Isogeometric Analysis: A practical introduction

Teachers: A. Reali and R. Vazquez

Abstract: Isogeometric Analysis (IGA) is a recent idea [Hughes, Cottrell, Bazilevs (2005); Cottrell, Hughes, Bazilevs (2009)] introduced to bridge the gap between Computational Mechanics and CAD. The key feature of IGA is to extend the finite element method representing geometry by functions which are typically used by CAD systems, and then invoking the isoparametric concept to define field variables. Thus, the computational domain exactly reproduces the CAD description of the physical domain. Numerical testing in different situations has shown that IGA holds great promises, with a substantial increase in the accuracy per degree of freedom with respect to standard finite elements.
Within this framework, this series of lectures mainly aims at giving a practical introduction to Isogeometric Analysis, covering some basic concepts of IGA and its implementation. Therefore, some basics of B-Splines and NURBS will be given first, along with a tutorial on the use of the “NURBS toolbox” to construct B-Spline and NURBS basis functions and geometries within a Matlab/Octave environment.
Then, the main ingredients toward the construction of simple isogeometric codes within the framework of Galerkin methods will be shown and discussed. This will include topics such as isoparametric mapping, numerical quadrature, boundary conditions, and other fundamental issues that have to be dealt with during the implementation of isogeometric methods.
As a first application, the approximation of structural vibrations on different examples will be studied. Within this framework, the use of consistent versus lumped mass matrix will be discussed. Some results on the dispersion properties of isogeometric methods in wave propagation will be shown, as well.
This part of the course is complemented by a series of lectures and tutorials on the actual implementation of IGA within a Matlab/Octave environment and, in particular, the free isogeometric software “GeoPDEs” ( will be presented and used to solve 2D and 3D benchmarks.
Finally, isogeometric collocation techniques will be introduced as an interesting high-order low-cost alternative to standard Galerkin approaches and applications to elastostatics and explicit dynamics will be in particular discussed.

Course schedule (all lectures will take place at the MS1 room of the Department of Civil Engineering and Architecture):
9-11: introduction to B-Splines and NURBS (AR)
11-13: tutorial on the “NURBS toolbox” and on the construction of B-Spline and NURBS basis functions and geometries (RV)
9-11: isogeometric Galerkin methods (AR)
11-13: implementation of isogeometric Galerkin methods within a Matlab/Octave environment (RV)
9-11: “GeoPDEs” software tutorial (RV)
11-13: isogeometric collocation methods (AR)

The course will be complemented by the discussion of small individual projects assigned to each participant at the end of the course.