Isogeometric Methods for Numerical Simulation

This chapter collects fundamental facts concerning theory and algorithms for rational spline functions. In addition, it addresses fundamental techniques for creating domain parameterizations for isogeometric analysis. All presented facts are covered in more detail by the the rich literature on Computer Aided Geometric Design, e.g., (Fa02; FHK02; HL96; PT95; PBP02).

Isogeometric analysis (IGA) is a fundamental step forward in computational mechanics that offers the possibility of integrating methods for analysis into Computer Aided Design (CAD) tools and vice versa. The benefits of such an approach are evident, since the time taken from design to analysis is greatly reduced leading to large savings in cost and time for industry. The tight coupling of CAD and analysis within IGA requires knowledge from both fields and it is one of the goals of the present paper to outline much of the commonly used notation. In this manuscript, through a clear and simple Matlab® implementation, we present an introduction to IGA applied to the Finite Element (FE) method and related computer implementation aspects. Furthermore, implementation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is also presented, where several fracture examples in both two-dimensions and three-dimensions are given as illustration. The open source Matlab® code which accompanies the present paper can be applied to one, two and threedimensional problems for linear elasticity, linear elastic fracture mechanics, structural mechanics (beams/plates/shells including large displacements and rotations) and Poisson problems with or without enrichment. It also implements the Bézier extraction concept that allows FE analysis to be performed on T-spline geometries. An attempt was also made to include a review of recent developments of IGA.

This chapter provides an introduction to the iso-geometric Boundary Element Method (BEM). The standard iso-geometric BEM is presented first and then isometric concepts are introduced. Both plane and 3-D problems are discussed and details of implementation given. The method is extended to non-homogeneous and non-linear problems. The main applications are in geotechnical engineering.

Within the framework of isogeometric analysis, collocation methods have been recently proposed as an interesting strong form alternative to standard Galerkin approaches, characterized by a significantly reduced computational cost, but still guaranteeing higher order convergence rates. In this chapter we provide a concise introduction to isogeometric collocation methods and propose a brief review of some of the most important results obtained so far in this context.

This chapter provides an introduction to the use of T-splines in isogeometric analysis. A simple definition of two-dimensional T-splines is given and Bézier extraction is introduced. The basic details for implementation of T-splines as finite element shape functions are given. Two examples of integrated analysis and design based on commercial tools are given to illustrate the utility of T-spline-based IGA in a design-through-analysis workflow.