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<li><a class="reference internal" href="#">Guide to PETSc Tutorial Examples, by Physics</a><ul>
<li><a class="reference internal" href="#poisson">Poisson</a></li>
<li><a class="reference internal" href="#elastostatics">Elastostatics</a></li>
<li><a class="reference internal" href="#stokes">Stokes</a></li>
<li><a class="reference internal" href="#euler">Euler</a></li>
<li><a class="reference internal" href="#heat-equation">Heat equation</a></li>
<li><a class="reference internal" href="#navier-stokes">Navier-Stokes</a></li>
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<div class="section" id="guide-to-petsc-tutorial-examples-by-physics">
<h1>Guide to PETSc Tutorial Examples, by Physics<a class="headerlink" href="#guide-to-petsc-tutorial-examples-by-physics" title="Permalink to this headline">¶</a></h1>
<p>Below we list examples which simulate particular physics problems so that users interested in a particular set of governing equations can easily locate a relevant example. Often PETSc will have several examples looking at the same physics using different numerical tools, such as different discretizations, meshing strategy, closure model, or parameter regime.</p>
<div class="section" id="poisson">
<h2>Poisson<a class="headerlink" href="#poisson" title="Permalink to this headline">¶</a></h2>
<p>The Poisson equation</p>
<div class="math">
\[-\Delta u = f\]</div>
<p>is used to model electrostatics, steady-state diffusion, and other physical processes. Many PETSc examples solve this equation.</p>
<blockquote>
<div><dl class="simple">
<dt>Finite Difference</dt><dd><dl class="field-list simple">
<dt class="field-odd">2D</dt>
<dd class="field-odd"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex5.c.html">SNES example 5</a></p>
</dd>
<dt class="field-even">3D</dt>
<dd class="field-even"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ksp/ksp/tutorials/ex45.c.html">KSP example 45</a></p>
</dd>
</dl>
</dd>
<dt>Finite Element</dt><dd><dl class="field-list simple">
<dt class="field-odd">2D</dt>
<dd class="field-odd"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex12.c.html">SNES example 12</a></p>
</dd>
<dt class="field-even">3D</dt>
<dd class="field-even"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex12.c.html">SNES example 12</a></p>
</dd>
</dl>
</dd>
</dl>
</div></blockquote>
</div>
<div class="section" id="elastostatics">
<h2>Elastostatics<a class="headerlink" href="#elastostatics" title="Permalink to this headline">¶</a></h2>
<p>The equation for elastostatics balances body forces against stresses in the body</p>
<div class="math">
\[-\nabla\cdot \bm \sigma = \bm f\]</div>
<p>where <span class="math">\(\bm\sigma\)</span> is the stress tensor. Linear, isotropic elasticity governing infinitesimal strains has the particular stress-strain relation</p>
<div class="math">
\[-\nabla\cdot \left( \lambda I \operatorname{trace}(\bm\varepsilon) + 2\mu \bm\varepsilon \right) = \bm f\]</div>
<p>where the strain tensor <span class="math">\(\bm \varepsilon\)</span> is given by</p>
<div class="math">
\[\bm \varepsilon = \frac{1}{2} \left(\nabla \bm u + (\nabla \bm u)^T \right)\]</div>
<p>where <span class="math">\(\bm u\)</span> is the infinitesimal displacement of the body.</p>
<dl class="simple">
<dt>Finite Element</dt><dd><dl class="field-list simple">
<dt class="field-odd">2D</dt>
<dd class="field-odd"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex17.c.html">SNES example 17</a></p>
</dd>
<dt class="field-even">3D</dt>
<dd class="field-even"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex17.c.html">SNES example 17</a></p>
</dd>
<dt class="field-odd">3D</dt>
<dd class="field-odd"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex56.c.html">SNES example 56</a></p>
</dd>
</dl>
</dd>
</dl>
<p>If we allow finite strains in the body, we can express the stress-strain relation in terms of the Jacobian of the deformation gradient</p>
<div class="math">
\[J = \mathrm{det}(F) = \mathrm{det}\left(\nabla u\right)\]</div>
<p>and the right Cauchy-Green deformation tensor</p>
<div class="math">
\[C = F^T F\]</div>
<p>so that</p>
<div class="math">
\[\frac{\mu}{2} \left( \mathrm{Tr}(C) - 3 \right) + J p + \frac{\kappa}{2} (J - 1) = 0\]</div>
<p>In the example itself, everything can be expressed in terms of determinants and cofactors of <span class="math">\(F\)</span>.</p>
<blockquote>
<div><dl class="simple">
<dt>Finite Element</dt><dd><dl class="field-list simple">
<dt class="field-odd">3D</dt>
<dd class="field-odd"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex77.c.html">SNES example 77</a></p>
</dd>
</dl>
</dd>
</dl>
</div></blockquote>
</div>
<div class="section" id="stokes">
<h2>Stokes<a class="headerlink" href="#stokes" title="Permalink to this headline">¶</a></h2>
<p>The Stokes equations</p>
<div class="math">
\[\begin{aligned}
-\nabla \cdot \left(\mu \left(\nabla u + \nabla u^T \right)\right) + \nabla p + f &= 0 \\
\nabla\cdot u &= 0 \end{aligned}\]</div>
<p>describe slow flow of an incompressible fluid with velocity <span class="math">\(u\)</span>, pressure <span class="math">\(p\)</span>, and body force <span class="math">\(f\)</span>.</p>
<blockquote>
<div><dl>
<dt>Finite Element</dt><dd><dl class="field-list simple">
<dt class="field-odd">2D</dt>
<dd class="field-odd"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex62.c.html">SNES example 62</a></p>
</dd>
<dt class="field-even">3D</dt>
<dd class="field-even"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/snes/tutorials/ex62.c.html">SNES example 62</a></p>
</dd>
</dl>
<p>See <a class="reference internal" href="physics/guide_to_stokes.html"><span class="doc">Guide to the Stokes Equations using Finite Elements in PETSc</span></a> for more.</p>
</dd>
</dl>
</div></blockquote>
</div>
<div class="section" id="euler">
<h2>Euler<a class="headerlink" href="#euler" title="Permalink to this headline">¶</a></h2>
</div>
<div class="section" id="heat-equation">
<h2>Heat equation<a class="headerlink" href="#heat-equation" title="Permalink to this headline">¶</a></h2>
<p>The heat equation</p>
<div class="math">
\[\frac{\partial u}{\partial t} - \Delta u = f\]</div>
<p>is used to model heat flow, time-dependent diffusion, and other physical processes.</p>
<blockquote>
<div><dl class="simple">
<dt>Finite Element</dt><dd><dl class="field-list simple">
<dt class="field-odd">2D</dt>
<dd class="field-odd"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex45.c.html">TS example 45</a></p>
</dd>
<dt class="field-even">3D</dt>
<dd class="field-even"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex45.c.html">TS example 45</a></p>
</dd>
</dl>
</dd>
</dl>
</div></blockquote>
</div>
<div class="section" id="navier-stokes">
<h2>Navier-Stokes<a class="headerlink" href="#navier-stokes" title="Permalink to this headline">¶</a></h2>
<p>The incompressible Navier-Stokes equations</p>
<div class="math">
\[\begin{aligned}
\frac{\partial u}{\partial t} + u\cdot\nabla u - \nabla \cdot \left(\mu \left(\nabla u + \nabla u^T\right)\right) + \nabla p + f &= 0 \\
\nabla\cdot u &= 0 \end{aligned}\]</div>
<p>are appropriate for flow of an incompressible fluid at low to moderate Reynolds number.</p>
<blockquote>
<div><dl class="simple">
<dt>Finite Element</dt><dd><dl class="field-list simple">
<dt class="field-odd">2D</dt>
<dd class="field-odd"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex46.c.html">TS example 46</a></p>
</dd>
<dt class="field-even">3D</dt>
<dd class="field-even"><p><a class="reference external" href="https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex46.c.html">TS example 46</a></p>
</dd>
</dl>
</dd>
</dl>
</div></blockquote>
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