nafems4/nafems4.md

dnl If you do not know what a Gauss point is, pause here, grab a classical FEM book and resume when you do.

dnl FEM is like magic to me. I can follow the whole derivation, from the strong, the weak and varitionals formulations down to the algebraic multigrid preconditioner for the inversion of the stiffness matrix passing through Sobolev spaces and the grid generatoin. Then I can sit down and code all these steps, including shape funcions, matrix assembly and computation of derivatives^{This is a task that if you can, you definitely have to do. Your time and effort will be highly rewarded in the not-so-far-away future.}. Yet, the fact that all these a-priori unconnected steps once gets a pretty picture that resembles reality is astonishing.


tomen este cpitulo como una charla de café - imginacion! ecuaciones numeros plot 3d drawing intrctive 3d model, but still you have to imagine dericatives nd stress tensors... "thermal expansion gives normal stresses" and then picture out three arrows pulling away




# Background and introduction

Take this as a chat between you an me, i.e. an average engineer that have already taken the journey from college to actual engineering.

Imagination! Try to picture how the results converge, and what they actually mean besides the pretty-colored figures.
We will need your imagination. Equations, numbers, plot, schematics, 3d views, interactive rotable 3d models... but 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.


We will dig into some math. If you do not like equations, just ignore them for now. Read the text skipping them. It is fine (for now).

 Practice math! prsctise math! hace diferencia de cuadrados una vez por mes como si hicieras crucigramas, buscá los libros de analisis y de fisica y hacé los ejercicios

## 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.