Shape Optimization of Shell Structure Acoustics

SM1. Introduction. This supplement contains details that were omitted from the main paper [SM2] because of page limitations. SM2. The Model for the Shell Structure Acoustic Interaction. To avoid switching between the main paper and this supplement, we have included the entire expanded version of Section 2, not only the details and proofs missing from Section 2 of the main paper. SM2.1. Naghdi Shells. When an elastic body is much smaller in one dimension than in the other two, it can be modeled using shell equations. The Naghdi shell model is derived from standard linear elasticity, reducing the original problem (1a-c) from three dimensions to two. The classical derivation makes use of a kinematic assumption and a mechanical assumption, which are not strictly necessary: Delfour has developed a version of the Naghdi model based on " intrinsic " differential calculus, which avoids recourse to these assumptions [SM13, SM14]. We follow the more classical approach proposed by Blouza and Le Dret [SM5, SM7] because it is well-known and closely tied to the implementation of the long-used MITC finite element methods; see [SM9, §6.3], [SM8, SM9]. In the derivation of Naghdi's model, we assume u to be real-valued. It will become complex-valued in §SM2.3, when we couple it with the Helmholtz equation. We consider " thin " domains, which are described by a so-called middle surface chart φ : Ω 0 ⊂ R 2 → R 3 defined on a reference domain Ω 0 ⊂ R 2 and a thickness function t : Ω 0 → R +. We make the following assumptions.