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update after call with angus on 2020-02-06

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      nafems4.md
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  6. Двоични данни
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nafems4.md Целия файл

@@ -91,7 +91,7 @@ An important part of the analysis that almost always applies to nuclear power pl

### Fatigue {#sec:fatigue}

Mechanical systems can fail for a wide variety of reasons. The effect known as fatigue can initiate and grow microscopic cracks at the grain level, called [dislocations](https://en.wikipedia.org/wiki/Dislocations). Cracks initiation grow at levels well below yield stresses. Once these cracks reach a critical size, then the material fails catastrophically. There are currently not 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:
Mechanical systems can fail for a wide variety of reasons. The effect known as fatigue can initiate and grow microscopic cracks at the grain level, called [dislocations](https://en.wikipedia.org/wiki/Dislocations). Cracks can grow at levels well below yield stresses. Once these cracks reach a critical size, the material fails catastrophically. There are currently not 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:

1. stress life, or
2. strain life
@@ -101,7 +101,7 @@ The first one is suitable for cases where the loads are nowhere near the yield s
For the case study, as the loads come principally from operational loads, the ASME\ stress-life approach should be used. The stress amplitude (AS) of a periodic cycle can be related to the number of cycles ($N$) where failure by fatigue is expected to occur. For each material, this dependence can be computed using standardised tests and a family of “fatigue curves” like the one depicted in\ [@fig:SN] (also called $S$-$N$ curve) for different temperatures can be obtained.


![A fatigue or $S$-$N$ curve for two steels.](SN.svg){#fig:SN width=75%}
![A fatigue or $S$-$N$ curve for two steels.](SN.svg){#fig:SN width=100%}

It should be noted that the fatigue curves are obtained in a particular load case, namely purely-periodic and one-dimensional, which cannot be directly generalised to other three-dimensional cases. Also, any real-life case will be subject to a mixture of complex cycles given by a stress time history and not to pure periodic conditions. The application of the curve data implies a set of simplifications and assumptions that are translated into different possible “rules” for composing real-life cycles. The ASME $S$-$N$ curve also adopt tow safety factors which increase the stress amplitude and reduce the number of cycles respectively. All these intermediate steps render the analysis of fatigue into a conservative computation scheme. Therefore, when a fatigue assessment performed using the fatigue curve method arrives at the conclusion that “fatigue is expected to occur after ten thousand cycles” what it actually means is “we are sure fatigue will not occur before ten thousand cycles, yet it may not occur before one hundred thousand or even more.”

@@ -232,10 +232,10 @@ That is all what we can say about an infinite pipe with uniform material propert
\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 where what we learned at college pretty much ends.


divert(-1)
## Finite elements, or solving actual problems

Besides infinite pipes (both thin and thick), spheres and a couple of other geometries, there are no 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 more complex 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.
Besides infinite pipes (both thin and thick), spheres and a couple of other geometries, there are no 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, constitutive and compatibility 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 more complex 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 {#sec:formulations}
@@ -268,18 +268,12 @@ Discretisation of a spatial domain, For the same number of cells, unstructured g
Back to the three numerical methods, we must say that finite differences is based on approximating derivatives (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 more strict in the mathematical sense. 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 king of mechanical analysis. But that does not mean that other methods may be employed. For instance, fluid mechanics are generally better solved using finite volumes. And further other combinations may be found in the literature.
divert(0)

[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. A detailed analysis of the analytical solution and how results obtained with finite elements converge with respect to the mesh size can be found in the report “On convergence of linearised stresses in an infinite pipe computed using the finite element method.” See [@sec:online].

![Comparison of $u_r(t)$ computed with finite elements and\ [@eq:ur].](ur.svg){#fig:ur width=90%}


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 notice the difference. There is nothing particular in both theories that can justify why FVM use “volumes” and FEM use “elements”. Actually volumes and elements are the same geometric constructions. As far as I know, the names were randomly assigned.



dnl 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 we solve an engineering problem with finite elements. Sure, there are some cases in which we simulate, such as using the [Monte Carlo method](https://en.wikipedia.org/wiki/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.”
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 we solve an engineering problem with finite elements. Sure, there are some cases in which we simulate, such as using the [Monte Carlo method](https://en.wikipedia.org/wiki/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.”

divert(-1)
### Kinds of finite elements {#sec:kinds}
@@ -359,6 +353,15 @@ So we need to address the issue of fatigue in nuclear reactor pipes that
c. seismic loads.

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 methods, 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. 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 displacement obtained with the finite element method. A detailed analysis of the analytical solution and how results obtained with finite elements converge with respect to the mesh size can be found in the report “On convergence of linearised stresses in an infinite pipe computed using the finite element method.” See [@sec:online].

![Comparison of $u_r(t)$ computed with finite elements and\ [@eq:ur].](ur.svg){#fig:ur width=90%}



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 (please [say “modelling” not “simulation.”](https://www.seamplex.com/blog/say-modeling-not-simulation.html)). And even if, in principle, it is true that more complex models should allow you 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 gigabytes of RAM to run) or because it takes more time to compute than you may have before the final report is expected.

First of all, we should note that we need to solve
@@ -367,7 +370,16 @@ First of all, we should note that we need to solve
ii. the natural frequencies and oscillation modes of the piping system to obtain the pseudo-accelerations generated by the design earthquake, and finally
iii. the elastic problem to obtain the stress tensor needed to compute the alternating stress to enter into the fatigue curve.

So for each time\ $t$ of the operational transient, the pipes are subject to
::::: {#fig:pt}
![Case #1](pt-1.svg){#fig:pt1 width=95%}

![Case #2](pt-2.svg){#fig:pt2 width=95%}

Typical transient pressure and temperature conditions for the pipes.
:::::

So for each time\ $t$ of the operational (or incidental) transient such as the ones depicted in [@fig:pt1;@fig:pt2], the pipes are subject to

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
@@ -375,7 +387,7 @@ So for each time\ $t$ of the operational transient, the pipes are subject to

### 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 300ºC down to 100ºC in a couple of minutes, stays at 100ºC for another couple of minutes and then gets back to 300ºC. 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 real 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 one part of the transients the temperature of the water inside the pipes falls from say 300ºC down to 100ºC in a couple of minutes, stays at 100ºC for another couple of minutes and then gets back to 300ºC (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 real 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. And even if we could, there are some tees that connect pipes with different nominal diameters that have a non-trivial geometry, such as the weldolet-type junction shown in\ [@fig:weldolet-cad;@fig:weldolet-mesh]. In this case, there are a number of SCLs (Stress Classification Lines) that go through the pipe’s thickness at both sides of the material interface as illustrated in\ [@fig:weldolet-scls]. It is in these locations that fatigue is to be evaluated.


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pt-1.dat Целия файл

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25 32.53268989 0.973532
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28 32.3171232 6.9604
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30 30.81182324 7.02684
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41 48.94346913 2.16444
47 49.89093543 2.16066
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71 64.91733842 2.20883
73 64.29932271 2.89197
76 65.03184011 3.17932
78 64.53629928 3.16819
82 66.9641054 3.17356
87 64.9455091 3.21538
107 65.80387905 3.21246
139 66.8807123 3.22283
155 67.62095531 3.22179
165 67.28983989 3.2223

Двоични данни
pt-1.pdf Целия файл


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Двоични данни
pt-2.pdf Целия файл


+ 483
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pt-2.svg Целия файл

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transients.ppl Целия файл

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set preamble "\usepackage{amsmath}"

set width 12*unit(cm)
#set size ratio 0.35

set axis x arrow nomirrored
set axis y arrow nomirrored
set key above
#set key top left

#set xrange [0:1]
#set yrange [1:1e20]

set grid x1 y1y2

set terminal pdf

set xlabel "Time~$t$ [hours]"
set y1label "Pressure~$p$ [MPa]"
set y2label "Temperature~$T$ [C]"

set output "pt-1.pdf"
plot \
"pt-1.dat" u 1:3 w lp lt 1 lw 2 pt 17 color green axes x1y1 ti "Pressure [MPa]",\
"pt-1.dat" u 1:2 w lp lt 1 lw 2 pt 16 color red axes x1y2 ti "Temperature [C]"

set output "pt-2.pdf"
plot \
"pt-2.dat" u 1:3 w lp lt 1 lw 2 pt 17 color green axes x1y1 ti "Pressure [MPa]",\
"pt-2.dat" u 1:2 w lp lt 1 lw 2 pt 16 color red axes x1y2 ti "Temperature [C]"

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