Isogeometric Analysis and Collocation

Isogeometric analysis (IGA) is a novel approach in computational analysis, based on the idea of integrating design and analysis into one model. By this, time-consuming model conversions (“meshing”) can be avoided and geometric approximation errors in the analysis eliminated. To this end, functions that are usually used in Computer Aided Design (CAD) are adopted as basis functions for analysis. Non-Uniform Rational B-Splines (NURBS) are the most widespread technology in CAD modeling and were the first basis used for isogeometric analysis. NURBS are a generalized form of B-Splines, which are piecewise polynomials with high inter-element continuity. Up to date, NURBS are the mostly used technology in isogeometric analysis, nevertheless, also alternative technologies, like, e.g., T-Splines, Subdivision Surfaces, LR-Splines, PHT-Splines, etc. have been investigated and successfully applied to different applications.
During the development of isogeometric methods, it was found that they not only improve the geometry modeling within analysis, they also appear to be preferable to standard finite elements on the basis of per-degree-of-freedom accuracy for several different applications.
In our group, we work on fundamental aspects of isogeometric analysis, such as development of element formulations and efficient numerical schemes, on the one side, and on its practical application to real-live problems, especially in Biomechanics, on the other side. The following list provides an overview over different research topics in the context of isogeometric analysis in our group:

• Structural vibrations and wave propagation
• Static structural problems in small and large deformations
• Efficient quadrature methods
• Locking-free structural element formulations
• Aortic valve simulations
• Carotid artery stenting
• Second gradient methods

Isogeometric collocation is a very recent technology where the high continuity of the isogeometric basis is used to solve the strong form of the corresponding partial differential equation. In this approach the strong form equations are discretized and are collocated on a set of suitable points such that a quadratic system of equations is obtained. Since the strong form equations for most problems involve second or higher derivatives, the method requires at least C1-continuity, which motivates the choice of NURBS for the discretization. The Greville abscissae, which are adopted from Spline modeling, are used to define the collocation points. In collocation methods, there are considerably less evaluation points necessary than in a Galerkin formulation, which makes these formulations much more efficient in terms of numerical costs and especially attractive for problems where the evaluation of the stiffness matrix is the time-critical part. In our group, we have applied isogeometric collocation methods to different static and dynamic problems, in particular, elastostatics and explicit dynamics, locking-free formulations for straight and curved beams, and plates.