Attached is a tar archive for a Fortran 90 library to compute maximum-entropy basis functions.  I have used the G95 compiler. The manual in PDF is also attached and a html version of the same is also available, which provide details on how to install the code and its capabilities. This library was tested by Mike Puso last year, who interfaced it to DYNA3D and NIKE3D codes.  Note that I have added a .doc extension to the tar file (can not upload files with .tar extension).

In the attached paper, we construct new generalized coordinates for arbitrary polytopes in d-dimensions (polygons and polyhedra in 2- and 3-dimensions, respectively) using the principle of maximum entropy. The paper is to appear in Computer Graphics Forum and will be presented at the SGP'08 Conference in Denmark.  

In the attached paper (accepted for publication), we present enriched FE formulations to impose Bloch-periodic boundary conditions. Bloch-periodic BCs arise in the description of wave-like phenomena in periodic media: periodic composites, Schrodinger equation in quantum mechanics, photonic band-gap materials, etc. For a perspective, see the J-Club on elastodynamic bandgaps and metamaterials that was organized by Biswajit Banerjee

Update: The position has been filled; thanks to all who responded.

A post-doctoral position is immediately available at UC Davis. The individual will work on a joint project led by myself and John Pask at LLNL on the development and application of a new finite-element based approach for large-scale quantum mechanical materials calculations.

I recently participated in a minisymposium (SIAM Conference ), where geometric modeling, graphics, and finite elements were the focus. Over the past 4 to 5 years, there has been a lot of interest in the construction of barycentric coordinates on polygons/polyhedra, and the minisymposium brought together many of us with common interests.

In the attached manuscript, we have coupled the extended finite element
method (X-FEM) to the fast marching method (FMM) for non-planar crack
growth simuations. Unlike the level set method, the FMM is
ideally-suited to advance a monotonically growing front. The FMM is a
single-pass algorithm (no iterations) without any time-step
restrictions. The perturbation crack solutions due to Gao and Rice
(IJF, 1987) and Lai, Movchan and Rodin (IJF, 2002) are used for the
purpose of comparisons. A few of the pertinent cited references can be
found off my X-FEM web page. The final version of the manuscript is now attached.

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.

A very short blurb on finite elements in one dimension.

An overview on meshfree approximation schemes that I recently posted can be found here. Following Zhigang's note indicating the limited accessibility of blogspot.com, a local version of the article is also provided. I am also attaching a PDF version of the html file. The conversion was done using PDF Online. An article that provides more details is also available online. JAVA applets for plotting basis functions can also be accessed [1D] [2D] [3D].