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auto07p.html
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<!DOCTYPE html>
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<title>S Ganga Prasath | Robotics, Behavior, Soft-matter</title>
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<body class="bg-slate-100 flex flex-col mx-auto bg-[url('/images/bg-full.png')] bg-repeat-x">
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<section id="auto07p">
<div class="absolute w-12 h-12 top-[20px] left-3 transform translate-x-px border-1 rounded-full">
<img src="images/icon-tutorial.svg" alt="" class="h-20 ml-2 mr-1 opacity-70" />
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<!-- Title -->
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Elastic instabilities using <code>auto-07p</code>
</div>
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<div
class="bg-slate-300 m-2 p-4 font-serif leading-relaxed border-[1px] space-y-2 border-slate-400 rounded-xl">
<p> Elastic materials are characterised by their ability to undergo local dilation, shear and twist under
externally applied load and relax to a neutral configuration when the load is removed. Often when the
applied load exceeds a critical value these materials undergo instabilities characterized by a smooth or
dramatic change in their morphology. These instabilities are captured by identifying the forces at play
and deriving governing equations for the deformation of the
material. The deformation field in the material is then related to the internal stress through a
constituent relation, Hooke's law being a famous one. These elastic instability problems, as we shall
see, are boundary value problems which are notorious to solve even numerically (analytical solutions are
available only in isolated instances).</p>
<p> In this set of 4 tutorials we will see how to solve for the deformation field of the material under
various active and passive forces and capture the instabilities using <a
href="https://github.com/auto-07p/auto-07p"
class="underline decoration-1 underline-offset-2 decoration-indigo-400 hover:decoration-indigo-500 hover:decoration-2">
<code>auto-07p</code></a>,
a numerical continuation package. The purpose of this tutorial really is to introduce you to the power
of <code>auto-07p</code> to approach these problems. Though the documentation is a very good place to
start, I realized during my learning process that there are a few important technical gaps that are
often not discussed if one were to go from the documentation to implementation.</p>
<p> In this 4 part series we will be discussing problems each with an unique
aspect in terms of implementation in <code>auto-07p</code>. It will become
clear what I mean by this as we go over the examples.</p>
<p>
We begin by introducing the framework <a href="./autoFw.html"
class="underline decoration-1 underline-offset-2 decoration-indigo-400 hover:decoration-indigo-500 hover:decoration-2">here</a>.
Please click this link and go over it before you start reading the tutorial problems. The relevant codes
used in these tutorials are available <a href="https://github.com/sgangaprasath/autoTutorial"
class="underline decoration-1 underline-offset-2 decoration-indigo-400 hover:decoration-indigo-500 hover:decoration-2">here</a>.
The problems we will be looking at (and the focus of each example in <code>auto-07p</code>
implementation) are
</p>
<ul class="list-disc mx-4">
<li> (18-Nov-21) <a href="./elastica.html"
class="underline decoration-1 underline-offset-2 decoration-indigo-400 hover:decoration-indigo-500 hover:decoration-2">
<strong>Bending of an <em>elastica</em></strong> </a> - bifurcation diagram, solution </li>
<li> (24-Nov-21) <a href="./fvk.html"
class="underline decoration-1 underline-offset-2 decoration-indigo-400 hover:decoration-indigo-500 hover:decoration-2">
<strong>Buckling of a sheet</strong> </a> - two parameter continuation </li>
<li> (04-Apr-22) <a href="./oscillating.html"
class="underline decoration-1 underline-offset-2 decoration-indigo-400 hover:decoration-indigo-500 hover:decoration-2">
<strong>Oscillating beam</strong> </a> - eigen-value problem, homotopy continuation </li>
<li> (05-Apr-22) <a href="./snake.html"
class="underline decoration-1 underline-offset-2 decoration-indigo-400 hover:decoration-indigo-500 hover:decoration-2">
<strong>Undulatory propulsion on land</strong> </a> - many parameter system, periodic solution,
homotopy continuation, adding new boundary boundary conditions </li>
</ul>
<p>
You can find the entire tutorial collection in pdf format <a
href="https://sgangaprasath.github.io/pubs/autoTutorial.pdf"
class="underline decoration-1 underline-offset-2 decoration-indigo-400 hover:decoration-indigo-500 hover:decoration-2">here</a>.
</p>
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