Material Point Method

Hi Nitin,

1. I will appreciate pointers to introductory or tutorial articles on MPM in SM (solid mechanics). Ideally, those that are written for a non-PIC non-MPM expert. Will appreciate presentation slides too. Could you please post any such material as attachment(s) here? (Also, similar articles on PIC.)

2. What is the computational complexity (or space and time costs) of the MPM technique? How sensitive is the complexity to the no. of Lagrangian particles? To the size of the background Eulerian mesh?

3. It seems that even after effecting any algorithmic reformulations for efficiency, the basic technique itself would be computationally very expensive. In what kind of applications is the "extra" cost justified? Are there any guidelines available regarding application areas? Or is it too early to tell?

Thanks in advance.

OK, thanks. Browsed MPM on the 'net for an hour or so, and then thought of dropping a note here in passing ... For whatever it is worth....

In MPM, Nairn has been researching woods as composites. Sulksy can already address granular materials fairly effectively because MPM makes it easy to include in the factor of contact-friction among individual grains. ... 

Now, putting two and two together, I think, it should be very easy for the MPM researchers to model the problem of interlocking of fibers in composite materials. The interlocking can occur both within the uncracked portion and near the crack-separated surfaces. The broken fibers (or other forms of reinforcements) come out protruding from the crack-surfaces behind the crack-tip, and may serve to impede further crack opening, thereby possibly enhancing toughness. ... As far as I remember, the modifications in deflection and stress fields due to reinforcement-interlocking has been well acknowledged as a factor in the composites literature. But I can't tell how serious is its contribution to the overall fracture toughness. And the overall impression that I carry (and it could be wrong) is that it has been difficult to address these kind of factors in numerical simulations using FEM. FEM analyses typically ignore the fiber-to-fiber friction and interlocking. The tendency in FEM analysis is to factor in all such factors right into the homogenization schemes, at least that for the zone near the crack-tip.

But, now, it looks like the problem could be attacked in a very *direct* manner using MPM.

Quantifying interlocking and friction may have other uses too, e.g., in making better particle-boards for acoustic applications. So, apart from rheological applications, MPM seems ideal for particle-boards.

I would appreciate knowing if any of my loud thinking above has been going wrong somewhere.

... And, yes, I still wish simplified slides/tutorials on MPM, suitable for independent reading by the UG students, were already available, ready-made, on the 'net. (I am lazy!)

The original version of MPM (developed and used by Sulsky and her group) runs into serious instability problems when particles cross cell boundaries.   If the motions/deformations are small then particles remain within cells and everything is hunky dory.  However, for large motions, you have to use alternative approaches such as GIMP (Bardenhagen and Kober) to get reasonable behavior.  That point was again made by Scott Bardenhagen and several other people who work with MPM in this year's 3rd MPM Workshop at Sandia.

Sulsky continues to use the old formulation and I'm not sure that I can trust her results for some of the things she does.  John Nairn has probably moved to GIMP by now but I'm not sure.  Unless I see convergence/completeness studies in their work, their results will at best be qualitative.  Please let me know if you find any convergence studies other than those coming out of Jim Guilkey/Mike Kirby/Martin Berzins' groups.

Getting frictional contact correct in GIMP turns out to be nontrivial whereas it's easy in Sulsky MPM.  Scott Bardenhagen and others published a paper on that in CMES  a few years ago.  Frictional contact in general GIMP  will be a good MS project for anyone interested in MPM. 

Thanks for the clarification on GIMP and MPM. ... You basically confirm that the original MPM approach does work alright for small deformations in SM and that the issue of numerical instability arises only when the deformation is large... Incidentally, this circumstance is remarkably like that for FDM!! ... So, that reminds me about posting a reply in another thread, FDM in SM. The discussion there, too, involves the aspect of Lagrangian and Eulerian viewpoints.

There are a few points I'd like to bring to our attention: 

1. One has to be careful when using the term "deformation".   I don't know why this terms was chosen by early continuum mechanicists.  It is confusing it also includes rigid body motions which are not really deformations in the colloquial sense of the word. 
2. When I say Sulsky MPM works for small deformations, I also mean that any rigid body motions have to be small enough that particles do not cross cells.  This is clearly not the case in many of the examples that she gives in her papers where she simulates bouncing balls and such.  The convergence behavior of such simulations with Sulsky MPM is often less than first order if they at all converge.
3. The main "purported" advantage of MPM is that it can deal with large deformations and large motions.  We would also like to see that we get more and more accurate results as we refine and at least first order convergence in both time and space.  Large deformations are made possible by the background grid and Largrangian particles.  However, if you use the delta function weights of the Sulsky method then convergence is not guaranteed unless an extra volume contribution is included in the grid basis function.  GIMP generalizes the weights so that any appropriate weighting function can be used - linear, quadratic, cubic spline etc.  We have found that cubic splines do very well though some boundary condition issues are yet to be resolved - including correct and fast contact in three dimensions.   Mike Steffan is probably going to publish some of that stuff in the near future.
4. One also has to be careful when talking about Eulerian/Lagrangian formulations.  MPM is purely Lagrangian.  The grid is not a true Eulerian grid - it's just a scratch pad for calculations.  Hence the statement "contact between materials is handled by the Eulerian formulation" is not accurate even though the balance of forces at grid points is what's used for doing contact in MPM.
5. Friction between interlocking fibers is not trivial in MPM.  You will need an extremely fine grid to get it right.  Finite elements will probably do the job better unless the fibers are being stretched significantly.

Hello everyone,

 From your discussing, it seems to me that MPM has been used succesfully in modeling microstructures. I do not see how this is done with MPM where a solid domain is discretised by a background grid and a set of Lagrangian particles. How material is set to a given particle? 

I would like to model fracture at the mesoscale of concrete where it is considered to be a three-phase material: matrix, aggregates and ITZ. This microsctructure is an output of a DEM packing simulation. As you know, the mesh generation step for this mesoscale specimen is time-consuming. So, I am looking for novel numerical methods for this kind of problem.

 I am very grateful to any suggestions from you.

 Thank you in advance.