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Here’s another fine mesh you’ve gotten me into!

Finite element analysis (FEA) is a versatile technique that can be used to simulate many physical situations, but here at Torr we mainly use FEA to simulate structural and thermal issues. The name says it all: in FEA, we create a computer model of something and split it into chunks. The chunks are called elements, and they are small enough to be meaningfully analysed: they are finite. The elements have discrete points at their vertices called nodes. FEA can create brightly coloured images that look great on posters and the like, but it can also provide great insights into physical problems. Being a bear of very little brain, I have not occupied myself unduly with the inner workings of FEA, but I think it works something like this. The software creates a series of equations containing mathematical matrices that represent every aspect of the model: the shape, dimensions, material properties, applied forces and pressures, thermal loads and anything else that you as a user specify. The software then solves the equations iteratively, over and over again, until all nodes have properties consistent with all their neighbours, a process known as convergence. This can take a while, depending on the model and the computational capacity.

 

In the previous paragraph, I hinted that I just let the software do all the work. There is some truth in this, but I take the creation of the model very seriously, because it matters. From the start I need to decide on a strategy: will it be a top-down approach in which I create the model in a 3D CAD kind of way and then later mesh it (chop it into chunks), or will I employ a bottom-up approach in which the model is created from the chunks? Sometimes it is a mixture of the two, but the model has to be good if the results are to be good. What, then, makes a good model? The elements need a low aspect ratio: they should not be too long and thin. If they are too long, the solution that approximates conditions within the elements might be inaccurate. For similar reasons, the elements must be small enough that the volume represented is not over-generalized. In regions where stress or temperature vary over large magnitudes, elements need to be smaller: a finer mesh. It is sometimes necessary to mesh very finely in some regions, but grade this to a coarser mesh elsewhere to reduce computational load. With too many nodes, not only does the analysis take longer but working with the model can become slow and laborious. In grading from a fine mesh to a coarser mesh, elements need to be able to communicate with their neighbours, so all elements should share corners: it is not good to have the corner node of an element sitting in the middle of an edge of its neighbour. That said, shared mid-edge nodes can improve the analysis at the expense of computational bandwidth: a cubic element suddenly has 20 nodes rather than the 8 nodes occupying only the corners. The illustration shows a model of a generic X-ray anode in which the smallest elements are 1 µm while the largest are 250 µm, a volume ratio of over 15 million. Unrealistically sharp internal corners can lead to a stress singularity, which sounds like a potentially catastrophic end to space and time, but worry not: we will cover this eventuality below.

Interpretation of an FEA solution is arguably more of a minefield than creating the model. I try to be aware of the flaws and limitations of the model, but equally the flaws and limitations of its creator: your dear writer. With ductile materials such as metals, life is generally much simpler and the region where the stress or temperature is highest is usually where the attention is focussed. For brittle material such as ceramics, life is more nuanced. Ceramics fail at a critical flaw that is generally (but not always) randomly located, so failure can occur in a region of lower stress. In interpreting the likelihood of failure, an integral approach is needed, where the stress is summed over the volume. This leads us to an interesting property of ceramics: failure probability is proportional to volume. An otherwise identical component with double the dimensions of its smaller twin is 8x more likely to fail under the same stress as it has 8x the volume. How does that work? Put simply, there is 8x the volume, so there is 8x the likelihood that there is a critical flaw.

But what about the stress singularity, I hear you ask? In most situations, a finer mesh gives a better solution, but when an internal corner is infinitely sharp, a finer mesh leads to an unrealistically elevated local stress level. In real life, internal corners are not infinitely sharp. Rounding off sharp corners in a model can lead to much more complex and computationally demanding mesh patterns, so a sensible approach is to be aware of such singularities and to disregard them as required. However, folks not versed in the vagaries of FEA are, quite understandably, not always comfortable with waving away such issues. Sometimes, for cosmetic effect, we bear the burden of an over-complex model to avoid the need to explain away the stress singularity.

Our FEA software of choice is Mecway, an inexpensive package with an intuitive interface. Its relative ease of use compared to its more expensive cousins does not equate to a lack of rigour. The support community contains some helpful but fierce intellects, including Victor Kemp, the creator and maintainer of Mecway. There is a very powerful open-source FEA package called Calculix, but its use is not straightforward, being text-based and not a million miles from coding. Along with its own formidable capabilities, Mecway offers the opportunity to access the additional power of the open-source package through a more forgiving graphical interface – a nice option. Your writer has no ties or interests with Mecway (other than a co-authored open access paper: see link below) so when I suggest you download the trial version and have a tinker I do so purely on merit.

FEA is a useful tool for analysis of many physical problems, but its application must be thoughtful and considered. No model is perfect, so the user must be aware of the limitations and the potential implications for the outcome. No user is perfect either, and in the case of your writer my own limitations often exceed those of the model, but as with the model an awareness of these limitations allows realistic and meaningful conclusions to be drawn from these computed simulations.

https://doi.org/10.1115/1.4040953