N. Sukumar's blog

Hello All,

Gianmarco Manzini (LANL) and I are teaching a short course entitled: "Virtual Element Method in Solid and Fluid Mechanics" at USNCCM17 in Albuquerque, which will be held on Sunday, July 23rd, 2023.  If you plan to be at the conference in-person, then do consider attending the short course.  A summary of the course follows:

We recently proposed a method that uses distance fields to exactly impose boundary conditions in physics-informed neural networks (PINN).  This contribution is available as an arXiv preprint.

Abstract: In this paper, we provide a retrospective examination of the developments and applications of the extended finite element method (X-FEM) in computational fracture mechanics. Our main attention is placed on the modeling of cracks (strong discontinuities) for quasistatic crack growth simulations in isotropic linear elastic continua.

As part of the 1st Pan-American Congress on Computational Mechanics (April 27-29, 2015 in Buenos Aires, Argentina), Marino Arroyo and I are organizing a minisymposium - ``Smooth Approximations on Unstructured Nodal Discretizations: Finite Elements, Spline-based Techniques and Meshfree.'' The deadline for abstract submissions is November 20, 2014.

Dear Colleagues:

We would like to invite you to submit a contribution to a minisymposium that we are organizing on Emerging Methods for Large-Scale Quantum-Mechanical Materials Calculations at the 12th US National Congress on Computational Mechanics, to be held July 22-25, 2013 in Raleigh, NC. This minisymposium aims to bring together leading researchers in this emerging area to discuss and exchange ideas on new methods developments for density-functional calculations, mathematical analysis, and applications of ab initio methods in electronic-structure calculations.

If you would like to include LaTeX equations in a presentation (Powerpoint, Keynote, or others), and have not found a good solution so far, then you might wan't to consider TeXclip.  It is on the web (no installation needed) and high-quality images of the TeX equations can be incorporated into a presentation rather seamlessly. I have been using TeXPoint so far, but do like the simple usage and the quality of the images that TeXclip generates.

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                                         Call for participation
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                  Workshop on Barycentric Coordinates in Geometry
                  Processing and Finite/Boundary Element Methods
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                  July 25-27, 2012, Columbia University, New York

I spotted this service on google.  It is easy to use, fast,  and is very accurate (though not perfect) in retrieving one's articles. It has some nice features; I particularly like the one that allows articles to be saved in BibTeX, which is attractive for those who use LaTeX.

I was pointed to this article, which was a good read. A link to the article in PDF is also provided here (author's home page).

In this manuscript (available at http://arxiv.org/abs/1004.1765), we present a systematically improvable, linear scaling formulation for the solution of the all-electron Coulomb problem in crystalline solids. In an infinite crystal, the electrostatic (Coulomb) potential is a sum of nuclear and electronic contributions, and each of these terms diverges and the sum is only conditionally convergent due to the long-range 1/r nature of the Coulomb interaction.

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.

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.

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.