Brown University School of Engineering

Engineering Seminar: Joint Materials/Solid Mechanics

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Monday, April 17, 2017 4:00pm - 5:00pm

Professor N. Sukumar, Dept. of Civil and Environmental Engineering at UC-Davis, will present his talk: "Maximum-Entropy Approximation Schemes in Computational Mechanics". Abstract: Over the past two decades, there have been significant advances in realizing computational methods that provide greater flexibility and versatility than traditional finite elements for the modeling and simulation of physical phenomena — for example, particle and meshfree methods for large deformation simulations, enriched partition-of-unity finite element methods for capturing strong and weak discontinuities on fixed meshes, embedded/immersed methods for computations on nonconforming meshes, and emergence of novel Galerkin discretizations on polytopal meshes. In this talk, I will present an overview of meshfree methods, with particular emphasis on maximum-entropy (max-ent) schemes. Given a set of scattered nodes, meshfree basis functions are constructed using just the nodal coordinates, without the need for any notion of nodal connectivity as provided via a finite element mesh. Among the family of meshfree approximants, a more recent development has been the emergence of smooth convex max-ent approximation schemes — the nonnegative basis functions are obtained by solving a constrained convex optimization problem in which the Shannon-Jaynes entropy functional is maximized, subject to the linear reproducing conditions as the constraints. The nonnegativity endows such approximations with the convex hull and variation-diminishing properties, which makes the numerical solution less susceptible to undershoot and overshoots, and for dynamic simulations result in positive-definite mass matrices that do not have any nonphysical negative entries. New applications of entropy approximation schemes in diverse fields continue to emerge: solid and fluid mechanics, computer graphics, bio-molecular simulations, and quantum mechanics to name a few. Applications such as large deformation simulations of incompressible solids, devising smooth approximants on unstructured meshes, use of max-ent for polygonal and polyhedral computations, and their links to enabling isogeometric analysis will be highlighted. Finally, I will discuss open problems and the outlook of such convex approximants in computational mechanics. Bio: Sukumar holds a B.Tech. from IIT Bombay in 1989, a M.S. from Oregon Graduate Institute in 1992, and a Ph.D. in Theoretical and Applied Mechanics from Northwestern University in 1998. He held post-doctoral appointments at Northwestern and Princeton University, before joining UC Davis in 2001, where he is currently a Professor in Civil and Environmental Engineering. Sukumar is a Regional Editor of International Journal of Fracture, and a member of the Executive Council of the USACM. He has spent sabbatical visits at Cornell University (2007) and SLAC National Accelerator Laboratory (2011). Sukumar's research focuses on smooth maximum-entropy approximation schemes, novel discretizations on polytopal meshes, fracture modeling with extended finite elements, and new methods development (enriched partition-of-unity methods) for large-scale ab initio quantum-mechanical materials calculations.