Contents

Thomas J.R. Hughes
University of Texas at Austin, USA

Isogeometric Analysis:  Introduction and Recent Developments
Designs are encapsulated in CAD (Computer Aided Design) systems and simulation is performed in FEA (Finite Element Analysis) systems.  FEA requires the conversions of CAD descriptions to analysis-suitable formats, leading to finite element meshes.  The conversion process involves many steps, is tedious and labor intensive, and is the major bottleneck in the engineering design-through-analysis process, accounting for more than 80% of overall analysis time.  This is a major impediment to the product development cycle.  The technical objectives are to create a new framework, simultaneously suitable for both design and analysis, and eliminate the bottleneck thereby, and leverage this framework to develop fundamentally new and improved computational mechanics methodologies to efficiently solve vexing problems. The key concept utilized is a new paradigm termed Isogeometric Analysis (IGA), based on rich geometric descriptions originating in CAD, resulting in one geometric model that is suitable for both design and analysis.  In the few short years since its inception , IGA has become a focus of research within both the fields of FEA and CAD. 

The purpose of this talk is to introduce and review recent progress to ward developing IGA procedures that do not involve traditional mesh generation and geometry clean-up steps, that is, the CAD file is directly utilized as the analysis input file, to summarize some of the mathematical developments within IGA that confirm the superior accuracy and robustness of spline-based approximations compared with traditional FEA, and to present some applications of IGA technology to problems of solids, structures and fluids that illustrate its advantages.

Alessandro Reali
University of Pavia, IT,

Isogeometric Analysis: basic concepts of isogeometric Galerkin formulations with some applications to structural vibrations and an extension to collocation methods.
This series of lectures mainly aims at covering some basic concepts of Isogeometric Analysis.Therefore, some basics of B-Splines and NURBS will be given first. 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. Also, some other numerical examples, solved with the free isogeometric software GeoPDEs (geopdes.sourceforge.net), will be presented.

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.

Tor Dokken
SINTEF, NO

Locally Refinable Splines
Local refinement of spline represented models is essential in isogeometric analysis to facilitate compact geometry and analysis models and avoid unnecessary growth in the dimension of the spline space used.

The three dominant approaches to locally refinable splines within isogeometric analysis are:

• Hierarchical B-splines based on a hierarchy of nested splines spaces

• T-splines where the refinement is specified  in the T-mesh relating the vertices (coefficients) of the B-splines to the knot line segments in the parameter domain of the B-splines

• Locally Refined B-splines (LR B-splines) where the refinement is specified directly in the parameter domain of the B-splines.

While refinement of Hierarchical B-splines and LR B-splines is addressed from the structure of the spline spaces generated, refinement of T-splines is addressed from the T-mesh. The basis of all approaches is the refinement of a spline surface with a rectangular domain.For all approaches a number of such rectangular parametric spline surfaces can be glued to form more complex patchworks of surfaces, where the number of surfaces meeting at a common vertex can be different from 4.The challenges of establishing proper continuity over the extraordinary points (vertices where 3, 5 or more surfaces meet) are the same for all approaches. For T-splines a T-mesh combining the T-meshes of all the rectangular T-splines surfaces is easy to construct, consequently providing one single composite T-mesh for the T-spline surface patch work. The vertices of the LR B-splines have the same geometric interpretation as the vertices of the T-splines. However, establishing adjacency relations between vertices in a T-spline fashion drastically reduces the modelling flexibility of the LR B-spline space, and this is not a feasible approach.

Published theory of Hierarchical B-splines and T-splines is dominantly addressing the 2-variate case. The theory of LR B-splines is for the d-variate case, both with respect to splines spaces, their spanning properties, and when the B-splines form a basis. The extended T-grid of T-splines corresponds to the LR-mesh of LR B-splines. Consequently LR B-splines also form an extended theory for T-splines over rectangular domains, and provide more general conditions for situations when T-splines form a basis than the current fairly restrictive notion of analysis suitable T-splines.

Trond Kvamsdal
NTNU - Trondheim, NO

Adaptive Isogeometric Analysis using LR B-splines
In any finite element analysis of real world problems, it is of great importance that the quality of the computed solution may be determined.Numerical simulation of many industrial problems often requires large computational resources. It is therefore, of utmost importance that computational resources are used as efficiently as possible, to make new results readily available and to expand the realm of which processes may be simulated. We thus identify reliability and efficiency as two challenges in simulation based engineering. These two challenges may be addressed by error estimation combined with adaptive refinements. A lot of research has been performed on error estimation and adaptive mesh refinement. However, adaptive methods are not yet an industrial tool, partly because the need for a link to traditional CAD-system makes this difficult in industrial practise. Here, the use of an isogeometric analysis framework may facilitate more widespread adoption of this technology in industry, as adaptive mesh refinement does not require any further communication with the CAD system.

We will herein address a posteriori error estimates applicable for isogeometric analysis as well adaptive refinements using LR B-splines. The efficiency and reliability of the techniques will be demonstrated for Poisson, elasticity and plate problems.

Clemens Verhoosel
Eindhoven University of Technology, NL

Nonlinear analysis of solids and structures using IGA
n these lectures we are concerned with the application of isogeometric analysis to a variety of problems in the non-linear analysis of solids and structures. We consider the discretization of the Kirchhoff-Love shell formulation, for which a smooth discretization is required. We demonstrate how Bézier extraction can be used to create a finite element compatible data structure for B-splines, NURBS and T-splines. We consider the application of IGA to the simulation of failure processes as a particular class of nonlinear solid mechanics problems. We show that, on the one hand, the higher-order continuity naturally inherited from splines facilitates the discretization of gradient theories in plasticity and smeared damage formulations and, on the other hand, the possibility to locally decrease inter-element continuity allows for the representation of discrete cracks. We demonstrate how this can be used in the context of cohesive zone formulations.

Kris van der Zee
Eindhoven University of Technology, NL

Shape calculus
These lectures address the connection between shape calculus and IGA. Shape Calculus is the study of changes with respect to geometry, with important applications in shape optimization, free- and moving boundary problems, (free-surface flow, fluid-structure interaction, solidification, etc.), geometric partial differential equations and evolving surfaces. Shape calculus is indispensible whenever the geometry of an underlying domain is a primary variable of interest, and the derivative with respect to this variable becomes important, e.g., in Newton schemes, gradient flows, or linearized-adjoint-based techniques. Shape derivatives generally require smooth geometries,which complicates or prevents the application to standard piecewise-linear discrete geometries in conventional FEM.IGA, whichprovides smooth geometries in a practical and unifying manner, has opened the door to the full arsenal of shape-calculus techniques in discrete settings.

Course Material

The course material consists of the book "Isogeometric Analysis" by J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs, (Wiley, 2009, ISBN: 978-0-470-74873-2), supplemented with auxiliary lecture notes. All course material will be provided free of charge to the participants at the start of the advanced school.