cosas

nafems4.md

@@ -34,6 +34,8 @@ After further years passed by, engineers (probably the same people that forked s
 
 ![A real-life piping system.](cad1.png)
 
+Actually, this article does not focus on a single case study but on some general ideas regarding analysis of fatigue in piping systems in nuclear power plants. There is no single case study but a compendium of ideas obtained by studying many different systems which are directly related to the safety of a real nuclear reactor.
+
 ## 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.
@@ -42,7 +44,9 @@ In each of the countries that have at least one nuclear power plant there exists
 
 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 stresses within 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 especially at the interfaces between materials with different thermal expansion coefficients.
+Nuclear power plants are not always working at 100% power. They need to be maintained and refueled, 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 stresses within 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 especially at the interfaces between materials with different thermal expansion coefficients.
+
+An important part of the analysis that almost always applies to nuclear power plants but usually also to other installations is the consideration of a possible seismic event. Given a postulated design earthquake, the civil structures that hold the pipes have a response spectra for each floor level. One has to combine these spectra with the natural oscillation modes and frequencies of the piping system using one of several available methodologies such as the “Square Root of Sum of Squares” or SRSS method. This way, an static-equivalent  distributed internal load can be computed and then applied as extra loads conservatively (see [@sec:kinds]) at the moment of highest mechanical demand.
 
 ## Fatigue {#sec:fatigue}
 
@@ -143,19 +147,19 @@ We can note that
  1. The stresses do not depend on the mechanical properties\ $E$ and\ $\nu$ of the material (the displacements do).
  2. All the stresses are linear with the pressure\ $p$, i.e. twice the pressure, twice the stress.
  3. The axial stress is uniform and does not depend on the radial coordinate\ $r$.
- 4. 
- 
+ 4. As the stress tensor is diagonal, these three stresses happen to also be the principal stresses.
+
 That is all what we can say about an infinite pipe with uniform material properties subject to an uniform internal pressure\ $p$. If
- 
+
  * the pipe was not infinite (say any real pipe that has to start and end somewhere), or
  * the cross-section of the pipe is not constant along the axis (say there is a reduction), or
  * there was more than one pipe (say there is a tee), or
  * the material properties are not uniform (say the pipe does not have an uniform temperature but a distribution), or
  * the pressure was not uniform (say because there is liquid inside and its weight cannot be neglected),
- 
-\noindent then we would no longer be able to fully solve the problem with paper and pencil and draw all the conclusions above. However, at least we have a start because we know that if the pipe is finite but long enough or the temperature is not uniform but almost, we still can use the analytical equations as approximations. But what happens is the pipe is short, there are branches and temperature changes like during a transient in a nuclear reactor? Well, that is why we have finite elements.
 
- 
+\noindent then we would no longer be able to fully solve the problem with paper and pencil and draw all the conclusions above. However, at least we have a start because we know that if the pipe is finite but long enough or the temperature is not uniform but almost, we still can use the analytical equations as approximations. After all, [Enrico Fermi](https://en.wikipedia.org/wiki/Enrico_Fermi) managed to reach criticality in the [Chicago Pile-1](https://en.wikipedia.org/wiki/Chicago_Pile-1) with paper and pencil. But what happens if the pipe is short, there are branches and temperature changes like during a transient in a nuclear reactor? Well, that is why we have finite elements. And this is were what we learned at college ends.
+
+
 
 # Finite elements, or solving an actual engineering problem
 
@@ -187,7 +191,7 @@ 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
+## Kinds of finite elements {#sec:kinds}
 
 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, for the mechanical problem, we can have different kinds of
 
@@ -211,7 +215,7 @@ This section is not (just) about different kinds of elements like tetrahedra, he
     - 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:
+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, electromagnetism, 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
@@ -229,13 +233,13 @@ Finally, when then uncertainties associated to the parameters, methods and model
   2. performing a large number of runs for different combination of parameters, and
   3. combining all the results into to obtain a best-estimate value plus uncertainty.
 
-This kind of computation is usually required by the nuclear regulatory authorities when power plant designers need to address the safety of the reactors. What is the heat capacity of uranium above 1000ºC? What is the heat transfer coefficient.   
+This kind of computation is usually required by the nuclear regulatory authorities when power plant designers need to address the safety of the reactors. What is the heat capacity of uranium above 1000ºC? What is the heat transfer coefficient when approaching the [critical heat flux](https://en.wikipedia.org/wiki/Critical_heat_flux) before the [Leidenfrost effect](https://en.wikipedia.org/wiki/Leidenfrost_effect)? A certain statistical analysis has to be done prior to actually parametrically swifting the input parameters so as to obtain a distribution of possible outcomes.
 
 **------**
 
 ## Five whys do you want to do FEA?
 
-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?
+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 discretise 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?
 
 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.
 
@@ -259,10 +263,17 @@ 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?
 
-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.
+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. But how much complexity should we add? We might do as little as axysimmetric linear steady-state conservative studies or as much as full three-dimensional non-linear transient best-estimate plus uncertainties computations. And here is where good engineers should appear: in putting their engineering judgment (call it experience or hunches) into defining what to solve. And it is not (just) because the first option is faster to solve than the latter. Involving many complex methods need more engineering time
+
+ 1. to prepare the input data and set up the algorithms, and
+ 2. to analyse the output data and write engineering reports.
 
+In the first years of the history of computer, when programs where written in decks and output results were printed in continuous paper sheets, it made sense for computer programs to calculate and write as much data as possible even if it was not needed. One would never know if it would not be needed in the future, and CPU time was so expensive that re-running engineering computations because a results was not included in the output was forbidden. But that is not remotely true in the XXI century anymore. Computing time is far cheaper than engineering time (result known as the [UNIX Rule of Economy](http://www.catb.org/~esr/writings/taoup/html/ch01s06.html)) that it should be neglected with respect to the time spent by a cognizant engineer searching and sorting thousands of hard-to-read floating-point numbers.
 
 divert(-1)
+
+-----------------------------------------------------------
+
 ## Computers, those little magic boxes
 
 When we think about finite elements, we automatically think about computers. Of
@@ -292,11 +303,30 @@ UNIX, scriptability, make programs to make programs (here a program is a calcula
 front and back
 
 avoid monolithic
+
+-----------------------------------------------------------
+
 divert(0)
 
 
 # Piping in nuclear rectors
 
+So we need to address the issue of fatigue in nuclear reactor pipes that
+
+ 1. have cross-section changes, branches, valves, etc.
+ 2. are made of different materials,
+ 3. are fixed at different locations to the wall through piping supports,
+ 4. are subject to
+    a. pressure transients,
+    b. heat transients, and
+    c. seismic loads.
+
+Due to 1--3, it is clear we need to use finite elements to solve the elastic problem to obtain the stress tensor which would go into the fatigue curves. Now, even though the nature of the load is time-dependent, the time scale of the phenomena is rather different and should be separately analyzed.
+
+First, the pressure transient 
+
+
+
 ## The infinite pipe revisited after college
 
 3D full
@@ -354,3 +384,8 @@ Back in College, we all learned how to solve engineering problems. But there is
 
  * use your imagination
  * practise math
+ * start with simple cases first
+ * grasp the dependence of results with independent variables
+ * keep in mind there are other methods beside finite elements
+ * within the finite element method, there is a wide variety of complexity in the problems that can be solved
+ * follow the “five whys rule” before compute anything, probably you do not need to