IGA. Robin Bouclier

3 with, in particular, the objective of simulating real aeronautical structures (i.e. made of stiffened panels, as depicted in Figure 1.17, right).

Figure 1.18. Connecting patches that are initially non-matching (a): the basis functions of both patches need to be modified in order to become identical along the interface (b). For a color version of this figure, see www.iste.co.uk/bouclier/IGA.zip

NOTE.– To truly bridge the gap between CAD and analysis in the case of arbitrary volumes, there is an additional challenge to overcome. Indeed, a volume is built in CAD through its boundary representation (B-rep) (Piegl 2005; Stroud 2006; Marussig and Hughes 2018), i.e. by grouping together intersected trimmed surface patches and filling the interior of the volume in the CAD rendering. Creating a solid analysis-suitable model or simulating solid structures directly from this CAD data also constitutes an active research topic in the field (Akhras et al. 2016; Massarwi and Elber 2016; Xu et al. 2017; Buffa et al. 2019; Massarwi et al. 2019). In this book, this issue is not addressed: we focus on either complex shell structures (thus the surface representation of the entity is directly available in CAD) or simple solid geometries.

Figure 1.19. Example of a non-conforming multipatch parameterization: the interface crosses the elements of the panel. The trace spaces along the interface of both sides are completely different. For a color version of this figure, see www.iste.co.uk/bouclier/IGA.zip

      

      1.4.3. Imposing shape variation

Developing advanced numerical schemes that are able to directly handle the trimmed and multipatch parameterizations of CAD models is all the more relevant, as the ultimate goal in this book is to perform the shape optimization of realistic structures. Indeed, we want to avoid modifying the parameterization of the whole complex model at every iteration of an optimization process. Illustrations are provided in Figures 1.20 and 1.21 for the case of the movement of a hole within a plate and of a stiffener on a panel, respectively. We seek to leave the discretization of the plate (respectively, panel) unchanged if only the position and shape of the hole (respectively, stiffener) are to be optimized. Thanks to trimming objects and non-conforming interfaces, each geometric sub-part has an independent parameterization, which facilitates the definition of its movements and its relations with a “good” analysis model (note that in the case of a plate with a hole, severe mesh distortion can occur with the boundary fitted alternative). Otherwise, this would lead to a method that is too cumbersome to be able to perform shape optimization properly, and in a reasonable time. Since it is of crucial importance for the computation and optimal design of complex structures, the discussion regarding the analysis-suitable model issue is not restricted to this section: additional insights will be given in the remaining chapters of this book.

Figure 1.20. The discretization of the plate needs to be updated at each modification of the hole when a boundary fitted model is used and mesh distortion may happen. Envisaging the trimmed model coming from CAD (i.e. the plate without the hole and the boundary of the hole) enables us to keep the mesh of the plate untouched, regardless of the position and shape of the hole. For a color version of this figure, see www.iste.co.uk/bouclier/IGA.zip

Figure 1.21. The discretization of the panel needs to be updated at each shape modification of the stiffener when a matching multipatch model is used. Envisaging a model with non-conforming discretization enables us to keep the mesh of the panel untouched, regardless of the position and the shape of the stiffener. For a color version of this figure, see www.iste.co.uk/bouclier/IGA.zip

1.5. Computation of non-conforming interfaces: a brief overview of usual weak coupling methods

As observed in the previous section, the development of advanced coupling methods that enable us to connect (possibly non-conforming) patches is crucial in IGA to simulate realistic structures and to answer our long-term objective of optimally designing such complex structures. This is also necessary in the search of a true bridge between CAD and analysis. To meet this goal, important studies have been performed in the last five years by the IGA community to revisit and extend the different weak domain coupling approaches that were initially developed for standard FEM. Basically, the resulting strategies can be divided into three classes: penalty coupling (Apostolatos et al. 2014a, 2014b; Herrema et al. 2019; Coradello et al. 2020; Leonetti et al. 2020), Mortar coupling (Brivadis et al. 2015; Dornisch et al. 2015, 2017; Bouclier et al. 2016, 2017; Coox et al. 2017a; Zou et al. 2018; Dittmann et al. 2019; Wunderlich et al. 2019) and Nitsche coupling (Nguyen et al. 2014; Ruess et al. 2014; Jiang et al. 2015; Schillinger et al. 2016a; Bouclier and Passieux 2018; Hu et al. 2018; Buffa et al. 2019; Elfverson et al. 2019). In the following, we attempt to present the principle for each approach. In order to do so, we place ourselves in the context of linear elasticity and stay general by only giving the minimal state of art, the basic continuum formulations and the main properties of each coupling technique. We indicate that the discussion reflects the author’s interpretation, which is not unique: it is performed in accordance with the works achieved in the next chapters, in particular the development of noninvasive and high-performance computing tools.

      

      1.5.1. Governing equations

We consider here, the simple case of a bi-domain linear elastic problem in Ω ⊂ ℝ^d, d = 2 or 3 being the parametric dimension of the domain. More precisely, Ω is divided into two disjoint, open and bounded subsets Ω₁ and Ω₂, joining at interface Γ (see Figure 1.22). In other words, we have Ω = Ω₁ ∪ Ω₂ ∪ Γ and Ω₁ ∩ Ω₂ = ∅. We assume that the two non-overlapping sub-domains Ω₁ and Ω₂ are subjected to body forces respectively. Furthermore, forces are associated with boundaries Γ_F1 and Γ_F2, and displacements are prescribed over boundaries Γ_u1 and Γ_u2, respectively. The boundaries satisfy the following relations for m ∈ {1, 2} (see Figure 1.22):

      [1.34]

      In each