# MPM vs SPH in hypervelocity impact problems

Particle methods are developed to overcome the mesh distortion issue. Methods like SPH by Gingold in 1977 and PIC (FEMPM) by Sulsky in 1994 are well-suited for modeling extremely large deformation problems with failure and fracture.

## 0. Post Details
- Reference: [Ma et al., Comparison study of MPM and SPH in modeling hypervelocity impact problems, 2008.](https://www.sciencedirect.com/science/article/abs/pii/S0734743X08001255)
- Post by: Jedd Yang
- Date: 2025-05-31
- Keywords: Computational Mechanics, Meshfree

## 1. FEMPM (Finite-Element Material Point Method)

FEMPM discretizes materials with particles that carry history-dependent data. The momentum equations are solved on a background grid, offering an Eulerian framework. It can be seen as a special Lagrangian FEM, where particles replace Gauss points as quadrature points—avoiding mesh distortion while retaining FEM robustness.

## 2. SPH (Smoothed Particle Hydrodynamics)

Originally developed for astrophysics, SPH also uses particles but solves the momentum equations directly on them. A major issue in solid mechanics is tensile instability: since astrophysical and general fluid flow problems typically lack tensile stress, SPH performs poorly when modeling solids—particles tend to clump under tension, leading to voids or numerical fractures.

## 3. FEMPM vs SPH

- Nodal connectivity (to evaluate the spatial derivatives)
- SPH dynamically finds neighbors with ~O(N log N) complexity due to frequent position updates.
- FEMPM solves on a fixed grid as FEM, reducing computational cost.

- Approximation functions
- SPH constructs a approximations via particle summation within a support domain—needs enough particles for accuracy.
- FEMPM uses traditional 𝐶^0 shape functions

- Consistency of approximation functions (to converge)
- SPH fails to satisfy constant/linear consistency when:
(1) Particles near boundary (truncated support)
(2) Particles irregularly distributed
- FEMPM, inheriting FEM, avoids such problem.

- Instability
- SPH's summation over only a limited particle number can cause clumping or computational blowup under tensile stress when the number is insufficiently large.
- FEMPM, altho uses the particles as well, constructs the uses shape function on the background grid. And the particle number is generally larger than that of the grid nodes, thereby reducing the instability. However, FEMPM's 𝐶^0 shape function causes cell-crossing instability when particles move across cell boundaries during deformation.

- Dirichlet Boundary conditions (BCs)
- SPH struggles with BCs imposition due to
(1) Non-vanishing residual boundary terms
(2) Approximation function lack of Kronecker delta property
- FEMPM, derived from the weak form, satisfies BCs naturally.

- Contact algorithm
- SPH handles contact implicitly—may cause penetration since velocity isn’t enforced as single-valued.
- FEMPM naturally ensures a single-valued velocity field, preventing interpenetration without extra cost.