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@@ -4,7 +4,17 @@ First of all, please take this text as a written chat between you an me, i.e. an

Please also note that I am not a mechanical engineer, although I shared many undergraduate courses with some of them. I am a nuclear engineer with a strong background on mathematics and computer programming. I went to college between 2002 and 2008. Probably a lot of things have changed since then---at least that is what these “millenials” guys and girls seem to be boasting about---but chances are we all studied solid mechanics and heat transfer with a teacher using a piece of chalk on a blackboard and students writing down notes with pencils on paper sheets. And there is really not much that one can do with pencil and paper regarding mechanical analysis. Any actual case worth the time of an engineer need to be more complex than an ideal canonical case with analytical solution.

We will be swinging back and forth between a case study about fatigue analysis in piping systems of a nuclear power plant and more generic and even romantic topics related to finite elements and computational mechanics. These latter regressions will not remain just as abstract theoretical ideas. Not only will they be directly applicable to the development of the main case, but they will also apply to a great deal of other engineering problems tackled with the finite element method.

::::: {#fig:pendulum}
![Simple pendulum](simple.svg){#fig:simple width=35%}
![Real pendulum](hamaca.jpg){#fig:hamaca width=60%}

A simple pendulum from college physics courses and a real-life pendulum. The same difference exists between theoretical FEM courses and actual engineering problems. Hint: the swing’s period _does_ depend on the hanging mass. See the [actual video](hamaca.webm).
:::::

Whether you are a student or a seasoned engineer with many years of experience, you might recall from first year physics courses the introduction of the [simple pendulum](https://en.wikipedia.org/wiki/Pendulum) as case study. You learned that the period does not depend on the hanging mass because the weight and the inertia exactly canceled each other. Also, that Galileo said that for small oscillations the period does not even depend on the amplitude. Someone showed you why it worked this way: because if\ $\sin \theta \approx \theta$ then the motion equations converge to an [harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator). It might have been a difficult subject for you back in those days. It also might seem a very simple case for you today. But it is _after_ college that you **xxxxxxxxxxxxxxxx**

Like the pendulums above, we will be swinging back and forth between a case study about fatigue analysis in piping systems of a nuclear power plant and more generic and even romantic topics related to finite elements and computational mechanics. These latter regressions will not remain just as abstract theoretical ideas. Not only will they be directly applicable to the development of the main case, but they will also apply to a great deal of other engineering problems tackled with the finite element method.

Finite elements are like magic to me. I mean, I can follow the whole derivation of the equations, from the strong, weak and variational formulations of the equilibrium equations for the mechanical problem (or the energy conservation for heat transfer) down to the algebraic multigrid preconditioner for the inversion of the stiffness matrix passing through Sobolev spaces and the grid generation. Then I can sit down and program all these steps into a computer, including the shape functions and its derivatives, the assembly of the discretised stiffness matrix assembly, the numerical solution of the system of equations and the computation of the gradient of the solution. Yet, the fact that all these a-priori unconnected steps once gets a pretty picture that resembles reality is still astonishing to me.

@@ -20,7 +30,11 @@ There are some useful tricks that come handy when trying to solve a mechanical p
One of the most important ones is using your _imagination_. You will need a lot of imagination to “see“ what it is actually going on when analysing an engineering problem. This skill comes from my background in nuclear engineering where I had not choice but to imagine a [positron-electron annihilation](https://en.wikipedia.org/wiki/Electron%E2%80%93positron_annihilation) or an [Spontaneous fission](https://en.wikipedia.org/wiki/Spontaneous_fission). But in mechanical engineering, it is likewise important to be able to imagine how the loads “press” one element with the other, how the material reacts depending on its properties, how the nodal displacements generate stresses (both normal and shear), how results converge, etc. And what these results actually mean besides the pretty-coloured figures.^[A former boss once told me “I need the CFD” when I handed in some results. I replied that I did not do computational fluid-dynamics but computed the neutron flux kinetics within a nuclear reactor core. He joked “I know, what I need are the _Colors For Directors_, those pretty coloured figures along with your actual results.”]
This journey will definitely need your imagination. We will see equations, numbers, plots, schematics, 3D geometries, interactive 3D views, etc. Still, when the theory says “thermal expansion produces linear stresses” you have to picture in your head three little arrows pulling away from the same point in three directions, or whatever mental picture you have about what you understand are thermally-induced stresses. What comes to your mind when someone says that out of the nine elements of the stress tensors there are only six that are independent? Whatever it is, try to practice that kind of graphical thoughts with every concept.

Another heads up is that we will dig into some math. Probably it would be be simple and you would deal with it very easily. But probably you do not like equations. No problem! Just ignore them for now. Read the text skipping them, it should work. It is fine to ignore math (for now). But, eventually, a time will come in which it cannot (or should not) be avoided. Here comes another experience tip: do not fear mathematics. Even more, keep exercising. You have used differences of squares in high school. You know (or at least knew) how to integrate by parts. Remember what Laplace transforms are used for? Once in a while, perform a division of polynomials using [Ruffini’s rule](https://en.wikipedia.org/wiki/Ruffini's_rule). Or compute the second derivative of the quotient of two functions. Whatever. It should be like doing crosswords on the newspaper. Grab those old physics college books and read the exercises at the end of each chapter. It will pay off later on.
Another heads up is that we will dig into some math. Probably it would be be simple and you would deal with it very easily. But probably you do not like equations. No problem! Just ignore them for now. Read the text skipping them, it should work.
Lukaz says [you do not need to know math to perform finite-element analysis](https://enterfea.com/math-behind-fea/). And he is right, in the sense that you do not need to know thermodynamics to drive a car. It is fine to ignore math for now.
But, eventually, a time will come in which it cannot (or should not) be avoided. If you want to go to space, you will definitely have to learn thermodynamics.

So here comes another experience tip: do not fear mathematics. Even more, keep exercising. You have used differences of squares in high school. You know (or at least knew) how to integrate by parts. Remember what Laplace transforms are used for? Once in a while, perform a division of polynomials using [Ruffini’s rule](https://en.wikipedia.org/wiki/Ruffini's_rule). Or compute the second derivative of the quotient of two functions. Whatever. It should be like doing crosswords on the newspaper. Grab those old physics college books and read the exercises at the end of each chapter. It will pay off later on.


# Case study: nuclear reactors, pressurised pipes and fatigue {#sec:case}
@@ -780,7 +794,7 @@ Let us focus on the first item and leave the second one for a separate discussio

**FIGURA 1D**

Math shows that the location where the derivatives of the interpolated displacements are closer to the real (i.e. the analytical in problem that have it) solution are the elements’ [Gauss points](https://en.wikipedia.org/wiki/Gaussian_quadrature). Even better, the material properties at these points are continuous (they are usually uniform but they can depend on temperature for example) because, unless we are using weird elements, there are no material interfaces inside elements. But how to manage a set of stresses given at the Gauss points instead of at the nodes? Should we use one mesh for the input and another one for the output? What happens when we need to know the stresses on a surface and not just in the bulk of the solid? There are still no one-size-fits-all answers.
Math shows that the location where the derivatives of the interpolated displacements are closer to the real (i.e. the analytical in problem that have it) solution are the elements’ [Gauss points](https://en.wikipedia.org/wiki/Gaussian_quadrature). Even better, the material properties at these points are continuous (they are usually uniform but they can depend on temperature for example) because, unless we are using weird elements, there are no material interfaces inside elements. But how to manage a set of stresses given at the Gauss points instead of at the nodes? Should we use one mesh for the input and another one for the output? What happens when we need to know the stresses on a surface and not just in the bulk of the solid? There are still no one-size-fits-all answers. There is a very interesting [blog post](http://tor-eng.com/2017/11/practical-tips-dealing-stress-singularities/) by Nick Stevens that addresses the issue of stresses computed at sharp corners. What does your favourite FEM program do with such a case?

In any case, this step takes a non-negligible amount of time. The most-common approach, i.e. the node-averaging method is driven mainly by the number of nodes of course. So all-in-all, these are the reasons to use the number of nodes instead of the numbers of elements as a basic parameter to measure the complexity of a FEM problem.

@@ -789,9 +803,9 @@ In any case, this step takes a non-negligible amount of time. The most-common ap

The main issue with fatigue in nuclear piping during operational transients is that at the welds between two materials with different thermal expansion coefficients there can appear potentially-high stresses during temperature changes. If these transients are repeated cyclically, fatigue may occur. We already have risen a warning flag about stresses at material interfaces. Besides all the open questions about computing stresses at nodes, this case also adds the fact that the material properties (say the Young Modulus\ $E$) is different in the elements that are at each side of the interface.

::::: {#fig:test}
::::: {#fig:two-cubes}
![Surface grid showing the fixed face (magenta), the load face (green) and the shared face in the middle.](two-cubes2.png){#fig:two-cubes2 width=48%}
![Warped displacements and Von Mises stresses.](two-cubes4.png){#fig:two-cubes2 width=48%}
![Warped displacements and Von Mises stresses.](two-cubes4.png){#fig:two-cubes4 width=48%}

Two cubes of different materials (the one in the left soft, the one in the right hard) share a face and a pressure is applied at the right-most face.
:::::
@@ -838,14 +852,15 @@ If you cannot get a clear answer for at least one of them, then start to be worr

What we as fully cognizant engineers that have to sign a report stating that a nuclear power plant will not collapse due to fatigue in its pipes is to fully understand what is going on with our stresses. [Richard Stallman](https://en.wikipedia.org/wiki/Richard_Stallman) says that the best way to solve a problem is to avoid it in the first place. In this case, we should avoid having to trust to a written manual and rely on software whose [source code](https://en.wikipedia.org/wiki/Source_code) is available. What we need is the capacity (RMS calls it _freedom_) to be able to see the detailed steps performed by the program so we can answer any question we (or other people) might have.

Without resorting into philosophical digressions about the difference between [free and open-source software](https://en.wikipedia.org/wiki/Free_and_open-source_software) (not because it is not worth it but because it would take a whole book), the programs that make their source code available for their users are called [open-source software](https://en.wikipedia.org/wiki/Open-source_software). If the users can also modify and re-distribute the modified versions are called
Without resorting into philosophical digressions about the difference between [free and open-source software](https://en.wikipedia.org/wiki/Free_and_open-source_software) (not because it is not worth it but because it would take a whole book), the programs that make their source code available for their users are called [open-source software](https://en.wikipedia.org/wiki/Open-source_software). If the users can also modify and re-distribute the modified versions are called

So you do not how to program and th

open source!
open source! not about price, free \neq gratis, it is about freedom in spanish this would be easier

que hacen los programas? NADIE SABE

two cubes
con angus tuvimos que adivinar, por suerte le pegamos pero sino estábamos atados de pies y manos. si hubiese sido open-source podíamos investigar o al menos tener la posibilidad de contratar a alguien

todo esto es para una interfaz abrupta, en realidad hay HAZ, cambio de estructura, etc
SCLs a una distancia (engineering)

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