Nonlinear finite elements/Homework 4

Problem 1: Axially Loaded Bar with Nonlinear Modulus

In the learning resource section we have discussed the solution of the problem of a nonlinear axially loaded bar with a body load. A Matlab code for solving the problem with finite elements can also be found on the relevant page.

In the code, we apply a tensile load at the end of the bar and use linear shape functions. Therefore, each element has two nodes.

In this problem, you are required to modify the Matlab code to use quadratic elements. Each element will have three nodes ($1$, $2$, and $3$). Assume that all elements have the same length $h$ and that node $2$ is at the middle of the element. The shape functions for the nodes are

${\begin{aligned}N_{1}(x)&={\cfrac {2(x_{2}-x)(x_{3}-x)}{h^{2}}}\\N_{2}(x)&={\cfrac {4(x-x_{1})(x_{3}-x)}{h^{2}}}\\N_{3}(x)&={\cfrac {2(x-x_{1})(x-x_{2})}{h^{2}}}\end{aligned}}$

Apply a compressive load $R$ at the free end of the bar.

1. Plot the shape functions for an element. Show that they add up to 1 at all points within the element.
2. Plot the displacement, stress, strain, and the error norm when you use 10 elements to discretize the bar. See the figure in the learning resource section for how you should plot these data. Discuss and comment on your observations.
3. Do the same for 100 elements. What are your observations on the effect of mesh refinement on the solution?
4. Now, increase the applied load. At what load does the solution fail to converge?Explain why the solution fails to converge at this load.

You could alternatively write your own Maple code or use ANSYS for these calculations depending on your interests.

Problem 2: Nonlinear Thermal Conduction

Consider the model chip system shown in Figure 1.

Figure 1. Schematic of a model chip.

Assume that heat is being generated in the silicon carbide region of the chip at the rate of 100 W/cm$^{2}$. The cooling channels contain a fluid at a constant 250 K and (supposedly) can dissipate heat at the rate of 800 W/cm$2$. The surroundings are at room temperature (300 K).

The thermal conductivity of the silicon carbide and the aluminum nitride are also given in Figure 1. The thermal conductivity of the epoxy resin is 1.5 W/m-K.

1. First run a linear simulation of the problem using room temperature thermal conductivities of the materials. This will act as a check of the model and setup and will help you determine any additional constraints that you will need. Do the cooling channels really dissipate heat at 800 W/cm$2$?
2. Fit curves to the thermal conductivity versus temperature data for Silicon Carbide and Aluminum Nitride.
3. Perform a nonlinear simulation using these temperature dependent properties to find the equilibrium temperature inside the silicon carbide region. State all simplifying assumptions that you have made.

Nonlinear finite elements