Scattering of elastic waves in materials with a single spherical inclusion is an archetype of many scattering problems in physics and geophysics with applications in material characterization, non-destructive testing, medical ultrasound, etc. The study of spherical inclusions, besides their simplicity, provides enlightening details of the scattering phenomenon and a good approximation for more realistic objects. The analytical solution of the scattering by a single isotropic sphere is addressed comprehensively in the literature. However, anisotropic behavior is observed in plenty of natural and synthetic materials and only a few studies have been performed on the investigation of mechanical wave scattering by anisotropic obstacles. In this regard, Jafarzadeh et al. (Wave Motion, 112 (2022) 102963) studied the scattering of elastic waves by a transversely anisotropic sphere . The aim of the present work is to study the more general case with an orthotropic spherical inclusion. Wave scattering by a single spherical obstacle with orthotropic anisotropy inside an infinite, three-dimensional, homogeneous, and isotropic elastic medium is considered. In the isotropic surrounding, the displacement field is constructed as a superposition of the incident and scattered waves. Using the classical approach, these waves are expressed as expansions in the regular and outgoing spherical vector wave functions, respectively. The objective is to find the transition (T) matrix elements that relate the expansion coefficients of the scattered wave to those of the incident wave. The spherical inclusion, on the other hand, has orthotropic symmetry in Cartesian coordinates. Transforming the anisotropic stress-strain relations and the elastodynamic equations into spherical coordinates shows that the governing equations are inhomogeneous due to the appearance of trigonometric functions in the polar and azimuthal coordinates. To deal with the inhomogeneous governing equations, the same methodology as in the prior studies of the authors for a sphere with hexagonal and cubic symmetry is followed. The displacement field is expanded into a series of vector spherical harmonics in the angular coordinates and each coefficient in turn is expanded into a power series in the radial coordinate. It follows from the stress-strain relation and the equation of motion that there is coupling among partial waves inside the sphere which is more complex than for the transversely isotropic and cubic cases. The equation of motion inside the sphere leads to recursion relations among the unknown coefficients in the power series. Then the rest of the unknowns are determined by the continuity of the displacement and traction on the surface of the sphere, and the T matrix elements are calculated. Specifically, in the low frequency limit, where the sphere is much smaller than the wavelength of the incident wave, these elements can be expressed explicitly. Furthermore, the T matrix of a single sphere is used in combination with Foldy theory, to study attenuation and phase velocity of polycrystalline materials. Comparisons are performed with other theories and with numerical FEM computations from the literature.

With the growing demand for renewable energy, an increased number of offshore wind farms are planned to be constructed in the coming decades. The monopile is the main foundation of offshore wind turbines in shallow waters while the installation process itself takes place with large hydraulic impact hammers. This process is accompanied by significant underwater noise pollution which can hinder the life of mammals and fish. To protect the marine ecosystem, strict sound thresholds are imposed by regulators in many countries. Among the various noise mitigation systems available, the air-bubble curtain is the most widely applied one. While there exist several models which aim to describe the mitigation performance of air-bubble curtains, these assume a cylindrically symmetric wave field. However, it is well known that the performance of the air-bubble curtains can vary significantly in azimuth due to the inherent variations in the airflow circulation through the perforated pipes positioned on the seabed surface. This paper presents a new model which is based on a multi-physics approach and considers the three-dimensional behavior of the air-bubble curtain system. The complete model consists of three modules: (i) a hydrodynamic model for capturing the characteristics of bubble clouds in varying development phases through depth; (ii) an acoustic model for predicting the sound insertion loss of the air-bubble curtain; and (iii) a vibroacoustic model for the prediction of underwater noise from pile driving which is coupled to the acoustic model in (iii) through a three-dimensional boundary integral formulation. The boundary integral model is validated against a three-dimensional finite element model. The model allows for a comparison of various mitigation scenarios including the perfectly deployed air bubble curtain system, i.e. no azimuth-dependent field, and an imperfect system due to possible leakage in the bubbly sound barrier along the circumference of the hose.

A method for the model order reduction of heavy-fluid cavities is proposed with the goal of building superelements in a finite element substructuring context. The non-symmetric displacement-pressure finite element formulation is considered. While there are numerous fluid-structure reduced order models in the literature for specific load cases, few are applicable to superelements which must be as unspecialized as possible. Typical vibroacoustic superelement methods include the widespread modal synthesis using uncoupled rigid-wall fluid modes and dry structure modes, sometimes considering the fluid added-mass effects. Vibroacoustic domains are composed of a structural domain and a fluid domain coupled on a fluid-structure boundary. By domain subdivision, the uncoupled structural and fluid subdomains are identified. The associated subproblems can be modeled by symmetric monophysics formulations. Only the fluid-structure subdomain at the intersection of the fluid and structural domains must be studied using a non-symmetric multiphysics formulation. Being defined on a submanifold, this coupled subproblem is typically significantly smaller than the uncoupled subproblems. It is proposed to reduce the uncoupled subdomains onto the coupling boundary and use a Petrov-Galerkin procedure to obtain a reduced order representation of this loaded boundary. The captured boundary dynamics is then propagated to each uncoupled subdomain. This leads to left and right global reduction bases. The Petrov-Galerkin procedure is also applied to the uncoupled subproblems to increase the reduction bases. Particular considerations have to be made regarding the nullspace when a fluid free surface is present. To that extent, the nullspace is determined analytically. Following the well-established Hurty/Craig-Bampton method, any set of retained structural degrees of freedom can be chosen before the reduction procedure without loss of generality. The superelement is then built by projection of the full order model operators onto the left and right reduction bases. The proposed method is applied to the study of a large industrial water tank in a seismic analysis context. Results show a significant improvement over the typical model order reduction methods at the same reduction basis size. While the projection onto uncoupled bases cannot describe the model past the sloshing regime, the proposed method accurately captures the dynamics of the system in the bandwidth of interest for seismic analysis.