MPM Diagrams is a collection of diagrams and intended to help in understanding concepts related to the material point method (MPM).
Thanks to Material point method after 25 years: theory, implementation and applications and The Material Point Method for Simulating Continuum Materials for being the source of much of the information contained in these diagrams.
The set of undeformed material points is called Ω^0 and the set of the "current position" material points called Ω^t.
The material or Lagrangian coordinate is the position vector in the original undeformed space of a material point and is denoted X.
The spatial or Eulerian coordinate is the position vector of the same material point in a deformed configuration and is denoted x.
The mapping between X and x is called Φ, and is given by
x = Φ(X, t)
The Jacobian of the deformation mapping Φ is given by F(X, t). It encodes the deformation at the material point originating at X, and the determinate of F is given by J and describes the change in volume at that material point. J > 1 means volume increase and J < 1 means decrease.
