# simulations in materials

Interetsed topics include:

• Material Point Method, eXtended Finite Element Method, and other mesh-free methods;
• Combined atomistic and continuum simulations;
• Multiscale homogenization.

I taught a computational course, Simulations in Materials, in the fall semester of 2002 for graduate students in the Louisiana State University.

### lecture notes on Simulations in Materials: Material Point Method

The Material Point Method are covered in my lecture, Simulations in Materials. Lecture notes 9 to 14 are on this topic.

lecture note 9:  Material Point Method: an introduction
lecture note 10: Material Point Method: grid equations
lecture note 11: Material Point Method: solution procedure
lecture note 12: Material Point Method: 2D problems
lecture note 13: Interpolating shape functions
lecture note 14: MPM homeworks

### re: Material Point Method

Have there been any works verifying MPM for unstructured grids?

### unstructured grids for MPM

John, any specific concern about the unstructured grids for MPM?

### Invitation to Dr. Banerjee for a lecture or comments in my Blog

Dear John,

One of my student did a MPM calculation on fracture problem (propagation of a 2D crack), by using unstructured Eulerian grids. But he did that only for fun in a homework project.

Actually I have very limited knowledge on MPM. One iMechanican here, Dr. Biswajit Banerjee (http://imechanica.org/user/1095), is more qualified to talk about this subject.

So, I hence courteously invite Dr. Banerjee to give us a lecture or write some comments. Thanks.

### Material Point Method

John's question was on MPM and unstructured grids. Let us briefly review MPM and the issues involved.

The main steps in developing an MPM algorithm for a problem involving a body are (I am describing the "GIMP" approach here):

• Find the weak form of the governing equations. Integrals are over the current configuration.
• Define particle characteristic functions that satisfy a partition of unity in the initial configuration. These functions help convert the integrals over the body into into a sum of integrals over particles.
• Next use standard finite element-like interpolation functions to discretize any continuous quantities. These interpolation functions are defined on a grid (which may be structured or unstructured).
• Finally, discretize the time derivatives (using some differencing scheme).
• The resulting system of algebraic equations are solved in the usual manner.

Let us now look at the schematic diagram of the method below.

The MPM algorithm usually follows the following steps :

• The body is discretized into material points (shown as green dots).
• Each material point has an associated volume in the initial configuration. The tiling of the body by these volumes is shown with the dashed red lines. Any compatible deformation leaves the tiling intact (when the volume associated with each each material point is determined from the deformation gradient). An explicit tiling is not required.
• A grid is overlayed on the body. This grid is only used to compute gradients and to solve the discretized governing equations. The grid is discarded at the end of each time step.
• An unstructured grid could be used if different regions of the body had significantly different densities of material points. However, the associated meshing and remeshing costs make this unattractive for large problems.
• The constitutive update is performed at the material points. All history variables are also stored at the material points.
• At this stage, a projection is performed to take information from the material points to the grid. This information is usually limited to momentum, internal force, and thermal inertia like quantities.
• After the discretized equations are solved on the grid, the velocities are interpolated back to the particles. The particles are updated and the cycle is repeated.

The main sources of error are

• The error in projecting quantities from material points to the grid.
• The error in evaluating the integrals over the current volume.

These errors make the original MPM method unstable. However, the GIMP method and its successors have much better behavior. There is an active research group in Utah who are exploring these issues.

Unstructured grids are better for minimizing the integration errors for problems that involve small deformations, i.e., if you don't have to remesh. Professor Rebecca Brannon looked at some such situations when she was at Sandia. The main problem with unstructure grids is that to do any large parallel simulation, you need a domain decomposition algorithm. This is not only more expensive but also involves considerable effort as far as software engineering is concerned.

Most of the problems I deal with involve large deformations in conjunction with fluid-structure interaction. For such problems, structured grids make life much easier, especially since most CFD codes use structured grids. But, as always, there's no free lunch. Adaptive mesh refinement can become quite complicated in a jiffy with structured grids.

I think there are many opportunities for exploration here. I would definitely like to know whether some nice features of MPM (no remeshing, large deformations, easy fluid-structure interaction, easy contact) can be retained if unstructured grids are used.

### Generalized Interpolation Material Point (GIMP)

Dear Biswajit,

Thanks for summarizing the recent developments of the Material Point Method.

GIMP was not introduced in my MPM courses which was taught in 2002. For people familiar with MPM but not with GIMP: GIMP stands for Generalized Interpolation Material Point that was developed by Bardenhagen and Kober (2004).

Using a C1 continuous weighting function, GIMP has solved the numerical noise problem associated with material points just crossing the grids.

### How many particles are needed in a grid cell?

How many particles are averagely needed in a grid cell, for 2D, and for 3D, in order to guarantee an accurate simulation?

In you schematic diagram, it seems ther are only 2 particles in a cell, averagely.

### Re: MPM and particles per cell

Henry,
There are two important purposes that particles serve in MPM (other than the fact that they carry the history of the material):

1. They can be thought of as integration points.
2. They fulfill a sampling density requirement for some applications.

If you think of particles as integration points, beyond a certain number of points the accuracy of integration will not increase (and may even deteriorate). Philip Walstedt and Mike Steffen at Utah are working on quantifying those limits at this time.
If the sampling density is too low, particles will not be able to see each other if the deformation becomes very large. This leads to unphysical effects such as cracks opening up when they should not. This problem has been alleviated to some extent by scaling the domain of influence of particles with the stretches (or more complicated book keeping in some cases). However, the simplest approach is to introduce a higher density of particles in regions where large deformations are expected.

### Combined atomistic and continuum simulations

Combined atomistic and continuum simulations are also included in the course.

http://www.imechanica.org/node/1108

### compare Material Point Method with other methods

I am trying to compare the Material Point Method with the eXtended Finite Element Method (XEFM), and many other mesh free methods (like Element -Free Galerkin, etc.), in simulating processes related to:

1) viscoelastic and other history-dependent material behaviors;
2) material separations, like in crack propagation, cutting, indenting processes;
3) fluid-solid interactions, and many other espects;
4) interface debonding in composite materials.

This comparison should be conducted by experts from MPM part, from XFEM part, from EFG part, and many other parts on mesh-free methods.

Mechanicians with different focuces are very well welcomed to contribute.

### MPM-XFEM

Dear Henry,

I am happy to participate, however, I think that we should identify first the key points that will be considered in the comparison, otherwise, we might work aimlessly.

I have developed a C++ XFEM code that we can use for failure and interface problems as well as non-linear materials.

Should we start by identifying the key points of interest?

- robustness

- convergence proofs

- versatility

- computational time

- accuracy

Let me remind you of the WCCM8 mini-symposium for which this topic would be very well suited:  Accuracy Assessment of the eXtended Finite Element Method: Adaptivity, Comparison with Competing Methods, Industrialisation [ID:141]
Stephane Bordas, Marc Duflot, Pierre-Olivier Bouchard

Dr Stephane Bordas

### strongest robustness, acceptable accuracy and computational time

Dear Stephane,

Which method provides strongest robustness, with acceptable accuracy and computational time, to simulate behaviors with many moving discontinuities, such as interacting cracks, debonding of interfaces in composite materials, and shockwaves?

Henry.

### arbitrary number of evolving discontinuities

Henry,

In my opinion, today, meshfree methods are the best alternative given your criteria.

See for instance the papers attached.

Sorry for the extremely late reply,

Stephane

Dr Stephane Bordas

### micromechanics versus multiscale finite element analysis

H. M. Inglis, P. H. Geubelle, K. Matous, H. Tan and Y. Huang, 2007.
Cohesive modeling of dewetting in particulate composites: micromechanics vs. multiscale finite element analysis. Mechanics of Materials, 39, 580–595.

We investigated the effect of damage due to particle debonding on the constitutive response of highly filled composites using two multiscale homogenization schemes: one based on a closed-form micromechanics solution, and the other on the finite element implementation of the mathematical theory of homogenization

In both cases, the particle debonding process is modeled using a bilinear cohesive law which relates cohesive tractions to displacement jumps along the particle–matrix interface. The analysis is performed in plane strain with linear kinematics.

A detailed comparative assessment between the two homogenization schemes is presented, with emphasis on the effect of volume fraction, particle size and particle-to-particle interaction.

### MPM development in CSAFE

The Material Point Method is adopted in one of DOE's Accelerated Strategic Computing Initiative (ASCI) center, Center for the Simulation of Accidental Fires and Explosion (CSAFE) at the University of Utah.

The method builds into its nature the ability to simulate large deformation, contact, fragmentation, fracture, solid-fluid-combustion interactions, and explosions.

Annual reports of CSAFE can be found from http://www.csafe.utah.edu/Publications/