Material Point Method

Why use Material Point Method? | Detailed Comparative Analyis

Introduction to the Material Point Method (MPM)

The Material Point Method (MPM) is a numerical technique used for simulating the behavior of continua, particularly in scenarios involving large deformations, complex material behaviors, and interactions between multiple phases. MPM is a hybrid Lagrangian-Eulerian method that combines the advantages of both approaches to overcome the limitations of traditional Finite Element Analysis (FEA).

Why Use the Material Point Method?

The Material Point Method (MPM) is a meshless computational method capable of simulating large deformations and history-dependent behavior of fluids and solids. While the industry has been dominated by mesh-based methods like the Finite Element Method (FEM) for the last few decades, FEM often encounters problems. These problems arise due to the dependence on a mesh in the FEM formulation and appear in the form of mesh tangling and the costly process of remeshing. As a meshless or meshfree method, MPM avoids these problems and is more capable of handling large deformations.

When compared to other meshless techniques like Smoothed Particle Hydrodynamics and the Element Free Galerkin method, MPM has been shown to be more stable in tension and more efficient in solving a wider range of problems. In comparison to pure particle methods, the calculation of gradients is trivial in MPM as the nodes remain fixed on a regular mesh. Since the particles interact with other particles via the grid, it is very easy to detect contact between entities.

What is the Material Point Method?

The Material Point Method (MPM) is a meshless technique that combines the benefits of Eulerian and Lagrangian mesh-based FEM. The method was developed in the mid-1990s as an extension of the fluid mechanics numerical method FLIP to solid mechanics. FLIP and MPM are descendants of the Particle-In-Cell (PIC) method that was invented and developed in the 1960s at Los Alamos National Laboratories. The development of the Fluid Implicit Particle method (FLIP) was prompted to improve the dissipation of PIC. This method was extended to solid mechanics to become known as the Material Point Method.

MPM is in active development in the academic community, being used to solve a variety of different problems and gaining maturity as a robust numerical method. While continuing improvements to the method's accuracy and stability are still being made 25 years since its inception, the general computational technique consists of 5 phases:

Phase 1 - Initialisation

As with all computational mechanics techniques, discretisation of a system is the first step in the analysis. Material points (or particles) are used to represent the geometry and a background mesh is placed behind the particles. This mesh is essentially arbitrary except that every particle must lie within an element. This mesh is commonly comprised of 2-node, 4-node, and 8-node isoparametric elements for 1, 2, and 3-dimensional analyses. The mesh is used only for calculation and temporary storage of information, as will be shown, and as such is up to the user to define. Generally, a Cartesian grid is used that is kept fixed and reset each time step for computational efficiency.

Left - Material object, Right - MPM discretisation with background grid

Figure 1: Materials are represented by particles in MPM surrounded by a grid.

Once the geometry has been constructed, variables are assigned to the particles such as mass and material properties. In addition, any initial conditions like velocity, stresses, temperatures, etc. are assigned. Boundary conditions are also applied at this point and can be applied to the background grid or the particles.

Phase 2 - Particle to Grid Extrapolation

In MPM, the background grid is used to solve the continuum equations of conservation. For this to occur, the nodes of the grid must contain the information of the material points. The material points carry the entire solution of the material throughout the simulation but use the grid to update values. An interpolation is made from the particles within each element to the nodes of their element. The masses (MP) and momenta (Pp) are extrapolated to the grid nodes using a sum over all the material points (NP).

Traditional FEM interpolation functions, Ni are used to interpolate values from material points to grid nodes. A consistent mass matrix is constructed using this methodology from the material point masses, however for computational efficiency this is commonly replaced with a lumped mass matrix. This is done by summing the row elements in the mass matrix to allow for trivial inversion when solving the equations of motion.

$$ m_{i} =\sum_{i=1}^{N_{p}} M_{p} N_{i} $$

$$ p_{i} =\sum_{i=1}^{N_{p}} P_{p} N_{i} $$

Phase 3 - Grid Calculations

The mesh in MPM essentially plays the role of a single step in a dynamic FEM code. The mesh is used to solve the weak form of the continuum equations of momentum and energy (not listed).

$$ \int_{\Omega} \rho \mathbf{w} \cdot \mathbf{a} d \Omega=-\int_{\Omega} \rho \sigma^{s}: \nabla \mathbf{w} d \Omega+\int_{\partial \Omega_{\tau}} \mathbf{w} \cdot \tau d S+\int_{\Omega} \rho \mathbf{w} \cdot \mathbf{b} d \Omega $$

The conservation of mass is not explicitly solved in MPM, instead each material point is assigned a mass in the initialisation phase using the formula:

$$ \rho(\mathbf{x}, t)=\sum_{p=1}^{N_{p}} M_{p} \delta\left(\mathbf{x}-\mathbf{X}_{p}(t)\right) $$

Since the material points are restrained to reside within the computational domain, this automatically solves the conservation of mass. Other variants of MPM, referred to as GIMP and CPDI, alter this equation for the material point mass. When the density equation is substituted for in the weak momentum equation, a discrete form is produced which is performed over all the elements.

$$ \begin{aligned} \sum_{p=1}^{N_{p}} M_{p} \mathbf{w}\left(\mathbf{X}_{\mathbf{p}}(t), t\right) \cdot \mathbf{a}\left(\mathbf{X}_{p}(t), t\right) = &-\sum_{p=1}^{N_{p}} M_{p} \sigma^{s}\left(\mathbf{X}_{p}(t), t\right):\left.\nabla \mathbf{w}(\mathbf{x}, t)\right|_{\mathbf{x}=\mathbf{X}_{p}(t)} \\ &+\int_{\partial \Omega_{\tau}} \mathbf{w} \cdot \tau d S+\sum_{p=1}^{N_{p}} M_{p} \mathbf{w}\left(\mathbf{X}_{p}(t), t\right) \cdot \mathbf{b}\left(\mathbf{X}_{p}(t), t\right) \end{aligned} $$

Phase 4 - Grid to Particle Extrapolation

The acceleration and velocities are now extrapolated to the particles within each element using the same interpolation functions as in Phase 2. This aspect of MPM is very important as the single-valued velocity field means that interpenetration of particles is forbidden by the MPM algorithm.

Phase 5 - Update Particle Variables and Reset Mesh

Particle velocities and positions are updated using the values extrapolated from the grid. Constitutive values are updated using the material equation for the material point. History variables are updated with no loss of information during deformation. This aspect makes MPM uniquely suited to modeling large deformations with history-dependent variables. These constitutive equations are used to update stress values, this along with the updated positions and velocities signifies the end of the time step. The mesh can be reset to the original orientation or moved to update with the material points as per the user's discretion.

Conclusion

The Material Point Method represents a significant advancement in computational mechanics, offering solutions to many challenges faced by traditional numerical methods. Through its unique combination of Lagrangian particles and Eulerian background grid, MPM provides robust solutions for problems involving large deformations, complex material behaviors, and multi-body interactions.

As explored in our comparison of computational methods, MPM bridges the gap between mesh-based and particle methods, combining the best aspects of both approaches. While traditional FEM remains valuable for many engineering applications, MPM excels in scenarios where mesh distortion would typically cause numerical difficulties.

The advantages of MPM, detailed in our analysis, make it particularly suitable for:

• Complex geotechnical problems

• Advanced manufacturing processes

• Impact and crash simulations

• Multi-phase material interactions

As computational methods continue to evolve, MPM's role in engineering simulation grows increasingly important, offering new possibilities for solving previously intractable problems in computational mechanics.