Numerical Methods ================= Discrete-Element Methods (DEM) ------------------------------ .. youtube:: 6XuyaMCjplI 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 :cite:`may2014ptatin,may2015scalable`, engineering mechanics :cite:`chen2002evaluation`, and computer graphics :cite:`jiang2016material`. This approach leads to field equations for density :math:`\rho` and momentum :math:`\rho \bm u`, .. math:: :label: field-mass-momentum \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 :math:`\bm \sigma`, defined at particles, is an objective function of the field quantities as well as particle properties, :math:`\mathcal F` is a projection from particles to fields (evaluated via quadrature in finite element methods), and :math:`\bm b` is any external body forces. The field conservation system :math:numref:`field-mass-momentum` is coupled with evolution equations for particles, which are advected in the velocity field :math:`\bm u` while retaining their composition, and may accumulate elastic strain, damage, etc. Time discretization ~~~~~~~~~~~~~~~~~~~ The field equations :math:numref:`field-mass-momentum` 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 :math:`\bm u`, though this can lead to instabilities that require short time steps or other mitigations :cite:`kaus2010stabilization`. .. note:: This section is a placeholder to demonstrate math, and does not yet properly describe the methods.