7 lat temu · 9a200940c4
--- a/nafems4.md
+++ b/nafems4.md
@@ -23,7 +23,7 @@ This journey will definitely need your imagination. We will see  equations, numb
 Another heads up is that we will dig into some math. Probably it would be be simple and you would deal with it very easily. But probably you do not like equations. No problem! Just ignore them for now. Read the text skipping them, it should work. It is fine to ignore math (for now). But, eventually, a time will come in which it cannot (or should not) be avoided. Here comes another experience tip: do not fear mathematics. Even more, keep exercising. You have used differences of squares in high school. You know (or at least knew) how to integrate by parts. Remember what Laplace transforms 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. Whatever. It should be like doing crosswords on the newspaper. Grab those old physics college books and read the exercises at the end of each chapter. It will pay off later on.


 # Case study: nuclear reactors, pressurised pipes and fatigue
 # Case study: nuclear reactors, pressurised 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.
 This code of practice (book) was born during the late\ XIX century, before finite-element methods for solving partial differential equations were even developed. And much longer before they were available for the general engineering community. Therefore, much of the code assumes design and verification is not necessarily performed numerically but with paper and pencil (yes, like in college). However, it still provides genuine guidance in order to ensure pressurised systems behave safely and properly without needing to resort to computational tools. Combining finite-element analysis with the ASME code gives the cognisant engineer a unique combination of tools to tackle the problem of designing and/or verifying pressurised piping systems.
@@ -268,7 +268,7 @@ Here is an [original example](https://www.toyota-global.com/company/toyota_tradi
 5. Why is the intake clogged with metal shavings?  
     Because there is no filter on the pump.

 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?
 You get the point, even though we know thanks to Richard Feynmann that to answer a “why” question at some point we need to rely on the questioner’s previous experience. 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. 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

@@ -349,7 +349,7 @@ Remember the main issue of the fatigue analysis in these systems is to analyse w
 dnl 33410 07-3-4D-29

 ::::: {#fig:weldolet-cad}
 ![Overall view](weldolet-cad1.png){#fig:weldolet-cad1}
 ![Overall view](weldolet-cad1.png){#fig:weldolet-cad1 width=80%}

 ![Detail of the weldolet-type junction](weldolet-cad2.png){#fig:weldolet-cad2}

@@ -358,7 +358,7 @@ CAD model of a piping system with a 3/4-inch weldolet-type fork (stainless steel


 ::::: {#fig:weldolet-mesh}
 ![Overall view](weldolet-mesh1.png){#fig:weldolet-mesh1}
 ![Overall view](weldolet-mesh1.png){#fig:weldolet-mesh1 width=65%}

 ![Detail of the grid around the junction, showing mesh refinement around the material interface (purple and green elements).](weldolet-mesh2.png){#fig:weldolet-mesh2}

@@ -525,24 +525,24 @@ Case C, combination & +10 & +15 & -15 &  +25 & +20 & -5   & -15 & +25 & +30 \\
 ````

 ````{=html}
 <table>
 <table class="table table-responsive">
 <tr>
  <th></th>
  <th colspan="3">face “right” ($x>0$)</th>
  <th colspan="3">face “right” ($x>0$)</th>
  <th colspan="3">face “right” ($x>0$)</th>
  <th colspan="3">face “right” (<span class="math inline">x&gt;0</span>)</th>
  <th colspan="3">face “back” (<span class="math inline">y&gt;0</span>)</th>
  <th colspan="3">face “top” (<span class="math inline">z&gt;0</span>)</th>
 </tr>  
 <tr>
  <th></th>
  <th>$F_x$</th>
  <th>$F_y$</th>
  <th>$F_z$</th>
  <th>$F_x$</th>
  <th>$F_y$</th>
  <th>$F_z$</th>
  <th>$F_x$</th>
  <th>$F_y$</th>
  <th>$F_z$</th>
  <td></td>
  <td><span class="math inline">F_x</span></td>
  <td><span class="math inline">F_y</span></td>
  <td><span class="math inline">F_z</span></td>
  <td><span class="math inline">F_x</span></td>
  <td><span class="math inline">F_y</span></td>
  <td><span class="math inline">F_z</span></td>
  <td><span class="math inline">F_x</span></td>
  <td><span class="math inline">F_y</span></td>
  <td><span class="math inline">F_z</span></td>
 </tr>
 <tr>
  <td>Case A, pure normal</td>
@@ -561,7 +561,7 @@ Case C, combination & +10 & +15 & -15 &  +25 & +20 & -5   & -15 & +25 & +30 \\



 In the first case, the principal stresses are uniform and equal to the three normal loads. As the forces are in Newton and the area of each face of the cube is 1mm², the usual sorting leads to
 In the first case, the principal stresses are uniform and equal to the three normal loads. As the forces are in Newton and the area of each face of the cube is 1\ mm$^2$, the usual sorting leads to

 divert(-1)
 $$
@@ -638,8 +638,6 @@ ans =
   5.767255
 ```

 octave:36>

 Did I convince you? More or less, right? The third eigenvalue seems to fit. Let us not throw all of our beloved linearity away and dig in further into the subject. There are still two important issues to discuss which can be easily addressed using fresh-year linear algebra (remember, do not fear math!). First of all, even though principal stresses are not linear with respect to the sum they are linear with respect to pure multiplication. Once more, think what happens to the the eigenvalues and eigenvectors of a single stress tensor as all its elements are scaled up or down by a real scalar. They are the same! So, for example, the [Von Mises stress](https://en.wikipedia.org/wiki/Von_Mises_yield_criterion) (which is a combination of the principal stresses) of a beam loaded with a force\ $\alpha \cdot \vec{F}$ is\ $\alpha$ times the stress of the beam loaded with a force\ $\vec{F}$. Please test this hypothesis by playing with your favorite FEM solver an play. Or even better, take a look at the stress invariants $I_1$, $I_2$ and $I_3$ (you can search online or peek into the source code of [Fino](https://www.seamplex.com/fino) and grep for the routine called `fino_compute_principal_stress()`) and see (using paper and pencil!) how they scale up if the individual elements of the stress tensor are scaled by a real factor\ $\alpha$.

 The other issue is that even though in general the eigenvalues of the sum of two matrices are not the same as the eigenvalues of the matrix sum, there are some cases when they are. In effect, if two matrices\ $A$ and\ $B$ commute, i.e. their product is commutative
@@ -671,15 +669,36 @@ The first thing that has to be said is that, as with any interesting problem, th
 ![Quarter-symmetry structured second-order incomplete hexahedra](quarter-struct.png){#fig:cube-struct width=48%}
 ![Quarter-symmetry unstructured second-order tetrahedra](quarter-caeplex.png){#fig:quarter-caeplex width=48%}

 Two of the hundreds of different ways the infinite pressurized pipe can be solved using FEM.
 Two of the hundreds of different ways the infinite pressurized pipe can be solved using FEM
 :::::


 You can get both the exponential nature of each added bullet and how easily we can add new further choices to solve a FEM problem. And each of these choices will reveal you something about the nature of either the mechanical problem or the numerical solution. It is not possible to teach any possible lesson from every outcome in college, so you will have to learn them by yourself getting hands at them. I have already tried to address the particular case of the infinite pipe in a [recent report](https://www.seamplex.com/fino/doc/pipe-linearized/) that is worth reading before carrying on with this article.
 You can get both the exponential nature of each added bullet and how easily we can add new further choices to solve a FEM problem. And each of these choices will reveal you something about the nature of either the mechanical problem or the numerical solution. It is not possible to teach any possible lesson from every outcome in college, so you will have to learn them by yourself getting hands at them. I have already tried to address the particular case of the infinite pipe in a [recent report](https://www.seamplex.com/fino/doc/pipe-linearized/)^[<https://www.seamplex.com/fino/doc/pipe-linearized/>] that is worth reading before carrying on with this article.


 ## ASME stress linearisation (not linearity!)

 We said in\ [@sec:case] that the ASME Boiler and Pressure Vessel Code was born long before modern finite-elements methods were developed and of course being massively available for general engineering analysis (democratised?). Yet the code provide a comprehensive, sound and, more importantly, a widely and commonly-accepted body of knowledge as for the regulatory authorities to require its enforcement to nuclear plant owners. One of the main issues of the ASME code refers to what is known as “membrane” and “bending” stresses, which are defined in section\ VIII (although widely used in other sections, particularly section\ III) annex 5-A. Briefly, they give the zeroth-order (membrane) and first-order (bending) moments of the stress distribution along a so-called Stress Classification Line or SCL, which should be chosen depending on the type of problem under analysis.

 The computation of these membrane and bending stresses are called [“stress linearisation”](https://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/stress-linearisation-for-practising-engineers/) because (I am guessing) it is like computing the first two terms of the Taylor expansion of a real stress distribution along a line, and retaining the first two terms. That is to say, to obtain a linear approximation.

 **figures**

 So what about the SCLs? Well, the ASME standard says that they are lines that go through a wall of the ipe (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 the iso-stress curves (those would be the lines that have the same color 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 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 normals, 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.

 ## ASME stress linearization (not linearity!)
 If you cannot wait to know, the expression for computing the $i$-$j$-th element of the membrane tensor is

 $$
 \text{M}_{ij} = \frac{1}{t} \cdot \int_0^t \sigma_{ij}(t^\prime) \, dt^\prime
 $$
 where $t$ is the length and $t^\prime$ is a parametrization of the SCL. The other linearised stress, namely the _membrane plus bending_ stress tensor \text{MB} again according to ASME VIII Annex 5-A is

 $$
 \text{MB}_{ij} = \text{M}_{ij} \pm \frac{6}{t^2} \cdot \int_0^t \sigma_{ij}(t^\prime) \cdot \left( \frac{t}{2}-t^\prime\right) \, dt^\prime
 $$
 where the sign should be taken such that the resulting combined scalar stress (i.e. Tresca, Von Mises, Principal\ 1, etc.) represents the worst-case scenario.

 No need to know or even understand these two integrals which for sure are not introduced to students in regular college courses. But it would be good to, as linearisation is a cornerstone subject for any serious mechanical analysis of pressurised components following the ASME code.



@@ -704,15 +723,18 @@ two cubes

 Back in College, we all learned how to solve engineering problems. But there is a real gap between the equations written in chalk on a blackboard (now probably in the form of beamer slide presentations) and actual real-life engineering problems. This chapter introduces a real case from the nuclear industry and starts by idealising the structure such that it has a known analytical solution that can be found in textbooks.  Additional realism is added in stages allowing the engineer to develop an understanding of the more complex physics and a faith in the veracity of the FE results where theoretical solutions are not available. Even more, a brief insight into the world of evaluation of low-cycle fatigue using such results further illustrates the complexities of real-life engineering analysis.

 * use your imagination
 * use and exercise 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
 * take into account that even 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
 * use engineering judgment and make sure understand the [“wronger than wrong”](https://en.wikipedia.org/wiki/Wronger_than_wrong) concept
 * play with your favourite FEM solver (mine is <https://caeplex.com>) solving simple cases, trying to predict the results and picturing the stress tensor and its eigenvectors in your imagination
 * grab any stress distribution from any of your FEM projects, compute the iso-stress curves and the draw normal lines to them to get acquainted with SCLs
 * first search online for “stress linearisation” (or “linearization” if you want) and then get a copy of ASME\ VIII Div\ 2 Annex 5-A



 dnl errors and uncertainties: model parameters (is E what we think? is the material linear?), geometry (does the CAD represent the reality?) equations (any effect we did not have take account), discretization (how well does the mesh describe the geometry?)