# Numerical Methods¶

## Discrete-Element Methods (DEM)¶

## Material-Point Methods (MPM)¶

Material-point methods solve conservation equations using Eulerian finite element methods while tracking materiel composition with Lagrangian particles. It is widely used in geodynamics [MBP14, MBP15], engineering mechanics [CHS+02], and computer graphics [JST+16]. This approach leads to field equations for density \(\rho\) and momentum \(\rho \bm u\),

where the stress \(\bm \sigma\), defined at particles, is an objective function of the field quantities as well as particle properties, \(\mathcal F\) is a projection from particles to fields (evaluated via quadrature in finite element methods), and \(\bm b\) is any external body forces. The field conservation system (1) is coupled with evolution equations for particles, which are advected in the velocity field \(\bm u\) while retaining their composition, and may accumulate elastic strain, damage, etc.

### Time discretization¶

The field equations (1) are typically solved using implicit or semi-implicit methods due to stiffness in low-speed experiments and nearly-incompressible materials. Particle positions are usually updated explicitly using the computed velocity \(\bm u\), though this can lead to instabilities that require short time steps or other mitigations [KMuhlhausM10].

Note

This section is a placeholder to demonstrate math, and does not yet properly describe the methods.