nafems4/nafems4.md

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

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 path 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


# 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