inf pipe

nafems4.md

@@ -9,9 +9,17 @@ Please also note that I am not a mechanical engineer, although I shared many und
 ![Simple pendulum](simple.svg){#fig:simple width=35%}\  
 ![Real pendulum](hamaca.jpg){#fig:hamaca width=60%}
 
-A simple pendulum from college physics courses and a real-life pendulum. The same difference exists between theoretical FEM courses and actual engineering problems. Hint: the swing’s period _does_ depend on the hanging mass. See the [actual video](hamaca.webm).
+A simple pendulum from college physics courses and a real-life pendulum. Hint: the swing’s period _does_ depend on the hanging mass. See the [actual video](hamaca.webm).
 :::::
 
+::::: {#fig:pipes}
+![College pipe](infinite-pipe.svg){#fig:infinite-pipe width=25%}\  
+![Real-life pipes](real-piping.png){#fig:real-life width=70%}
+
+An infinitely-long pressurised thick pipe as taught in college and a section of real-life piping system.
+:::::
+
+
 Whether you are a student or a seasoned engineer with many years of experience, you might recall from first year physics courses the introduction of the [simple pendulum](https://en.wikipedia.org/wiki/Pendulum) as case study\ ([@fig:simple]). You learned that the period does not depend on the hanging mass because the weight and the inertia exactly canceled each other. Also, that Galileo said (and [Newton proved](https://www.seamplex.com/wasora/realbook/real-012-mechanics.html)) that for small oscillations the period does not even depend on the amplitude. Someone showed you why it worked this way: because if\ $\sin \theta \approx \theta$ then the motion equations converge to an [harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator). It might have been a difficult subject for you back in those days when you were learning physics and calculus at the same time. You might later study the [Lagrangian](https://en.wikipedia.org/wiki/Lagrangian_mechanics) and even the [Hamiltonian](https://en.wikipedia.org/wiki/Hamiltonian_mechanics) formulations, added a [parametric excitation](https://en.wikipedia.org/wiki/Parametric_oscillator) and studied the [chaotic double pendulum](https://www.seamplex.com/wasora/realbook/real-017-double-pendulum.html). But it was probably after college, say when you took your first son to a swing on a windy day\ ([@fig:hamaca]), that you were faced with a real pendulum worth your full attention. Ok, this is my personal story but could easily be yours as well. My point is that the very same distance between what I imagined as a student what studying a pendulum was and what I saw that day at the swing (namely that the period does depend on the hanging mass) is the same distance between the mechanical problems studied in college and the actual cases encountered during a professional engineer’s lifetime. In this regard, I am referring only to technical issues. The part of dealing with clients, colleagues, bosses, etc. which is definitely not taught in engineering schools (you can get a heads up in business schools, but again it would be a theoretical pendulum) is way beyond the scope of both this article and my own capacities.
 
 Like the pendulums above, we will be swinging back and forth between a case study about fatigue analysis in piping systems of a nuclear power plant and more generic and even romantic topics related to finite elements and computational mechanics. These latter regressions will not remain just as abstract theoretical ideas. Not only will they be directly applicable to the development of the main case, but they will also apply to a great deal of other engineering problems tackled with the finite element method.
@@ -19,7 +27,7 @@ Like the pendulums above, we will be swinging back and forth between a case stud
 Finite elements are like magic to me. I mean, I can follow the whole derivation of the equations, from the strong, weak and variational formulations of the equilibrium equations for the mechanical problem (or the energy conservation for heat transfer) down to the algebraic multigrid preconditioner for the inversion of the stiffness matrix passing through Sobolev spaces and the grid generation. Then I can sit down and program all these steps into a computer, including the shape functions and its derivatives, the assembly of the discretised stiffness matrix assembly, the numerical solution of the system of equations and the computation of the gradient of the solution. Yet, the fact that all these a-priori unconnected steps once gets a pretty picture that resembles reality is still astonishing to me.
 
 
-Again, take all this information as coming from a fellow that has already taken such a journey from college’s pencil and paper to real engineering cases involving complex numerical calculations. And developing, in the meantime, both an actual working finite-element [back-end](https://www.seamplex.com/fino) and [front-end](https://www.caeplex.com) from scratch (the dean of the engineering school I attended used to say “It is not the same to read than to write manuals, we should aim at writing.”).
+Again, take all this information as coming from a fellow that has already taken such a journey from college’s pencil and paper to real engineering cases involving complex numerical calculations. And developing, in the meantime, both an actual working finite-element [back-end](https://www.seamplex.com/fino) and [front-end](https://www.caeplex.com) from scratch (the dean of the engineering school I attended used to say “It is not the same to read than to write manuals, andwe should aim at writing.”).
 
 
 ## Tips and tricks
@@ -27,7 +35,7 @@ Again, take all this information as coming from a fellow that has already taken
 There are some useful tricks that come handy when trying to solve a mechanical problem. Throughout this text, I will try to tell you some of them.
 
 
-One of the most important ones is using your _imagination_. You will need a lot of imagination to “see“ what it is actually going on when analysing an engineering problem. This skill comes from my background in nuclear engineering where I had not choice but to imagine a [positron-electron annihilation](https://en.wikipedia.org/wiki/Electron%E2%80%93positron_annihilation) or an [Spontaneous fission](https://en.wikipedia.org/wiki/Spontaneous_fission). But in mechanical engineering, it is likewise important to be able to imagine how the loads “press” one element with the other, how the material reacts depending on its properties, how the nodal displacements generate stresses (both normal and shear), how results converge, etc. And what these results actually mean besides the pretty-coloured figures (a former boss once told me “I need the CFD” when I handed in some results. I replied that I did not do computational fluid-dynamics but computed the neutron flux kinetics within a nuclear reactor core. He joked “I know, what I need are the _Colors For Directors_, those pretty coloured figures along with your actual results.”).
+One of the most important ones is using your _imagination_. You will need a lot of imagination to “see” what it is actually going on when analysing an engineering problem. This skill comes from my background in nuclear engineering where I had not choice but to imagine a [positron-electron annihilation](https://en.wikipedia.org/wiki/Electron%E2%80%93positron_annihilation) or an [Spontaneous fission](https://en.wikipedia.org/wiki/Spontaneous_fission). But in mechanical engineering, it is likewise important to be able to imagine how the loads “press” one element with the other, how the material reacts depending on its properties, how the nodal displacements generate stresses (both normal and shear), how results converge, etc. And what these results actually mean besides the pretty-coloured figures (a former boss once told me “I need the CFD” when I handed in some results. I replied that I did not do computational fluid-dynamics but computed the neutron flux kinetics within a nuclear reactor core. He joked “I know, what I need are the _Colors For Directors_, those pretty coloured figures along with your actual results.”).
 This journey will definitely need your imagination. We will see  equations, numbers, plots, schematics, 3D geometries, interactive 3D views, etc. 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, or whatever mental picture you have about what you understand are thermally-induced stresses. What comes to your mind when someone says that out of the nine elements of the stress tensors there are only six that are independent? Whatever it is, try to practice that kind of graphical thoughts with every concept.
 
 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.
@@ -48,8 +56,6 @@ In the years following [Enrico Fermi](https://en.wikipedia.org/wiki/Enrico_Fermi
 
 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.
 
-![A section of a real-life piping system.](real-piping.png){#fig:real-life}
-
 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
@@ -410,7 +416,7 @@ An example case where the SCLs are located around the junction between stainless
 
 
 
-As an example, let us consider the system depicted in\ [@fig:valve-cad1;fig:valve-scls1] where there is a stainless-carbon steel interface at the discharge of the valves. Instead of solving the transient heat-conduction problem with the internal temperature of the pipes equal to the temperature of the water in the reference transient condition of the power plant and an external condition of natural convection to the ambient temperature in the whole mesh of\ [@fig:valve-mesh1], a reduced model consisting of half of one of the two valves and a small length of the pipes at both the valve inlet and outlet is used. Once the temperature distribution\ $\hat{T}(\vec{x},t)$ for each time is obtained in the reduced mesh ([@fig:valve-temp], which has the origin at the center of the valve), the actual temperature distribution\ $T(\vec{x},t)$ is computed by an algebraic genearalisation of $\hat{T}(\vec{x},t)$ in the full coordinate system (where the origin is shown in\ [@fig:valve-cad1]). As stated above, those locations which are not covered by the reduced model are generalised with a time-dependent uniform temperature which is the average of the inner and outer temperatures at the inlet and outlet of the reduced mesh. The result is illustrated in figure\ [@fig:valve-gen].
+As an example, let us consider the system depicted in\ [@fig:valve-cad1;@fig:valve-scls1] where there is a stainless-carbon steel interface at the discharge of the valves. Instead of solving the transient heat-conduction problem with the internal temperature of the pipes equal to the temperature of the water in the reference transient condition of the power plant and an external condition of natural convection to the ambient temperature in the whole mesh of\ [@fig:valve-mesh1], a reduced model consisting of half of one of the two valves and a small length of the pipes at both the valve inlet and outlet is used. Once the temperature distribution\ $\hat{T}(\vec{x},t)$ for each time is obtained in the reduced mesh ([@fig:valve-temp], which has the origin at the center of the valve), the actual temperature distribution\ $T(\vec{x},t)$ is computed by an algebraic genearalisation of $\hat{T}(\vec{x},t)$ in the full coordinate system (where the origin is shown in\ [@fig:valve-cad1]). As stated above, those locations which are not covered by the reduced model are generalised with a time-dependent uniform temperature which is the average of the inner and outer temperatures at the inlet and outlet of the reduced mesh. The result is illustrated in figure\ [@fig:valve-gen].
 
 ::::: {#fig:valve-temp-gen}
 ![Reduced mesh around the valve refined around the interface where the transient heat conduction problem is solved.](valve-temp.png){#fig:valve-temp width=80%}

upload.sh

@@ -1 +1 @@
-rsync -avz --progress . seamplex.com:html/jimbo/docs/nafems4
+rsync -avz --progress --exclude=".git" --exclude="refs" . seamplex.com:html/jimbo/docs/nafems4