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..