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Tutorials, by Physics
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.. highlight:: none

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.


Poisson
=======

The Poisson equation

.. math::

  -\Delta u = f

is used to model electrostatics, steady-state diffusion, and other physical processes. Many PETSc examples solve this equation.

  Finite Difference
    :2D: `SNES example 5 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex5.c.html>`_
    :3D: `KSP example 45 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ksp/ksp/tutorials/ex45.c.html>`_

  Finite Element
    :2D: `SNES example 12 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex12.c.html>`_
    :3D: `SNES example 12 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex12.c.html>`_

Elastostatics
=============

The equation for elastostatics balances body forces against stresses in the body

.. math::

  -\nabla\cdot \bm \sigma = \bm f

where :math:`\bm\sigma` is the stress tensor. Linear, isotropic elasticity governing infinitesimal strains has the particular stress-strain relation

.. math::

  -\nabla\cdot \left( \lambda I \operatorname{trace}(\bm\varepsilon) + 2\mu \bm\varepsilon \right) = \bm f

where the strain tensor :math:`\bm \varepsilon` is given by

.. math::

  \bm \varepsilon = \frac{1}{2} \left(\nabla \bm u + (\nabla \bm u)^T \right)

where :math:`\bm u` is the infinitesimal displacement of the body. The resulting discretizations use PETSc's nonlinear solvers

Finite Element
  :2D: `SNES example 17 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex17.c.html>`_
  :3D: `SNES example 17 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex17.c.html>`_
  :3D: `SNES example 56 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex56.c.html>`_

If we allow finite strains in the body, we can express the stress-strain relation in terms of the Jacobian of the deformation gradient

.. math::

  J = \mathrm{det}(F) = \mathrm{det}\left(\nabla u\right)

and the right Cauchy-Green deformation tensor

.. math::

  C = F^T F

so that

.. math::

  \frac{\mu}{2} \left( \mathrm{Tr}(C) - 3 \right) + J p + \frac{\kappa}{2} (J - 1) = 0

In the example everything is expressed in terms of determinants and cofactors of :math:`F`.

  Finite Element
    :3D: `SNES example 77 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex77.c.html>`_


Stokes
======

:doc:`physics/guide_to_stokes`

Euler
=====

Not yet developed

Heat equation
=============

The time-dependent heat equation

.. math::

  \frac{\partial u}{\partial t} - \Delta u = f

is used to model heat flow, time-dependent diffusion, and other physical processes.

  Finite Element
    :2D: `TS example 45 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex45.c.html>`_
    :3D: `TS example 45 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex45.c.html>`_

Navier-Stokes
=============

The time-dependent incompressible Navier-Stokes equations

.. 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}

are appropriate for flow of an incompressible fluid at low to moderate Reynolds number.

  Finite Element
    :2D: `TS example 46 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex46.c.html>`_
    :3D: `TS example 46 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex46.c.html>`_
