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.
