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N. Sukumar's blog

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Exact imposition of boundary conditions in physics-informed neural networks

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.

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Special Issue on GBCs in CAGD

In 2019, Elsevier CAGD will be publishing a special issue on ``Generalized Barycentric Coordinates,'' edited by Michael S. Floater (University of Oslo), Kai Hormann (University of Lugano), and myself.

If you expect to have any new contribution in this area, then please do consider submitting a paper to this special issue.

All the details about the special issue can be found in the attached ``Call for Papers.''

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Perspective article on the X-FEM in computational fracture

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.

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Minisymposium at PANACM 2015: Smooth Approximations on Unstructured Meshes

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.

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Gradient bounds for Wachspress shape functions on polytopes

In this contribution (see, we derive lower and upper bounds for Wachspress coordinates over any simple d-dimensional simple convex polytope.  Numerical results for the Poisson equation on nontrivial polyhedral meshes are presented that affirm the linear rate of convergence in the energy seminorm of the polyhedral finite element method.  Matlab code to compute the Wachspress shape functions and its gradient on convex polygonal and polyhedral elements is also provided.

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Minisymposium on Electronic-Structure Methods at USNCCM12

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.

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Education via the Internet

I was pointed to this article on an emerging model for education/universities, which I found to be a good read.  

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TeX-based equation image generator

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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Workshop on Generalized Barycentric Coordinates in Geometry Processing and FEM/BEM

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

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Google scholar citations

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.

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How To Choose a Good Scientific Problem

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).

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Linear scaling solution of the all-electron Coulomb problem in solids

In this manuscript (available at, 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.

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PUFE homework exercise

As a follow-up to the discussion here, I am attaching a PUFE homework that I have assigned in 2006 and 2008 when I have offered a meshfree/pufe course. This might be of some help to those who have an interest in partition-of-unity enriched finite element methods.

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Fortran 90 library for maximum-entropy basis functions

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.

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Maximum entropy coordinates for arbitrary polytopes

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.  

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Bloch-periodic boundary conditions in classical and enriched FEM

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

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Postdoctoral Position at UC Davis in Computational Materials Science

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.

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Geometric Design, Computer Graphics, and FEM

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.

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Non-planar crack growth (X-FEM and fast marching)

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.

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

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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FEM in one dimension

A very short blurb on finite elements in one dimension.

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