prof. Reali's seminar announcement

[Emanuele Tuillier Illingworth]

MATHLAB SEMINAR ANNOUNCEMENT


Title: Isogeometric collocation methods
Speaker: Alessandro Reali (University of Pavia)
Date: Wednesday, 29 May, 2013 - 14:00
Room: SISSA - Santorio A - room 133

Abstract
Isogeometric Analysis (IGA) is a recent idea, firstly introduced by 
Hughes et al. [1], to bridge the gap between Computational Mechanics and 
Computer Aided Design (CAD). The key feature of IGA is to extend the 
finite element method representing geometry by functions, such as 
Non-Uniform Rational B-Splines (NURBS), which are typically used by CAD 
systems, and then invoking the isoparametric concept to define field 
variables. Thus, the computational domain exactly reproduces the NURBS 
description of the physical domain. Numerical testing in different 
situations has shown that IGA holds great promises, with a substantial 
increase in the accuracy-to-number-of-degrees-of-freedom ratio with 
respect to standard finite elements, also thanks to the high regularity 
properties of the employed functions. In the framework of NURBS-based 
IGA, collocation methods have been proposed in [2], constituting a 
viable and interesting high-order low-cost alternative to standard 
Galerkin approaches (cf. [3]). Recently, such techniques have also been 
successfully applied to elastostatics and explicit elastodynamics (see 
[4]). In this presentation, after an introduction to isogeometric 
collocation methods, we move to the solution of elasticity problems and 
present in detail the results discussed in [4]. A special attention is 
devoted to the development of explicit high-order collocation methods 
for elastodynamics. Several numerical experiments are presented in order 
to show the good behavior of these approximation techniques. We also 
report some interesting results on the use of isogeometric collocation 
in the framework of thin structures. In particular, we focus on both 
initially straight and spatial curved Timoshenko beams and show how 
shear locking is avoided in the context of mixed methods, independently 
on the selected approximation orders (see [5,6]). We finally present 
some recent applications of these methods to the solution of the 
Cahn-Hilliard equation, for which isogemetric collocation represents 
indeed an accurate, efficient, and geometrically flexible option [7]. We 
then conclude proposing some snapshots on the extension of isogeometric 
collocation to adaptive hierarchical NURBS discretizations (cf. [3]), as 
well as on further possible developments and applications.

References
[1] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs (2005). Isogeometric 
analysis: CAD, finite elements, NURBS, exact geometry, and mesh 
refinement. Comp. Meth. Appl. Mech. Eng., 194, 4135-4195.
[2] F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Reali and G. 
Sangalli (2010). Isogeometric Collocation Methods. Math. Mod. Meth. 
Appl. Sci., 20, 2075-2107.
[3] D. Schillinger, J.A. Evans, A. Reali, M.A. Scott and T.J.R. Hughes 
(2013). Isogeometric Collocation: Cost Comparison with Galerkin Methods 
and Extension to Adaptive Hierarchical NURBS Discretizations. ICES 
Report 13-03 (submitted to Comp. Meth. Appl. Mech. Eng.).
[4] F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Reali and G. 
Sangalli (2012). Isogeometric collocation for elastostatics and explicit 
dynamics. Comp. Meth. Appl. Mech. Eng., 249-252, 2-14.
[5] L. Beirao da Veiga, C. Lovadina and Reali (2012). Avoiding shear 
locking for the Timoshenko beam problem via isogeometric collocation 
methods. Comp. Meth. Appl. Mech. Eng., 241-244, 38-51.
[6] F. Auricchio, L. Beirao da Veiga, J. Kiendl, C. Lovadina and A. 
Reali (2013). Locking-free isogeometric collocation methods for spatial 
Timoshenko rods. Submitted to Comp. Meth. Appl. Mech. Eng..
[7] H. Gomez, A. Reali and G. Sangalli (2013). Accurate, efficient, and 
(iso)geometrically flexible collocation methods for phase-field models. 
Submitted to Journ. Comp. Phys..