cosas

.gitignore

@@ -32,3 +32,4 @@ nafems4-emf.pdf
 nafems4-emf.epub
 nafems4-emf.docx
 nafems4-emf.odt
+nafems4.nafems4.md

default.tex

@@ -0,0 +1,500 @@
+% Options for packages loaded elsewhere
+\PassOptionsToPackage{unicode$for(hyperrefoptions)$,$hyperrefoptions$$endfor$}{hyperref}
+\PassOptionsToPackage{hyphens}{url}
+$if(colorlinks)$
+\PassOptionsToPackage{dvipsnames,svgnames*,x11names*}{xcolor}
+$endif$
+$if(dir)$
+$if(latex-dir-rtl)$
+\PassOptionsToPackage{RTLdocument}{bidi}
+$endif$
+$endif$
+$if(CJKmainfont)$
+\PassOptionsToPackage{space}{xeCJK}
+$endif$
+%
+\documentclass[
+$if(fontsize)$
+  $fontsize$,
+$endif$
+$if(lang)$
+  $babel-lang$,
+$endif$
+$if(papersize)$
+  $papersize$paper,
+$endif$
+$if(beamer)$
+  ignorenonframetext,
+$if(handout)$
+  handout,
+$endif$
+$if(aspectratio)$
+  aspectratio=$aspectratio$,
+$endif$
+$endif$
+$for(classoption)$
+  $classoption$$sep$,
+$endfor$
+]{$documentclass$}
+$if(beamer)$
+$if(background-image)$
+\usebackgroundtemplate{%
+  \includegraphics[width=\paperwidth]{$background-image$}%
+}
+$endif$
+\usepackage{pgfpages}
+\setbeamertemplate{caption}[numbered]
+\setbeamertemplate{caption label separator}{: }
+\setbeamercolor{caption name}{fg=normal text.fg}
+\beamertemplatenavigationsymbols$if(navigation)$$navigation$$else$empty$endif$
+$for(beameroption)$
+\setbeameroption{$beameroption$}
+$endfor$
+% Prevent slide breaks in the middle of a paragraph
+\widowpenalties 1 10000
+\raggedbottom
+$if(section-titles)$
+\setbeamertemplate{part page}{
+  \centering
+  \begin{beamercolorbox}[sep=16pt,center]{part title}
+    \usebeamerfont{part title}\insertpart\par
+  \end{beamercolorbox}
+}
+\setbeamertemplate{section page}{
+  \centering
+  \begin{beamercolorbox}[sep=12pt,center]{part title}
+    \usebeamerfont{section title}\insertsection\par
+  \end{beamercolorbox}
+}
+\setbeamertemplate{subsection page}{
+  \centering
+  \begin{beamercolorbox}[sep=8pt,center]{part title}
+    \usebeamerfont{subsection title}\insertsubsection\par
+  \end{beamercolorbox}
+}
+\AtBeginPart{
+  \frame{\partpage}
+}
+\AtBeginSection{
+  \ifbibliography
+  \else
+    \frame{\sectionpage}
+  \fi
+}
+\AtBeginSubsection{
+  \frame{\subsectionpage}
+}
+$endif$
+$endif$
+$if(beamerarticle)$
+\usepackage{beamerarticle} % needs to be loaded first
+$endif$
+$if(fontfamily)$
+\usepackage[$for(fontfamilyoptions)$$fontfamilyoptions$$sep$,$endfor$]{$fontfamily$}
+$else$
+\usepackage{lmodern}
+$endif$
+$if(linestretch)$
+\usepackage{setspace}
+$endif$
+\usepackage{amssymb,amsmath}
+\usepackage{ifxetex,ifluatex}
+\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
+  \usepackage[$if(fontenc)$$fontenc$$else$T1$endif$]{fontenc}
+  \usepackage[utf8]{inputenc}
+  \usepackage{textcomp} % provide euro and other symbols
+\else % if luatex or xetex
+$if(mathspec)$
+  \ifxetex
+    \usepackage{mathspec}
+  \else
+    \usepackage{unicode-math}
+  \fi
+$else$
+  \usepackage{unicode-math}
+$endif$
+  \defaultfontfeatures{Scale=MatchLowercase}
+  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
+$if(mainfont)$
+  \setmainfont[$for(mainfontoptions)$$mainfontoptions$$sep$,$endfor$]{$mainfont$}
+$endif$
+$if(sansfont)$
+  \setsansfont[$for(sansfontoptions)$$sansfontoptions$$sep$,$endfor$]{$sansfont$}
+$endif$
+$if(monofont)$
+  \setmonofont[$for(monofontoptions)$$monofontoptions$$sep$,$endfor$]{$monofont$}
+$endif$
+$for(fontfamilies)$
+  \newfontfamily{$fontfamilies.name$}[$for(fontfamilies.options)$$fontfamilies.options$$sep$,$endfor$]{$fontfamilies.font$}
+$endfor$
+$if(mathfont)$
+$if(mathspec)$
+  \ifxetex
+    \setmathfont(Digits,Latin,Greek)[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$}
+  \else
+    \setmathfont[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$}
+  \fi
+$else$
+  \setmathfont[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$}
+$endif$
+$endif$
+$if(CJKmainfont)$
+  \ifxetex
+    \usepackage{xeCJK}
+    \setCJKmainfont[$for(CJKoptions)$$CJKoptions$$sep$,$endfor$]{$CJKmainfont$}
+  \fi
+$endif$
+$if(luatexjapresetoptions)$
+  \ifluatex
+    \usepackage[$for(luatexjapresetoptions)$$luatexjapresetoptions$$sep$,$endfor$]{luatexja-preset}
+  \fi
+$endif$
+$if(CJKmainfont)$
+  \ifluatex
+    \usepackage[$for(luatexjafontspecoptions)$$luatexjafontspecoptions$$sep$,$endfor$]{luatexja-fontspec}
+    \setmainjfont[$for(CJKoptions)$$CJKoptions$$sep$,$endfor$]{$CJKmainfont$}
+  \fi
+$endif$
+\fi
+$if(beamer)$
+$if(theme)$
+\usetheme[$for(themeoptions)$$themeoptions$$sep$,$endfor$]{$theme$}
+$endif$
+$if(colortheme)$
+\usecolortheme{$colortheme$}
+$endif$
+$if(fonttheme)$
+\usefonttheme{$fonttheme$}
+$endif$
+$if(mainfont)$
+\usefonttheme{serif} % use mainfont rather than sansfont for slide text
+$endif$
+$if(innertheme)$
+\useinnertheme{$innertheme$}
+$endif$
+$if(outertheme)$
+\useoutertheme{$outertheme$}
+$endif$
+$endif$
+% Use upquote if available, for straight quotes in verbatim environments
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\IfFileExists{microtype.sty}{% use microtype if available
+  \usepackage[$for(microtypeoptions)$$microtypeoptions$$sep$,$endfor$]{microtype}
+  \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
+}{}
+$if(indent)$
+$else$
+\makeatletter
+\@ifundefined{KOMAClassName}{% if non-KOMA class
+  \IfFileExists{parskip.sty}{%
+    \usepackage{parskip}
+  }{% else
+    \setlength{\parindent}{0pt}
+    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
+}{% if KOMA class
+  \KOMAoptions{parskip=half}}
+\makeatother
+$endif$
+$if(verbatim-in-note)$
+\usepackage{fancyvrb}
+$endif$
+\usepackage{xcolor}
+\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
+\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
+\hypersetup{
+$if(title-meta)$
+  pdftitle={$title-meta$},
+$endif$
+$if(author-meta)$
+  pdfauthor={$author-meta$},
+$endif$
+$if(lang)$
+  pdflang={$lang$},
+$endif$
+$if(subject)$
+  pdfsubject={$subject$},
+$endif$
+$if(keywords)$
+  pdfkeywords={$for(keywords)$$keywords$$sep$, $endfor$},
+$endif$
+$if(colorlinks)$
+  colorlinks=true,
+  linkcolor=$if(linkcolor)$$linkcolor$$else$Maroon$endif$,
+  filecolor=$if(filecolor)$$filecolor$$else$Maroon$endif$,
+  citecolor=$if(citecolor)$$citecolor$$else$Blue$endif$,
+  urlcolor=$if(urlcolor)$$urlcolor$$else$Blue$endif$,
+$else$
+  hidelinks,
+$endif$
+  pdfcreator={LaTeX via pandoc}}
+\urlstyle{same} % disable monospaced font for URLs
+$if(verbatim-in-note)$
+\VerbatimFootnotes % allow verbatim text in footnotes
+$endif$
+$if(geometry)$
+$if(beamer)$
+\geometry{$for(geometry)$$geometry$$sep$,$endfor$}
+$else$
+\usepackage[$for(geometry)$$geometry$$sep$,$endfor$]{geometry}
+$endif$
+$endif$
+$if(beamer)$
+\newif\ifbibliography
+$endif$
+$if(listings)$
+\usepackage{listings}
+\newcommand{\passthrough}[1]{#1}
+\lstset{defaultdialect=[5.3]Lua}
+\lstset{defaultdialect=[x86masm]Assembler}
+$endif$
+$if(lhs)$
+\lstnewenvironment{code}{\lstset{language=Haskell,basicstyle=\small\ttfamily}}{}
+$endif$
+$if(highlighting-macros)$
+$highlighting-macros$
+$endif$
+$if(tables)$
+\usepackage{longtable,booktabs}
+$if(beamer)$
+\usepackage{caption}
+% Make caption package work with longtable
+\makeatletter
+\def\fnum@table{\tablename~\thetable}
+\makeatother
+$else$
+% Correct order of tables after \paragraph or \subparagraph
+\usepackage{etoolbox}
+\makeatletter
+\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{}
+\makeatother
+% Allow footnotes in longtable head/foot
+\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
+\makesavenoteenv{longtable}
+$endif$
+$endif$
+$if(graphics)$
+\usepackage{graphicx}
+\makeatletter
+\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi}
+\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi}
+\makeatother
+% Scale images if necessary, so that they will not overflow the page
+% margins by default, and it is still possible to overwrite the defaults
+% using explicit options in \includegraphics[width, height, ...]{}
+\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
+% Set default figure placement to htbp
+\makeatletter
+\def\fps@figure{htbp}
+\makeatother
+$endif$
+$if(links-as-notes)$
+% Make links footnotes instead of hotlinks:
+\DeclareRobustCommand{\href}[2]{#2\footnote{\url{#1}}}
+$endif$
+$if(strikeout)$
+\usepackage[normalem]{ulem}
+% Avoid problems with \sout in headers with hyperref
+\pdfstringdefDisableCommands{\renewcommand{\sout}{}}
+$endif$
+\setlength{\emergencystretch}{3em} % prevent overfull lines
+\providecommand{\tightlist}{%
+  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+$if(numbersections)$
+\setcounter{secnumdepth}{$if(secnumdepth)$$secnumdepth$$else$5$endif$}
+$else$
+\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
+$endif$
+$if(beamer)$
+$else$
+$if(block-headings)$
+% Make \paragraph and \subparagraph free-standing
+\ifx\paragraph\undefined\else
+  \let\oldparagraph\paragraph
+  \renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}}
+\fi
+\ifx\subparagraph\undefined\else
+  \let\oldsubparagraph\subparagraph
+  \renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}}
+\fi
+$endif$
+$endif$
+$if(pagestyle)$
+\pagestyle{$pagestyle$}
+$endif$
+$for(header-includes)$
+$header-includes$
+$endfor$
+$if(lang)$
+\ifxetex
+  % Load polyglossia as late as possible: uses bidi with RTL langages (e.g. Hebrew, Arabic)
+  \usepackage{polyglossia}
+  \setmainlanguage[$polyglossia-lang.options$]{$polyglossia-lang.name$}
+$for(polyglossia-otherlangs)$
+  \setotherlanguage[$polyglossia-otherlangs.options$]{$polyglossia-otherlangs.name$}
+$endfor$
+\else
+  \usepackage[shorthands=off,$for(babel-otherlangs)$$babel-otherlangs$,$endfor$main=$babel-lang$]{babel}
+$if(babel-newcommands)$
+  $babel-newcommands$
+$endif$
+\fi
+$endif$
+$if(dir)$
+\ifxetex
+  % Load bidi as late as possible as it modifies e.g. graphicx
+  \usepackage{bidi}
+\fi
+\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
+  \TeXXeTstate=1
+  \newcommand{\RL}[1]{\beginR #1\endR}
+  \newcommand{\LR}[1]{\beginL #1\endL}
+  \newenvironment{RTL}{\beginR}{\endR}
+  \newenvironment{LTR}{\beginL}{\endL}
+\fi
+$endif$
+$if(natbib)$
+\usepackage[$natbiboptions$]{natbib}
+\bibliographystyle{$if(biblio-style)$$biblio-style$$else$plainnat$endif$}
+$endif$
+$if(biblatex)$
+\usepackage[$if(biblio-style)$style=$biblio-style$,$endif$$for(biblatexoptions)$$biblatexoptions$$sep$,$endfor$]{biblatex}
+$for(bibliography)$
+\addbibresource{$bibliography$}
+$endfor$
+$endif$
+$if(csl-refs)$
+\newlength{\cslhangindent}
+\setlength{\cslhangindent}{1.5em}
+\newenvironment{cslreferences}%
+  {$if(csl-hanging-indent)$\setlength{\parindent}{0pt}%
+  \everypar{\setlength{\hangindent}{\cslhangindent}}\ignorespaces$endif$}%
+  {\par}
+$endif$
+
+$if(title)$
+\title{$title$$if(thanks)$\thanks{$thanks$}$endif$}
+$endif$
+$if(subtitle)$
+$if(beamer)$
+$else$
+\usepackage{etoolbox}
+\makeatletter
+\providecommand{\subtitle}[1]{% add subtitle to \maketitle
+  \apptocmd{\@title}{\par {\large #1 \par}}{}{}
+}
+\makeatother
+$endif$
+\subtitle{$subtitle$}
+$endif$
+\author{$for(author)$$author$$sep$ \and $endfor$}
+\date{$date$}
+$if(beamer)$
+$if(institute)$
+\institute{$for(institute)$$institute$$sep$ \and $endfor$}
+$endif$
+$if(titlegraphic)$
+\titlegraphic{\includegraphics{$titlegraphic$}}
+$endif$
+$if(logo)$
+\logo{\includegraphics{$logo$}}
+$endif$
+$endif$
+\include{syntax}
+
+\renewcommand\vec[1]{\mathbf{#1}}
+
+\begin{document}
+$if(has-frontmatter)$
+\frontmatter
+$endif$
+$if(title)$
+$if(beamer)$
+\frame{\titlepage}
+$else$
+\maketitle
+$endif$
+$if(abstract)$
+\begin{abstract}
+$abstract$
+\end{abstract}
+$endif$
+$endif$
+
+$for(include-before)$
+$include-before$
+
+$endfor$
+$if(toc)$
+$if(toc-title)$
+\renewcommand*\contentsname{$toc-title$}
+$endif$
+$if(beamer)$
+\begin{frame}[allowframebreaks]
+$if(toc-title)$
+  \frametitle{$toc-title$}
+$endif$
+  \tableofcontents[hideallsubsections]
+\end{frame}
+$else$
+{
+$if(colorlinks)$
+\hypersetup{linkcolor=$if(toccolor)$$toccolor$$else$$endif$}
+$endif$
+\setcounter{tocdepth}{$toc-depth$}
+\tableofcontents
+}
+$endif$
+$endif$
+$if(lot)$
+\listoftables
+$endif$
+$if(lof)$
+\listoffigures
+$endif$
+$if(linestretch)$
+\setstretch{$linestretch$}
+$endif$
+$if(has-frontmatter)$
+\mainmatter
+$endif$
+$body$
+
+$if(has-frontmatter)$
+\backmatter
+$endif$
+$if(natbib)$
+$if(bibliography)$
+$if(biblio-title)$
+$if(has-chapters)$
+\renewcommand\bibname{$biblio-title$}
+$else$
+\renewcommand\refname{$biblio-title$}
+$endif$
+$endif$
+$if(beamer)$
+\begin{frame}[allowframebreaks]{$biblio-title$}
+  \bibliographytrue
+$endif$
+  \bibliography{$for(bibliography)$$bibliography$$sep$,$endfor$}
+$if(beamer)$
+\end{frame}
+$endif$
+
+$endif$
+$endif$
+$if(biblatex)$
+$if(beamer)$
+\begin{frame}[allowframebreaks]{$biblio-title$}
+  \bibliographytrue
+  \printbibliography[heading=none]
+\end{frame}
+$else$
+\printbibliography$if(biblio-title)$[title=$biblio-title$]$endif$
+$endif$
+
+$endif$
+$for(include-after)$
+$include-after$
+
+$endfor$
+\end{document}

ieee.csl

@@ -0,0 +1,353 @@
+<?xml version="1.0" encoding="utf-8"?>
+<style xmlns="http://purl.org/net/xbiblio/csl" class="in-text" version="1.0" demote-non-dropping-particle="sort-only">
+  <info>
+    <title>IEEE</title>
+    <id>http://www.zotero.org/styles/ieee</id>
+    <link href="http://www.zotero.org/styles/ieee" rel="self"/>
+    <link href="https://www.ieee.org/content/dam/ieee-org/ieee/web/org/conferences/style_references_manual.pdf" rel="documentation"/>
+    <author>
+      <name>Michael Berkowitz</name>
+      <email>mberkowi@gmu.edu</email>
+    </author>
+    <contributor>
+      <name>Julian Onions</name>
+      <email>julian.onions@gmail.com</email>
+    </contributor>
+    <contributor>
+      <name>Rintze Zelle</name>
+      <uri>http://twitter.com/rintzezelle</uri>
+    </contributor>
+    <contributor>
+      <name>Stephen Frank</name>
+      <uri>http://www.zotero.org/sfrank</uri>
+    </contributor>
+    <contributor>
+      <name>Sebastian Karcher</name>
+    </contributor>
+    <contributor>
+      <name>Giuseppe Silano</name>
+      <email>g.silano89@gmail.com</email>
+      <uri>http://giuseppesilano.net</uri>
+    </contributor>
+    <category citation-format="numeric"/>
+    <category field="engineering"/>
+    <category field="generic-base"/>
+    <updated>2019-01-02T15:01:14+00:00</updated>
+    <rights license="http://creativecommons.org/licenses/by-sa/3.0/">This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License</rights>
+  </info>
+  <locale xml:lang="en">
+    <terms>
+      <term name="chapter" form="short">ch.</term>
+      <term name="presented at">presented at the</term>
+      <term name="available at">available</term>
+    </terms>
+  </locale>
+  <!-- Macros -->
+  <macro name="status">
+    <choose>
+      <if variable="page issue volume" match="none">
+        <text variable="status" text-case="capitalize-first" suffix="" font-weight="bold"/>
+      </if>
+    </choose>
+  </macro>
+  <macro name="edition">
+    <choose>
+      <if type="bill book chapter graphic legal_case legislation motion_picture paper-conference report song" match="any">
+        <choose>
+          <if is-numeric="edition">
+            <group delimiter=" ">
+              <number variable="edition" form="ordinal"/>
+              <text term="edition" form="short"/>
+            </group>
+          </if>
+          <else>
+            <text variable="edition" text-case="capitalize-first" suffix="."/>
+          </else>
+        </choose>
+      </if>
+    </choose>
+  </macro>
+  <macro name="issued">
+    <choose>
+      <if type="article-journal report" match="any">
+        <date variable="issued">
+          <date-part name="month" form="short" suffix=" "/>
+          <date-part name="year" form="long"/>
+        </date>
+      </if>
+      <else-if type="bill book chapter graphic legal_case legislation motion_picture paper-conference song thesis" match="any">
+        <date variable="issued">
+          <date-part name="year" form="long"/>
+        </date>
+      </else-if>
+      <else>
+        <date variable="issued">
+          <date-part name="day" form="numeric-leading-zeros" suffix="-"/>
+          <date-part name="month" form="short" suffix="-" strip-periods="true"/>
+          <date-part name="year" form="long"/>
+        </date>
+      </else>
+    </choose>
+  </macro>
+  <macro name="author">
+    <names variable="author">
+      <name and="text" et-al-min="7" et-al-use-first="1" initialize-with=". "/>
+      <label form="short" prefix=", " text-case="capitalize-first"/>
+      <et-al font-style="italic"/>
+      <substitute>
+        <names variable="editor"/>
+        <names variable="translator"/>
+      </substitute>
+    </names>
+  </macro>
+  <macro name="editor">
+    <names variable="editor">
+      <name initialize-with=". " delimiter=", " and="text"/>
+      <label form="short" prefix=", " text-case="capitalize-first"/>
+    </names>
+  </macro>
+  <macro name="locators">
+    <group delimiter=", ">
+      <text macro="edition"/>
+      <group delimiter=" ">
+        <text term="volume" form="short"/>
+        <number variable="volume" form="numeric"/>
+      </group>
+      <group delimiter=" ">
+        <number variable="number-of-volumes" form="numeric"/>
+        <text term="volume" form="short" plural="true"/>
+      </group>
+      <group delimiter=" ">
+        <text term="issue" form="short"/>
+        <number variable="issue" form="numeric"/>
+      </group>
+    </group>
+  </macro>
+  <macro name="title">
+    <choose>
+      <if type="bill book graphic legal_case legislation motion_picture song" match="any">
+        <text variable="title" font-style="italic"/>
+      </if>
+      <else>
+        <text variable="title" quotes="true"/>
+      </else>
+    </choose>
+  </macro>
+  <macro name="publisher">
+    <choose>
+      <if type="bill book chapter graphic legal_case legislation motion_picture paper-conference song" match="any">
+        <group delimiter=": ">
+          <text variable="publisher-place"/>
+          <text variable="publisher"/>
+        </group>
+      </if>
+      <else>
+        <group delimiter=", ">
+          <text variable="publisher"/>
+          <text variable="publisher-place"/>
+        </group>
+      </else>
+    </choose>
+  </macro>
+  <macro name="event">
+    <choose>
+      <if type="paper-conference speech" match="any">
+        <choose>
+          <!-- Published Conference Paper -->
+          <if variable="container-title">
+            <group delimiter=", ">
+              <group delimiter=" ">
+                <text term="in"/>
+                <text variable="container-title" font-style="italic"/>
+              </group>
+              <text variable="event-place"/>
+            </group>
+          </if>
+          <!-- Unpublished Conference Paper -->
+          <else>
+            <group delimiter=", ">
+              <group delimiter=" ">
+                <text term="presented at"/>
+                <text variable="event"/>
+              </group>
+              <text variable="event-place"/>
+            </group>
+          </else>
+        </choose>
+      </if>
+    </choose>
+  </macro>
+  <macro name="access">
+    <choose>
+      <if type="webpage">
+        <choose>
+          <if variable="URL">
+            <group delimiter=". ">
+              <text term="online" prefix="[" suffix="]" text-case="capitalize-first"/>
+              <group delimiter=": ">
+                <text term="available at" text-case="capitalize-first"/>
+                <text variable="URL"/>
+              </group>
+              <group prefix="[" suffix="]" delimiter=": ">
+                <text term="accessed" text-case="capitalize-first"/>
+                <date variable="accessed">
+                  <date-part name="day" form="numeric-leading-zeros" suffix="-"/>
+                  <date-part name="month" form="short" suffix="-" strip-periods="true"/>
+                  <date-part name="year" form="long"/>
+                </date>
+              </group>
+            </group>
+          </if>
+        </choose>
+      </if>
+    </choose>
+  </macro>
+  <macro name="page">
+    <group>
+      <label variable="page" form="short" suffix=" "/>
+      <text variable="page"/>
+    </group>
+  </macro>
+  <macro name="citation-locator">
+    <group delimiter=" ">
+      <choose>
+        <if locator="page">
+          <label variable="locator" form="short"/>
+        </if>
+        <else>
+          <label variable="locator" form="short" text-case="capitalize-first"/>
+        </else>
+      </choose>
+      <text variable="locator"/>
+    </group>
+  </macro>
+  <!-- Citation -->
+  <citation collapse="citation-number">
+    <sort>
+      <key variable="citation-number"/>
+    </sort>
+    <layout delimiter=", ">
+      <group prefix="[" suffix="]" delimiter=", ">
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markuw.m4

@@ -1,5 +1,6 @@
 divert(`-1')
 changequote(`[[', `]]')
+changecom(`/*', `*/')
 define(lorem_ipsum, [[Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nullam elit orci, semper sit amet augue sed, vehicula tristique mauris. Quisque tincidunt felis ex, et sodales nisl fermentum eget. Vestibulum nec lacinia diam. Vivamus condimentum porta volutpat. Nam eu malesuada augue. Vestibulum in volutpat massa. Proin nisi magna, lobortis et tincidunt nec, scelerisque sed quam. Phasellus rutrum scelerisque felis vitae vestibulum. Curabitur egestas nulla ac nibh ullamcorper rhoncus. Nam sit amet justo vulputate, consectetur urna eget, mollis nisl. Nunc vitae suscipit urna. Cras fermentum purus lorem, nec rutrum elit faucibus at. Nullam nec maximus orci. Quisque aliquet et metus vel gravida. Vivamus egestas, elit in ullamcorper semper, massa orci dictum tellus, at rhoncus leo dui eget risus.]])
 # forloop(var, from, to, stmt) - simple version
 define([[forloop]], [[pushdef([[$1]], [[$2]])_forloop($@)popdef([[$1]])]])

meta.yaml

@@ -3,8 +3,17 @@ title: The NAFEMS Case Studies Volumes\ 1
 author: Angus Ramsay
 fontsize: 10pt
 lang: en-GB
+geometry:
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 link-citations: true
 colorlinks: true
 linkReferences: true
@@ -12,8 +21,12 @@ nameInLink: true
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nafems4.md

@@ -9,25 +9,26 @@
 
 # CS4: Fatigue Assessment in Nuclear Piping
 
-## From college theory to an actual engineering problem {#sec:intro}
+## From college theory to actual engineering {#sec:intro}
 
 First of all, please take this text as a written chat between you an me, i.e. a practising engineer that has already taken the journey from college to performing actual engineering works using finite element analysis and has something to say about it. Picture yourself in a coffee bar, talking and discussing concepts and ideas with me. Maybe needing to go to a blackboard (or notepad?). Even using a tablet to illustrate some three-dimensional results. But always as a chat between colleagues.
 
-Please also note that I am not a mechanical engineer, although I shared many undergraduate courses with some of them. 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 paper sheets. 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.
+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.
 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)
+ 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 hanging mass) 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.
+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.
 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}
 ![Simple pendulum](simple.svg){#fig:simple width=35%}\  
 ![Real pendulum](hamaca.jpg){#fig:hamaca width=60%}
 
-A [simple pendulum from college physics courses](https://www.seamplex.com/wasora/doc/realbook/012-mechanics/) and a real-life pendulum. Hint: the swing’s period _does_ depend on the hanging mass as shown in\ <https://youtu.be/Q-lKK4A2OzA>
+A [simple pendulum from college physics](https://www.seamplex.com/wasora/doc/realbook/012-mechanics/) and a real-life pendulum.
 :::::
 
 ::::: {#fig:pipes}
@@ -56,13 +57,14 @@ There are some useful hints that come in handy when trying to solve a mechanical
 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.
 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. What comes to your mind when someone says that out of the nine elements of the [stress tensor](https://en.wikipedia.org/wiki/Cauchy_stress_tensor) ([@sec:tensor]) there are only six that are independent? 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 then and again.
+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.
 
 
 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.
-Anyhow, here comes another experience tip: 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 pay off later on.
+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.
 
-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 it 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.
+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.
 
 
 ## Case study: reactors, pipes and fatigue {#sec:case}
@@ -72,56 +74,61 @@ This code of practice was born during the late\ 19th century, before finite-elem
 
 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).
 
-After further years passed by, engineers (probably the same people that wrote 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 were 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 (sometimes) knowledge evolves, and it is these kinds of complexities that engineers are faced with during their professional lives. We have to admit it, it would be a very difficult task to re-write everything from scratch every time something changes.
-
-![Three-dimensional CAD model for the piping system of\ [@fig:isometric]](real-piping.png){#fig:real-life width=100%}
+After further years passed by, engineers (probably the same people that wrote 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 were 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 (sometimes) knowledge evolves, and it is these kinds of complexities that engineers are faced with during their professional lives. We have to admit it, it would be a very difficult task to re-write everything from scratch every time something changes. And, even though sometimes one would like to change how the world works, most of the times there are sounds reasons not to do so.
 
 
 ### Nuclear reactors
 
-In each of the countries that have at least one nuclear power plant there exists a national regulatory body who is responsible for allowing the owner to operate the reactor. These operating licenses are time-limited, with a range that can vary from 25 to 60 years, depending on the design and technology of the reactor. Once expired, the owner might be entitled to an extension, which the regulatory authority can grant provided it can be shown that a certain (and very detailed) set of safety criteria are met. One particular example of requirements is that for fatigue in pipes, especially those that belong to systems that are directly related to the reactor safety.
+In each of the countries that have at least one nuclear power plant there exists a national regulatory body who is responsible for licensing the owner to operate the reactor. These operating licenses are time-limited, with a range that can vary from 25 to 60 years, depending on the design and technology of the reactor. Once expired, the owner might be entitled to an extension, which the regulatory authority can grant provided it can be shown that a certain (and very detailed) set of safety criteria are met. One particular example of requirements is that for fatigue in pipes, especially those that belong to systems that are directly related to the reactor safety.
 
 ### Pressurised pipes
 
-Why are pipes are subject to fatigue? Well, on the one hand and without getting into many technical details, the most common nuclear reactor design uses liquid water as coolant and moderator. On the other hand, nuclear power plants cannot by-pass the thermodynamics of the [Carnot cycle](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 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.
+
+[@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.
+
+![3D\ CAD model for the (non-real) piping system analysed in this study case](cad-figure-annotated.svg){#fig:cad-figure width=85%}
 
-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. These transient cases involve temperatures and pressures changes in the pipes, which in turn give rise to changes in the stresses within the pipes. 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, especially at the interfaces between dissimilar materials e.g. with different thermal expansion coefficients.
+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.
 
-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.
+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}
 
-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:
+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:
 
  1. stress life, or
  2. strain life
 
-The first one is suitable for cases where the loads are nowhere near the yield stress of the material. When plastic deformation is expected to occur, strain-life methods ought to be employed.
+The first one is suitable for cases where the stresses are nowhere near the yield stress of the material. When plastic deformation is expected to occur, strain-life methods ought to be employed.
 
-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.
+For the preset case study, as the loads come principally from operational loads which are not expected to cause plastic deformation, the ASME\ stress-life approach should be used. The stress amplitude ($S$) 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=100%}
+![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 $S$-$N$ curves also adopt two 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.”  
+It should be noted that the fatigue curves are obtained for a particular load case, usually 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 S-N 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 curves also adopt two 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.”  
 
-## Solid mechanics, or what we are taught at college
+## 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 displacement restrictions and loads (i.e. to solve the solid mechanics problem). 
+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. 
+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
 
  1. equilibrium equations: the fact that external loads need to be balanced by internal stresses,
- 2. constitutive equations: how strains generate stresses within the solid’s material(s), and
- 3. compatibility equations: make sure that the deformed body satisfies the displacement restrictions.
+ 2. constitutive equations: the relationship between stresses and strains within the solid’s material(s), and
+ 3. compatibility equations: make sure that the deformed body house continuous satisfies the support conditions.
 
  
-### Cauchy’s stress tensor {#sec:tensor}
+### Cauchy’s stress tensor 
 
 In any case, what we need to understand (and imagine) from point\ #1 above is that external forces lead to internal stresses. And in any three-dimensional body subject to such external loads, the best way to represent internal stresses is through a $3 \times 3$ _stress tensor_. This is the first point in which we should not fear mathematics. Trust me, let’s follow good old Augustin-Louis. It will pay back later on.
 
 Does the term _tensor_ scare you? It should not. A [tensor](https://en.wikipedia.org/wiki/Tensor) is a general mathematical object and might get complex when dealing with many dimensions (as those encountered in weird stuff like [string theory](https://en.wikipedia.org/wiki/String_theory)), but we will stick here to second-order tensors. They are slightly more complex than a vector, and I assume you are not afraid of vectors, are you? If you recall fresh-year algebra courses, a [vector](https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)) somehow generalises the idea of a [scalar](https://en.wikipedia.org/wiki/Scalar_(mathematics)) in the following sense: a given vector $\mathbf{v}$ can be projected into any direction $\mathbf{n}$ to obtain a scalar $p$. We call this scalar $p$ the “projection” of the vector $\mathbf{v}$ in the direction $\mathbf{n}$. Well, a tensor can be also projected into any direction $\mathbf{n}$. The difference is that instead of a scalar, a vector is now obtained.
 
 So, let me introduce then the three-dimensional [Cauchy’s stress tensor](https://en.wikipedia.org/wiki/Cauchy_stress_tensor):
+divert(0)dnl
+What does this mean? It means finding the components of the [Cauchy’s stress tensor](https://en.wikipedia.org/wiki/Cauchy_stress_tensor)
 
 $$
 \begin{bmatrix}
@@ -131,32 +138,60 @@ $$
 \end{bmatrix}
 $$
 
-It looks (and works) like a regular $3 \times 3$ matrix. Indeed, we will take advantage of this matrix-like behaviour in [@sec:linearity] below. Some brief comments about it:
+\noindent 
+which looks (and works) like a regular $3 \times 3$ matrix.
+divert(-1)Indeed, we will take advantage of this matrix-like behaviour in [@sec:linearity] below.divert(0)
+Some brief comments about it:
 
  * The $\sigma$s are normal stresses, i.e. they try to stretch or tighten the material (depending on the sign).
  * The $\tau$s are shear stresses, i.e. they try to twist the material.
  * Due to rotational equilibrium requirements the conjugate shear stresses should be equal: $\tau_{xy} = \tau_{yx}$, $\tau_{yz} = \tau_{zy}$,  and $\tau_{zx} = \tau_{xz}$. Therefore, the stress tensor is symmetric i.e. there are only six independent elements.
  * The elements of the tensor depend on the orientation of the coordinate system.
- * There exists a particular coordinate system in which the stress tensor is diagonal, i.e. where all the shear stresses are zero. In this case, the three diagonal elements are called the [principal stresses](https://en.wikipedia.org/wiki/Principal_stresses), which happen to be the three [eigenvalues](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) of the stress tensor. The basis of the coordinate system that renders the tensor diagonal are the eigenvectors.
-
+ * There exists a particular coordinate system in which the stress tensor is diagonal, i.e. where all the shear stresses are zero. In this case, the three diagonal elements are called the [principal stresses](https://en.wikipedia.org/wiki/Principal_stresses) $\sigma_1$, $\sigma_2$ and $\sigma_3$---which happen to be the three [eigenvalues](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) of the stress tensor. The basis of the coordinate system that renders the tensor diagonal are the eigenvectors.
 
+divert(-1)
 What does this all have to do with mechanical engineering? Well, once we know what the stress tensor is for every point of a solid, in order to obtain the internal forces per unit area acting in a plane passing through that point and with a normal given by the direction $\mathbf{n}$, all we have to do is “project” the stress tensor through $\mathbf{n}$. In plain simple words:
 
 > If you can compute the stress tensor at each point of your geometry, then... Congratulations! You have solved the solid mechanics problem.
+divert(0)
+
+The full solution involve the nine stresses, out of which there are only six that are different. If we manage to compute the principal stresses $\sigma_1$, $\sigma_2$ and $\sigma_3$ we reduce the number to three. We can go further and obtain a single scalar stress “intensity” value by using one of several material yield theories. The two most common ones are those by [Tresca](https://en.wikipedia.org/wiki/Henri_Tresca) and [von\ Mises](https://en.wikipedia.org/wiki/Richard_von_Mises). The former is the maximum absolute difference between all possible combination of principal stresses
+
+$$
+\sigma_\text{Tr} = \max \Big[
+\left| \sigma_1 - \sigma_2 \right|,
+\left| \sigma_2 - \sigma_3 \right|,
+\left| \sigma_3 - \sigma_1 \right|
+\Big]
+$$
 
+\noindent
+and the latter is 
+
+$$
+\sigma_\text{vM} = \sqrt{\frac{\left(\sigma_1 - \sigma_2 \right)^2+
+\left(\sigma_2 - \sigma_3 \right)^2+
+\left(\sigma_3 - \sigma_1 \right)^2}{2}}
+$$
+
+
+Up to 2010, both sections\ III (nuclear components) and VIII (general pressurised components) of ASME were based on Tresca theory. In newer revisions, section\ VIII switched to von\ Mises while section\ III kept using Tresca.
 
 ### An infinitely-long pressurised pipe {#sec:infinite-pipe}
 
-Let us proceed 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 a pipe. 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 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.
 
 ::::: {#fig:timoshenko}
 ![Thick cylinder under pressure](fig41.svg){#fig:timoshenko-cyl width=48%}
 ![Equilibrium in cylindrical coordinates.](fig40.svg){#fig:timoshenko-eq width=48%}
 
-Figures from [Semyon Timoshenko](https://en.wikipedia.org/wiki/Semyon_Timoshenko) classical book “Theory of elasticity.”
+Figures from [Timoshenko](https://en.wikipedia.org/wiki/Semyon_Timoshenko) seminal book “Theory of elasticity” [@timoshenko]
 :::::
 
-Given the cylindrical symmetry of the problem, 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 ([plane strain condition](https://www.ramsay-maunder.co.uk/downloads/NBR03.pdf)). 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
+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
 
 $$ \frac{d\sigma_r}{dr} + \frac{\sigma_r(r) - \sigma_\theta(r)}{r} = 0 $${#eq:equilibrium}
 
@@ -178,26 +213,30 @@ u(r) = c_1 \cdot r + \frac{c_2}{r}
 $$
 
 For the boundary conditions of the particular problem that the radial stress should be equal to the negative of the internal pressure at $r=a$ and null at $r=b$, the axial displacement has the particular solution
+divert(0)
+
+#### Displacements
+
+Remember that when any solid body is subject to external forces, it has to react in such a way to satisfy the equilibrium conditions. The way solids do this is by deforming a little bit in such a way that the whole body acts as a compressed (or elongated) spring balancing the load.
+Starting from [@fig:timoshenko-eq] and after some mathematics (detailed shown in reference\ [@pipe-linearized]), which is what most of us have already done in college, we can find that the displacement field has the following analytical solution:
 
 $$
 u_r(r) = p \cdot \frac{1+\nu}{E} \cdot \frac{a^2}{b^2-a^2} \cdot \left[ 1-2\nu + \frac{b^2}{r^2} \right]\cdot r
 $${#eq:ur}
 
+divert(-1)
 Exercises for the reader:
 
  1. Think (imagine) and answer why the radial stress should be equal to the negative of the internal pressure at $r=a$ and not the actual pressure\ $p$. Hint: it does not have anything to do with actions and reactions.
  2. Verify that the radial displacement $u_r(r)$ satisfies the equilibrium and constitutive equations.
+divert(0)
 
 
-#### Displacements
-
-Remember that when any solid body is subject to external forces, it has to react in such a way to satisfy the equilibrium conditions. The way solids do this is by deforming a little bit in such a way that the whole body acts as a compressed (or elongated) spring balancing the load. So it is worth to ask how a pressurised pipe deforms to counteract the internal pressure\ $p$.
-
- * There are no longitudinal displacements\ $u_l$ because the pipe is infinite in the axial direction.
- * There are no azimuthal displacements\ $u_\theta$ because the pipe is fully symmetric around the axis.
+ * There are no longitudinal displacements\ $u_l$ because the pipe is infinite in the axial direction so it makes no sense to assume it will have a finite displacement in that direction.
+ * There are no azimuthal displacements\ $u_\theta$ because the pipe is fully symmetric around the axis and since all angles are equal it make no sense to have any dependence on\ $\theta$.
  * There are only radial displacements\ $u_r$ and they depend only on the radial coordinate\ $r$ and not on the axial position\ $z$ or on the azimuthal angle\ $\theta$. 
 
-What does [@eq:ur] mean? Well, that overall the whole pipe expands a little bit radially with the inner face being displaced more than the external surface (use your imagination or wait until we get to [@fig:ur]). How does it expand?
+What does [@eq:ur] mean? Well, that overall the whole pipe expands a little bit radially with the inner face being displaced more than the external surface (use your imagination or wait until we get to [@fig:ur]). But the important thing here is that we have an expression that explicitly tells us how it expands:
 
  1. linearly with the pressure, i.e. twice the pressure means twice the displacement, and
  2. inversely proportional to the Young’s Modulus\ $E$ divided by $1+\nu$, i.e. the more resistant the material, the less radial displacements.
@@ -221,7 +260,7 @@ We can note that
  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 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\ $p$. 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
@@ -229,7 +268,7 @@ That is all what we can say about an infinite pipe with uniform material propert
  * the material properties were not uniform (say the pipe does not have an uniform temperature but a distribution), or
  * the pressure was not uniform (say because there is liquid inside and its weight cannot be neglected),
 
-\noindent then we would no longer be able to fully solve the problem with paper and pencil and draw all the conclusions above. However, at least we have a start because we know that if the pipe is finite but long enough or the temperature is not uniform but almost, we still can use the analytical equations as approximations. 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.
+\noindent then we would no longer be able to fully solve the problem with paper and pencil, have an explicit equation that tells us how the system reacts to the external loads nor draw any of 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)
@@ -340,7 +379,7 @@ Getting back to the case study: do we need to do FEM analysis? Well, it does not
 In the first years of the [history of computers](https://en.wikipedia.org/wiki/History_of_computing_hardware), when programs were written in decks and output results were printed in continuous paper sheets, it made sense for computer programs to calculate and write as much data as possible even if it was not needed. One would never know if it would not be needed in the future, and CPU time was so expensive that re-running engineering computations because a particular result was not included in the output was forbidden. But that is not remotely true in the XXI century anymore. Computing time is far cheaper than engineering time (result known as the [UNIX Rule of Economy](http://www.catb.org/~esr/writings/taoup/html/ch01s06.html)) that it should be neglected with respect to the time spent by a cognisant engineer searching and sorting thousands of hard-to-read floating-point numbers.
 divert(0)
 
-## Piping in nuclear rectors {#sec:piping-nuclear}
+## Piping in nuclear rectors, or a real-life case {#sec:piping-nuclear}
 
 So we need to address the issue of fatigue in nuclear reactor pipes that
 
@@ -354,79 +393,84 @@ 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 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.
+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.
 
-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].
+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.
 
-![Comparison of $u_r(t)$ computed with finite elements and\ [@eq:ur].](ur.svg){#fig:ur width=90%}
+:::: {#fig:pipe-linearized}
+![$u_r(r)$ [@eq:ur].](ur.svg){#fig:ur width=90%}
 
+![$\sigma_r(r)$ [@eq:sigmar].](sigmar.svg){#fig:sigmar width=90%}
 
+Comparison between analytical and FEA displacements and stresses
+::::
 
-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.
+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.
 
-First of all, we should note that we need to solve
-
- i. the heat transfer equation to get the temperature distribution within the pipes,
- 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.
+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.]
 
 ::::: {#fig:pt}
-![Case #1: postulated transient data.](pt-1.svg){#fig:pt1 width=95%} 
+![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%} 
 
-![Case #2: actual plant data from a real transient.](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%} 
 
-Typical transient pressure and temperature conditions for the pipes. 
+![Transient #4: power reduction and turbine trip](pt-4.svg){#fig:pt4 width=95%} 
+
+The four transient operational conditions for the case study
 :::::
 
+Then we note that we need to solve
+
+ i. the transient heat transfer equation to get the temperature distribution within the pipes for all times,
+ 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 (or incidental) transient such as the ones depicted in [@fig:pt1;@fig:pt2], the pipes are subject to
+### Stress classification lines {#sec:scl}
+ 
+For each time\ $t$ of the operational (or incidental) transients, 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
  c. internal distributed forces\ $\mathbf{f}=\rho \cdot \mathbf{a}$ at those times where the design earthquake is assumed to occur.
 
-### 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 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.
+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.
 
-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.
+![Location of four SCLs defined to analyse fatigue around 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.
 
-::::: {#fig:weldolet}
-![Overall view](weldolet-cad1-commented.png){#fig:weldolet-cad width=65%}
 
-![Unstructured tetrahedra-based grid](weldolet-mesh2.png){#fig:weldolet-mesh width=85%}
 
-CAD model of a piping system with a 3/4-inch weldolet-type fork (stainless steel) from a main 12-inch pipe (carbon steel)
-:::::
+ 
+ 
+### 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.
 
-![Location of six SCLs defined to analyse fatigue around a junction.](weldolet-scls-n.png){#fig:weldolet-scls width=90%}
+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 (it is the number of nodes that defines the problem size, not the number of elements) in order to get a decent grid is around 200k 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.
+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) where he introduces the idea that even if something cannot be computed exactly, there are different levels of error. For instance, believing that the Earth is a sphere is less wrong than believing that the Earth is flat, but wrong nonetheless, since it really deviates from a perfect sphere and resembles more an oblate spheroid.
+> 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 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) temperature equal to the average of the inner and outer temperatures in the pipe at each side of the reduced mesh.
 
-![An example case where the SCLs are located around the junction between stainless-steel valves and carbon steel pipes at both sides of the material interface in the vertical plane both at the top and at the bottom of the pipe.](valve-scls1-n.png){#fig:valve width=85%}
 
-As an example, let us consider the system depicted in\ [@fig:valve] 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, 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\ $\hat{T}(\mathbf{x},t)$ for each time is obtained in the reduced mesh ([@fig:valve-temp], which has the origin at the centre of the valve), the actual temperature distribution\ $T(\mathbf{x},t)$ is computed by an algebraic generalisation of $\hat{T}(\mathbf{x},t)$ in the full coordinate system. 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.
+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.
 
-![Mesh of half-a-symmetric-valve refined around the interface where the transient heat conduction problem is solved.](valve-temp-commented.png){#fig:valve-temp width=100%}
+![Transient temperature distribution in a reduced mesh around the valve](temp-valve-commented.svg){#fig:valve-temp width=100%}
 
 
-Note that there is no need to have a one-to-one correspondence between the elements from the reduced mesh with the elements from the original one. Actually, the reduced mesh contains first-order elements whilst the former has second-order elements. Also the grid density is different. Nevertheless, the finite-element solver [Fino](https://www.seamplex.com/fino) used to solve both the heat and the mechanical problems, allows to read functions of space and time defined over one mesh and continuously evaluate and use them into another one even if the two grids have different elements, orders or even dimensions. In effect, in the system from\ [@fig:real-life] the material interface is between a orifice plate made in stainless steel that is welded to a carbon-steel pipe ([@fig:real]). The thermal problem can be modelled using a two-dimensional axi-symmetric grid and then generalised to the full three-dimensional mesh using the algebraic manipulation capabilities provided by [Fino](https://www.seamplex.com/fino) (actually by [wasora](https://www.seamplex.com/wasora)) as shown in\ [@fig:real].
+Note that there is no need to have a one-to-one correspondence between the elements from the reduced mesh with the elements from the original one. Actually, the reduced mesh contains first-order elements whilst the former has second-order elements. Also the grid density is different, yet both of them are locally refined around the material interface. Nevertheless, the finite-element solver [Fino](https://www.seamplex.com/fino)---used to solve both the heat and the mechanical problems---allows to read functions of space and time defined over one mesh and continuously evaluate and use them into another one even if the two grids have different elements, orders or even dimensions.
 
-dnl ![Full 3D mesh, carbon steel is magenta, stainless steel is green.](real-mesh2.png){#fig:real-mesh2 width=70%}
-dnl ![Temperature distribution for a certain time within the transient computed on a reduced two-dimensional axi-symmetric mesh modelling half the orifice plate and a length of the carbon pipe.](real-temp.png){#fig:real-temp width=34%}
-
-![Temperature distribution for a certain time within the transient computed on a reduced two-dimensional axi-symmetric mesh modelling half the orifice plate and a length of the carbon pipe and generalisation of the temperature to the full three-dimensional grid.](real-gen.png){#fig:real width=100%}
 
 
 ### Seismic loads {#sec:seismic}
@@ -465,19 +509,22 @@ These first modes, shown in\ [@fig:modes], that take up most of the energy are t
 divert(0)
 
 
+Since the computation of the loads that a certain earthquake gives rise to would add a significant amount of complexity to this already-complex case study (after all this is the main point of this chapter!), the details are skipped. In any case, we need to compute the first natural modes of oscillation of the piping section under study ([@fig:modes]) and then, after some cumbersome mathematics (which I encourage those brave readers to dig into), obtain an statically-equivalent load distribution. In effect, [@fig:mods] shows the first six natural modes of oscillation of the piping system unders study. This information can be used to compute an acceleration distribution which, when multiplied by the density, give a load distribution which is, in a certain sense, “statically equivalent” to the dynamic solicitations created by the earthquake.
+
 ::::: {#fig:modes}
-![$f_1 \approx 30$\ [Hz](https://www.seamplex.com/docs/nafems4/mode1.webm)](mode1-commented.png){width=48%}\  
-![$f_2 \approx 60$\ [Hz](https://www.seamplex.com/docs/nafems4/mode2.webm)](mode2.png){width=48%}
+![$f_1 \approx 20$\ [Hz](https://www.seamplex.com/docs/nafems4/mode1.webm)](mode1.0000.png){width=48%}\  
+![$f_2 \approx 35$\ [Hz](https://www.seamplex.com/docs/nafems4/mode2.webm)](mode2.0000.png){width=48%}
 
-![$f_3 \approx 70$\ [Hz](https://www.seamplex.com/docs/nafems4/mode3.webm)](mode3.png){width=48%}\  
-![$f_4 \approx 75$\ [Hz](https://www.seamplex.com/docs/nafems4/mode5.webm)](mode4.png){width=48%}
+![$f_3 \approx 50$\ [Hz](https://www.seamplex.com/docs/nafems4/mode3.webm)](mode3.0000.png){width=48%}\  
+![$f_4 \approx 70$\ [Hz](https://www.seamplex.com/docs/nafems4/mode5.webm)](mode4.0000.png){width=48%}
 
-![$f_5 \approx 100$\ [Hz](https://www.seamplex.com/docs/nafems4/mode5.webm)](mode5.png){width=48%}\  
-![$f_6 \approx 130$\ [Hz](https://www.seamplex.com/docs/nafems4/mode6.webm)](mode6.png){width=48%}
+![$f_5 \approx 90$\ [Hz](https://www.seamplex.com/docs/nafems4/mode5.webm)](mode5.0000.png){width=48%}\  
+![$f_6 \approx 90$\ [Hz](https://www.seamplex.com/docs/nafems4/mode6.webm)](mode6.0000.png){width=48%}
 
-First six natural oscillation modes for a piping section between axial supports. Blue-red colour scale shows magnitude of deformation, as compared to the original geometry (grey). Full animations available online ([@sec:online]).
+First natural oscillation modes. Videos available online ([@sec:online]).
 :::::
 
+divert(-1)
 ::::: {#fig:acceleration}
 ![$a_x$](ax.png){width=80%}
 
@@ -487,11 +534,11 @@ First six natural oscillation modes for a piping section between axial supports.
 
 Equivalent accelerations for a certain piping section
 :::::
+divert(0)
 
-Since the computation of the loads that a certain earthquake gives rise to would add a significant amount of complexity to this already-complex case study, the details are skipped. In any case, we need to compute the first natural modes of oscillation of the piping section under study ([@fig:modes]) and then, after some cumbersome maths, obtain an statically-equivalent load distribution. In effect, [@fig:acceleration] shows a sample distribution of acceleration distribution within a certain piping system which, when multiplied by the density, give a load distribution which is statically equivalent to the dynamic solicitations created by the earthquake.
-The ASME code says that these accelerations ought to be applied twice: once with the original sign and once with all the elements with the opposite sign, as SRSS “looses signs.” The application of each of these equivalent loads should last two seconds in the original time domain.
-
+The ASME code says that these accelerations ought to be applied twice: once with the original sign and once with all the elements with the opposite sign. The application of each of these equivalent loads should last two seconds in the original time domain.
 
+divert(-1)
 ### Linearity (not yet linearisation) {#sec:linearity}
 
 Even though we did not yet discuss it in detail, we want to solve an elastic problem subject to an internal pressure condition, with a non-uniform temperature distribution that leads to both thermal stresses and variations in the mechanical properties of the materials. And as if this was not enough, we want to add during a couple of seconds a statically-equivalent distributed load arising from a design earthquake. This last point means that at the transient instant where the stresses (from the fatigue’s point of view) are maximum we have to add the distributed loads that we computed from the seismic spectra to the other thermal and pressure loads. But we have a linear elastic problem (well, we still do not have it but we will in\ [@sec:break]), so we might be tempted to exploit the problem’s linearity and compute all the effects separately and then sum them up to obtain the whole combination. We may thus compute just the stresses due to the seismic loads and then add these stresses to the stresses at  any time of the transient, in particular to the one with the highest ones. After all, in linear problems the result of the sum of two cases is the result of the sum of the cases, right? Not always.
@@ -514,9 +561,11 @@ Now we are going to create and compare three load cases:^[The provided links lea
 
 The loads in each cases are applied to the three remaining faces, namely “right” ($x>0$), “back” ($y>0$) and “top,” ($z>0$). Their magnitude in Newtons are:
 
-::::::{=latex}
-\rowcolors{2}{black!10}{black!0}
-::::::
+
+dnl ::::::{=latex}
+dnl \rowcolors{2}{black!10}{black!0}
+dnl ::::::
+
 
 |        |       | “right” |     |       | “back” |      |       | “top” |       |
 | ------ |:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|
@@ -609,16 +658,20 @@ then the sums (in plural because there are three eigenvalues) of their eigenvalu
 \medskip
 
 The moral of this fable is that if we have a case that is the combination of two other simpler cases (say one with only surface loads and one with only volumetric loads), in general we cannot just add the principal stresses (or Von Mises) of two cases and expect to obtain a correct answer. We have to solve the full case again (both the surface and the volumetric loads at the same time). In case we are stubborn enough and still want to stick to solving the cases separately, there is a trick we can resort to. Instead of summing principal stresses, what we can do is to sum either displacements or the individual stress components, which are fully linear. So we might pre-deform (or pre-stress) case B with the results from case A and then expect the FEM program to obtain the correct stresses for the combined case. However, this scheme is actually far more complex than just solving the combined case in a single run and it also needs an educated guess and/or trial and error to know at what time the pre-deformation or pre-stressing should be applied to take into account the seismic load.
+divert(0)
 
+### ASME stress linearisation divert(-1)(not linearity!)divert(0)
 
-### ASME stress linearisation (not linearity!)
+divert(-1) After discussing linearity, let us now dig into linearisation. The name is similar but these two animals are very different beasts.divert(0)
+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 provides a comprehensive, sound and---more importantly---widely and commonly-accepted body of knowledge such that the regulatory authorities require its enforcement to nuclear plant owners.
 
-After discussing linearity, let us now dig into linearisation. The name is similar but these two animals are very different beasts. 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 provides 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. These are defined in [section\ VIII](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code#ASME_BPVC_Section_VIII_-_Rules_for_Construction_of_Pressure_Vessels) annex 5-A, although they are widely used in other sections, particularly [section\ III](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code#ASME_BPVC_Section_III_-_Rules_for_Construction_of_Nuclear_Facility_Components). Briefly, they give the zeroth-order (membrane) and first-order (bending) [moments](https://en.wikipedia.org/wiki/Moment_(mathematics)) (in the statistical sense) 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.
+One of the main issues of the ASME code refers to what is known as “membrane” and “bending” stresses. These are defined in [section\ VIII](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code#ASME_BPVC_Section_VIII_-_Rules_for_Construction_of_Pressure_Vessels) annex 5-A, although they are generally used in other sections, particularly [section\ III](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code#ASME_BPVC_Section_III_-_Rules_for_Construction_of_Nuclear_Facility_Components). divert(-1)Briefly, they give the zeroth-order (membrane) and first-order (bending) [moments](https://en.wikipedia.org/wiki/Moment_(mathematics)) 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.
+divert(0)
 
-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 it is like computing the [Taylor expansion](https://en.wikipedia.org/wiki/Taylor_series) (or for the case, expansion in [Legendre polynomials](https://en.wikipedia.org/wiki/Legendre_polynomials)) of an arbitrary stress distribution along a line, and retaining the first two terms. That is to say, to obtain a _linear approximation_. There are physical interpretations for both membrane and bending stresses, which are beyond the scope of this chapter. As for the ASME requirements, they are a way of having the mean and linear contributions of a certain stress distribution along the pipe’s wall thickness.
+Briefly, the membrane stress gives a measure of the internal stresses that are needed to balance the external loads. The bending stress gives a measure of the internal stresses which are not self-balanced and arise due to the external load operating on the solid under study. These measures are computed along the already-introduced Stress Classification Lines. There are very interesting physical interpretations of what these stresses mean which. They are beyond the scope of this chapter but reference [@angus-linearization] provides a very interesting study on the subject.
 
+The computation of these membrane and bending stresses is called “stress linearisation” because it is like computing the [Taylor expansion](https://en.wikipedia.org/wiki/Taylor_series) (or to differentiate between balanced and non-balanced stresses, expansion in [Legendre polynomials](https://en.wikipedia.org/wiki/Legendre_polynomials)) of an arbitrary stress distribution along a line, and retaining the first two terms. That is to say, to obtain a _linear approximation_ of the fully-detailed stress distribution along the SCL.  As for the ASME requirements, they are a way of having the mean and linear contributions of a certain stress distribution along the pipe’s wall thickness.
 
-So what about the SCLs? Well, the ASME standard says that they are 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 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 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. 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.
 
 divert(-1)
 \medskip
@@ -670,10 +723,13 @@ No need to know or even understand these integrals which for sure are not introd
 
 divert(0)
 
+
+divert(-1)
 ## The infinite pipe revisited after college {#sec:infinite-pipe-fem}
 
 Let us now make a (tiny) step from the general and almost philosophical subject from the last section down to the particular case study, and reconsider the infinite pressurised pipe once again. It is time to solve the problem with a computer using finite elements and to obtain some funny coloured pictures instead of just equations (like we did in [@sec:infinite-pipe]).
 
+
 The first thing that has to be said is that, as with any interesting problem, there are literally hundreds or different ways of solving it, each of them throwing particular conclusions. For example, one can:
 
  1. solve a real 3D problem or a 2D axi-symmetric case (or even a 1D case using the [collocation method](https://en.wikipedia.org/wiki/Collocation_method) to solve the radial dependence),
@@ -798,7 +854,6 @@ Now proceed to picturing the general three-dimensional cases with unstructured t
 Detailed mathematics show that the location where the derivatives of the interpolated displacements are closer to the real (i.e. the analytical ones in problems that have it) solution are the elements’ [Gauss points](https://en.wikipedia.org/wiki/Gaussian_quadrature). Even better, the material properties at these points are continuous (they are usually uniform but they can depend on temperature for example) because, unless we are using weird elements, there are no material interfaces inside elements. But how to manage a set of stresses given at the Gauss points instead of at the nodes? Should we use one mesh for the input and another one for the output? What happens when we need to know the stresses on a surface and not just in the bulk of the solid? There are still no one-size-fits-all answers. There is a very interesting [blog post](https://nickjstevens.netlify.com/post/2019/stress-singularities/) by Nick Stevens that addresses the issue of stresses computed at sharp corners. What does your favourite FEM program do with such a case?
 
 In any case, this step takes a non-negligible amount of time. The most-common approach, i.e. the node-averaging method is driven mainly by the number of nodes of course. So all-in-all, these are the reasons to use the number of nodes instead of the numbers of elements as a basic parameter to measure the complexity of a FEM problem.
-divert(0)
 
 ## Adding complexity: the truth is out there
 
@@ -806,13 +861,15 @@ Let us review some issues that appear when solving our case study and that might
 
 ### Two (or more) materials {#sec:two-materials}
 
-The main issue with fatigue in nuclear piping during operational transients is that at the welds between two materials with different thermal expansion coefficients there can appear potentially-high stresses during temperature changes. If these transients are repeated cyclically, fatigue may occur. We already have risen a warning flag about stresses at material interfaces in\ [@sec:two-materials]. Besides all the open questions about computing stresses at nodes, this case also adds the fact that the material properties (say the Young’s Modulus\ $E$) is different in the elements that are at each side of the interface.
+The main issue with fatigue in nuclear piping during operational transients is that at the welds between two materials with different thermal expansion coefficients there can appear potentially-high stresses during temperature changes. If these transients are repeated cyclically, fatigue may occur.
+dnl We have already risen a warning flag about stresses at material interfaces in\ [@sec:two-materials].
+Besides all the open questions about [computing stresses at nodes](https://www.seamplex.com/fino/cases/050-tet10/), this case also adds the fact that the material properties (say the Young’s Modulus\ $E$) are different in the elements that are at each side of the interface.
 
 ::::: {#fig:two-cubes}
-![Surface grid showing the fixed face (magenta), the load face (green) and the shared face in the middle](two-cubes2.png){#fig:two-cubes2 width=48%}\  
-![Warped displacements and Von\ Mises stresses](two-cubes4.png){#fig:two-cubes4 width=48%}
+![Mesh with shared surface](two-cubes2.png){#fig:two-cubes2 width=48%}\  
+![Stress on  warped displacements](two-cubes4.png){#fig:two-cubes4 width=48%}
 
-Two cubes of different materials (the one in the left soft, the one in the right hard) share a face and a pressure is applied at the right-most face
+Two cubes of different materials share a face and a pressure is applied at the right-most face
 :::::
 
 
@@ -822,7 +879,7 @@ $$
 \sigma = E \cdot \epsilon
 $$
 
-Faced with the problem of computing the stress\ $\sigma$ at one node shared by many elements, we might:
+Faced with the problem of computing the stress\ $\sigma$ at one node shared by many elements, we (actually our favourite FEM program) might:
 
  1. compute the (weighted?) averages\ $\langle E \rangle$ and $\langle \epsilon \rangle$ and then compute the stress as\ $\langle \sigma \rangle = \langle E \rangle \cdot \langle \epsilon \rangle$. This would be like having a meta-material at the interface with average properties, or
  2. compute the stress as the (weighted?) average of the product $E \cdot \epsilon$ in each node\ $\langle \sigma \rangle = \langle E \cdot \epsilon \rangle$. This would be like forcing a non-differentiable function to behave smoothly, or
@@ -831,30 +888,38 @@ Faced with the problem of computing the stress\ $\sigma$ at one node shared by m
 
 There might be other choices as well. Do you know what your favourite FEM program does? Now follow up with these questions:
 
- a. Does the manual say?
- b. Does it tell you the details like how it weighs the averages?
- c. Does it discard values that are beyond a number of standard deviations away?
- d. How many standard deviations?
+ a. Does the manual say what it does?
+ b. Does the manual say how it does what what it does?
+ c. Does it provide the user (i.e. you) with different choices?
+ d. Can you tell what these options entail?
  e. ...
 
 
-You can still add a lot of questions that you should be having right now. If you cannot get a clear answer for at least one of them, then start to worry. After you do, add the following question:
+You can still add a lot of questions that you should be having right now.
+
+divert(-1)
+If you cannot get a clear answer for at least one of them, then start to worry. After you do, add the following question:
 
 > Do you believe your favourite FEM program’s manual?
 
-What we as responsible engineers who have to sign a report stating that a nuclear power plant will not collapse due to fatigue in its pipes, is to fully understand what is going on with our stresses. [Richard Stallman](https://en.wikipedia.org/wiki/Richard_Stallman) says that the best way to solve a problem is to avoid it in the first place. In this case, we should avoid having to trust a written manual and rely on software whose [source code](https://en.wikipedia.org/wiki/Source_code) is available. What we need is the capacity (RMS calls it _freedom_) to be able to see the detailed steps performed by the program so we can answer any question we (or other people) might have.
+What we as responsible engineers who have to sign a report stating that a nuclear power plant will not collapse due to fatigue in its pipes, is to fully understand what is going on with our stresses.
+[Richard Stallman](https://en.wikipedia.org/wiki/Richard_Stallman) says that the best way to solve a problem is to avoid it in the first place. In this case, we should avoid having to trust a written manual and rely on software whose [source code](https://en.wikipedia.org/wiki/Source_code) is available. What we need is the capacity (RMS calls it _freedom_) to be able to see the detailed steps performed by the program so we can answer any question we (or other people) might have.
 
 Without resorting into philosophical digressions about the difference between [free and open-source software](https://en.wikipedia.org/wiki/Free_and_open-source_software) (not because it is not worth it, but because it would take a whole book), the programs that make their source code available for their users are called [open-source software](https://en.wikipedia.org/wiki/Open-source_software). If the users can also modify and re-distribute the modified versions, they are called [free software](https://www.fsf.org/about/what-is-free-software). Note that the important concept here is freedom, not price. In Spanish (my native language) it would have been easier because there are two separate words for free as in freedom (“libre”) and for free as in price (“gratis”).
 
-In effect, a couple of years ago Angus Ramsay noted [a weird behaviour](https://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/asmeansys-potential-issue-with-thermal-expansion-calculations/) in the results given by a certain commercial [non-free](https://en.wikipedia.org/wiki/Proprietary_software) FEA software regarding the handling of expansion coefficients from ASME data. To understand what was going on, Angus and I had to guess what the program was doing to [reproduce the allegedly weird results](https://www.seamplex.com/docs/SP-WA-17-TN-F38B-A.pdf). Finally, it was a [matter how the data was rounded up to be presented in a paper table](https://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/accuracy-of-thermal-expansion-properties-in-asme-bpv-code/) rather than a programming flaw. Nevertheless, we were lucky our guesses lead us to a reasonable answer. If we had access to the program’s source code, we could have thoroughly analysed the issue in a more efficient way. Sure, we might not have the same programming skills the original authors of the software have, but if it had been [free software](https://en.wikipedia.org/wiki/Free_software) we would have had the _freedom_ to hire a programmer to help us out. That is what _free_ means. In [Eric Raymond](https://en.wikipedia.org/wiki/Eric_S._Raymond)’s words, [“given enough eyeballs, all bugs are shallow.”](http://www.catb.org/~esr/writings/cathedral-bazaar/) This is rather important in engineering software where [verification and validation](https://en.wikipedia.org/wiki/Software_verification_and_validation) is a must, especially in regulated fields like the nuclear industry. First, think how can a piece of software be verified if the source code is not available for independent analysis. And then, ask yourself another question:
+In effect, a couple of years ago Angus Ramsay noted [a weird behaviour](https://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/asmeansys-potential-issue-with-thermal-expansion-calculations/) in the results given by a certain commercial [non-free](https://en.wikipedia.org/wiki/Proprietary_software) FEA software regarding the handling of expansion coefficients from ASME data. To understand what was going on, Angus and I had to guess what the program was doing to [reproduce the allegedly weird results](https://www.seamplex.com/docs/SP-WA-17-TN-F38B-A.pdf). Finally, it was a [matter how the data was rounded up to be presented in a paper table](https://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/accuracy-of-thermal-expansion-properties-in-asme-bpv-code/) rather than a programming flaw. Nevertheless, we were lucky our guesses lead us to a reasonable answer. If we had access to the program’s source code, we could have thoroughly analysed the issue in a more efficient way. Sure, we might not have the same programming skills the original authors of the software have, but if it had been [free software](https://en.wikipedia.org/wiki/Free_software) we would have had the _freedom_ to hire a programmer to help us out. That is what _free_ means. In [Eric Raymond](https://en.wikipedia.org/wiki/Eric_S._Raymond)’s words, [“given enough eyeballs, all bugs are shallow.”](http://www.catb.org/~esr/writings/cathedral-bazaar/) This is rather important in engineering software where [verification and validation](https://en.wikipedia.org/wiki/Software_verification_and_validation) is a must, especially in regulated fields like the nuclear industry. First, think how can a piece of software be verified if the source code is not available for independent analysis.
+
+So now, ask yourself another question:
 
 > Do you trust your favourite FEM program?
 
 Back to the two-material problem, all the discussion above in\ [@sec:two-materials] about non-continuous derivatives applies to a sharp abrupt interface. In the study case the junctions are welded so there is a [heat-affected zone](https://en.wikipedia.org/wiki/Heat-affected_zone) with changes in the material micro structure. Therefore, there exists a smooth transition from the mechanical properties of one material to the other one in a way that is very hard to predict and to model. In principle, the assumption of a sharp interface is conservative in the sense that it is expected the computed stresses to be larger than the actual ones. There cannot be an SCL exactly on a material interface so there should be at least two SCLs, one at each side of the junctions as\ [@fig:weldolet-scls] illustrates. The actual distance would have to be determined first as an educated guess, then via trial and error and finally in accordance with the regulator.
+divert(0)
 
+divert(-1)
 ### A parametric tee {#sec:parametric}
 
-Time for another experiment. We know (more or less) what to expect from an infinite pressurised pipe from\ [@sec:infinite-pipe] and [@sec:infinite-pipe-fem]. What if we added a branch to such pipe? Even more, what if we successively varied the diameter of the branch to see what happens? This is called parametric analysis, and sooner or later (if not now) you will find yourself performing this kind of computations more often than any other one.
+Time for another experiment. We know (more or less) what to expect from an infinite pressurised pipe from\ [@sec:infinite-pipe]. What if we added a branch to such pipe? Even more, what if we successively varied the diameter of the branch to see what happens? This is called parametric analysis, and sooner or later (if not now) you will find yourself performing this kind of computations more often than any other one.
 
 divert(-1)
 So here come the five Feynmann-Ohno questions:
@@ -873,7 +938,7 @@ So here come the five Feynmann-Ohno questions:
 
  5. Why do you want to understand what is going on?  
      For the same reason you are now reading this chapter.
-divert(0)
+divert(-1)
 
 Let us solve the following mock-up case: a long main 12-inch schedule\ 80 pipe has an orthogonal branch of a certain nominal diameter of\ $d_b$\ inches (it seems that the [SI](https://en.wikipedia.org/wiki/International_System_of_Units) did not do well amongst the piping engineering community). Both the main line and the branch are pressurised with\ $p=10$\ MPa. The main pipe is aligned with the\ $x$ axis and the branch coincides with the\ $z$ axis. Thus, the $x$-$z$ plane acts as a symmetry plane and we only need to model two octants of the full geometry, as shown in\ [@fig:tee-geo]. Note that the tee is modelled as the boolean intersection of two cylinders. There are no filleted edges nor rounded corners or any other smoothing. A real CAD file containing the appropriate geometry needs to be built for the real case study.
 
@@ -931,10 +996,11 @@ Most of the time at college we would barely do what is needed to be approved in
 
 One further detail: it is always a sane check to try to explain the numerical results based on physical reasoning (i.e. “with your fingers”) as we did two paragraphs above. Most of the time you will be solving problems whilst already knowing what the result would (or ought to) be.
 
+divert(0)
 
 ### Bake, shake and break {#sec:break}
 
-A fellow mechanical engineer who went to the same high school I did, who went to the same engineering school I did and who worked at the same company I did, but who earned a PhD in Norway once told me two interesting things about finite-elements graduate courses. First, that in Trondheim the classes were taught by faculty from the the mathematics department rather than from the mechanical engineering department. It made complete sense to me, as I always have thought finite elements mainly as a maths subject. And even though engineers might know some maths, it is nothing compared to actual mathematicians. Secondly, that they called the thermal, natural oscillations and elastic problems as the rhyming trio “bake, shake and break” (they also had “wake” for fluids, but that is a different story). These are just the three problems listed in section\ [@sec:piping-nuclear] that we need to solve in our nuclear power plant.
+A fellow mechanical engineer who went to the same high school I did, who went to the same engineering school I did and who worked at the same company I did, but who earned a PhD in Norway once told me two interesting things about finite-elements graduate courses. First, that in Trondheim the classes were taught by faculty from the the mathematics department rather than from the mechanical engineering department. It made complete sense to me, as I always have thought finite elements mainly as a maths subject. And even though some engineers might know some maths, it is nothing compared to actual mathematicians. Secondly, that they called the thermal, natural oscillations and elastic problems as the rhyming trio “bake, shake and break” (they also had “wake” for fluids, but that is a different story). These are just the three problems listed in section\ [@sec:piping-nuclear] that we need to solve in our nuclear power plant.
 
 So here we are again with the case study where we have to compute the linearised stresses at certain SCLs located near the interface between two different kinds of steels during operational and incidental transients of the plant. The first part is then to “bake” the pipes, modelling a thermal transient with time-dependent temperature or convection boundary conditions (it depends on the system). This steps gives a time and space-dependent temperature\ $T(x,y,z,t)$.
 
@@ -947,7 +1013,8 @@ Finally we attempt to “break” the pipes successively solving many steady-sta
   3. The temperature distribution\ $T(x,y,t,z)$ for bullets 1 & 2 is the generalisation of the reduced-model procedure explained in\ [@sec:thermal].
   4. The internal faces of the pipes are subject to an uniform pressure\ $p(t)$ given by the definition of the transient.
   5. There are mechanical supports throughout the piping system. Depending on the type of the support (i.e. vertical, lateral, axial, full, etc.) one or more degrees of freedom (i.e. $u$, $v$ and/or $w$) are fixed to zero. The ends of the CAD models are chosen always to have axially-null displacements.
-  6. The earthquake-equivalent volumetric force\ $\mathbf{f}(x,y,z)$ is only be applied at the time\ $t$ where the maximum stresses occur. Note that due to the discussion from\ [@sec:linearity] we cannot compute the stresses raised just by\ $\mathbf{f}(x,y,z)$ and then add them to the main stresses. The force has to be included into the “break” step. An educated guess of the time where the maximum stress occur is usually enough. Anyway, it might be necessary a trial and error scheme to find the sweet spot.
+  6. The earthquake-equivalent volumetric force\ $\mathbf{f}(x,y,z)$ is only be applied at the time\ $t$ where the maximum stresses occur.
+dnl Note that due to the discussion from\ [@sec:linearity] we cannot compute the stresses raised just by\ $\mathbf{f}(x,y,z)$ and then add them to the main stresses. The force has to be included into the “break” step. An educated guess of the time where the maximum stress occur is usually enough. Anyway, it might be necessary a trial and error scheme to find the sweet spot.
   7. According to ASME\ III, the seismic load has to be applied during two seconds with the two possible signs. That is to say, apply $\mathbf{f}(x,y,z)$ during two seconds and then  $-\mathbf{f}(x,y,z)$ during two further seconds when the main stresses are maximums.
   8. A number of stress classification lines have to be defined. The locations should be previously accorded with the plant owner and/or the regulator.
   9. The stress linearisation has to be performed individually for each principal stress\ $\sigma_1$, $\sigma_2$ and $\sigma_3$ to fulfil the requirements ASME\ III\ NB-3126 (see [@sec:in-air] below).
@@ -966,7 +1033,7 @@ $$ \sigma = E(T) \cdot \epsilon $$
 
 What changes with temperature is the slope of\ $\sigma$ with respect to\ $\epsilon$ (think and imagine!), but the relationship between them is still _linear_.
 
-On the other hand, we have a given non-trivial temperature distribution\ $T(\mathbf{x}, t)$ within the pipes that is a snapshot of a transient heat conduction problem at a certain time\ $t$ (think and picture yourself taking photos of the temperature distribution changing in time). Let us now forget about the time, as after all we are solving a steady-state elastic problem. Now you can trust me or ask a FEM teacher, but the continuous displacement formulation can be loosely written as
+On the other hand, we have a non-trivial temperature distribution\ $T(\mathbf{x}, t)$ within the pipes that is a snapshot of a transient heat conduction problem at a certain time\ $t$ (think and picture yourself taking photos of the temperature distribution changing in time). Let us now forget about the time, as after all we are solving a steady-state elastic problem. Now you can trust me or ask a FEM teacher, but the continuous displacement formulation can be loosely written as
 
 $$ K\big[E\left(T(\mathbf{x})\right), \mathbf{x}\big] \cdot \mathbf{u}(\mathbf{x}) = \mathbf{b}(\mathbf{x})$$
 
@@ -980,47 +1047,9 @@ And this last expression is linear in\ $\mathbf{u}$! In effect, the discretisati
  2. ask a teacher, or
  3. go through with the maths.
 
-
-::::: {#fig:case}
-![General view. Carbon steel is grey and stainless steel is green.](case-cad1.png){#fig:case-cad1 width=75%}
-
-![Detail of the mesh refinement around the interface.](case-mech-mesh2.png){#fig:case-mesh2 width=100%}
-
-A section of a piping system in a nuclear power plant
-:::::
-
-![Location of the 15\ SCLs](case-scls-n.png){#fig:case-scls width=75%}
-
-::::: {#fig:case-temp}
-![Simplified axi-symmetric domain](case-temp-4-0015.png){#fig:case-temp1 width=70%}
-
-![Generalisation](case-temp-gen-4-0015.png){#fig:case-temp2 width=100%}
-
-Temperature distribution for a certain instant of the transient, computed in the simplified two-dimensional axi-symmetric domain and its generalisation to the three-dimensional mechanical domain as discussed in\ [@sec:thermal]
-:::::
-
-::::: {#fig:case-mode}
-![$f_1 \approx 60$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode1.webm)](case-mode1.png){#fig:case-mode-1 width=30%}\  
-![$f_1 \approx 130$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode2.webm)](case-mode2.png){#fig:case-mode-2 width=30%}\  
-![$f_1 \approx 200$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode3.webm)](case-mode3.png){#fig:case-mode-3 width=30%}
-
-![$f_1 \approx 270$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode4.webm)](case-mode4.png){#fig:case-mode-4 width=30%}\  
-![$f_1 \approx 330$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode5.webm)](case-mode5.png){#fig:case-mode-5 width=30%}\  
-![$f_1 \approx 560$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode6.webm)](case-mode6.png){#fig:case-mode-6 width=30%}
-
-![$f_1 \approx 660$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode7.webm)](case-mode7.png){#fig:case-mode-7 width=30%}\  
-![$f_1 \approx 730$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode8.webm)](case-mode8.png){#fig:case-mode-8 width=30%}\  
-![$f_1 \approx 930$\ [Hz](https://www.seamplex.com/docs/nafems4/case-mode9.webm)](case-mode9.png){#fig:case-mode-9 width=30%}
-
-First nine natural modes of oscillation of the piping system subject to the boundary conditions the supports provide.
-Animations are available online ([@sec:online]).
-:::::
-
-dnl ![Floor response spectrum.](case-spectrum.svg){#fig:case-spectrum width=70%}
-
 To recapitulate, the figures in this section show some partial non-dimensional results of an actual system of a certain nuclear power plant. The main issues to study were the interfaces between a carbon-steel pipe and a stainless-steel orifice plate used to measure the (heavy) water flow through the line. The steps discussed so far include
 
- 1. building a CAD model of the piping section under study, which will be the main domain ([@fig:case-cad1] or [@fig:real-life;@fig:weldolet-cad;@fig:valve])
+ 1. building a CAD model of the piping section under study, which will be the main domain ([@fig:case-cad1] or [@fig:cad-figure;@fig:weldolet-cad;@fig:valve])
  2. creating a mesh for the main domain refining locally around the material interfaces ([@fig:case-mesh2] or [@fig:weldolet-mesh;@fig:real])
  3. defining the number and locations of the SCLs ([@fig:case-scls] or [@fig:weldolet-scls;@fig:valve])
  4. computing a heat conduction (bake) transient problem with temperatures as a function of time from the operational transient in a simple domain using temperature-dependent thermal conduction coefficients from ASME\ II\ part D ([@fig:case-temp] or [@fig:valve-temp])
@@ -1039,14 +1068,6 @@ To recapitulate, the figures in this section show some partial non-dimensional r
 A pretty nice list of steps, which definitely I would not have been able to tackle when I was in college. Would you?
 
 
-::::: {#fig:case-acceleration}
-![$a_x$](case-ax.png){width=33%}
-![$a_y$](case-ay.png){width=33%}
-![$a_z$](case-az.png){width=33%}
-
-The static equivalent accelerations computed using the SRSS method
-:::::
-
 ## Cumulative usage factors {#sec:usage}
 
 Strictly speaking, finite elements are not needed anymore at this point of the analysis. But even though we are (or want to be) FEM experts, we have to understand that if the objective of a work is to evaluate fatigue (or fracture mechanics or whatever), finite elements are just a mean and not and end. If we just mastered FEM and nothing else, our field of work would be highly reduced. We need to use all of our computational knowledge to perform actually engineering tasks and to be able to tell our bosses and clients whether the pipe would fail or not. This tip is induced in college but it is definitely reinforced afterwards when working with actual clients and bosses.
@@ -1215,12 +1236,14 @@ Here is a list of sub-problems and stuff to play with.
 
  * The pendulum-swing video from [@sec:intro]
     - <https://youtu.be/Q-lKK4A2OzA>
- * “On convergence of linearized stresses in an infinite pipe computed using the finite element method” from [@sec:infinite-pipe;@sec:linearity;@sec:infinite-pipe;@sec:infinite-pipe-fem]
+ * “On convergence of linearized stresses in an infinite pipe computed using the finite element method” from [@sec:infinite-pipe;@sec:infinite-pipe;@sec:infinite-pipe-fem]
     - <https://www.seamplex.com/fino/doc/pipe-linearized/> 
+divert(-1)    
  * The three cubes from [@sec:linearity]
     - case A: pure normal loads (<https://caeplex.com/p/d8fe>)
     - case B: pure shear loads (<https://caeplex.com/p/b494>)
     - case C: the combination of A & B (<https://caeplex.com/p/9899>)
+divert(0)
  * The animations of natural oscillations in [@fig:modes]
     - <https://www.seamplex.com/docs/nafems4/mode1.webm>
     - <https://www.seamplex.com/docs/nafems4/mode2.webm>
@@ -1237,4 +1260,6 @@ Here is a list of sub-problems and stuff to play with.
 See <https://www.seamplex.com/nafems> for new material, updated links and the full version of this case with many more details about the case and the associated mathematics.
 
 
-# Concluding Remarks
+dnl # Concluding Remarks
+
+# References

pipe-dist-1-1.dat

@@ -0,0 +1,101 @@
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+0	161.04	0	0.0447834	60.97	18.1977	-0.325379
+0	161.255	0	0.0447575	60.8894	18.199	-0.240548
+0	161.47	0	0.0447318	60.8092	18.2004	-0.155833
+0	161.685	0	0.0447062	60.7294	18.2018	-0.0712322
+0	161.9	0	0.0446802	60.635	18.1992	0.00589281

pipe-dist-2-6.dat

@@ -0,0 +1,101 @@
+0	140.4	0	0.0478735	70.6596	18.2055	-9.97559
+0	140.615	0	0.0478342	70.5341	18.2027	-9.85868
+0	140.83	0	0.0477951	70.4092	18.2001	-9.7418
+0	141.045	0	0.0477561	70.2852	18.1977	-9.62493
+0	141.26	0	0.0477173	70.1618	18.1955	-9.50809
+0	141.475	0	0.0476787	70.0391	18.1934	-9.39127
+0	141.69	0	0.0476402	69.9171	18.1916	-9.27448
+0	141.905	0	0.0476019	69.7959	18.1899	-9.15771
+0	142.12	0	0.0475638	69.6775	18.1899	-9.03962
+0	142.335	0	0.0475258	69.56	18.1902	-8.92134
+0	142.55	0	0.047488	69.4429	18.1906	-8.80305
+0	142.765	0	0.0474503	69.3267	18.1913	-8.68454
+0	142.98	0	0.0474129	69.2108	18.1921	-8.56608
+0	143.195	0	0.0473755	69.0953	18.193	-8.44773
+0	143.41	0	0.0473384	68.9801	18.1939	-8.32949
+0	143.625	0	0.0473014	68.8652	18.1948	-8.21136
+0	143.84	0	0.0472646	68.7506	18.1958	-8.09335
+0	144.055	0	0.0472279	68.6364	18.1969	-7.97544
+0	144.27	0	0.0471914	68.5224	18.198	-7.85765
+0	144.485	0	0.0471551	68.4098	18.1988	-7.74246
+0	144.7	0	0.0471189	68.2992	18.1991	-7.63183
+0	144.915	0	0.0470829	68.1885	18.1993	-7.52144
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+0	148.355	0	0.0465273	66.4419	18.2014	-5.77578
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+0	161.9	0	0.0446814	60.6538	18.1965	0.00291815

pipe-linearized.ppl

@@ -0,0 +1,110 @@
+# problem parameters for
+# 12"-inch schedule 100
+b = 323.8/2    # external radius [ mm ]
+a = b-21.5     # internal radius [ mm ]
+l = 2*(b-a)    # axial length [ mm ]
+n = 4          # number of elements through thickness 
+
+E = 200e3      # Young modulus [ MPa ]
+nu = 0.3       # Poisson's ratio [ non-dimensional ]
+p = 10         # internal pressure [ MPa ]
+
+
+# definition of analytical solutions for comparison from
+# <http://eprints.whiterose.ac.uk/110536/1/art%253A10.1007%252Fs00707-016-1762-7.pdf>
+ur(x,y,z) = (p*a**2*sqrt(y**2+z**2))/(E*(b**2-a**2)) * ((1-2*nu)*(1+nu) + (1+nu)*b**2/(y**2+z**2))
+sigmal(x,y,z)     = 2*nu*p*a**2/(b**2-a**2)
+sigmar(x,y,z)     = p*a**2/(b**2-a**2) * (1 - b**2/(y**2+z**2))
+sigmatheta(x,y,z) = p*a**2/(b**2-a**2) * (1 + b**2/(y**2+z**2))
+
+
+# analytical results in [ MPa ]	
+M_tresca    = 	69.9467
+M_vonmises  = 	61.7785
+M_1         = 	65.3023
+M_2         = 	18.1974
+M_3         = 	-4.64439
+
+MB_tresca   = 	79.9062
+MB_vonmises = 	70.2562
+MB_1        = 	70.2821
+MB_2        = 	18.1974
+MB_3        = 	-9.62415
+
+
+ur1(x) = ur(0,x,0)
+sigmal1(x) = sigmal(0,x,0)
+sigmar1(x) = sigmar(0,x,0)
+sigmatheta1(x) = sigmatheta(0,x,0)
+
+delta = 0.05
+
+set preamble "\usepackage{amsmath}"
+
+set width 14*unit(cm)
+
+set axis x arrow nomirrored
+set axis y arrow nomirrored
+
+
+set grid
+set terminal pdf
+
+
+set output "ur.pdf"
+set xrange [a:b]
+set yrange [*:*]
+set key top right
+set xlabel "$r$ [mm]"
+set ylabel "$u_r(r)$ [mm]"
+plot ur1(x)                        w l  lw 3  lt 1   color olivegreen   ti "Analytical solution $u_r(r)$",\
+     "pipe-dist-1-3.dat" u 2:4 w l  lw 1  lt 2   color maroon       ti "",\
+     "pipe-dist-1-3.dat" u 2:4 w p  pt 18 ps 1.5 color maroon       every (100)/(1*3) ti "$n=3$ first order",\
+     "pipe-dist-1-6.dat" u 2:4 w l  lw 1  lt 4   color tan          ti "",\
+     "pipe-dist-1-6.dat" u 2:4 w p  pt 16 ps 1.5 color tan          every (100)/(1*6) ti "$n=6$ first order",\
+     "pipe-dist-2-2.dat" u 2:4 w l  lw 2  lt 3   color cerulean     ti "",\
+     "pipe-dist-2-2.dat" u 2:4 w p  pt 17 ps 1.5 color cerulean     every (100)/(2*2) ti "$n=2$ second order"
+
+set output "sigmatheta.pdf"
+set xrange [a:b]
+set yrange [*:*]
+set key top right
+set xlabel "$r$ [mm]"
+set ylabel "$\sigma_\theta(r)$ [mm]"
+plot sigmatheta1(x)                w l  lw 3  lt 1   color springgreen ti "Analytical solution $\sigma_\theta(r)$",\
+     "pipe-dist-1-3.dat" u 2:5 w l  lw 1  lt 2   color melon       ti "",\
+     "pipe-dist-1-3.dat" u 2:5 w p  pt 18 ps 1.5 color melon       every (100)/(1*3) ti "$n=3$ first order",\
+     "pipe-dist-1-6.dat" u 2:5 w l  lw 1  lt 2   color redorange   ti "",\
+     "pipe-dist-1-6.dat" u 2:5 w p  pt 16 ps 1.5 color redorange   every (100)/(1*6) ti "$n=6$ first order",\
+     "pipe-dist-2-2.dat" u 2:5 w l  lw 2  lt 3   color turquoise   ti "",\
+     "pipe-dist-2-2.dat" u 2:5 w p  pt 17 ps 1.5 color turquoise   every (100)/(2*2) ti "$n=2$ second order"
+
+set output "sigmal.pdf"
+set xrange [a:b]
+set yrange [*:*]
+set key top right
+set xlabel "$r$ [mm]"
+set ylabel "$\sigma_l(r)$ [mm]"
+plot sigmal1(x)                    w l  lw 3  lt 1   color forestgreen    ti "Analytical solution $\sigma_l(r)$",\
+     "pipe-dist-1-3.dat" u 2:6 w l  lw 1  lt 2   color peach          ti "",\
+     "pipe-dist-1-3.dat" u 2:6 w p  pt 18 ps 1.5 color peach          every (100)/(1*3) ti "$n=3$ first order",\
+     "pipe-dist-1-6.dat" u 2:6 w l  lw 1  lt 2   color dandelion      ti "",\
+     "pipe-dist-1-6.dat" u 2:6 w p  pt 16 ps 1.5 color dandelion      every (100)/(1*6) ti "$n=6$ first order",\
+     "pipe-dist-2-3.dat" u 2:6 w l  lw 2  lt 3   color periwinkle     ti "",\
+     "pipe-dist-2-3.dat" u 2:6 w p  pt 17 ps 1.5 color periwinkle     every (100)/(2*2) ti "$n=2$ second order"
+
+set output "sigmar.pdf"
+set xrange [a:b]
+set yrange [*:*]
+set key top left
+set xlabel "$r$ [mm]"
+set ylabel "$\sigma_r(r)$ [mm]"
+plot sigmar1(x)                    w l  lw 3  lt 1   color emerald      ti "Analytical solution $\sigma_r(r)$",\
+     "pipe-dist-1-3.dat" u 2:7 w l  lw 1  lt 2   color bittersweet  ti "",\
+     "pipe-dist-1-3.dat" u 2:7 w p  pt 18 ps 1.5 color bittersweet  every (100)/(1*3) ti "$n=3$ first order",\
+     "pipe-dist-1-6.dat" u 2:7 w l  lw 1  lt 2   color orange       ti "",\
+     "pipe-dist-1-6.dat" u 2:7 w p  pt 16 ps 1.5 color orange       every (100)/(1*6) ti "$n=6$ first order",\
+     "pipe-dist-2-3.dat" u 2:7 w l  lw 2  lt 3   color royalblue    ti "",\
+     "pipe-dist-2-3.dat" u 2:7 w p  pt 17 ps 1.5 color royalblue    every (100)/(2*2) ti "$n=2$ second order"
+
+

pt-1.dat

@@ -1,4 +1,4 @@
-0 261.111 11.2729
-60 55.5556 0.27579
-120 26.6667 0.27579
-200 26.6667 0.27579
+0 40 0.1
+200 40 10
+300 300 10
+600 300 10

pt-1.svg

@@ -1,5 +1,5 @@
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pt.dat

@@ -0,0 +1,19 @@
+0 40 0.1
+200 40 10
+300 300 10
+600 300 10
+600 300 10
+3600 40 10
+3650 40 0.3
+4250 40 0.1
+4250 250 10
+4330 300 10
+4410 280 10
+4490 250 5
+6990 320 10
+7190 320 10
+7190 300 10
+7195 200 6.5
+8895 200 10
+8975 300 10
+9175 300 10

references.bib

@@ -110,3 +110,45 @@
  language = "english",
  hyphenation = "english", 
 }
+
+
+@book{timoshenko,
+ author = "S. Timoshenko and J. N. Goodier",
+ title = "Theory of elasticity",
+ publisher = "McGraw-Hill",
+ edition = "2nd",
+ year = "1951",
+ language = "english",
+ hyphenation = "english", 
+}
+
+@report{pipe-linearized,
+ author = "Jeremy Theler",
+ institution = "Seamplex",
+ title = "On convergence of linearized stresses in an infinite pipe computed using the finite element method",
+ year = "2019",
+ url = "https://seamplex.com/docs/pipe-linearized",
+ language = "english",
+ hyphenation = "english", 
+}
+
+@report{angus-linearization,
+ author = "Angus Ramsay",
+ title = "Stress Linearisation for Practising Engineers",
+ year = "2016",
+ url = "https://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/stress-linearisation-for-practising-engineers/",
+ language = "english",
+ hyphenation = "english", 
+}
+
+
+@book{relativity-wrong,
+ author = "Asimov, Isaac",
+ year = "1988",
+ title = "The relativity of wrong",
+ location = "New York",
+ publisher = "Doubleday",
+ ISBN = "0-385-24473-8",
+ language = "english",
+ hyphenation = "english", 
+}

simple2020.tex

@@ -0,0 +1,551 @@
+% Options for packages loaded elsewhere
+\PassOptionsToPackage{unicode$for(hyperrefoptions)$,$hyperrefoptions$$endfor$}{hyperref}
+\PassOptionsToPackage{hyphens}{url}
+$if(colorlinks)$
+\PassOptionsToPackage{dvipsnames,svgnames*,x11names*}{xcolor}
+$endif$
+$if(dir)$
+$if(latex-dir-rtl)$
+\PassOptionsToPackage{RTLdocument}{bidi}
+$endif$
+$endif$
+$if(CJKmainfont)$
+\PassOptionsToPackage{space}{xeCJK}
+$endif$
+%
+\documentclass[
+$if(fontsize)$
+  $fontsize$,
+$endif$
+$if(lang)$
+  $babel-lang$,
+$endif$
+$if(papersize)$
+  $papersize$paper,
+$endif$
+$if(beamer)$
+  ignorenonframetext,
+$if(handout)$
+  handout,
+$endif$
+$if(aspectratio)$
+  aspectratio=$aspectratio$,
+$endif$
+$endif$
+$for(classoption)$
+  $classoption$$sep$,
+$endfor$
+]{$documentclass$}
+$if(beamer)$
+$if(background-image)$
+\usebackgroundtemplate{%
+  \includegraphics[width=\paperwidth]{$background-image$}%
+}
+$endif$
+\usepackage{pgfpages}
+\setbeamertemplate{caption}[numbered]
+\setbeamertemplate{caption label separator}{: }
+\setbeamercolor{caption name}{fg=normal text.fg}
+\beamertemplatenavigationsymbols$if(navigation)$$navigation$$else$empty$endif$
+$for(beameroption)$
+\setbeameroption{$beameroption$}
+$endfor$
+% Prevent slide breaks in the middle of a paragraph
+\widowpenalties 1 10000
+\raggedbottom
+$if(section-titles)$
+\setbeamertemplate{part page}{
+  \centering
+  \begin{beamercolorbox}[sep=16pt,center]{part title}
+    \usebeamerfont{part title}\insertpart\par
+  \end{beamercolorbox}
+}
+\setbeamertemplate{section page}{
+  \centering
+  \begin{beamercolorbox}[sep=12pt,center]{part title}
+    \usebeamerfont{section title}\insertsection\par
+  \end{beamercolorbox}
+}
+\setbeamertemplate{subsection page}{
+  \centering
+  \begin{beamercolorbox}[sep=8pt,center]{part title}
+    \usebeamerfont{subsection title}\insertsubsection\par
+  \end{beamercolorbox}
+}
+\AtBeginPart{
+  \frame{\partpage}
+}
+\AtBeginSection{
+  \ifbibliography
+  \else
+    \frame{\sectionpage}
+  \fi
+}
+\AtBeginSubsection{
+  \frame{\subsectionpage}
+}
+$endif$
+$endif$
+$if(beamerarticle)$
+\usepackage{beamerarticle} % needs to be loaded first
+$endif$
+$if(fontfamily)$
+\usepackage[$for(fontfamilyoptions)$$fontfamilyoptions$$sep$,$endfor$]{$fontfamily$}
+$else$
+\usepackage{lmodern}
+$endif$
+$if(linestretch)$
+\usepackage{setspace}
+$endif$
+\usepackage{amssymb,amsmath}
+\usepackage{ifxetex,ifluatex}
+\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
+  \usepackage[$if(fontenc)$$fontenc$$else$T1$endif$]{fontenc}
+  \usepackage[utf8]{inputenc}
+  \usepackage{textcomp} % provide euro and other symbols
+\else % if luatex or xetex
+$if(mathspec)$
+  \ifxetex
+    \usepackage{mathspec}
+  \else
+    \usepackage{unicode-math}
+  \fi
+$else$
+  \usepackage{unicode-math}
+$endif$
+%   \defaultfontfeatures{Scale=MatchLowercase}
+  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
+$if(mainfont)$
+  \setmainfont[$for(mainfontoptions)$$mainfontoptions$$sep$,$endfor$]{$mainfont$}
+$else$
+  \setmainfont{LinLibertineO}
+$endif$
+$if(sansfont)$
+  \setsansfont[$for(sansfontoptions)$$sansfontoptions$$sep$,$endfor$]{$sansfont$}
+$else$
+  \setsansfont{Carlito}
+$endif$
+$if(monofont)$
+  \setmonofont[$for(monofontoptions)$$monofontoptions$$sep$,$endfor$]{$monofont$}
+$else$
+  \setmonofont[Mapping=tex-ansi]{DejaVuSansMono}
+$endif$
+$for(fontfamilies)$
+  \newfontfamily{$fontfamilies.name$}[$for(fontfamilies.options)$$fontfamilies.options$$sep$,$endfor$]{$fontfamilies.font$}
+$endfor$
+$if(mathfont)$
+$if(mathspec)$
+  \ifxetex
+    \setmathfont(Digits,Latin,Greek)[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$}
+  \else
+    \setmathfont[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$}
+  \fi
+$else$
+  \setmathfont[$for(mathfontoptions)$$mathfontoptions$$sep$,$endfor$]{$mathfont$}
+$endif$
+$endif$
+$if(CJKmainfont)$
+  \ifxetex
+    \usepackage{xeCJK}
+    \setCJKmainfont[$for(CJKoptions)$$CJKoptions$$sep$,$endfor$]{$CJKmainfont$}
+  \fi
+$endif$
+$if(luatexjapresetoptions)$
+  \ifluatex
+    \usepackage[$for(luatexjapresetoptions)$$luatexjapresetoptions$$sep$,$endfor$]{luatexja-preset}
+  \fi
+$endif$
+$if(CJKmainfont)$
+  \ifluatex
+    \usepackage[$for(luatexjafontspecoptions)$$luatexjafontspecoptions$$sep$,$endfor$]{luatexja-fontspec}
+    \setmainjfont[$for(CJKoptions)$$CJKoptions$$sep$,$endfor$]{$CJKmainfont$}
+  \fi
+$endif$
+\fi
+$if(beamer)$
+$if(theme)$
+\usetheme[$for(themeoptions)$$themeoptions$$sep$,$endfor$]{$theme$}
+$endif$
+$if(colortheme)$
+\usecolortheme{$colortheme$}
+$endif$
+$if(fonttheme)$
+\usefonttheme{$fonttheme$}
+$endif$
+$if(mainfont)$
+\usefonttheme{serif} % use mainfont rather than sansfont for slide text
+$endif$
+$if(innertheme)$
+\useinnertheme{$innertheme$}
+$endif$
+$if(outertheme)$
+\useoutertheme{$outertheme$}
+$endif$
+$endif$
+% Use upquote if available, for straight quotes in verbatim environments
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\IfFileExists{microtype.sty}{% use microtype if available
+  \usepackage[$for(microtypeoptions)$$microtypeoptions$$sep$,$endfor$]{microtype}
+  \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
+}{}
+$if(indent)$
+$else$
+\makeatletter
+\@ifundefined{KOMAClassName}{% if non-KOMA class
+  \IfFileExists{parskip.sty}{%
+    \usepackage{parskip}
+  }{% else
+    \setlength{\parindent}{0pt}
+    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
+}{% if KOMA class
+  \KOMAoptions{parskip=half}}
+\makeatother
+$endif$
+$if(verbatim-in-note)$
+\usepackage{fancyvrb}
+$endif$
+\usepackage{xcolor}
+\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
+\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
+\hypersetup{
+$if(title-meta)$
+  pdftitle={$title-meta$},
+$endif$
+$if(author-meta)$
+  pdfauthor={$author-meta$},
+$endif$
+$if(lang)$
+  pdflang={$lang$},
+$endif$
+$if(subject)$
+  pdfsubject={$subject$},
+$endif$
+$if(keywords)$
+  pdfkeywords={$for(keywords)$$keywords$$sep$, $endfor$},
+$endif$
+$if(colorlinks)$
+  colorlinks=true,
+  linkcolor=$if(linkcolor)$$linkcolor$$else$Maroon$endif$,
+  filecolor=$if(filecolor)$$filecolor$$else$Maroon$endif$,
+  citecolor=$if(citecolor)$$citecolor$$else$Blue$endif$,
+  urlcolor=$if(urlcolor)$$urlcolor$$else$Blue$endif$,
+$else$
+  hidelinks,
+$endif$
+  pdfcreator={LaTeX via pandoc}}
+\urlstyle{same} % disable monospaced font for URLs
+$if(verbatim-in-note)$
+\VerbatimFootnotes % allow verbatim text in footnotes
+$endif$
+$if(geometry)$
+$if(beamer)$
+\geometry{$for(geometry)$$geometry$$sep$,$endfor$}
+$else$
+\usepackage[$for(geometry)$$geometry$$sep$,$endfor$]{geometry}
+$endif$
+$endif$
+$if(beamer)$
+\newif\ifbibliography
+$endif$
+$if(listings)$
+\usepackage{listings}
+\newcommand{\passthrough}[1]{#1}
+\lstset{defaultdialect=[5.3]Lua}
+\lstset{defaultdialect=[x86masm]Assembler}
+$endif$
+$if(lhs)$
+\lstnewenvironment{code}{\lstset{language=Haskell,basicstyle=\small\ttfamily}}{}
+$endif$
+$if(highlighting-macros)$
+$highlighting-macros$
+$endif$
+$if(tables)$
+\usepackage{longtable,booktabs}
+$if(beamer)$
+\usepackage{caption}
+% Make caption package work with longtable
+\makeatletter
+\def\fnum@table{\tablename~\thetable}
+\makeatother
+$else$
+% Correct order of tables after \paragraph or \subparagraph
+\usepackage{etoolbox}
+\makeatletter
+\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{}
+\makeatother
+% Allow footnotes in longtable head/foot
+\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
+\makesavenoteenv{longtable}
+$endif$
+$endif$
+$if(graphics)$
+\usepackage{graphicx}
+\makeatletter
+\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi}
+\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi}
+\makeatother
+% Scale images if necessary, so that they will not overflow the page
+% margins by default, and it is still possible to overwrite the defaults
+% using explicit options in \includegraphics[width, height, ...]{}
+\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
+% Set default figure placement to htbp
+\makeatletter
+\def\fps@figure{htbp}
+\makeatother
+$endif$
+
+% headers y footers a medida
+\usepackage{lastpage}
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+\fancyhf{} % clear all header and footer fields
+\renewcommand{\headrulewidth}{0.0pt}
+\fancyhead[RO]{\slshape \rightmark}
+\fancyhead[LE]{\slshape \leftmark}
+\fancyfoot[LE,RO]{\scriptsize{$if(date)$ $date$ $endif$---$if(hash)$ \texttt{$hash$} $endif$}}
+\fancyfoot[C]{\thepage}
+
+$if(links-as-notes)$
+% Make links footnotes instead of hotlinks:
+\DeclareRobustCommand{\href}[2]{#2\footnote{\url{#1}}}
+$endif$
+$if(strikeout)$
+\usepackage[normalem]{ulem}
+% Avoid problems with \sout in headers with hyperref
+\pdfstringdefDisableCommands{\renewcommand{\sout}{}}
+$endif$
+\setlength{\emergencystretch}{3em} % prevent overfull lines
+\providecommand{\tightlist}{%
+  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+$if(numbersections)$
+\setcounter{secnumdepth}{$if(secnumdepth)$$secnumdepth$$else$5$endif$}
+$else$
+\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
+$endif$
+$if(beamer)$
+$else$
+$if(block-headings)$
+% Make \paragraph and \subparagraph free-standing
+\ifx\paragraph\undefined\else
+  \let\oldparagraph\paragraph
+  \renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}}
+\fi
+\ifx\subparagraph\undefined\else
+  \let\oldsubparagraph\subparagraph
+  \renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}}
+\fi
+$endif$
+$endif$
+$if(pagestyle)$
+\pagestyle{$pagestyle$}
+$endif$
+$for(header-includes)$
+$header-includes$
+$endfor$
+$if(lang)$
+\ifxetex
+  % Load polyglossia as late as possible: uses bidi with RTL langages (e.g. Hebrew, Arabic)
+  \usepackage{polyglossia}
+  \setmainlanguage[$polyglossia-lang.options$]{$polyglossia-lang.name$}
+$for(polyglossia-otherlangs)$
+  \setotherlanguage[$polyglossia-otherlangs.options$]{$polyglossia-otherlangs.name$}
+$endfor$
+\else
+  \usepackage[shorthands=off,$for(babel-otherlangs)$$babel-otherlangs$,$endfor$main=$babel-lang$]{babel}
+$if(babel-newcommands)$
+  $babel-newcommands$
+$endif$
+\fi
+$endif$
+$if(dir)$
+\ifxetex
+  % Load bidi as late as possible as it modifies e.g. graphicx
+  \usepackage{bidi}
+\fi
+\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
+  \TeXXeTstate=1
+  \newcommand{\RL}[1]{\beginR #1\endR}
+  \newcommand{\LR}[1]{\beginL #1\endL}
+  \newenvironment{RTL}{\beginR}{\endR}
+  \newenvironment{LTR}{\beginL}{\endL}
+\fi
+$endif$
+$if(natbib)$
+\usepackage[$natbiboptions$]{natbib}
+\bibliographystyle{$if(biblio-style)$$biblio-style$$else$plainnat$endif$}
+$endif$
+$if(biblatex)$
+\usepackage[$if(biblio-style)$style=$biblio-style$,$endif$$for(biblatexoptions)$$biblatexoptions$$sep$,$endfor$]{biblatex}
+$for(bibliography)$
+\addbibresource{$bibliography$}
+$endfor$
+$endif$
+
+\usepackage{subfig}
+% SIunits
+\usepackage{siunitx}
+
+% set default figure placement to htbp
+\makeatletter
+%\def\fps@figure{htbp}
+\pretocmd\env@cases{\def\@rowc@lors{}}{}{}
+\pretocmd\start@align{\def\@rowc@lors{}}{}{}
+\makeatother
+
+% fuente para los captions de figuras
+\usepackage{caption}
+\renewcommand{\captionfont}{\sf\small}
+\renewcommand{\captionlabelfont}{\bf\sf}
+
+$if(csl-refs)$
+\newlength{\cslhangindent}
+\setlength{\cslhangindent}{1.5em}
+\newenvironment{cslreferences}%
+  {$if(csl-hanging-indent)$\setlength{\parindent}{0pt}%
+  \everypar{\setlength{\hangindent}{\cslhangindent}}\ignorespaces$endif$}%
+  {\par}
+$endif$
+
+$if(title)$
+\title{$title$$if(thanks)$\thanks{$thanks$}$endif$}
+$endif$
+$if(subtitle)$
+$if(beamer)$
+$else$
+\usepackage{etoolbox}
+\makeatletter
+\providecommand{\subtitle}[1]{% add subtitle to \maketitle
+  \apptocmd{\@title}{\par {\large #1 \par}}{}{}
+}
+\makeatother
+$endif$
+\subtitle{$subtitle$}
+$endif$
+\author{$for(author)$$author$$sep$ \and $endfor$}
+\date{$date$}
+$if(beamer)$
+$if(institute)$
+\institute{$for(institute)$$institute$$sep$ \and $endfor$}
+$endif$
+$if(titlegraphic)$
+\titlegraphic{\includegraphics{$titlegraphic$}}
+$endif$
+$if(logo)$
+\logo{\includegraphics{$logo$}}
+$endif$
+$endif$
+\include{syntax}
+
+\renewcommand\vec[1]{\mathbf{#1}}
+
+\begin{document}
+% \thispagestyle{empty}
+% \thispagestyle{plain}
+$if(has-frontmatter)$
+\frontmatter
+$endif$
+$if(title)$
+$if(beamer)$
+\frame{\titlepage}
+$else$
+\maketitle
+$endif$
+$if(abstract)$
+\begin{abstract}
+$abstract$
+\end{abstract}
+$endif$
+$endif$
+
+$for(include-before)$
+$include-before$
+
+$endfor$
+$if(toc)$
+$if(toc-title)$
+\renewcommand*\contentsname{$toc-title$}
+$endif$
+$if(beamer)$
+\begin{frame}[allowframebreaks]
+$if(toc-title)$
+  \frametitle{$toc-title$}
+$endif$
+  \tableofcontents[hideallsubsections]
+\end{frame}
+$else$
+{
+$if(colorlinks)$
+\hypersetup{linkcolor=$if(toccolor)$$toccolor$$else$$endif$}
+$endif$
+\setcounter{tocdepth}{$toc-depth$}
+\tableofcontents
+\newpage
+}
+$endif$
+$endif$
+$if(lot)$
+\listoftables
+\pagebreak
+$endif$
+$if(lof)$
+\listoffigures
+\pagebreak
+$endif$
+$if(linestretch)$
+\setstretch{$linestretch$}
+$endif$
+$if(has-frontmatter)$
+\mainmatter
+$endif$
+% vectors done right
+\renewcommand{\vec}[1]{\ensuremath\mathbf{#1}}
+
+
+% \setcounter{topnumber}{2}
+% \setcounter{bottomnumber}{2}
+% \setcounter{totalnumber}{4}
+% \renewcommand{\topfraction}{0.80}
+% \renewcommand{\bottomfraction}{0.80}
+% \renewcommand{\textfraction}{0.20}
+\renewcommand{\floatpagefraction}{0.65}
+$body$
+
+$if(has-frontmatter)$
+\backmatter
+$endif$
+$if(natbib)$
+$if(bibliography)$
+$if(biblio-title)$
+$if(has-chapters)$
+\renewcommand\bibname{$biblio-title$}
+$else$
+\renewcommand\refname{$biblio-title$}
+$endif$
+$endif$
+$if(beamer)$
+\begin{frame}[allowframebreaks]{$biblio-title$}
+  \bibliographytrue
+$endif$
+  \bibliography{$for(bibliography)$$bibliography$$sep$,$endfor$}
+$if(beamer)$
+\end{frame}
+$endif$
+
+$endif$
+$endif$
+$if(biblatex)$
+$if(beamer)$
+\begin{frame}[allowframebreaks]{$biblio-title$}
+  \bibliographytrue
+  \printbibliography[heading=none]
+\end{frame}
+$else$
+\printbibliography$if(biblio-title)$[title=$biblio-title$]$endif$
+$endif$
+
+$endif$
+$for(include-after)$
+$include-after$
+
+$endfor$
+\end{document}

uno_solo_alphazero.dat

@@ -0,0 +1,14 @@
+1	0	0.73	0.63	0.44	-0.10	0.59	0.49	-0.07	-0.10
+1	10	11.00	9.24	2.80	-1.76	10.11	8.39	2.35	-1.72
+1	30	31.55	26.46	7.52	-5.09	29.15	24.17	7.19	-4.97
+1	60	62.37	52.30	14.60	-10.07	57.71	47.85	14.44	-9.85
+1	70	62.32	52.25	14.52	-10.07	57.33	47.49	14.32	-9.84
+1	90	62.37	52.30	14.51	-10.07	57.13	47.28	14.19	-9.85
+1	130	62.43	52.36	14.48	-10.07	56.80	46.95	14.02	-9.84
+1	134	62.42	52.35	14.49	-10.07	56.85	47.00	14.03	-9.85
+1	140	62.35	52.28	14.51	-10.07	56.91	47.07	14.08	-9.84
+1	150	62.55	52.48	14.58	-10.07	57.12	47.27	14.09	-9.85
+1	170	62.64	52.57	14.63	-10.07	57.23	47.38	14.08	-9.85
+1	210	62.70	52.62	14.66	-10.07	57.31	47.45	14.07	-9.85
+1	290	62.72	52.64	14.68	-10.08	57.34	47.49	14.07	-9.85
+1	380	62.72	52.64	14.68	-10.08	57.34	47.49	14.07	-9.85

uno_solo_kinf.dat

@@ -0,0 +1,14 @@
+1	0	8.30	5.29	0.77	-3.01	14.78	-0.03	-1.18	-14.80
+1	10	15.10	13.63	0.20	-1.47	12.97	4.00	-2.36	-8.97
+1	30	35.43	30.62	5.17	-4.81	21.62	16.14	-0.61	-5.48
+1	60	66.14	56.34	12.36	-9.80	48.59	38.18	8.32	-10.40
+1	70	73.47	64.18	7.96	-9.29	40.32	28.57	-8.75	-11.75
+1	75	77.40	68.42	5.52	-8.98	45.81	25.84	-12.22	-19.98
+1	90	89.18	80.53	-1.06	-8.65	78.28	21.74	-13.82	-56.54
+1	130	142.08	112.87	3.27	-29.21	178.32	18.59	-19.75	-159.73
+1	140	156.46	121.34	4.62	-35.12	203.47	17.86	-21.65	-185.61
+1	150	156.49	121.36	4.63	-35.13	203.53	17.86	-21.66	-185.67
+1	170	156.49	121.36	4.63	-35.13	203.53	17.86	-21.66	-185.67
+1	210	156.49	121.36	4.63	-35.13	203.53	17.86	-21.66	-185.67
+1	290	156.49	121.36	4.63	-35.13	203.53	17.86	-21.66	-185.67
+1	380	156.49	121.36	4.63	-35.13	203.53	17.86	-21.66	-185.67

uno_solo_posta.dat

@@ -0,0 +1,14 @@
+1	0	8.30	5.32	0.77	-2.98	14.77	0.02	-1.16	-14.75
+1	10	15.12	13.66	0.23	-1.47	12.97	4.05	-2.34	-8.92
+1	30	35.45	30.64	5.20	-4.80	21.66	16.19	-0.56	-5.46
+1	60	66.15	56.36	12.38	-9.79	48.61	38.22	8.35	-10.39
+1	70	68.76	11.87	-18.90	-56.89	60.30	-49.07	-73.20	-109.36
+1	90	110.69	-15.90	-16.92	-126.60	169.40	-65.15	-159.14	-234.55
+1	130	183.18	-18.81	-26.72	-201.99	337.55	-75.99	-241.62	-413.55
+1	134	185.09	-18.11	-26.08	-203.19	351.23	-78.11	-246.72	-429.33
+1	140	191.33	-17.72	-24.82	-209.06	374.55	-77.92	-255.18	-452.47
+1	150	165.10	18.71	-2.31	-146.39	318.72	-42.25	-157.73	-360.97
+1	170	143.53	50.08	0.31	-93.45	271.36	-30.86	-94.35	-302.22
+1	210	130.35	79.37	4.10	-50.98	227.29	-25.58	-39.55	-252.87
+1	290	130.92	100.13	6.64	-30.78	215.79	-5.66	-25.53	-221.45
+1	380	136.05	108.61	6.93	-27.45	212.49	2.19	-24.14	-210.30