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autoFw.html
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<body>
<h1 id="auto-07p-framework"><code>auto-07p</code> framework</h1>
<p>The framework that we use to solve elastic instability problems has 3 major steps.</p>
<ol>
<li><p>To identify the model equations that describe the deformation field in elastic structure, derived either phenomenologically or from first principles. This is the part that captures the physics of the problem.</p></li>
<li><p>Using <code>auto-07p</code> to solve the system with the appropriate boundary conditions and continue the solution along a physical parameter in the system to see if the morphology changes with this parameter. <code>auto-07p</code> accepts first-order differential equations of the form <span class="math display">u'(t) = f(u(t), p), \quad f(.,.), u(.) \in \text{R}^n,</span> where <span class="math inline">p</span> are the parameters in the problem.If we have a higher order system, we need to convert and represent it in this form. <code>auto-07p</code> finds solution to this equations i.e. <span class="math inline">f(u(t),p)=0</span> and will continue it along the different parameters <span class="math inline">p</span> in the model. We will then look at the process of segregating the solution from the files that <code>auto-07p</code> spits out, post-processing them to understand the results.</p></li>
<li><p>Of course, the last step is to interpret the results and compare with experiments if any. And then go over to step <span class="math inline">(i)</span> if there is mismatch between experiments and theoretical results.</p></li>
</ol>
<p>The step I will be focusing in these tutorials is <span class="math inline">(ii)</span> where we assume that the equations required to describe the elastic structure are already available. We will then extract the bifurcation diagram as well as plot the solution files.</p>
<p><img src="./figs/fig1.jpeg" alt="image" /></p>
</body>
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