IGA. Robin Bouclier

of freedom (DOF)) and f is the force vector. We do not provide more information here on the expressions of these operators. In the rest of this chapter, these quantities will either be the very classic ones (context of linear elasticity, for example), or detailed in more specific situations (e.g. context of thin shell analysis).

      1.2.4.2. Performance

Beyond the reinforced link between CAD and numerical simulation, IGA turned out to be a superior scientific computing technology. It is now often seen as a high-performance computing method. The reason for this is the higher regularity of spline-based functions that enables us to capture physical solutions with fewer DOF than in standard FEM (Evans et al. 2009). In this respect, IGA has now been successfully applied to numerous disciplines of computational mechanics, such as shell analysis (Kiendl et al. 2009; Echter et al. 2013; Bouclier et al. 2015b), contact problems (Seitz et al. 2016; Antolin et al. 2018), fluid–structure interaction (Kamensky et al. 2017; Apostolatos et al. 2019), biomechanics (Vilanova et al. 2013; Morganti et al. 2015; Patelli et al. 2017), geometrical and material nonlinear solid mechanics (Elguedj et al. 2008; Lipton et al. 2010; Bouclier et al. 2015a; Ambati et al. 2018), fluid dynamics (Bazilevs et al. 2007; Akkerman et al. 2011) and structural vibration (Cottrell et al. 2006; Shojaee et al. 2012), to name a few. In this book, the idea is to benefit from the capabilities of IGA and to arrive at the shape optimization of complex structures, which undoubtedly constitutes one of the most valuable applications for IGA, as both its superior analysis and attractive geometrical properties can magnificently express.

NOTE.– When dealing with the performance of IGA, another important aspect to mention concerns the numerical integration used to evaluate the stiffness operator K (see equation [1.33]). In this work, we proceed, as originally, in IGA, which is the same as in the FEM standard: we perform a classic looping over the knot-span elements, and we take p + 1 Gauss points per element and per parametric direction (with p being the polynomial degree of the shape functions in the corresponding direction). However, we quote that this procedure is not optimal given the higher regularity of the spline functions. Numerous works that aimed to reduce the cost of the construction of isogeometric operators have emerged since the advent of the concept. In this respect, we may cite the development of advanced quadrature rules, more or less related to the entire isogeometric patch (Hughes et al. 2010; Auricchio et al. 2012a; Bouclier et al. 2012; Schillinger et al. 2014; Adam et al. 2015). In the same idea, it may be preferred to focus on collocated-IGA (Auricchio et al. 2010, 2012b) when the polynomial degree is very high. Finally, it was recently noted in the standard Galerkin framework that a revisit of the standard looping over elements in the assembly process could allow IGA to meet its full efficiency (Calabro et al. 2017; Sangalli and Tani 2018; Hiemstra et al. 2019). These innovative implementation techniques are not considered in this work.

1.3. Improved CAD-CAE integration for robust optimization

      With the isogeometric concept and the underlying spline technologies in hand, let us now step back and briefly retrace the original motivation behind IGA, i.e. the opportunity to reunify computer-aided design (CAD) and computer-aided engineering (CAE). Pushing forward the reasoning also enables us to envisage the power of the approach for shape optimization, as discussed at the end of this section.

      1.3.1. Returning to the original motivations behind IGA

As previously mentioned, IGA was originally introduced by Hughes et al. (2005) with a principal motivation: to reunify the worlds of CAD and CAE, which, in past decades have evolved independently (although they are closely related during engineering design and certification). Engineering design involves, on one side, the designer who generates geometric models through CAD files, and on the other side, the stress analyst who builds the inputs according to a numerical code (that calls upon FEM most of the time). The problem is that the translation of the CAD file into the FEM inputs is not straightforward in practice: an important analysis preparation step is necessary, as illustrated in Figure 1.13(a).

Note that the issue to move from design and analysis does not only concern the mesh generation, but also the efficient creation of appropriate, analysis-suitable geometries (Cottrell et al. 2009). In fact, the designer endeavors to represent, as faithfully as possible, a conceptual geometry. It means that the generated CAD model usually presents many details, for example holes, fillets and numerous sub-parts, such as bearings. Highly detailed CAD models are rarely exploitable for the simulation since it usually contains details that may be viewed as irrelevant by the analyst. As a result, in addition to the mesh generation step, a lot of time is spent generating analysis-suitable geometries, where the initial CAD model is defeatured. For sophisticated engineering systems, things can turn out to be very tricky. Different simulations such as, for example, structural mechanics, fluid dynamics, heat transfer and many others are performed. Each type of analysis has its own particular method, and thus different models are required for each of them. Figure 1.14 illustrates this point: for sophisticated engineering systems, for example, here an airplane, different visions of the same product may exist depending on the physical field. It follows that appropriate simulation-specific geometries are needed. Still looking at the example of Figure 1.14, the aerodynamicist may focus on the outer geometry of the aircraft, whereas the stress analyst is mainly concerned by the internal structure. Therefore, there is not a unique CAD model of the airplane. A lot of geometric models are required, depending on the simulation to be performed.

Figure 1.13. The original motivation of IGA is to improve engineering design and certification by integrating Design and Analysis into a single model: henceforth, designers and engineers can collaborate more intimately. For a color version of this figure, see www.iste.co.uk/bouclier/IGA.zip

Figure 1.14. Engineering systems: each engineer has their own vision and requirements for the same product (adapted from artwork by C.W. Miller)

According to Cottrell et al. (2009), the translation to analysis-suitable geometries, in addition to the meshing and the input setting for simulation codes, take de facto over 80% of the overall analysis time. It is thus clear that a stronger dialog between CAD and CAE would be of significant benefit. The engineer should be aware of what the designer does during the whole design and certification process, and vice versa. IGA brings this issue back to the table and has fostered fruitful discussions regarding a better CAD-CAE integration. As explained in Figure 1.13(b), the goal of IGA can be seen as an attempt to ease the collaboration between the designers and the engineers by integrating efficient geometric modeling possibilities and accurate simulation capabilities into a single model. As of now, it is nevertheless important to state that the dream process of Figure 1.13(b) is still not truly possible in practice, even with IGA. Indeed, we will see in section