Online Colloquium and PhD Course
Structure Preserving Methods in Isogeometric Analysis
Origins, Status, Recent Progress and Structure Preserving Methods.
April 26, H. 17.00
Thomas J.R. Hughes
Oden Institute for Computational Engineering and Sciences
University of Texas at Austin, 201 East 24th Street, Austin,
Texas 78712, USA
Teams link for the Colloquium
The vision of Isogeometric Analysis (IGA) was first presented in a paper published October 1, 2005 . Since then it has become a focus of research within both the fields of Finite Element Analysis (FEA) and Computer Aided Geometric Design (CAGD) and has become a mainstream analysis methodology and provided a new paradigm for geometric design [2-4]. The key concept utilized in the technical approach is the development of a new foundation for FEA, based on rich geometric descriptions originating in CAGD, more tightly integrating design and analysis. Industrial applications and commercial software developments have expanded recently. In this presentation, I will describe the origins of IGA, its status, recent progress, areas of current activity, and the development of isogeometric structure preserving methods.
Key Words: Computational Mechanics, Computer Aided Design, Finite Element Analysis, Computer Aided Engineering
 T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement, Computer Methods in Applied Mechanics and Engineering, 194, (2005) 4135-4195.
 J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Chichester, U.K., 2009.
 Special Issue on Isogeometric Analysis, (eds. T.J.R. Hughes, J.T. Oden and M. Papadrakakis), Computer Methods in Applied Mechanics and Engineering, 284, (1 February 2015), 1-1182.
 Special Issue on Isogeometric Analysis: Progress and Challenges, (eds. T.J.R. Hughes, J.T. Oden and M. Papadrakakis), Computer Methods in Applied Mechanics and Engineering, 316, (1 April 2017), 1-1270.
Online PhD Course
Structure-preserving methods: finite element exterior
calculus and isogeometric generalizations
April 27-30, 2021
Assistant Professor, Delft Institute of Applied Mathematics, Delft University of Technology
of Mathematics, University of Rome Tor Vergata
Teams link to the PhD course
Finite element exterior calculus is a framework for designing stable and accurate finite element discretizations for a wide variety of systems of PDEs. The involved finite element spaces are constructed using piecewise polynomial differential forms, and stability of the discrete problems is established by preserving at the discrete level the geometric, topological, algebraic and analytic structures that ensure well-posedness of the continuous problem. The framework achieves this using methods from differential geometry, algebraic topology, homological algebra and functional analysis.
This short course will focus on the foundations of finite element exterior calculus as well as its extension to isogeometric spaces. The latter allow exact geometric descriptions and enable the construction of piecewise polynomial spaces that are smoother than those provided by classical finite elements. Applications are made to the discretization of a variety of problems, including the Hodge Laplacian and elliptic eigenvalue problems.
Introduction to exterior calculus, the de Rham complex, basics of homological algebra
Hilbert space setting and the abstract Hodge-Laplacian
Structure preservation and well-posedness of the abstract Hodge-Laplacian
Extensions to isogeometric spaces