Deformation Gradient and Particle State Update

In MPM, each particle carries a deformation gradient $F_{p}$ , which tracks how the local material volume attached to the particle is stretched, rotated, or sheared over time. It plays a central role in computing stress based on constitutive models.

At the beginning of each time step, the grid lies on a regular, undeformed lattice. Let $x_{i}^{n}$ be the position of grid node $i$ at time $t^{n}$ . The deformation gradient is updated by observing how the grid locally moves over the time step, assuming a given grid velocity field $\hat{v}_{i}^{n + 1}$ .

Deformation Gradient Update

The most common approach is to compute $F_{p}^{n + 1}$ using a 1st-order approximation:

$F_{p}^{n + 1} = (I + Δ t i \sum v_{i}^{n + 1} \nabla (w_{i p}^{n})^{T}) F_{p}^{n} .$

In MLS-MPM [Hu et al. 2018] and APIC [Jiang et al. 2015], the deformation gradient update can be rewritten more compactly using the affine velocity matrix $C_{p}$ that is already computed during P2G and G2P transfers.

Recall the local affine velocity field:

$v_{p} (x) = v_{p} + C_{p} (x - x_{p}) .$

From this, the grid motion induces a local velocity gradient $C_{p}$ , and the deformation gradient can be updated as:

$F_{p}^{n + 1} = (I + Δ t C_{p}) F_{p}^{n} . (3)$

This bypasses the need to explicitly evaluate $\nabla w_{i p}$ , and instead uses the already-aggregated affine behavior stored in $C_{p}$ .

Plasticity flow is also applied at this stage to project the updated deformation gradient back onto the admissible space defined by the material's yield criterion. This process, known as return mapping, ensures that the material obeys plastic limits, and will be discussed in detail in the next lecture.

Position Update

The particle position at time $t^{n + 1}$ is then advected as:

$x_{p}^{n + 1} = x_{p}^{n} + Δ t v_{p}^{n + 1} .$

Physics-Based Simulation