# Tutorials, by Physics#

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

$-\Delta u = f$

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

Finite Difference
2D:

SNES example 5

3D:

KSP example 45

Finite Element
2D:

SNES example 12

3D:

SNES example 12

## Elastostatics#

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

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

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

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

where the strain tensor $$\bm \varepsilon$$ is given by

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

where $$\bm u$$ is the infinitesimal displacement of the body. The resulting discretizations use PETSc’s nonlinear solvers

Finite Element
2D:

SNES example 17

3D:

SNES example 17

3D:

SNES example 56

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

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

and the right Cauchy-Green deformation tensor

$C = F^T F$

so that

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

In the example everything is expressed in terms of determinants and cofactors of $$F$$.

Finite Element
3D:

SNES example 77

## Stokes#

Guide to the Stokes Equations using Finite Elements

## Euler#

Not yet developed

## Heat equation#

The time-dependent heat equation

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

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

Finite Element
2D:

TS example 45

3D:

TS example 45