7 år sedan · 556eb528b5
--- a/nafems4.md
+++ b/nafems4.md
@@ -32,7 +32,12 @@ In the years following Enrico Fermi’s demonstration that a self-sustainable fi

 After further years passed by, engineers (probably the same people that forked section\ III) 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 previously amended body of knowledge. This is how knowledge evolves, and it is this kind of complexities that engineers are faced with during their professional lives. We have to face it, it would be a very hard work to re-write everything from scratch every time something changes.

 **figure of a CAD pipe system**
 **figure of a CAD pipe system RO-02 12D-24 33410**

 ## Case study

 In each of the countries that have at least one nuclear power plant there exists a national regulatory body who is responsible for allowing the owner to operate the reactor. These operating licenses are time-limited, with a range that can vary from 25 to 60 years, depending on the design and technology of the reactor. Once expired, the owner might be entitled to an extension, which the regulatory authority can accept provided it can be shown that a certain (and very detailed) set of safety criteria are met. One particular example of requirements is that of fatigue in pipes, especially those that belong to systems that are directly related to the reactor safety.



 # Solid mechanics, or what we are taught at college
@@ -92,7 +97,7 @@ dnl google thin walled pressure vessel strain

 # Finite elements, or solving an actual engineering problem

 Besides infinite pipes (both thin and thick), spheres and a couple of other geometries, there are not other cases for which we can obtain analytical expressions for the elements of the stress tensor. To get results for a solid with real engineering interest, we need to use numerical methods to solve the equilibrium equations. It is not that the equations are hard _per se_. It is that the part we engineers like to design (which are of course better than cylinders and spheres) are so intrincate that render the simple equations into monster which are unsolvable with pencil and paper. Hence, finite elements enter into the scene.
 Besides infinite pipes (both thin and thick), spheres and a couple of other geometries, there are not other cases for which we can obtain analytical expressions for the elements of the stress tensor. To get results for a solid with real engineering interest, we need to use numerical methods to solve the equilibrium equations. It is not that the equations are hard _per se_. It is that the mechanical parts we engineers like to design (which are of course better than cylinders and spheres) are so intrincate that render simple equations into monsters which are unsolvable with pencil and paper. Hence, finite elements enter into the scene.

 ## The name of the game

@@ -104,27 +109,56 @@ There are literally dozens of ways to numerically solve the equilibrium equation
 2. Finite volumes
 3. Finite elements
 
 Each of these methods (also called schemes) have of course their own features, pros and cons. They all exploit the fact that the equations are easy to solve in simple geometries (say a cube). Then the actual geometry is divided into a yuxtaposition of these cubes, the equations are solved in each one and then a global solution is obtained by sewing the little simple solutions one to another. The process of dividing the original domain into simple geometries is called _discretization_, and the resulting collection of these simple geometries is called a mesh or grid. They are composed of volumes, called cells (or elements) and vertices called nodes. 
 Each of these methods (also called schemes) have of course their own features, pros and cons. They all exploit the fact that the equations are easy to solve in simple geometries (say a cube). Then the actual geometry is divided into a yuxtaposition of these cubes, the equations are solved in each one and then a global solution is obtained by sewing the little simple solutions one to another. The process of dividing the original domain into simple geometries is called _discretization_, and the resulting collection of these simple geometries is called a mesh or grid. They are composed of volumes, called cells (or elements) and vertices called nodes. Now, grids can be either

 a. structured, or
 b. unstructured
 
 [Figure@fig:grids] illustrate how the same domain can be discretized using these two kind of grids. In the first case, we could identify any single cell by using just two indexes. We could even tell which nodes define each cell just from these indexes. In the second case, we need an explicit list first to know how many cells there are. Even more, there is no way to link the nodes with the cells (back and forth) other than having a list of nodes and cells. Again, there are pros and cons for each of the grid types such as simplicity, flexibility, etc. In general, unstructured grids and better represent a certain geometry with the same number of cells. Structured grids suffer the so-called “staircase effect” that makes the unusable for discretizing mechanical parts.  
 

 The first of the three methods is based on approximating derivative (i.e. differentials) by incremental quotients (i.e. differences). The second one heavily relies on geometrical ideas rather than on pure mathematical grounds. Finally, our beloved finite elements are the most “mathematical” ones. Actually, a complete derivation of the finite element method can be written in a textbook without requiring a single figure, just like D’Alembert did more than two centuries ago. In any case, it is important to note that finite differences and elements compute results at the _nodes_ of a mesh, whilst finite volumes compute results at the _cells_ of a mesh. 
 The first of the three methods is based on approximating derivative (i.e. differentials) by incremental quotients (i.e. differences). The second one heavily relies on geometrical ideas rather than on pure mathematical grounds. Finally, our beloved finite elements are the most “mathematical” ones. Actually, a complete derivation of the finite element method can be written in a textbook without requiring a single figure, just like D’Alembert did more than two centuries ago. In any case, it is important to note that finite differences and elements compute results at the _nodes_ of a mesh, whilst finite volumes compute results at the _cells_ of a mesh. Finally, any method may be used in structured grids but only finite elements and volumes are especially suited for working with unstructured grids. 

 There are technical reasons that justify why the finite element method is the kings for mechanical analysis. But that does not mean that other methods may be employed. For instance, fluid mechanics are better solved using finite volumes. And further other combinations may be found in the literature. 
 There are technical reasons that justify why the finite element method is the king of mechanical analysis. But that does not mean that other methods may be employed. For instance, fluid mechanics are better solved using finite volumes. And further other combinations may be found in the literature. 

 Before proceeding, I would like to make two comments about common nomenclature. The first one is that if we exchanged the words “volumes” and “elements” in all the written books and articles, nobody would note the difference. There is nothing particular in both theories that can justify why finite volumes use volumes and finite elements use elements. Actually volumes and elements are the same geometric constructions. The names were randomly assigned.

 The second one is more philosophical and refers to the word “simulation” which is often used to refer to solving a problem using a numerical scheme such as the finite element method. [I am against at using this word for this endeavour](https://www.seamplex.com/blog/say-modeling-not-simulation.html). The term simulation has a connotation of both “pretending” and “faking” something, that is definitely not what we are doing when solving an engineering problem with finite elements. Sure there are some cases in which we simulate, such as using the Monte Carlo method (originally used by Fermi as an attempt to understand how neutrons behave in the core of nuclear reactors). But when solving deterministic mechanical engineering problems I would rather say “modelling” than “simulation.”


 ## So, why do you want to do FEA?
 ## Five whys do you want to do FEA?

 solid mechanics, because we ant to have a degree.
 So we know we need a numerical scheme to solve our mechanical problem because anything slightly more complex than an infinite pipe does not have analytical solution. We need an unstructured grid because we would not use Legos to discretize pipes. We selected the finite elements method over the finite volumes method, because FEM is the king. Can we pause again and ask ourselves why is it that we want to do finite-element analysis?

 five whys
 There exists a very useful problem-solving technique coined by [Taiichi Ohno](https://en.wikipedia.org/wiki/Taiichi_Ohno), the father of the [Toyota production system](https://en.wikipedia.org/wiki/Toyota_Production_System), known as the [Five-whys rule](https://en.wikipedia.org/wiki/5_Whys). It is based on the fact people make decisions following a certain reasoning logic that most of the time is subjective and biased and not purely rational and neutral. By recursively asking (at least five times) the cause of a certain issue, it might possible to understand what the real nature of the problem (or issue being investigated) is. And it might even be possible to to take counter-measures in order to fix what seems wrong.

 Here is an [original example](https://www.toyota-global.com/company/toyota_traditions/quality/mar_apr_2006.html):


 1. Why did the robot stop?  
     The circuit has overloaded, causing a fuse to blow.
    
 2. Why is the circuit overloaded?  
     There was insufficient lubrication on the bearings, so they locked up.
     
 3. Why was there insufficient lubrication on the bearings?  
     The oil pump on the robot is not circulating sufficient oil.
     
 4. Why is the pump not circulating sufficient oil?  
     The pump intake is clogged with metal shavings.
     
 5. Why is the intake clogged with metal shavings?  
     Because there is no filter on the pump.

 You get the point. We usually assume we have to do what we usually do (i.e. perform finite element analysis). But do we? Do we add a filter or just replace the fuse?

 **justify FEM in the nuclear fatigue case**

 divert(-1)
 ## Computers, those little magic boxes

 When we think about finite elements, we automatically think about computers. Of 

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

 ENIAC
@@ -150,10 +184,12 @@ UNIX, scriptability, make programs to make programs (here a program is a calcula
 front and back

 avoid monolithic
 divert(0)


 # Nuclear-grade piping and ASME
 # Piping in nuclear rectors

 ## The infinite pipe revisited
 ## The infinite pipe revisited after college

 3D full

@@ -172,6 +208,13 @@ struct vs unstruct

 complete vs incomplete (hexa)


 ## 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?)

 ## Linearity of displacements and stresses

 cantilever beam, principal stresses, linearity of von mises
@@ -180,11 +223,6 @@ cantilever beam, principal stresses, linearity of von mises



 ### 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