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# simulations in materials

Fri, 2007-03-23 01:36 - Henry Tan

Interetsed topics include:

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

Links to other blogs:

- Where can I read about the basic ideas of the meshfree methods?
- The eXtended Finite Element Method (XFEM)
- A posteriori error estimation (indication) for extended finite element methods (XFEM)
- Fracture in Abaqus

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

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## Comments

## 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):

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

The MPM algorithm usually follows the following steps :

The main sources of error are

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 Pointthat 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):

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

- need for adaptive remeshing

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 BouchardDr Stephane Bordas

http://www.civil.gla.ac.uk/~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

http://people.civil.gla.ac.uk/~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/