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

Navier-Stokes#

The time-dependent incompressible Navier-Stokes equations

\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

3D:

TS example 46