Discrete approach for shape optimization of 2D time-harmonic acoustic radiation problems solved by BEM using the fully-analytical adjoint method

The Boundary Element Method (BEM) is widely employed to address acoustic problems like radiation and scattering. When combined with shape optimization, it provides engineers with the ability to design components, cavities, and rooms with the desired acoustic characteristics. However, obtaining the sensitivities of acoustic measures via this technique is not as simple as it is with the Finite Element Method. This study presents a node-based shape optimization procedure in which the coordinates of each node of a predefined 2D mesh (consisting of isoparametric elements) are considered as design variables aiming to minimize the radiation efficiency of an infinite-length cylinder. To accomplish this objective, sequential convex programming is employed and a detailed mathematical framework is proposed for the sensitivity analysis of acoustic measures obtained by BEM, taking advantage of the properties of the resulting discretized systems. Derivatives of the BEM matrices with respect to the nodal coordinates are analytically computed, even for a large number of variables, allowing an accurate use of the adjoint method, and avoiding semi-analytical methods or direct differentiation methods. A regularization scheme is proposed to circumvent undesirable results with very irregular shapes. The examples demonstrate the potential of the procedure, highlighting the efficacy of the method.