Some partial derivatives may be obtained from the results of Lecture $13$ slides$)$

Instructions: For questions $(3)$&$(4),$ complete each part separately, separate the code using $"$#$\%\%\%",$

$(3)$ Using relative errors in the continuing condition, apply the multivariate Newton$-$Raphson code to solve the above problem. Print the displacement values ${}_1,{}_2,$ and ${}_3.$ The initial values of ${}_1,{}_2$ and $u_3$ can be determined using trail and error.

$(4)$ Instead of using relative errors in the continuing condition, the residuals are commonly used in practical computational FEA software packages, e$.$g$.$ ANSYS, Abaqus. In this problem, force residuals are $\frac{?}{\text{b}\text{a}\text{r}}(f)=\left[\begin{array}{c}{f}_1(u_1,u_2,u_3)\\ f_2(u_1,u_2,u_3)\\ f_3(u_1,u_2,u_3)\end{array}\right],$ ie$.,$ the remaining force errors at the three nodes. If we converge to the right solution, then the force residual should be small numbers. Modify the continuing condition and use the maximum of force residuals $>{10}^{-4}$ instead. Print the displacement values $u_1,u_2,$ and $u_3.$ Plot the residuals in $i$ th iteration vs iteration number.