Seminars at the Faculty of Informatics

Higher order polygonal finite element methods

The Faculty of Informatics is pleased to announce a seminar given by N. Sukumar

DATE: Friday, September 21st, 2012
PLACE: USI Università della Svizzera italiana, room SI-003, Informatics building (Via G. Buffi 13)
TIME: 15.30

Emanation from the work of Wachspress in 1975, linearly precise generalized barycentric coordinates on planar polygons have been widely adopted for applications in geometry processing and solid mechanics. In a Galerkin implementation, these barycentric coordinates (shape
functions) are used to construct the discrete finite element space. In the spirit of three-node triangle and four-node quadrilateral elements, Wachspress shape functions are first constructed on a regular polygon and through an isoparametric mapping, the shape functions are defined on any convex polygon. Wachspress, harmonic, mean value (MVC) and maximum-entropy coordinates (MEC) are admissible for planar convex polygons; for meshes with nonconvex polygons, harmonic, MVC and MEC are conforming.

In this talk, I will first give an overview of the use of linearly precise generalized barycentric coordinates in Galerkin computations. Then, I will present an extension of MEC to approximations that possess second-order precision on arbitrary planar polygons. To this end, we adopt a relative entropy measure for signed shape functions in conjunction with nodal prior weight functions that have the appropriate zero-set on the boundary of the polygon.
We maximize the objective functional subject to the linear constraints for quadratic precision. Along an edge of a polygon, the approximation is identical to univariate Bernstein polynomials: the choice of the prior weight function ensures that the shape functions satisfy a weak Kronecker-delta property on each edge. Some preliminary tests on the numerical implementation of quadratic MEC will be presented, and the promise of the approach will be discussed.

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 has spent sabbatical visits at Cornell University in Ithaca, NY, and SLAC National Accelerator Laboratory in Menlo Park.
Sukumar's current research focuses on maximum-entropy meshfree methods, mesh-independent modeling of discontinuities with extended finite elements, and development of a new partition-of-unity finite element method for large-scale ab initio quantum-mechanical calculations.

HOST: Prof. Kai Hormann

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