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The self-developed Boundary Element code BEMCUP-3D solves structural-dynamic and acoustic problems as well as fluid-structure-interaction-phenomena in the frequency domain. Attainable outputs of this program are e.g. the system matrices. The inverse acoustic problem (sound source identification) is considered without inversion of matrices. The envelope surface (measurement surface), which encloses the entire arbitrarily shaped sound source, is treated as an exterior problem. The Dirichlet data on this surface or boundary are given from the proper sound pressure distribution of the sound source, which is also an exterior problem. This ensures that the corresponding velocity values on the measurement surface are exactly the same for both problems. Next the region between the sound source (an arbitrarily vibrating structure) and the envelope surface (a measurement surface in experimental investigations) is treated as an interior problem. The boundary conditions on the outer surface (measurement surface) for this problem are of Dirichlet type and the already available Neumann data, the sound pressure and the velocity distributions. An algorithm makes sure that after solving the unknown sound pressure and velocity values of the sound source are situated in the solving vector. Simple sound sources enable to investigate the stability, an optimal shape and an optimal position of the measurement surface.

In this paper, an algorithm is derived to solve a problem of inverse acoustics. It considers the damped acoustic boundary value problem, i.e. the Helmholtz equation and admittance boundary condition, in order to approximate the boundary admittance of interior domains. The algorithm is implemented by using a finite element method and tested for two-dimensional cavities with arbitrary shapes. The admittance condition is reconstructed based on sound pressure measurements. The solution of the arising nonlinear system of equations is obtained by applying the Newton method following a presetting method for finding reasonable initial boundary admittance values. A residual norm accounts for the objective function. Its first- and second-order sensitivities are determined analytically by using a modal decomposition in order to avoid direct inversion of the system matrix. The experiment is simulated by taking sound pressure data of the forward solution as inputs for the inverse problem. Test examples show that very few measurement points are necessary to reproduce piecewise constant boundary admittance values very accurately. Then, the admittance boundary condition is applied to reproduce the sound pressure distribution in the cavity. Again, it becomes obvious that only a few measurement points are required to reconstruct the sound pressure field.

Interior acoustic problems require accurately representing the boundary conditions of all acoustically interacting surfaces to achieve precise acoustic predictions. The complex-valued boundary admittance fully characterizes these properties. Yet, conventional approaches to determine boundary admittances, such as the impedance tube, have limitations which do not accurately represent real-world conditions. This motivates in situ methods, where the acoustic boundary admittance is estimated in the actual mounting condition based on sound pressure measurements at certain observation points within the domain. In contrast to existing deterministic methods, a Bayesian approach is employed in this work, which provides probability distributions for the boundary admittances rather than point estimates. This offers valuable insights into the uncertainty associated with the estimation, proving beneficial for applications where a comprehensive understanding of uncertainty is desired. A finite element model is utilized to generate sound pressure data and serves as the forward model during the inference process. This makes it particularly suited for applications that involve pre-existing geometrical models, such as digital twin applications and model updating. The proposed method is applied to a two-dimensional car cabin model, demonstrating the framework’s efficacy in accurately inferring the acoustic boundary admittance using just ten observation points.