Substructuring approaches are nowadays widely used to predict numerically the vibroacoustic behavior of complex mechanical systems. Some of these methods are based on admittance or mobility frequency transfer functions at the coupling interfaces. They have already been used intensively to couple subsystems linked by point contacts and enable to solve problems at higher frequency while saving computation costs. In the case of subsystems coupled along lines, a Condensed Transfer Function method is developed in the present paper. The admittances on the coupling line are condensed in order to reduce the number of coupling forces evaluated. Three variants are presented, where the transfer functions are condensed using three different functions. After describing the principle of the CTF method, simple structures will be given as test cases for validation.

International audienceThe vibroacoustic behavior of axisymmetric stiffened shells immersed in water has been intensively studied in the past. On the contrary, little attention has been paid to the modeling of these shells coupled to non-axisymmetric internal frames. Indeed, breaking the axisymmetry couples the circumferential orders of the Fourier series and considerably increases the computational costs. In order to tackle this issue, we propose a sub-structuring approach called the Condensed Transfer Function (CTF) method that will allow assembling a model of axisym-metric stiffened shell with models of non-axisymmetric internal frames. The CTF method is developed in the general case of mechanical subsystems coupled along curves. A set of orthonormal functions called condensation functions, which depend on the curvilinear abscissa along the coupling line, is considered. This set is then used as a basis for approximating and decomposing the displacements and the applied forces at the line junctions. Thanks to the definition and calculation of condensed transfer functions for each uncoupled subsystem and by using the superposition principle for passive linear systems, the behavior of the coupled subsystems can be deduced. A plane plate is considered as a test case to study the convergence of the method with respect to the type and the number of condensation functions taken into account. The CTF method is then applied to couple a submerged non-periodically stiffened shell described using the Circumferential Admittance Approach (CAA) with internal substructures described by Finite Element Method (FEM). The influence of non-axisymmetric internal substructures can finally be studied and it is shown that it tends to increase the radiation efficiency of the shell and can modify the vibrational and acoustic energy distribution

The nonlinear parabolic equation (NPE) is a time-domain method widely used in underwater sound propagation applications. It allows simulation of weakly nonlinear sound propagation within an inhomogeneous medium. So that this method can be used for outdoor sound propagation applications it must account for the effects of an absorbing ground surface. The NPE being formulated in the time domain, complex impedances cannot be used and, hence, the ground layer is included in the computational system with the help of a second NPE based on the Zwikker-Kosten model. A two-way coupling between these two layers (air and ground) is required for the whole system to behave correctly. Coupling equations are derived from linearized Euler's equations. In the frame of a parabolic model, this two-way coupling only involves spatial derivatives, making its numerical implementation straightforward. Several propagation examples, both linear or nonlinear, are then presented. The method is shown to give satisfactory results for a wide range of ground characteristics. Finally, the problem of including Forchheimer's nonlinearities in the two-way coupling is addressed and an approximate solution is proposed.

A nonlinear parabolic equation (NPE) model for weakly nonlinear sound propagation in an inhomogeneous medium is described. The model being formulated in the time domain, complex impedances cannot be used to simulate ground surfaces. A second NPE model is thus derived to include the medium in the computational system. Based on a nonlinear extension of the Zwikker-Kosten model for rigidly-framed porous media, it allows to include Forchheimer's nonlinearities. Both models are then adapted to terrain-following coordinates, and used together with an interface condition, allow to simulate finite-amplitude sound propagation over a non-flat, finite-impedance ground surface. Numerical examples show that the NPE model is in good agreement with the solutions of the frequency domain boundary element method. Applications of this model to the simulation of sound propagation from explosions in air are then discussed.

A computational model for long-range non-linear sound propagation over urban environments is described. First the probability model of the geometrical parameters of an urban environment are determined using Information Theory and the Maximum Entropy Principle. The propagation model is then presented: it is based on the non-linear parabolic equation (NPE) and its extension to propagation in porous media, in which the urban layer of the real system is represented by a porous ground layer. The uncertainties introduced by the use of this simplified model and the presence of the variability of the real system are taken into account with a probabilistic model. Reference solutions are obtained thanks to the boundary element method (BEM); these experimental observations are then used to identify the parameters of the probability model. This inverse stochastic problem is solved using an evolutionary algorithm which involves both the mean-square method and the maximum likelihood method. Applications and model validation are then presented for two different urban environment morphologies. It is shown that the identification method provides an accurate and robust way for identifying the stochastic model parameters, independently of the variability of the real system. Constructed confidence regions are in good agreement with the numerical observations.