Prof. Natarajan Sukumar from the University of California, invited by Prof. Yongxing Shen, gave us a talk on Aug. 9th, 2017.  Prof. Natarajan Sukumar presented the joint work with Jean B. Lasserre (CNRS-LAAS, Institute of Mathematics, Toulouse) and Eric B. Chin (UC Davis), a new approach to accurately integrate polynomial functions over polytopes that uses Stokes's theorem and the property of homogeneous functions, whereby it suffices to integrate such functions on the boundary facets of the polytope.  For homogeneous polynomials, this approach is used to further reduce the integration to just evaluation of the function and its partial derivatives at the vertices of the polytope.  This results in an exact cubature rule for a homogeneous polynomial.  For weakly singular integrands in 2D, accurate and efficient numerical integration is realized. Numerical examples in two and three dimensions were presented to demonstrate the efficacy of the integration scheme, and as applications he considered elastic fracture in 2D and the modeling of holes with higher-order finite elements within the framework of the X-FEM.

Time & Date: 10:00 a.m.-11:00 a.m., Aug. 9th, 2017 (Wednesday)

Location: Room 228, conference room, JI Building

Title: Numerical integration of homogeneous functions on convex and non convex polytopes: Applications in the extended finite element method

Abstract

Accurate integration of polynomial functions over arbitrarily-shaped polygonal and polyhedral domains is required in methods such as the extended finite element method (X-FEM), embedded interface methods, virtual element method, and discontinuous and weak Galerkin methods, just to name a few. The most common approaches to perform this integration have been: tessellation of the domain into simplices; application of Stokes's theorem to reduce the volume integral to a surface integral; and use of moment-fitting methods to design suitable cubature rules.

In this talk, I will present a new approach that uses Stokes's theorem and the property of homogeneous functions, whereby it suffices to integrate such functions on the boundary facets of the polytope.  For homogeneous polynomials, this approach is used to further reduce the integration to just evaluation of the function and its partial derivatives at the vertices of the polytope.  This results in an exact cubature rule for a homogeneous polynomial.  For weakly singular integrands in 2D, accurate and efficient numerical integration is realized. Numerical examples in two and three dimensions will be presented to demonstrate the efficacy of the integration scheme, and as applications we will consider elastic fracture in 2D and the modeling of holes with higher-order finite elements within the framework of the X-FEM.  This is joint-work with Jean B. Lasserre (CNRS-LAAS, Institute of Mathematics, Toulouse) and Eric B. Chin (UC Davis)

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 is a member of the Executive Council of the US Association of Computational Mechanics. 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 element methods, and new methods development (enriched partition-of-unity methods) for large-scale quantum-mechanical materials calculations.