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Epi-convergence (max-ent bases), crack growth

N. Sukumar's picture

In the attached paper, we have used Variational Analysis techniques (in particular, the theory of epi-convergence) to prove the continuity of maximum-entropy basis functions. In general, for non-smooth functionals, moving objectives and/or constraints, the tools of Newton-Leibniz calculus (gradient, point-convergence) prove to be insufficient; notions of set-valued mappings, set-convergence, etc., are required. Epi-convergence bears close affinity to Gamma- or Mosco-convergence (widely used in the mathematical treatment of martensitic phase transformations). The introductory material on convex analysis and epi-convergence had to be omitted in the revised version; hence the material is by no means self-contained. Here are a few more pointers that would prove to be helpful. Our main point of reference is Variational Analysis by RTR and RJBW; the Princeton Classic Convex Analysis by RTR provides the important tools in convex analysis. For convex optimization, the text Convex Optimization by SB and LV (available online) is excellent. The lecture slides provide a very nice (and gentle) introduction to some of the important concepts in convex analysis. The epigraphical landscape is very rich, and many of the applications would resonate with mechanicians.

On a different topic (non-planar crack growth), we have coupled the x-fem to a new fast marching algorithm. Here are couple of animations on growth of an inclined penny crack in tension (unstructured tetrahedral mesh with just over 12K nodes): larger `time' increment and smaller `time' increment. This is joint-work with Chopp, Bechet and Moes (NSF-OISE project). I will update this page as and when more relevant links are available.

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Comments

Sukumar,

I tried to read your paper but didn't quite understand a few things (and your recommended reading list is truly daunting :)

My questions are fairly basic:

1) What is the motivation for yet another space of basis functions?

2) What advantage do maximum entropy basis functions provide over other basis functions?

I'm sure these questions are answered in the literature. However, I also think that one of the benefits of a forum such as iMechanica is that we can get such questions answered by authors without having to spend inordinate amounts of time trying to understand the extensive literature on a particular subject. Therefore, could you explain in simple (and unrigorous) terms why you are exploring maximum entropy bases and why the continuity of these bases is so important.

On a slightly different but related note, I think our readers will find the following discussion interesting. The discussion is by John Baez and is called "Why Mathematics is Boring".

 

Biswajit 

N. Sukumar's picture

Biswajit,  Have barely scratched the surface, and hence be rest assured, I have a hunch that barring the authors of the Variational Analysis (VA) book, most folks (including experts on convex analysis) would concur with you.  My intent was just to provide some context, and hence the links. In regard to max-ent, someone who knows the meshfree landscape should probably comment since it'll be from an unbiased viewpoint.  My biased take is as follows.  In regard to your questions: (1)&(2)  if you trace the roots of basis functions used in meshfree methods and the developments therein over the past 10+ years (be it SPH, radial basis functions, moving least squares, etc.), a few needs have come to the forefront. (A) smooth transition from FEM-to-meshfree approximation (since meshfree in the entire domain tends to become costly); and (B) FEM-like behavior on the boundary is desirable to readily impose essential boundary conditions.  The use of convex approximants (phi_i >= 0) is the key, as was revealed in the article by Arroyo and Ortiz (2006). The essential boundary condition is taken care of by any convex approximant and they used a modified entropy functional (with links to statistical mechanics) to also obtain basis functions with compact support. When seen through the eyes of the Kullback-Leibler distance, one can generalize this construction----via choosing a prior (weight function) and then using entropic regularization to obtain basis functions that satisfy the constraint (linear reproducing conditions).  If you look back, trying to `correct' SPH (constant consistency to linear consistency) as was done in RKPM has the same motivation. Now, with max-ent, this approach is generalized----you can correct any prior (weight function) that has compact support to construct basis functions that are linearly complete, compactly-supported, and will interpolate on the boundary (bdry of the convex hull).  This is the main appeal when compared to other meshfree basis functions. Many other attributes of convex approximants can be found in A&O's paper.  In this document, I have provided more details, connections, and also links to the above article as well as many others that are pertinent. In reading this, the appeal and place of convex optimization will hopefully come through and ergo the usefulness of tools (such as VA) from convex analysis.  Mechanics is replete with variational formulations (constrained, convex, nonconvex, saddle-point, etc.), and it appears only befitting that like Delaunay interpolants, meshfree basis functions are also obtained via a (constrained) variational route.  Hope this has provided some answers?

Jinxiong Zhou's picture

Dear Prof. Sukumar,

I read your paper and share the similar questions with Prof. Biswajit.  After reading your reply, I still have two specific questions:

1) Is any weight function which is compactly supported can be used to construct basis function? The only requirement is that it has compact support?

2) We all know one of the shorcomings of meshfree method is its higher cost for construction of basis functions as compared with FEM.  Then how about your max-ent basis evaluation efficiency? As compared to FEM or present meshfree method, is it more timeconsuming or more efficient? At least I learned from your paper that this method is more flexible and versatile.

Thank you!

Jinxiong

N. Sukumar's picture

Jinxiong, In regard to your questions: 1) Yes, any continuous weight function that has compact-support can be used. Need at least continuity on the weight to get C0 basis functions; can get smoother basis functions if smoother weights are used (similar to mls). If a constant weight is used () then all the basis functions are non-zero at any point in the entire domain, and hence they are not compactly-supported. Plots of basis functions for couple of different weights appear in the paper that I attached. 2) No scheme can beat fem if just a straight-up comparison of basis function computations is done; a more realistic comparison would be the overall run-time in light of the accuracy that can be achieved and also if capabilities can be extended beyond the known limits of fem. I have not done a straight-up comparison with mls (don't have a mls-code on-hand). Marino presented some results on such a comparison last year at wccm, and as I recall, the timings of mls and max-ent were of the same order. He'd be able to answer your question (with precise numbers), and also provide more details on the computations and tests he has performed. Of course, the max-ent timings will have some dependence on the tolerance that is set for satisfying the reproducing (linear consistency) conditions (an extra Newton iteration will be needed if 12-digits instead of 6-digits precision is required). A point I did not mention was that (in my opinion) implementing max-ent for use in a meshfree method is easier than most meshfree basis since one need not worry about the imposition of essential boundary conditions: no need to distinguish between different types of nodes and different regions (FE, MLS, and blended/interface regions) if essential boundary conditions are to be implemented via the FE interpolant on the essential boundary. As in mls, one has to assign the support sizes of the nodes a priori. I have a max-ent implementation in Fortran 90 and the same code can be used for 1D, 2D, and 3D, with the choice of selecting different weight functions as the prior. Overall, this provides flexibility and ease of usage.

Jinxiong Zhou's picture

Dear Sukumar,

Thank you for your quick response and detailed clarification! I totally agree with you that the max-ent method is more flexible and possessing  Kronecker delta property is really a unqie feature as compared with standard meshfree method. It is very useful for treating bimaterial composites. I will try it!

Jinxiong

Suku,

Very interested in the 3D penny crack extension -- very impressive work. Will appreciate if you could email me or point me to the literature.

Qingda

 

 

Sukumar,

Thanks for the clarifications and pointers.  Your writeup is a great introduction to the issues facing meshfree methods.   Since Arroyo reads iMechanica, perhaps he can provide further insights?

Biswajit 

 

Marino Arroyo's picture

Hi all,
Biswajit, I understand your questions, why yet another meshfree method? First I would like to point out that there are many meshfree methods (at least many acronyms) but not so many meshfree approximants. The moving-least squares idea is very nice and is at the root of most methods. The maxent approximants provide an alternative method and rationale to construct approximants from a scattered set of points. 
I think Suku gave a very good account on the advantages of these particular shape functions as compared to others. I would summarize them as: (1) efficient and robust calculation in any spacial dimension, not just 1, 2 or three, (2) dirichlet BCs are straightforward, (3) since they are nonnegative they do not display Gibbs-type oscillations near shocks, (4) for PDE with smooth solutions, the accuracy of Galerkin methods based on these approximants is very good, hence for a given accuracy they are much more efficient than FE. But to be honest, I am most excited about the connection it establishes between different areas such as convex geometry, information theory, approximation theory and numerical methods, statistical learning, etc. As a matter of fact a very similar method has been devised for unsupervised statistical learning, and it is argued that max-ent approximants have very robust in the presence of noisy data and perform particularly well in high-D. 
http://idl.ee.washington.edu/publications/thesis.pdf
Best regards,
Marino

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