IGA: Non-Invasive Coupling with FEM and Regularization of Digital Image Correlation Problems, Volume 2

1IGA: A Projection of FEM onto a Powerful Reduced Basis

1.1. Introduction

Isogeometric analysis (IGA) was originally introduced by Hughes et al. (2005) and formalized in Cottrell et al. (2009) in order to reunify geometric modeling and computational mechanics. The main idea is to resort to the same bases for analysis as the ones employed to describe the geometry in computer-aided design (CAD), so that a common geometrical model can be used by both the designers and analysts. In this framework, the method can be viewed as a generalization of the finite element method (FEM) that considers smooth and higher-order functions, for example, non-uniform-rational-B-spline (NURBS) functions (Cohen et al. 1980; Piegl and Tiller 1997; Rogers 2000; Farin 2002), to replace typical piecewise Lagrange polynomials in the computations. Other geometry descriptions include, for example, subdivision surfaces (Cirak et al. 2002), T-splines (Bazilevs et al. 2010), U-splines (Thomas et al. 2018) and spline composition (Elber 2017; Antolin et al. 2019). Within this work, only NURBS (which constitute the most commonly used technology in CAD) and simpler B-splines are used. In this chapter, we employ the terminologies spline or isogeometric indifferently to denote an NURBS or a B-spline object. Beyond the reinforced link between CAD and numerical simulation, IGA turned out to be a superior approximation technology, which on a per-degree-of-freedom basis exhibits increased accuracy and robustness in ...