Fig. 1

Is FE Man telling the truth? Let’s pretend somebody has sued him for making false claims. We’re in the High Court, and the lawyers pose two carefully-worded questions:

And let’s hear some answers:

These answers are glib, unthinking, superficial, off-hand, dismissive, dangerous, and wrong. If you gave these answers in evidence in the High Court under cross-examination, the first answer could be said to be downright reckless, while the second one would be disrespectful to the court (and it would reflect badly on the you, the witness). The lawyer might raise an eyebrow but would probably make no comment in reply. Instead, he or she would set about the painstaking process of proving your answers to be incorrect. 

“Proving” (in the mathematical sense) is maybe asking a bit too much in a court case where the standard of proof is “beyond reasonable doubt”. But let’s try to establish an argument, by analysing the precise meaning of the claims.

Is this pedantic? Does it belong in a technical article? Yes and Yes. Sometimes the extreme precision that is possible in English grammar is dismissed as pedantry — quite rightly, most of the time, because if engineers “went around talking like that” they would not “get on very well”: they do much better with the informal idiomatic English that we speak all day, every day. Bad English good.¹ But in court cases, things that might seem pedantic in ordinary life may acquire critical importance for the case. Let’s think.

Back to the question(s). The answers “Yes” and “Who cares?” might be the ones we would like to hear. They seem to get rid of the nasty stupid questions quickly and easily.

I believe that any engineer with a deep knowledge of the finite-element method and its applications would give these answers:

These answers are rational, modest, and defensible; whereas the previous answers were hype(rbolic) and clipped, reflecting a shallow understanding of the topic. In cross-examinations I understand that if the lawyer asks a “Yes/No” question, the best answer is a polite, considered “Yes” or “No”; nothing more should be offered gratuitously. That’s because the lawyers like to lead the argument themselves as they construct their case. It’s not the job of the witness to lead the argument. It’s out of order. The words “absolutely” and “Who cares?” in the first set of answers make the witness vulnerable, whereas “Yes” and “No” force the lawyer to work the case in smaller increments (until he or she arrives at a very big “Yes/No” question that might summarily decide the case). The witness can only hope to get the chance to explain the answers in the most advantageous way: the most technically-correct way, one hopes.

Let’s leave the courtroom drama behind (with a sigh of relief) and look at some examples that may help to justify the second set of answers.

For the first question, consider this. I can manufacture a motor in my workshop using scrap material of unknown origin, and I can record the dimensions with reasonable precision. Can I analyse that motor with the finite-element method? No — or at least, not correctly — because I do not know the material properties of that “scrap material of unknown origin”.

However, the finite-element method can still help me. I could analyse the motor using hypothetical material property data that might be typical of what I used to make the motor, and I can do this with several different material property datasets. By matching the calculation with measured characteristics of the motor, I can deduce with some reasonable certainty what kind of material I might have used. This is an example of using the finite-element method as a diagnostic tool. Beyond that, if my motor is not performing well, finite-element analysis will give me ideas as to how to improve it. It will do this effectively once I have built the finite-element models for the material-property tests. In this mode, the finite-element method is a design tool.

But this good work with the finite-element method does not mean that it can analyse every structure. It simply means that it is useful with my particular structure.

In another scenario, the local farmer might bring me a broken motor from his cow-shed and ask me to “sort it out”. Can I analyse that motor with the finite-element method? No, because (again, for the same reason) I do not know the material properties, even if I measure all the dimensions precisely. At the risk of stating the obvious: if the motor has a burned-out winding, I don’t need finite-element analysis: I need a hammer and chisel, and maybe a blow-torch (just to get started by stripping out the old winding).

In this second example, I would not dare to mention the finite-element method to the local farmer. I might show him into my workshop to see my tools and instruments, but if he came with a burned-out motor in the back of his Land-Rover, I would not open the door of my study, even if he offered to take his muddy boots off. This is not disrespectful to the farmer (or even to the finite-element method), but in those circumstances the finite-element method simply would not be the right place to start. I would be insulting the farmer, and opening myself to ridicule, if I tried to throw Maxwell’s equations at him. If I tried to use only the finite-element method to resolve the farmer’s problem, I would end up (very quickly) with “paralysis by analysis”. His motor would not be “sorted”, and my supply of free eggs might well be in danger.

Now to the second question. Please consider Fig. 2, which shows part of an electric machine modelled in finite-elements. This is a true story. The analysis proceeded smoothly, but the results seemed odd. Zooming into the air-gap region, we discovered a discontinuity in the normal component of B at an air/iron interface — a clear violation of Gauss’ law. At first we thought we had found an error in the finite-element software, but it soon turned out that an incorrect boundary condition had been applied at the boundary. User error (as usual).²

Fig. 2

The finite-element analysis went along without any hint of trouble until we checked the results against test data. The first conclusion to be drawn is that, Yes, the finite-element method can indeed analyse a structure that cannot be manufactured, for it is impossible to manufacture a structure that violates Gauss’ law.³ But there is a more subtle lesson to learn from this exercise: Check your results against test data, because it is possible to set up a finite-element calculation with errors in the formulation, in such a way that the results can be interpreted as describing a structure that cannot be manufactured.

It might be interesting for readers to try to think of other examples which illustrate answers to the questions (even if — or particularly if — they turn out to be debatable). If you have “lunchtime banter”, or “study sessions” in your company, maybe the questions would be a good topic.

On a darker note, one has to consider the possibilities of undetected errors not only in finite-element calculations but in all calculations. A problem I have with AI is to know how it checks for errors, if it cannot go into the test laboratory and make measurements with real instruments. The possibilities for errors would seem to multiply, if software is built on software built on software built on software built on software built on software. . .without the tests that we make in the laboratory or the workshop, and even without the “sanity checks” that we make by instinct and experience. One even wonders if it is possible to introduce deliberate undetectable corruption into numerical analysis or its results — a chilling prospect. Fortunately as engineers we are naturally always on guard for erroneous data (usually in light of comparison with test data, even though that too can contain errors); but we might need to “up our game” in view of the “climate change” in methods of doing engineering calculations.

Notes

¹ A quotation from a long-time Japanese colleague, passed on to me with friendly humour by another long-time colleague. I don’t know if other languages have the same dichotomy between the colloquial and the exact, between the vernacular and the grammatically correct, between the chatter and the grandiloquent. In English it is both a curse and a blessing.

² Readers may be reassured to learn that this story dates back about 30 years, and the finite-element software in question was “another brand” (not JMAG) running under MS-DOS. The PhD student who noticed the problem may be reading this, and if so I hope it gives him an amusing reminiscence. But we should beware that the same problem can arise again, anytime, anywhere.

³ I certainly hope that is true, and that there are no physicists waiting around the corner to refute it!