nafems4/nafems4.md

# Background and introduction

Take this text as a written chat between you an me, i.e. an average engineer that have already taken the journey from college to performing actual engineering using finite element analysis and has something to say about it. Picture yourself with me in a coffee bar, taling and discussing concepts and ideas. Maybe needing to go to a blackboard (or notepad?). Even using a tablet to illustrate some three-dimensional results. But always as a chat between colleagues.

College, paper and pencil or chalk and blackboard.

## Tips and tricks

FEM is like magic to me. I mean, I can follow the whole derivation of the equations, from the strong, weak and varitional 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 discretized 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.

There are some useful tricks that come handy when trying to solve a mechanical problem. Throughout this text, I will try to tell you some of them.

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 analyzing an engineering problem. 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-colored figures.^[A former manager 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 said “I know, what I need are the _Colors For Directors_, those pretty colored 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 math cannot be avoided. Here comes another experience tip: do not fear math. Even more, keep exercising. You have used differences of squares in high school. You know (or at least knew) how to integrate by parts. 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.


## Nuclear reactors, pressurized pipes and fatigue

Piping systems in sensitive industries like nuclear or oil & gas should be designed and analysed following the recommendations of an appropriate set of codes and norms, such as the ASME\ Boiler and Pressure Vessel Code.
This code of practice (book) was born during the late XIX century, before finite-element methods for solving partial differential equations were even developed, and much longer before they were available for the general engineering community. Therefore, much of the code assumes design and verification is not necessarily performed numerically but with paper and pencil. However, it still provides genuine guidance in order to ensure pressurised systems behave safely and properly without needing to resort to computational tools. Combining finite-element analysis (even plain linear equations) with the ASME code gives the cognizant engineer a unique combination of tools to tackle the problem of designing and/or verifying pressurised piping systems.

In the years following Enrico Fermi’s demonstration that a self-sustainable fission reaction chain was possible, people started to build plants in order to transform the energy stored within the atoms nuclei into usable electrical power. They quickly reached the conclusion that high-pressure heat exchangers and turbines were needed. So they started to follow the ASME\ Boiler and Pressure Vessel Code. They also realised that some requirements did not fit the needs of the nuclear industry, but instead of writing a new code from scratch they added a new section to the existing body of knowledge: the celebrated ASME Section\ III\ [1].

After further years passed by, people (probably the same characters as before) noticed that fatigue in nuclear power plants was not exactly the same as in other piping systems. There were some environmental factors directly associated to the power plant that was not taken into account by the regular ASME code. Again, instead of writing a new code from scratch, people decided to add correction factors to the previous body of knowledge. This is how knowledge evolves, and it is this kind of complexities that engineers are faced with during their professional lives. And, yes, it would be a very hard work to re-write everything from scratch every time something changes.

# Solid mechanics, or what we are taught at college

An infinite pipe subject to uniform internal pressure

ecuación diferencial 1D -> appendix

# Finite elements, or solving an actual engineering problem

## The name of the game

FEM, FVM and FDM

Simulation

## Why do you want to do FEA?

five whys

## Computers, those little magic boxes

https://www.springfieldspringfield.co.uk/view_episode_scripts.php?tv-show=the-simpsons&episode=s05e03

ENIAC

### A brief review of history

FEM, Computers

graphics cards

### Hardware

### Software

FOSS

Avoid black boxes

Reflections on trusting trust

UNIX, scriptability, make programs to make programs (here a program is a calculation)

front and back

avoid monolithic

# Nuclear-grade piping and ASME

## The infinite pipe revisited

3D full

Quarter

2 grados

2D axysimmetric

1D collocation


struct vs unstruct

1st vs 2nd

complete vs incomplete (hexa)

## Linearity of displacements and stresses

cantilever beam, principal stresses, linearity of von mises

### ASME stress linearization



### The relativity of wrong

citar a asimov y al report de convergencia

errors and uncertainties: model parameters (is E what we think? is the material linear?), geometry (does the CAD represent the reality?) equations (any effect we did not have take account), discretization (how well does the mesh describe the geometry?)

## Two (or more) materials

### Young and Poisson

two cubes

## A parametric tee

## Temperature

# Fatigue

## In air

## In water

# Conclusions

Back in College, we all learned how to solve engineering problems. But there is a real gap between the equations written in chalk on a blackboard (now probably in the form of beamer slide presentations) and actual real-life engineering problems. This chapter introduces a real case from the nuclear industry and starts by idealising the structure such that it has a known analytical solution that can be found in textbooks.  Additional realism is added in stages allowing the engineer to develop an understanding of the more complex physics and a faith in the veracity of the FE results where theoretical solutions are not available. Even more, a brief insight into the world of evaluation of low-cycle fatigue using such results further illustrates the complexities of real-life engineering analysis.