# Numerical Methods¶

## 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 $$\rho \bm u$$,

(1)\begin{aligned} \rho_t + \nabla\cdot (\rho \bm u) = 0 \\ (\rho \bm u)_t + \nabla\cdot \mathcal F(\bm \sigma) &= \bm b \\ \end{aligned}

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 .

Note

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