fixes, mech mesh

nafems4.md

@@ -15,13 +15,13 @@ First of all, please take this text as a written chat between you an me, i.e. a
 
 Please also note that I am not a mechanical engineer, although I took many undergraduate courses in this discipline. I am a nuclear engineer with a strong background in mathematics and computer programming. I went to college between 2002 and 2008. Probably a lot of things have changed since then---at least that is what these “millennial” guys and girls seem to be boasting about---but chances are we all studied solid mechanics and heat transfer with a teacher using a piece of chalk on a blackboard while we as students wrote down notes with pencils on sheets of paper. And there is really not much that one can do with pencil and paper regarding mechanical analysis. Any actual case worth the time of an engineer needs to be more complex than an ideal canonical case with a closed-form solution.
 
-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 a case study\ ([@fig:simple]). You learned that the period does not depend on the hanging mass because the weight and the inertia exactly cancelled each other. Also, that [Galileo](https://en.wikipedia.org/wiki/Galileo_Galilei) said (and [Newton proved](https://seamplex.com/wasora/doc/realbook/012-mechanics/)) that for small oscillations the period does not even depend on the amplitude. Someone showed you why it worked this way: because if\ $\sin \theta \approx \theta$ (in radians) then the motion equations are identical 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.
+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 a case study\ ([@fig:simple]). You learned that the period does not depend on the hanging mass because the weight and the inertia exactly cancel each other. Also, that [Galileo](https://en.wikipedia.org/wiki/Galileo_Galilei) said (and [Newton proved](https://seamplex.com/wasora/doc/realbook/012-mechanics/)) that for small oscillations the period does not even depend on the amplitude. Someone showed you why it worked this way: because if\ $\sin \theta \approx \theta$ (in radians) then the motion equations are identical 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.
 divert(-1)
  You might have later studied 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 analysed the [chaotic double pendulum](https://seamplex.com/wasora/doc/realbook/017-double-pendulum).
 divert(0)dnl
 But it was probably after college, say when you took your first child to a swing on a windy day ([@fig:hamaca]), that you were faced with a real pendulum worth your full attention.
 dnl Yes, this is my personal story but it could have easily been yours as well.
-The very same difference between what I imagined studying a pendulum was and what I saw that day at the swing (namely that the period _does_ depend on the mass of the blob^[See\ <https://youtu.be/Q-lKK4A2OzA>]) is the same difference between the mechanical problems studied in college and the actual cases encountered during a professional engineer’s lifetime ([@fig:pipes]). Real-life engineering problems are far more complex and involve far more information than what professors could write on a blackboard during a class.
+The very same difference between what I imagined studying a pendulum was and what I saw that day at the swing (namely that the period _does_ depend on the mass of the bob^[See\ <https://youtu.be/Q-lKK4A2OzA> and [@sec:online].]) is the same difference between the mechanical problems studied in college and the actual cases encountered during a professional engineer’s lifetime ([@fig:pipes]). Real-life engineering problems are far more complex and involve far more information than what professors could write on a blackboard during a class.
 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 an understanding in business schools, but again it would be a theoretical pendulum) is way beyond the scope of both this article and my own capacities.
 
 ::::: {#fig:pendulum}
@@ -46,7 +46,7 @@ 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](https://en.wikipedia.org/wiki/Multigrid_method) [preconditioner](https://en.wikipedia.org/wiki/Preconditioner) for the inversion of the [stiffness matrix](https://en.wikipedia.org/wiki/Stiffness_matrix) passing through [Sobolev spaces](https://en.wikipedia.org/wiki/Sobolev_space) and the [grid generation](https://en.wikipedia.org/wiki/Mesh_generation). Then I can sit down and program all these steps into a computer, including the [shape functions](https://en.wikipedia.org/wiki/Finite_element_method_in_structural_mechanics#Interpolation_or_shape_functions) and its derivatives, the assembly of the discretised stiffness matrix ([@sec:building]), the numerical solution of the system of equations ([@sec:solving]) and the computation of the gradient of the solution (@sec:stress-computation). Yet, the fact that all these a-priori unconnected steps give rise to pretty pictures that resemble reality is still astonishing to me.
 divert(0)
 
-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 [solver](https://www.seamplex.com/fino) and a [web-based pre and post-processor](https://www.caeplex.com) from scratch.dnl^[The dean of the engineering school I attended used to say “It is not the same to read than to write manuals, and we should aim at writing them.”]
+When finishing college, we feel we can solve and fix the world (if you did not finish yet, you will feel it shortly). But the thing is that we cannot (yet). Once 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 [solver](https://www.seamplex.com/fino) and a [web-based pre and post-processor](https://www.caeplex.com) from scratch.dnl^[The dean of the engineering school I attended used to say “It is not the same to read than to write manuals, and we should aim at writing them.”]
 
 
 ### Tips and hints
@@ -54,23 +54,25 @@ Again, take all this information as coming from a fellow that has already taken
 There are some useful hints that come in 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 to use 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 no 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.
+One of the most important ones is to use 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 no 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) of an uranium nuclei. 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 as the mesh gets denser, etc.
 dnl 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 to convince the managers.”]
 
-This journey will definitely need your imagination. We will peek a little bit into equations, numbers, plots, schematics, CAD geometries, 3D\ views, etc. Still, when the theory says “thermal expansion produces normal 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 thermally-induced stresses are. Whatever it is, try to practice that kind of graphical thoughts with every new concept. Nevertheless, there will be particular locations of the text where imagination will be most useful. I will bring the subject up now and again through the text.
+This journey will definitely need your imagination. We will peek a little bit into equations, numbers, plots, schematics, CAD geometries, 3D\ views, etc. Still, when the theory says “thermal expansion produces normal 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 thermally-induced stresses are. Whatever it is, try to practice that kind of graphical thoughts with every new concept. Nevertheless, there will be particular locations of the text where imagination will be most useful. I will bring the subject up now and again throughout the text.
 
 
 Another point to observe is that we will be digging into some mathematics. Probably they would be simple and you would deal with them very easily. But chances are you do not like equations. No problem! Just ignore them for now. Read the text skipping them, it should work as well.
-But very much like learning to drive does not involve a lecture on thermodynamics, it is true that solving problems with finite elements does not require to learn complex mathematics. But it is a “nice-to-have,” in the same sense that people who know harmony theory enjoy very much more good music than those who do not.
-So my experience tip is this one: even though you do not strictly need them, keep exercising mathematics. You have used [differences of squares](https://en.wikipedia.org/wiki/Difference_of_two_squares) in high school, didn’t you? You know (or at least knew) how to [integrate by parts](https://en.wikipedia.org/wiki/Integration_by_parts). Do you remember what [Laplace transforms](https://en.wikipedia.org/wiki/Laplace_transform) are used for? 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](https://en.wikipedia.org/wiki/Quotient_rule). Whatever. It should be like doing crosswords on the newspaper. Grab those old physics college books and solve the exercises at the end of each chapter. All the effort  will, for sure, pay off later on.
+Very much like learning to drive does not involve a lecture on thermodynamics, it is true that solving problems with finite elements does not require to learn complex mathematics. But both thermodynamics and mathematics are “nice-to-have,” in the same sense that people who know harmony theory enjoy very much more good music than those who do not.
+So my experience tip is this one: even though you do not strictly need them, keep exercising mathematics. You have used [differences of squares](https://en.wikipedia.org/wiki/Difference_of_two_squares) in high school, didn’t you? You know (or at least knew) how to [integrate by parts](https://en.wikipedia.org/wiki/Integration_by_parts). Do you remember what [Laplace transforms](https://en.wikipedia.org/wiki/Laplace_transform) are used for? 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](https://en.wikipedia.org/wiki/Quotient_rule). Whatever. It should be like doing crosswords on the newspaper. Grab those old physics college books and solve the exercises at the end of each chapter. All the effort  will, trust me, pay off later on.
 
-One final comment: throughout the text I will be referring to “your favourite FEM program.” I bet you do have one. Mine is [CAEplex](https://caeplex.com) (it works on top of [Fino](https://www.seamplex.com/fino), which is free and open source). We will be using your favourite program in this test study to perform some tests and play a little bit. And we will also use it to think about what it means to use a FEM program to generate results that will eventually end up in a written project with your signature. Keep that in mind.
+divert(-1)
+One final comment: throughout the text I will be referring to “your favourite FEM program.” I bet you do have one. Mine is [CAEplex](https://caeplex.com) (it works on top of [Fino](https://www.seamplex.com/fino), which is free and open source). We will be using your favourite program in this case study to perform some tests and play a little bit. And we will also use it to think about what it means to use a FEM program to generate results that will eventually end up in a written project with your signature. Keep that in mind.
+divert(0)
 
 
 ## Case study: reactors, pipes and fatigue {#sec:case}
 
 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](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code).
-This code of practice was born during the late\ 19th 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). It provides 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.
+This code of practice was born during the late\ 19th 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). It provides guidance in order to ensure pressurised systems behave safely and properly without necessarily needing to resort to computational tools. Yet 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.
 
 In the years following [Enrico Fermi](https://en.wikipedia.org/wiki/Enrico_Fermi)’s demonstration that a self-sustainable [fission reaction](https://en.wikipedia.org/wiki/Nuclear_fission) chain was possible (actually, in fact, after WWII was over), people started to build plants in order to transform the energy stored within an atom’s nucleus into usable electrical power. They quickly reached the conclusion that high-pressure heat exchangers and turbines were needed. So they started to follow codes of practise like the aforementioned ASME\ B&PVC. They also realised that some requirements did not fit the needs of the nuclear industry. But instead of writing a new code from scratch, they added a new chapter to the existing body of knowledge: the celebrated [ASME Section\ III](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code#ASME_BPVC_Section_III_-_Rules_for_Construction_of_Nuclear_Facility_Components).
 
@@ -83,19 +85,19 @@ In each of the countries that have at least one nuclear power plant there exists
 
 ### Pressurised pipes
 
-Why are pipes 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](https://en.wikipedia.org/wiki/Carnot_cycle), and in order to maximise the efficiency of the conversion of the energy stored in the uranium nuclei into electricity we 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 temperatures and pressures near those values in many systems of the plant, especially in the primary circuit and 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.
+Why are pipes subject to fatigue? Well, on the one hand and without getting into many technical details, the most common nuclear reactor design uses liquid water too extract the heat generated in the fuel rods (coolant) and to slow down fast neutrons born in the fission process (moderator). Nuclear power plants cannot by-pass the thermodynamics of the [Carnot cycle](https://en.wikipedia.org/wiki/Carnot_cycle) thus in order to maximise the efficiency of the conversion of the energy stored in the uranium nuclei into electricity we 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 temperatures and pressures near those values in many systems of the plant, especially in the primary circuit (which is in contact with the reactor core) and 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.
 
-[@Fig:cad-figure] shows the three-dimensional CAD model of a non-real piping system of an imaginary nuclear power plant which will serve as our case study to illustrate the complexities that arise in real-life engineering projects as compared to theoretical pipes drawn on a blackboard. There is a valve with a 10-inch inlet and a 12-inch outlet, there are elbows and a tee, there are supports at the end of the pipes and at intermediate locations, etc. And, more importantly, the 12-inch pipe, the valve body and nozzles are made of stainless steel while the 10-inch pipe is made of carbon steel. So _differential_ thermal expansion leading to non-trivial mechanical stresses are expected to occur if temperature and pressure change in time, which happens a lot because nuclear power plants are not always working at 100% of their maximum power capacity. They need to be maintained and refuelled, they may undergo operational (and some incidental) transients, they might operate at a lower power due to load following conditions, etc.
+[@Fig:cad-figure] shows the three-dimensional CAD model of a non-real piping system of an imaginary nuclear power plant which will serve as our case study to illustrate the complexities that arise in real-life engineering projects as compared to theoretical pipes drawn on a blackboard. There is a valve with a 10-inch inlet and a 12-inch outlet. There are elbows and a tee. There are supports at the end of the pipes and at intermediate locations, etc. And, more importantly, the 12-inch pipe, the valve body and the inlet and outlet nozzles are made of stainless steel while the 10-inch pipe is made of carbon steel. So _differential_ thermal expansion leading to non-trivial mechanical stresses are expected to occur if the temperature distribution changes in time. This indeed happens a lot because nuclear power plants are not always working at 100% of their maximum power capacity. They need to be maintained and refuelled, they may undergo operational (and some incidental) transients, they might operate at a lower power due to load following conditions, etc.
 
-![3D\ CAD model for the (non-real) piping system analysed in this study case](cad-figure-annotated.svg){#fig:cad-figure width=85%}
+![3D\ CAD model for the (non-real) piping system used as the study case](cad-figure-annotated.svg){#fig:cad-figure width=85%}
 
-It should be noted that this case study is still a simplification of real-life piping systems, which are still more complex that [@fig:cad-figure]. Also, the analysis which we will perform in this chapter is also far more simple than what nuclear regulatory bodies require in order to grant lifetime extension licenses to plant operators. For instance, we will not discuss the analysis of ASME’s primary stresses nor go deep into the design-based earthquake analysis which should be supposed to occur during the operational transients.
+It should be noted that this case study is still a simplification of real-life piping systems, which are far more complex that [@fig:cad-figure]. Also, the analysis which we will perform in this chapter is also far more simple than what nuclear regulatory bodies require in order to grant lifetime extension licenses to plant operators. For instance, we will not discuss the analysis of ASME’s primary stresses nor go deep into the design-based earthquake analysis which should be supposed to occur during the operational transients.
 
 dnl 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, both the civil structures and the piping system itself need to be able to withstand such a load, even if it occurs at the moment of highest mechanical demand during one of the operational transients. 
 
 ### Fatigue {#sec:fatigue}
 
-As the transients are postulated to occur cyclically during a number of times throughout the life-time of the plant (plus its extension period), mechanical fatigue in these piping systems may arise. This effect can initiate and grow microscopic cracks at the grain level, called [dislocations](https://en.wikipedia.org/wiki/Dislocations). Cracks can grow at stresses well below the yield level. Once these cracks reach a critical size, the material fails catastrophically. There are currently no complete mechanical models describing fatigue from first principles. Therefore an empirical approach is used. There are two main ways to approach practical fatigue assessment problems using experimental data:
+As the transients are postulated to occur cyclically during a number of times throughout the life-time of the plant (plus its extension period), mechanical fatigue in these piping systems may arise. This effect can initiate and grow microscopic cracks at the grain level, called [dislocations](https://en.wikipedia.org/wiki/Dislocations). Cracks can grow at stresses well below the yield level. Once these cracks reach a critical size, the material fails catastrophically [@schijve]. There are currently no complete mechanical models describing fatigue from first principles, thus an empirical approach is used. There are two main ways to approach practical fatigue assessment problems using experimental data:
 
  1. stress life, or
  2. strain life
@@ -111,7 +113,7 @@ It should be noted that the fatigue curves are obtained for a particular load ca
 
 ## Solid mechanics, or what we are taught at college {#sec:tensor}
 
-So, let us start our journey. Our starting place: undergraduate solid mechanics courses. Our goal: to obtain the internal state of a solid subject to a set of support conditions and loads, i.e. to solve the solid mechanics problem. 
+Let us start our journey. Our starting place: undergraduate solid mechanics courses. Our goal: to obtain the internal state of a solid subject to a set of support conditions and loads, i.e. to solve the solid mechanics problem. 
 divert(-1)
 It was [Augustin-Louis Cauchy](https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy) who formulated for the first time the elasticity equations  we use today. We need to simultaneously solve
 
@@ -179,7 +181,7 @@ Up to 2010, both sections\ III (nuclear components) and VIII (general pressurise
 
 ### An infinitely-long pressurised pipe {#sec:infinite-pipe}
 
-Let us take to our second step, and consider the infinite pipe subject to uniform internal pressure already introduced in\ [@fig:infinite-pipe]. Actually, we are going to solve the mechanical problem on an infinite hollow cylinder, which _looks_ like an imaginary construction called “infinite pipe” that clearly does not exist. This case is usually tackled in college courses, and chances are you already solved it. In fact, the first (and simpler) problem is the “thin cylinder problem.” Then, the “thick cylinder problem” is introduced (the one we solve below), which is slightly more complex. Nevertheless, it has an analytical solution. For the present case, let us consider an infinite pipe (i.e. a hollow cylinder) of internal radius $a$ and external radius $b$ with uniform mechanical properties---Young’s modulus $E$ and Poisson’s ratio $\nu$---subject to an internal uniform pressure $p>0$. [@Fig:timoshenko-cyl] provides a classical illustration of this problem. What follows is more or less what we are taught in school: some equations with a brief explanation of the results. And then we move on to the next subject.
+Let us take to our second step, and consider the infinite pipe subject to uniform internal pressure already introduced in\ [@fig:infinite-pipe]. Actually, we are going to solve the mechanical problem on an infinite hollow cylinder, which _looks_ like an imaginary construction called “infinite pipe” that clearly does not exist. This case is usually tackled in college courses, and chances are you already solved it. In fact, the first (and simpler) problem is the “thin cylinder problem.” Then, the “thick cylinder problem” is introduced (the one we solve below), which is slightly more complex. Nevertheless, it does have an analytical solution. For the present case, let us consider an infinite pipe (i.e. a hollow cylinder) of internal radius $a$ and external radius $b$ with uniform mechanical properties---Young’s modulus $E$ and Poisson’s ratio $\nu$---subject to an internal uniform pressure $p>0$. [@Fig:timoshenko-cyl] provides a classical illustration of this problem. What follows is more or less what we are taught in school: some equations with a brief explanation of the results. And then we move on to the next subject.
 
 ::::: {#fig:timoshenko}
 ![Thick cylinder under pressure](fig41.svg){#fig:timoshenko-cyl width=48%}
@@ -188,7 +190,7 @@ Let us take to our second step, and consider the infinite pipe subject to unifor
 Figures from [Timoshenko](https://en.wikipedia.org/wiki/Semyon_Timoshenko) seminal book “Theory of elasticity” [@timoshenko]
 :::::
 
-Given the cylindrical symmetry of the problem, there can be no dependence on the angular coordinate\ $\theta$ (i.e. there can be no torsion). Also, due to the assumption that the pipe is infinite, no result can depend on the axial direction (i.e. it cannot bend along its axis) so there is only one independent variable, namely the radial coordinate $r$. Moreover, there are only two displacement fields that need to be considered: the axial $u_a(r)$ and the radial $u_r(r)$. The former is identically zero due to the fact that the cylinder is infinite so it makes no sense to assume the pipe can move any finite value along its axis, rendering a [plane strain condition](https://www.ramsay-maunder.co.uk/downloads/NBR03.pdf).
+dnl Given the cylindrical symmetry of the problem, there can be no dependence on the angular coordinate\ $\theta$ (i.e. there can be no torsion). Also, due to the assumption that the pipe is infinite, no result can depend on the axial direction (i.e. it cannot bend along its axis) so there is only one independent variable, namely the radial coordinate $r$. Moreover, there are only two displacement fields that need to be considered: the axial $u_a(r)$ and the radial $u_r(r)$. The former is identically zero due to the fact that the cylinder is infinite so it makes no sense to assume the pipe can move any finite value along its axis, rendering a [plane strain condition](https://www.ramsay-maunder.co.uk/downloads/NBR03.pdf).
 
 divert(-1)
 The equilibrium equation along the radial direction $r$, also known as the [Lamé](https://en.wikipedia.org/wiki/Gabriel_Lam%C3%A9) equation, can be derived with the aid of [@fig:timoshenko-eq] as
@@ -255,12 +257,12 @@ As the solid is deformed, that is to say that different parts are relatively dis
 
 We can note that
 
- 1. The stresses do not depend on the mechanical properties\ $E$ and\ $\nu$ of the material (the displacements do).
+ 1. The stresses do not depend on the value of the Young’s modulus\ $E$ and\ $\sigma_l$ is the only one that depends on\ $\nu$ (the displacements do).
  2. All the stresses are linear with the pressure\ $p$, i.e. twice the pressure means twice the stress.
  3. The axial stress is uniform and does not depend on the radial coordinate\ $r$.
  4. As the stress tensor is diagonal, these three stresses happen to also be the principal stresses.
 
-That is all that what we can say about an infinite pipe with uniform material properties subject to an uniform internal pressure\ $p$. If
+That is all that what we can say about an infinite pipe with uniform material properties subject to an uniform internal pressure\ $p$. Note that if
 
  * the pipe was not infinite (say any real pipe that has to start and end somewhere), or
  * the cross-section of the pipe was not constant along the axis (say there is an elbow or even a reduction), or
@@ -393,7 +395,7 @@ So we need to address the issue of fatigue in nuclear reactor pipes that
 
 dnl As I wanted to illustrate in [@sec:five], it is very important to decide what kind of problem (actually problems) we should be dealing with.
 
-Since we already agreed there is no way to obtain analytical expressions for the stresses in this general case, we need to employ a numerical scheme to solve the equations. Of course we are choosing the finite element method, but keep in mind that there are a lot of other methods for solving partial differential equations: finite differences, finite volumes, modal methods, etc. In particular, finite elements compute nodal values (i.e. displacements and stresses at discrete points in space) and then provide a way to interpolate back the results into any other arbitrary point of the domain. If the method is applied correctly, a mesh refinement will lead to improved results---at the cost of needing an exponentially-increasing computing power, measured in both CPU and RAM. In the limit of infinite number of nodes, the FEM results converge to the actual solution of the original PDEs.
+Since we already agreed there is no way to obtain analytical expressions for the stresses in this general case, we need to employ a numerical scheme to solve the equations. Of course we are choosing the finite element method, but keep in mind that there are a lot of other methods for solving partial differential equations: finite differences, finite volumes, modal methods, etc. Even within finite elements there are many variation, such as displacement-based or mixed formulations, Galerkin or least-squares weighting, and so on. In particular, finite elements compute nodal values (i.e. displacements and stresses at discrete points in space) and then provide a way to interpolate back the results into any other arbitrary point of the domain. If the method is applied correctly, a mesh refinement will lead to improved results---at the cost of needing an exponentially-increasing computing power, measured in both CPU and RAM. In the limit of infinite number of nodes, the FEM results converge to the actual solution of the original PDEs.
 
 For instance, [@fig:ur] shows how some results obtained with the finite element method using different number and order of elements compare to the analytical solution from the previous section. The bullets correspond to the nodal values of the radial displacements and stresses obtained with the finite element method [@pipe-linearized]. The more number of nodes employed, the more accurate the results are---at the expense of an increase of time and computational effort needed to solve the problem.
 
@@ -402,23 +404,23 @@ For instance, [@fig:ur] shows how some results obtained with the finite element
 
 ![$\sigma_r(r)$ [@eq:sigmar].](sigmar.svg){#fig:sigmar width=90%}
 
-Comparison between analytical and FEA displacements and stresses
+Comparison between analytical and FEA results [@pipe-linearized]
 ::::
 
-As a nuclear engineer, I learned (theoretically in college but practically after college) that there are some models that let you see some effects and some that let you see other effects. And even if, in principle, it is true that more complex models should allow you to compute more stuff, they definitely might show you nothing at all if the model is so big and complex that it does not fit into a computer (say because it needs hundreds of terabytes of RAM) or because it takes more time to compute than you may have before the final report is expected.
+As a nuclear engineer, I learned (theoretically in college but practically after college) that there are some models that let you see some effects and some that let you see other effects. And even if, in principle, it is true that more complex models should allow you to compute more stuff, they definitely might show you nothing at all if the model is so big and complex that it does not fit into a computer cluster (say because it needs hundreds of terabytes of RAM) or because it takes more time to compute than you may have before the final report is expected.
 
-Therefore, it is worth to take some time to think about what we need to do, what our choices are to build finite-element models and which one is the most cost-effective one. First of all, we need to define which transients are going to be taken into account. For the current case study, we define that the piping system from [@fig:cad-figure] will be subject to the the four simple time histories for the internal pressure\ $p$ and the fluid temperature\ $T$ as a function of time shown in [@fig:pt].^[Actual real piping systems might be subject to dozens of more complex transients.]
+Therefore, it is worth to take some time to think about what we need to do, what our choices are to build finite-element [models](https://www.seamplex.com/blog/say-modeling-not-simulation.html) and which one is the most convenient in terms of costs and efficiency. First of all, we need to define which transients are going to be taken into account. For the current imaginary case study, we define that the piping system from [@fig:cad-figure] will be subject to the the four simple (and again imaginary) time histories for the internal pressure\ $p$ and the fluid temperature\ $T$ as a function of time shown in [@fig:pt].^[Actual real piping systems might be subject to dozens of more complex transients.]
 
 ::::: {#fig:pt}
 ![Transient #1: heating from cooldown state](pt-1.svg){#fig:pt1 width=95%} 
 
-![Transient #2: cooling from hot pressurized state](pt-2.svg){#fig:pt2 width=95%} 
+![Transient #2: cooling from hot pressurised state](pt-2.svg){#fig:pt2 width=95%} 
 
 ![Transient #3: cooldown from full power to zero in hot condition](pt-3.svg){#fig:pt3 width=95%} 
 
 ![Transient #4: power reduction and turbine trip](pt-4.svg){#fig:pt4 width=95%} 
 
-The four transient operational conditions for the case study
+The four (imaginary) transient operational conditions for the case study
 :::::
 
 Then we note that we need to solve
@@ -430,17 +432,18 @@ Then we note that we need to solve
  
 ### Stress classification lines {#sec:scl}
  
-For each time\ $t$ of the operational (or incidental) transients, the pipes are subject to
+For each time\ $t$ of the operational (or incidental) transients, the pipes are subject to water flowing with 
 
- a. an uniform internal pressure\ $p_i(t)$ that depends on time,
- b. a non-uniform internal temperature $T_i(t)$ that gives rise to a non-trivial time-dependent temperature distribution\ $T(\mathbf{x},t)$ in the bulk of the pipes, and
- c. internal distributed forces\ $\mathbf{f}=\rho \cdot \mathbf{a}$ at those times where the design earthquake is assumed to occur.
+ a. an internal pressure\ $p_i(t)$ that depends on time,
+ b. a certain time-dependent temperature $T_i(t)$ that gives rise to another non-trivial time-dependent temperature distribution\ $T(\mathbf{x},t)$ in the bulk of the pipes.
+ 
+Also, at those times where the designed earthquake is assumed to occur, there are internal distributed forces\ $\mathbf{f}=\rho \cdot \mathbf{a}$ acting on both the water and the pipes’ steel.
 
-All these effects will give rise to stresses that, if repeated over time, will create and growth microscopic cracks which might end in failure by fatigue. The ASME standard gives a code of conduct on how to estimate this damage, and it starts by asking to define _stress classification lines_ (SCLs). ASME says that they are straight lines that go through a wall of the pipe (or vessel or pump, which is what the ASME code is for) from the inside to the outside and ought to be normal to the iso-stress curves. Stop. Picture yourself a stress field, draw in your head the iso-stress curves (those would be the lines that have the same colour in your picture) and then imagine a set of lines that travel in a perpendicular direction to them. Finally, choose the one that seems the prettiest (which most of the time is the one that seems the easiest). There you go! You now have an SCL. But there is a catch. So far, we have referred to a generic concept of “stress.” Which of the several stresses out there should you picture? One of the three normal, the three shear, von\ Mises, Tresca? Well, actually you will have to imagine tensors instead of scalars. And there might not be such a thing as “iso-stress” curves, let alone normal directions. So pick any radial straight line through the pipe wall at a location that seems relevant and now you are done. In our case study, there will be a few different locations around the material interfaces where high stresses due to differential thermal expansion are expected to occur. Just keep this though with you: it is very important to define where the SCLs are located, as they will define the “quality” of the obtained results.
+All these effects will give rise to stresses that, if repeated over time, will create and growth microscopic cracks which might end in failure by fatigue. The ASME standard gives a code of conduct on how to estimate this damage, and it starts by asking to define _stress classification lines_ (SCLs). ASME says that they are straight lines that go through a wall of the pipe (or vessel or pump, which is what the ASME code is for) from the inside to the outside and ought to be normal to the iso-stress curves. Stop. Picture yourself a stress field, draw in your head the iso-stress curves (those would be the lines that have the same colour in your picture) and then imagine a set of lines that travel in a perpendicular direction to them. Finally, choose the one that seems the prettiest (which most of the time is the one that seems the easiest). There you go! You now have an SCL. But there is a catch. So far, we have referred to a generic concept of “stress.” Which of the several stresses out there should you picture? One of the three normal, the three shear, von\ Mises, Tresca? Well, actually you would have to imagine tensors instead of scalars. And there might not be such a thing as “iso-stress” curves, let alone normal directions. So pick any radial straight line through the pipe wall at a location that seems relevant and now you are done. In our case study, there will be a few different locations around the material interfaces where high stresses due to differential thermal expansion are expected to occur. Just keep this though with you: it is very important to define where the SCLs are located, as they will define the “quality” of the obtained results.
 
-![Location of four SCLs defined to analyse fatigue around a material interface](scls.svg){#fig:scls width=90%}
+![Location of four SCLs to asses fatigue at a material interface](scls.svg){#fig:scls width=90%}
 
-For the present case study, four SCLs were defined as illustrated in [@fig:scls]: at a distance of three millimeters from the carbon-stainless steel material interface at the valve inlet, on the vertical $x$-$z$ plane, two on each material, two at the bottom and two at the top of the pipes. It is at the internal point of these four SCLs that fatigue resistance is to be assessed.
+For the present case study, four SCLs were defined as illustrated in [@fig:scls]: at a distance of three millimetres from the carbon-stainless steel material interface at the valve inlet, on the vertical $x$-$z$ plane, two on each material, two at the bottom and two at the top of the pipes. It is at the internal point of these four SCLs that fatigue resistance is to be assessed.
 
 
 
@@ -448,23 +451,34 @@ For the present case study, four SCLs were defined as illustrated in [@fig:scls]
  
 ### Thermal transient {#sec:thermal}
 
-Let us invoke our imagination once again. Assume in one part of the transients the temperature of the water inside the pipes falls from say 250ºC down to room temperature in a couple of minutes, stays cool for another couple of minutes and then gets back to 250ºC (similarly to [@fig:pt]). The temperature within the bulk of the pipes changes as times evolves. The internal wall of the pipes follow the transient temperature (it might be exactly equal or close to it through the [Newton’s law of cooling](https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling)). If the pipe was in a state of uniform temperature, the ramp in the internal wall will start cooling the bulk of the pipe creating a transient thermal gradient. Due to thermal inertia effects, the temperature can have a non-trivial dependence when the ramps start or end (think about it!). So we need to compute a transient heat transfer problem with convective boundary conditions because any other usual tricks like computing a sequence of steady-state computations for different times would not be able to recover these non-trivial distributions.
+Let us invoke our imagination once again. Assume in a certain time interval of the transients the temperature of the water inside the pipes fell abruptly from say 250ºC down 250ºC in a few seconds, stayed cool for half an hour and then went back to 250ºC (similarly to [@fig:pt4]). The internal wall of the pipes would follow the transient temperature (it might be exactly equal or close to it through the [Newton’s law of cooling](https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling))
+If the pipe was in a state of uniform temperature, the ramp in the internal wall would start cooling the bulk of the pipe creating a transient thermal gradient. Due to thermal inertia effects, the temperature might have a non-trivial dependence when the ramps started or ended. First try to think and  picture it! Then see @fig:valve-temp and the videos referenced at @sec:online. So we need to compute a transient heat transfer problem with convective boundary conditions because any other usual tricks like computing a sequence of steady-state computations for different times would not be able to recover these non-trivial distributions.
 
 Remember the main issue of the fatigue analysis in these systems is to analyse what happens around the location of changes of piping classes where different materials (i.e. different expansion coefficients) are present, potentially causing high stresses due to differential thermal expansion (or contraction) under transient conditions. Therefore, even though we are dealing with pipes we cannot use beam or circular shell elements, because we need to take into account the three-dimensional effects of the temperature distribution along the pipe thickness, let alone to model what happens within the body of the valve.
 
-On the one hand, a reasonable number of nodes^[Remember it is the number of degrees of freedom that defines the problem size, which in the finite element method is given by the number of nodes and not by the number of elements. Conversely, if we used finite volumes, it would be given by the number of elements and not by the number of nodes. The two meshes below have the same number of nodes but the one on the right has more nodes and will thus give far more accurate results.\newline![](distorted.png){width=100%}] in order to get a decent grid is around a couple of thousand for each system. On the other hand, solving a couple of dozens of transient heat transfer problems (which we cannot avoid due to the large thermal inertia of the pipes) during a few thousands of seconds over a couple hundred of thousands of nodes might take more time and storage space to hold the results than we might expect.
+On the one hand, a reasonable number of nodes^[Remember it is the number of degrees of freedom that defines the problem size, which in the finite element method is given by the number of nodes and not by the number of elements. Conversely, if we used finite volumes, it would be given by the number of elements and not by the number of nodes. The two meshes below have the same number of nodes but the one on the right has more nodes and will thus give far more accurate results.\newline![](distorted.png){width=100%}] in order to get a decent grid is around a couple of thousand for each piping system under study (@fig:mech). On the other hand, solving dozens of transient heat transfer problems during a few thousands of seconds over a couple hundred of thousands of nodes might take more time and storage space to hold the results than we might have.
+
+::: {#fig:mech}
+![Full view](mech.png){widih=90% #fig:mech-msh}
+
+
+![Detail around the material interface](mech-zoom.png){widih=90% #fig:mech-zoom}
+
+Unstructured volumetric mesh for the CAD of [@fig:cad-figure]
+:::
+
 
-There is a wonderful essay by [Isaac Asimov](https://en.wikipedia.org/wiki/Isaac_Asimov) called [“The Relativity of Wrong”](https://en.wikipedia.org/wiki/The_Relativity_of_Wrong) [@relativity-wrong] where he introduces the idea that even if something cannot be computed exactly, there are different levels of error.
+There is a wonderful essay by [Isaac Asimov](https://en.wikipedia.org/wiki/Isaac_Asimov) called [“The Relativity of Wrong”](https://en.wikipedia.org/wiki/The_Relativity_of_Wrong) [@relativity-wrong] where he introduces the idea that even if something cannot be computed exactly, there are still different levels of error.
 
 > When people thought the Earth was flat, they were wrong. When people thought the Earth was spherical, they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.
 
 We can then merge this idea by Asimov with an adapted version of the [Saint-Venant's principle](https://en.wikipedia.org/wiki/Saint-Venant%27s_principle) and note that the detailed transient temperature distribution is important only around the location of the SCLs. We can then make an engineering approximation and
 
  1. compute the transient thermal problem using a reduced mesh around the SCLs, and
- 2. assume the part of the full system which is not contained in the reduced mesh is at an uniform (though not constant) temperature equal to the average of the inner and outer temperatures in the pipe at each side of the reduced mesh.
+ 2. assume the part of the full system which is not contained in the reduced mesh is at an uniform (though not constant in time) temperature equal to the average of the inner and outer temperatures in the pipe at each side of the reduced mesh.
 
 
-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 geometry of [@fig:cad-figure], a reduced model consisting of half the valve body a small length of the pipes at both the valve inlet and outlet is used as shown in [@fig:valve-temp]. Once the temperature distribution\ $\hat{T}(\mathbf{x},t)$ for each time is obtained in the reduced mesh, the actual temperature distribution\ $T(\mathbf{x},t)$ is computed by an algebraic generalisation of $\hat{T}(\mathbf{x},t)$ in the full mesh of [@fig:cad-figure]. 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.
+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 geometry of [@fig:cad-figure], a reduced model consisting of half the valve body a small length of the pipes at both the valve inlet and outlet is used as shown in [@fig:valve-temp]. Once the temperature distribution\ $\hat{T}(\mathbf{x},t)$ for each time is obtained in the reduced mesh, the actual temperature distribution\ $T(\mathbf{x},t)$ is computed by an algebraic generalisation of $\hat{T}(\mathbf{x},t)$ in the full mesh of [@fig:mech]. 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.
 
 ![Transient temperature distribution in a reduced mesh around the valve](temp-valve-commented.svg){#fig:valve-temp width=100%}
 

pt-1.dat

@@ -1,4 +1,5 @@
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references.bib

@@ -152,3 +152,14 @@
  language = "english",
  hyphenation = "english", 
 }
+
+@book{schijve,
+ author = "Jaap Schijve",
+ title = "Fatigue of Structures and Materials",
+ publisher = "Springer",
+ edition = "2nd",
+ isbn = "978-1-4020-6807-2",
+ year = "2009",
+ langid = "english",
+ language = "english",
+}

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