Now the time has come to select which “final project” to do, and I’ve gone for “Stress Concentration due to Geometric Imperfections”. I’ve mainly dealt with wave equations when using the FDM, so I thought a structural problem would be a nice change. The task is to find the shear stress in a long cylinder which is subjected to torsional forces (as if you’re twisting each end of the cylinder), and to use simulations to find how the shear stress changes when the cylinder contains a crack of varying size.

The shear stress is expressed as \(||\nabla u||\). That is, it’s the magnitude of the gradient of \(u\), where \(u\) is the solution to the Poisson equation \(\nabla^2 u = -2\) with \(u = 0\) on the boundary of the domain. It’s important to note that the solution \(u\) does not have a physical meaning in itself, however one can think of it as a “stress potential”.

One of the strengths of the FEM compared to the FDM is that instead of operating on a regularized grid it operates on generalized elements. Typical elements in 2D are quadrilaterals and triangles, however more complex elements can also be used. The main point is however that the shape of each element can be arbitrary. This means that it can easily deal with complex geometry by simply making the elements follow the contour of the geometry. In addition one can make the mesh denser near regions of interest. In my assignment I will use this property to generate adaptive meshes, by increasing the mesh density in regions where the solution changes a lot ie where the shear stress is high. Fortunately for me, Diffpack has a module for handling the gritty details of this.

In part 2 I’ll share some more details about the project.

I started simple with a drop hitting the floor, and rendered it using LuxRender. My poor Q6600 spent roughly 80 cpu-hours on it.

Now I just have to start to write stuff…

Patient Bear - Will be ready when you are