Читать книгу IGA - Robin Bouclier - Страница 32

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 sub-domain, the kinematic constraints, the equilibrium equations and the constitutive relations have to be verified. Using again, the subscript m to denote a quantity that is valid over region Ω_m (with m ∈ {1, 2}), the corresponding governing equations read:

[1.35a]

[1.35b]

[1.35c]

[1.35d]

Figure 1.22. Reference domain coupling problem. For a color version of this figure, see www.iste.co.uk/bouclier/IGA.zip

In the above equations, ε (u_m) denotes the infinitesimal strain tensors, σ_m denotes the Cauchy stress tensors, C_m denotes the Hooke tensors and n_m represents the outward unit normal to Ω_m. We specify here that we perform, as follows, the notations in the document: continuous quantities are in normal type, while discrete quantities (i.e. vectors and matrices) are in boldface type (see equation [1.33]). To complete the formulation of the boundary value problem, the interface condition has to be added:

[1.36a]

[1.36b]

It ensures the kinematic compatibility and equilibrium of the tractions, respectively, along the coupling interface Γ between the two sub-domains. Combining conditions [1.36] with local equations [1.35] is strictly equivalent to a linear elasticity problem posed over the whole initial domain Ω. In the case of arbitrary non-matching and non-conforming interfaces, the coupling conditions cannot be imposed in a strong way. Instead, those coupling conditions are prescribed in the weak sense, and thus, the coupling methods differ by the way those weak conditions are formulated.