cosas

nafems4.md

@@ -23,7 +23,7 @@ This journey will definitely need your imagination. We will see  equations, numb
 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 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, pressurised pipes and fatigue
+# Case study: nuclear reactors, pressurised 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 (yes, like in college). 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 with the ASME code gives the cognisant engineer a unique combination of tools to tackle the problem of designing and/or verifying pressurised piping systems.
@@ -34,10 +34,21 @@ After further years passed by, engineers (probably the same people that forked s
 
 **figure of a CAD pipe system RO-02 12D-24 33410**
 
-## Case study
+## Nuclear reactors
 
 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.
 
+## Pressurised pipes 
+
+How come that pipes are subject to fatigue? Well, on the one hand and without getting into many technical details, the most common nuclear reactor design uses liquid water as coolant and moderator. On the other hand, nuclear power plants cannot by-pass the thermodynamics of the Carnot cycle, and in order to maximise the efficiency of the conversion between the energy stored in the uranium nuclei into electricity they need to reach temperatures as high as possible. So, if we want to have liquid water in the core as hot as possible, we need to increase the pressure. The limiting temperature and pressure are given by the [critical point of water](https://en.wikipedia.org/wiki/Critical_point_(thermodynamics)), which is around 374ºC and 22\ MPa. It is therefore expected to have temperature and pressures near those values in many systems of the plant, especially in the primary circuit those that directly interact with it, such as pressure and inventory control system, decay power removal system, feedwater supply system, emergency core-cooling system, etc. 
+
+Nuclear power plants are not always working at 100% power. They need to be maintained and refuelled, they may undergo operational transients, they might operates at a lower power due to load following conditions, etc. These transient cases involved changes both in temperatures and in pressures that the pipes are subject to, which in turn give rise to changes in the stress tensor of the pipes. As the transients are postulated to occur conservatively cyclically during a number of times during the life-time of the plant (plus its extension period), mechanical fatigue in these piping systems arise. 
+
+## Fatigue
+
+**explain how fatigue is estimated**
+
+**conservative**
 
 
 # Solid mechanics, or what we are taught at college
@@ -97,7 +108,7 @@ 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 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.
+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 intricate 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
 
@@ -125,6 +136,35 @@ Before proceeding, I would like to make two comments about common nomenclature.
 
 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.”
 
+## Kinds of finite elements
+
+This section is not (just) about different kinds of elements like tetrahedra, hexahedra, pyramids and so on. It is about the different kinds of analysis there are. Indeed, there are a whole plethora of particular types of calculations we can perform, all of which can be called “finite element analysis.” For instance, just for the steady-state mechanical problem, we can have different kinds of
+
+ * main elements
+    - 1D beam elements
+    - 2D shell elements
+    - 3D bulk elements
+ * mathematical models   
+    - pure linear
+    - material non-linear
+    - geometrical non-linear
+ * particular studies
+    - buckling
+    - modal
+ * element features
+    - full elements
+    - sub-integrated elements
+    - incomplete elements
+
+And then there exist different pre-processors, meshers, solvers, pre-conditioners, post-processing steps, etc. A similar list can be made for the heat conduction problem, electromagnetics, the Schröedinger equation, neutron transport, etc.  But there is also another level of “kind of problem,” which is related to how much accuracy and precision we are to willing sacrifice in order to have a (probably very much) simpler problem to solve. Again, there are different combinations here but a certain problem can be solved using any of the following three approaches, listed in increasing amount of difficulty and complexity:
+
+ i. conservative
+ ii. best-estimate
+ iii. probabilistic
+ 
+The first one is the easiest way because we can choose parameters and make engineering decisions that simplify the computation in the worse-case scenario. Estimation. This is actually how fatigue results are obtained, as discussed in\ [@sec:fatigue]. 
+
+**------**
 
 ## Five whys do you want to do FEA?
 
@@ -152,7 +192,8 @@ Here is an [original example](https://www.toyota-global.com/company/toyota_tradi
 
 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**
+Getting back to the case study: do we need to do FEM analysis? Well, it does not look like we can obtain the stresses the transient cases with just pencil and paper. 
+
 
 divert(-1)
 ## Computers, those little magic boxes
@@ -234,7 +275,7 @@ two cubes
 
 ## Temperature
 
-# Fatigue
+# Fatigue {#sec:fatigue}
 
 ## In air