IGA. Robin Bouclier

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.

      1.5.2. Penalty coupling

      Let us start by defining the functional spaces _m and _m over domain Ω_m that will contain the displacement solution and test functions, respectively:

      [1.37]

      To account for the coupling in the penalty approach, the interface Dirichlet condition [1.36a] is weighted with its test counterpart, multiplied with a penalty parameter α_pen and added to the standard virtual work formulation of the uncoupled problem. In this way, we obtain the following variational formulation for the penalty coupling: find (u₁, u₂) ∈ ₁ × ₂, such that:

[1.38]

      where the standard bilinear form a_m and the linear form l_m associated with domain Ω_m (m ∈ {1, 2}) can be written as:

      [1.39]

      and denotes the jump operator as follows:

[1.40]

      In the above equations, we note that we use the notations ・ and : to refer to the scalar product of vector fields and of second-order tensor fields, respectively.

It is important to note that the accuracy of the method strongly depends on the choice of the penalty parameter. Indeed, a large value for α_pen is necessary to enforce the kinematic compatibility [1.36a]; however, making it too large makes the discrete coupled system ill-conditioned. A related drawback with this basic penalty method is that it is variationally inconsistent, in the sense that we cannot recover the interface conditions [1.36] of the strong form by using the weak form [1.38], especially the equilibrium of the tractions [1.36b]. Finally, it has to be noted that the stiffness operators of the sub-domains are merged together, which may complicate the separation of the computational resources between the different sub-domains. For all of these reasons, we will not consider this method in this book. Nevertheless, given its simplicity, we quote that the penalty approach remains of interest in IGA. Some efforts are currently performed to enhance such penalty techniques for isogeometric non-conforming coupling, in particular, with respect to the choice of the penalty parameter (see, for example, Herrema et al. (2019) and Coradello et al. (2020)).

      

      1.5.3. Mortar coupling

      A second option to weakly formulate the coupling problem [1.35]-[1.36] is to resort to the class of Lagrange multiplier methods. We note that the Lagrange multiplier approach is also often referred to as the Mortar approach in literature (Wohlmuth 2000; Brivadis et al. 2015; Dornisch et al. 2015; Zou et al. 2018; Wunderlich et al. 2019). We will use the two terminologies indifferently in the whole document. In this context, a mixed formulation is set up to impose the coupling constraints [1.36]. Classically, a single Lagrange multiplier λ ∈ (where is an appropriate space) is introduced, as the dual unknown, to represent both of the interface traction forces, i.e. −σ₁n₁ = σ₂n₂ = −λ in equation [1.36b]. Then, the interface Dirichlet condition [1.36a] is imposed in a weak sense over Γ, using the Lagrange multiplier. This leads to the formulation of the following Lagrangian of the coupled problem:

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