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