Read this notebook first to learn about the purpose of this section of the User Guide and navigate to the other notebooks.

The mesh in Underworld is one of the basic components of any model. The mesh holds the unknowns of velocity, pressure, temperature and so on.

Mesh Variables are unknowns discretised on the structured mesh but they also know about how they should be interpolated, differentiated, integrated.

Lagrangian (material) points are carried through the mesh by deformation of the background fluid. They record their history along their path. Collectively, they form a 'particle swarm' and need some careful management in a parallel code (which Underworld understands and does correctly)

In Underworld, we need a mechanism to evaluate arbitrary mathematical functions at various times during the solve that we cannot predict in advance. We provide this feature through an interface that allows meshVariables to be combined, processed and evaluated when needed.

We provide templates that use the mesh, swarm, boundary conditions and variables to solve particular systems of equations. For example, there is a solver that is pre-configured to solve the incompressible Stokes equation \begin{align*} \frac{\partial \tau_{ij}}{\partial x_j} - \frac{\partial p}{\partial x_i} &= f_i \\ \frac{\partial u_i}{\partial x_i} &= 0 \end{align*} with various hooks for complicated constitutive behaviour and non-linearity already enabled. (See Stokes solver notebook for details)

Including ways to integrate functions across the domain / on a surface, how to store variables and restore them for checkpointing.

The gLucifer module is a parallel version of the fast, particle-based lavavu visualisation toolbox that we use to render images and create object databases during large parallel runs.

The Stokes solver is the engine of Underworld and may require some tuning up to perform at its best. This notebook delves into some of the options available.