Admittance Boundary Conditions Research Articles

Admittance boundary conditions and sound pressure field estimation of vibro-acoustic systems using an extended Kalman filter and parametric model order reduction

Time-domain vibro-acoustic finite element simulations have been recently used in auralization and virtual sensing. However, when comparing the transient response based on a finite element model to actual measurements, often discrepancies are seen due to, amongst others, the presence of inaccurate admittance boundary conditions. To achieve accurate time-domain simulations, these uncertainties need to be included in the time-domain simulation. In this paper, the admittance boundary conditions are modelled using the superposition of several passive functions whose parameters are considered as the additional states in an extended Kalman filter scheme. Finite element models for vibro-acoustic systems including admittance boundary conditions have a high computational cost and result in frequency-dependent damping matrices. To get a smaller time-domain model, stability-preserving model order reduction is employed to reduce the number of degrees of freedom while maintaining a good accuracy and stability. Furthermore, to preserve the parameter dependency, parametric model order reduction based on a Taylor series expansion is employed. Experiments are performed on a vibro-acoustic system including a foam layer which can be represented by admittance boundary conditions. These conditions are identified and it is shown that the predicted sound field pressure is improved significantly as compared to the direct time-domain simulation.

Impact of compliant coating on Mack-mode evolution in hypersonic boundary layers

This paper focuses on the linear evolution of Mack instability modes in a hypersonic boundary layer over a flat plate that is partially coated by a compliant section. The compliant section is a thin, flexible membrane covering on a porous wall consisting of micro holes. The instability pressure could induce a vibration of the membrane, leading to a feedback to the boundary-layer fluids through the transverse velocity fluctuation. Such a process is formulated by an admittance boundary condition for the boundary-layer perturbation, which is dependent on the thickness and tension of the membrane, the properties of the porous wall, and the frequency of the Mack-mode perturbation. Using this admittance condition, the impact of the compliant coating on the Mack growth rate is studied systematically by solving the compressible Orr–Sommerfeld equations. It is found that the compliant coating could suppress the Mack instability with a frequency band in the neighbourhood of the most unstable frequency, and the stabilising frequency band widens as the membrane thickness and tension decrease, indicating a more favourable effect of a softer membrane. For a Mack mode with a specified dimensional frequency – since its dimensionless frequency, normalised by the local boundary-layer thickness and oncoming velocity, increases as it propagates downstream – the second-mode frequency band usually appears in downstream locations, and so does the stabilising effect of the membrane. Thus it is favourable to apply a compliant panel at a downstream region. In this situation, the solid–compliant junction could produce an additional scattering effect on the evolution of the Mack mode due to the sudden change of its boundary condition. The scattering effect is quantified by a transmission coefficient defined by the equivalent amplitude of the compliant-wall perturbation to the solid-wall perturbation, which can be obtained by the harmonic linearised Navier–Stokes (HLNS) approach. If the admittance is weak, then the transmission coefficient can also be predicted by an analytical solution based on the residue theorem. It is found that most of the second modes are suppressed by the scattering effect as long as the argument of the admittance is in the interval $[150^\circ, 210^\circ ]$ , agreeing with most of the physical situation. The analytical predictions agree well with the HLNS calculations when the modulus of the admittance is less than $O(0.1)$ .

Model order reduction of time-domain vibro-acoustic finite element simulations with admittance boundary conditions for virtual sensing applications

Virtual sensing employs an estimator that combines a numerical model and a limited set of measurements to estimate the full field pressure of a vibro-acoustic system in the time domain. A key aspect of virtual sensing is to create a reduced-order model of the vibro-acoustic system, such that the numerical model can be effectively employed in the virtual sensing scheme. This paper aims to create a reduced-order vibro-acoustic finite element model with frequency-dependent admittance boundary conditions for virtual sensing applications using a Kalman filter. The frequency-dependent components in admittance boundary conditions always result in a frequency-dependent damping matrix which hinders the modeling in the time domain. This paper first treats the vibro-acoustic system as a negative feedback interconnection of two subsystems: a state-space model of admittance transfer functions and a vibro-acoustic system with rigid boundary conditions. Stacking the states of these two subsystems gives the final descriptor system. Then, due to the passivity of the admittance transfer function and the definiteness of the system matrices, the second-order form of the descriptor model is modified to give a full-order model which is proven to be stability-preserving under one-sided projection methods. A coupled vibro-acoustic system is experimentally presented which demonstrates that the proposed methodology can provide a stable reduced order model and allows the full field estimation of the pressure in the presence of frequency-dependent boundary conditions.

Anapole state revealed by cloaking metallic cylinders with split ring resonators

This paper proposes the experimental demonstration of an anapole-based cylindrical electromagnetic cloaking scheme. An anapole state is excited by arranging around a cylindrical metallic target vertical split-ring resonators, forming an equivalent surface admittance boundary condition able to suppress the scattering. Using Mie formalism and multipole scattering theory, we identify the actual reason behind the cloaking operation, characterizing the anapole condition by the scattering contributions from toroidal and electric dipole moments. Numerical results are verified using full-wave simulation softwares and subsequently validated with back-scattering measurements inside an anechoic chamber.

Nonlinear broadband time-domain admittance boundary condition for duct acoustics. Application to perforated plate liners

The behavior of perforated plates at high excitation level is generally modeled by a surface impedance that depends on the rms velocity in the perforations. A time-domain admittance boundary condition (TDABC) is developed to account for this variation using a multipole model. Two formulations are considered, based on the interpolation either of the admittance or of the multipole coefficients from a data set of reference values. These TDABC are implemented in a finite-difference time-domain solver of the linearized Euler equations and are validated by comparison with experimental results on an impedance tube. Application to a two-dimensional lined duct corresponding to the reference geometry of the NASA Grazing Incidence Tube is then performed. The spatial variation of the perforated plate liner impedance is highlighted and it is shown that assuming a uniform impedance can lead to an unacceptable prediction of the liner attenuation. These results are confirmed both for a harmonic or broadband excitation.

Mode coalescence and the Green’s function in a two-dimensional waveguide with arbitrary admittance boundary conditions

This study focuses on sound attenuation in a two-dimensional waveguide with arbitrary admittance boundary conditions on both sides of the guide. The emphasis is on understanding the formation and potential applications of the exceptional points (EPs) which arise when two (EP2) or three (EP3) modes degenerate into a single mode. A perturbation approach is used to obtain asymptotic expressions for the trajectories of the axial wavenumbers in the complex plane as they coalesce to form an EP. The numerical results presented herein suggest that the first triple root (EP3) assures maximum modal attenuation along the waveguide. Further, it is demonstrated that the classical Green’s function is degenerate at an EP. Modified Green’s functions which are valid at EP2 and EP3 are presented.

Distribution Design of Porous Layer Inside Muffler by Boundary Element Analysis

This study proposes an optimization approach to maximize the transmission loss (TL) of a muffler by optimizing the thickness of the embedded porous layer. The objective function, i.e. the TL value, is computed by the four-pole parameters based on the boundary element analysis, and the thicknesses assigned to the discretized elements are naturally chosen as the design variables, resulting in a continuous optimization problem. The sound-absorbing property of the porous layer is simulated using the local admittance boundary condition in the boundary element method (BEM) to link the objective function with the design variable. An efficient sensitivity analysis is developed on the basis of the adjoint variable method and the BEM. Finally, the method of moving asymptotes is applied to solve the optimization problem with the assistance of sensitivity information. As the design variable only affects the (local) admittance value, resulting in a diagonal admittance matrix, and the coefficient matrices of the boundary element analysis will not be influenced by the design variables, we can regard the overall optimization as highly efficient. Moreover, the proposed and previously developed (density-based) optimization approaches are compared. Results show high similarities in the optimization results, thus validating the presented optimization approach. The numerical results also indicate the frequency-dependency of the optimized design. The frequency-averaged TL in the frequency band of interest is selected as the objective function to achieve a multi-frequency optimization.

Distribution Optimization for Acoustic Design of Porous Layer by the Boundary Element Method

In this work, we develop an optimization approach to optimize the distribution of porous material layer inside cavity. The optimization seeks to improve the absorbing effects of the porous material, decreasing the noise level at regions of interest or increasing the sound energy dissipated by the porous material. To achieve the preset optimization aim, two different objective functions are accordingly defined. The acoustic absorption characteristics of porous materials are numerically described using the Delany–Bazley–Miki empirical model and modeled by the admittance boundary conditions in the boundary element analysis. Based on the solid isotropic material with penalization method, an admittance interpolation scheme is established between the element admittance and artificial element density. This transforms the discrete optimization into a continuous optimization problem, which can be solved by a gradient solver with the sensitivity information. As a key treatment in this study, we develop a fast sensitivity analysis approach based on an adjoint variable method and the fast multipole method to calculate the sensitivities of the objective function with respect to a large number of design variables. Finally, we validate the proposed optimization approach through a cabin example.

Sound radiation of a line source moving above an absorbing plane with a frequency-dependent admittance

This paper is concerned with the derivation of an analytical solution for the acoustic pressure field generated by a line source moving at a constant speed and height above an absorbing plane and summarizes the results obtained in a recent article [Dragna and Blanc-Benon, J. Sound Vib. 349, 259-275, 2015]. As an extension of previous studies, the frequency dependence of the admittance is accounted for. First, a Lorentz transformation is used to obtain an equivalent stationary problem. A special attention is paid to the translation of the admittance boundary condition in the Lorentz space. An analytical solution is obtained as a Fourier transform. Excitation of surface waves is investigated depending on the Mach number. In the far field, an asymptotic expression is sought using the modified saddle point method. The solution is expressed under the form of a Weyl-Van der Pol formula, in which the admittance is evaluated at the Doppler frequency. Comparison of the pressure field obtained with the analytical s...

Fourier methods for harmonic scalar waves in general waveguides

A set of semi-analytical techniques based on Fourier analysis is used to solve wave scattering problems in variously shaped waveguides with varying normal admittance boundary conditions. Key components are newly developed conformal mapping methods, wave splitting, Fourier series expansions in eigen-functions to non-normal operators, the building block method or the cascade technique, Dirichlet-to-Neumann operators, and reformulation in terms of stable differential equations for reflection and transmission matrices. For an example the results show good correspondence with a finite element method solution to the same problem in the low and medium frequency domain. The Fourier method complements finite element analysis as a waveguide simulation tool. For inverse engineering involving tuning of straight waveguide parts joining complicated waveguide elements, the Fourier method is an attractive alternative including time aspects. The prime motivation for the Fourier method is its added physical understanding primarily at low frequencies.

Discrete Geometric Formulation of Admittance Boundary Conditions for Frequency Domain Problems Over Tetrahedral Dual Grids

In this work, it is shown how admittance boundary conditions for electromagnetic boundary value problems in the frequency domain can be formulated for the Discrete Geometric Approach. The details are presented for primal grids composed of tetrahedra and barycentric dual grids.

FLUID STRUCTURE INTERACTION AND ADMITTANCE BOUNDARY CONDITIONS: SETUP OF AN ANALYTICAL EXAMPLE

Often, acoustic simulation techniques suffer from errors of the computational model and its parameters. Quantification of the boundary condition is a crucial point for simulations. In particular, the boundary admittance is often unknown and hard to quantify. This article demonstrates how to reduce a fluid-structure interaction model to a pure fluid model with local or nonlocal admittance boundary conditions. Starting point is a BEM formulation for the fluid and a FEM formulation for the structure. An admittance matrix is derived from this formulation. Then, the multidimensional BEM–FEM formulation is adjusted to a one-dimensional example, a duct with structural elements at both ends. Two configurations are investigated, one with local admittance boundary conditions and one with nonlocal admittance boundary conditions which result in a diagonal and in a fully populated admittance matrix, respectively.

Acoustic modelling in view of a determination of the Boltzmann constant within 1 ppm for the redefinition of the kelvin

iMERA/Euromet Project 885 is co-ordinating European effort towards a new determination of the Boltzmann constant kB to within 1 ppm with the aim of redefining the unit of thermodynamic temperature. This project will enable the National Physical Laboratory to perform primary thermometry in the region of −40 °C (Hg) to 156 °C (In) with sub-millikelvin uncertainties by 2012. The chosen technique relies on determining the speed of sound in a monatomic gas. Using the radial acoustic modes of a spherical resonator, consisting of a copper shell and filled with argon or helium, the speed of sound can be measured with great precision and from this measurement the Boltzmann constant can be inferred. This project draws on expertise in dimensional, density, microwave and acoustic measurements at the state-of-the-art. In order to gain further understanding of the experimental configuration a vibro-acoustic model has been developed using the finite element method. Initial calculations were carried out to ensure that predictions of the resonant frequency could be made with the required precision by comparing against an analytical model of a spherical shell filled with a gas. A more elaborate model better representing the experimental configuration was then developed. Thermo-viscous effects close to the fluid-structure boundary were accounted for using a linear acoustic formulation, from which a normal incidence admittance boundary condition was derived and imposed on the inner surface of the resonator. Acoustic pressure, particle velocity and temperature variation as a function of position may be obtained within the gas as a function of frequency. It is therefore possible to investigate how changes in the configuration affect the frequency of radial modes. It is hoped that this approach will shed a better understanding of the underlying complex physical phenomena allowing a minimization of the overall uncertainty.

NONLINEAR APPROACH TO APPROXIMATE ACOUSTIC BOUNDARY ADMITTANCE IN CAVITIES

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.

FEM-based reconstruction of sound pressure field damped by partially absorbing boundary conditions

This paper presents an inverse formulation of the acoustic boundary value problem featuring arbitrary admittance boundary conditions. The problem is discretised by using finite elements to reconstruct the sound pressure field of a cavity based on a preferably small number of measurements. For that, a modal approach is investigated to handle the generally under-determined and ill-conditioned system of equations. The viability of the algorithm is tested to reconstruct sound pressure field inside a two-dimensional sedan passenger compartment.

Transient finite-element analysis using a surface admittance boundary condition to incorporate eddy-current effects

We present a time-domain finite-element formulation for modeling electromagnetic fields in two-dimensional regions containing conductors, dielectrics, and magnetic materials. Such an analysis is needed to determine terminal parameters (equivalent to resistance and inductance per unit length) for multiconductor transmission lines excited with time-varying signals. We use a novel separation of the excitation and response within the formulation and a surface admittance boundary condition to provide a time-domain solution capable of incorporating both proximity-effect and skin-effect eddy currents over a wide range of material parameters. We use a Prony expansion to convert the inherent convolution integral into a recursive expression, resulting in an efficient numerical procedure that can incorporate nonlinear material effects.

Treatment of frequency-dependent admittance boundary conditions in transient acoustic finite/infinite-element models.

The paper addresses the handling of frequency-dependent, local admittance boundary conditions in acoustic transient finite/infinite-element models. The proposed approach avoids the evaluation of a convolution integral along the related boundary. Based on a similar technique developed in an aeroacoustic/finite difference context, the spatially local boundary condition is rewritten in a discrete form that involves normal accelerations and pressure time derivatives at the current time step and few steps before. The incorporation of such a discrete (in time) boundary condition in a finite/infinite-element context is addressed. The infinite-element scheme selected for that purpose relies on the conjugated Astley-Leis formulation. Implementation aspects cover the handling of frequency-dependent boundary conditions along both finite- and infinite-element edges. Numerical examples (waveguide, single source in a half-space bounded by an impedance plane, diffraction by an acoustically treated screen) are presented in order to demonstrate the efficiency of the proposed approach.

A conjugated infinite element method for half-space acoustic problems.

Many acoustic problems (especially in environmental acoustics) involve half-space domains bounded by a plane subjected to normal admittance boundary conditions. In the "low" frequency domain, the numerical treatment of such problems usually relies on boundary element methods based on a particular Green's function suited for the half-(admittance) plane. In the present paper, an alternative hybrid finite/infinite element scheme is proposed. The method relies on a direct treatment of nonhomogeneous boundary conditions along infinite element edges (or faces). The procedure is validated through comparisons with an available reference solution.

Exact surface impedance/admittance boundary conditions for complex geometries: theory and applications

A methodology useful to derive exact and higher order surface impedance/admittance boundary conditions (HOI/ABC's) for complex geometries is presented. It is shown that exact surface boundary conditions are always expressed through dyadic integral operators involving the tangential magnetic and electric fields all over the surface of the body. Quasi-local surface boundary conditions that include curvature effects are shown to be computable through an asymptotic approximation of the integral operators. Finally, an example of a surface admittance boundary condition useful to analyze a structure exhibiting discontinuities along its surface boundary is presented. Practical examples to demonstrate the feasibility of the proposed methodology, as well as the accuracy of the resulting surface boundary conditions are also presented.

Application of the concept of acoustic influence coefficients for the optimization of a vehicle roof

To improve the acoustic behaviour of vehicles is an increasing challenge for every car manufacturer. The sound pressure at the drivers ear due to an arbitrary excitation of the structure can be calculated by a harmonic analysis first of the structure and then of the fluid. Using the concept of influence coefficients one has to carry out the harmonic analysis of the fluid only once to determine the sound pressure at the drivers ear. If the influence coefficients are available one has to solve an easy algebraic relation instead of a full harmonic analysis. This requires that geometry modifications are small compared with the acoustic wave lengths. So, the computational expenditure is mainly confined to the harmonic analysis of the structure. In this paper, the principal way of carrying out an optimization of a vehicle body is presented for the example of a vehicle roof. The parametric geometry based model of the roof is time harmonically excited. All other parts of the body are considered to remain rigid. Admittance boundary conditions are included. Results are presented for different loads and frequency domains, for limited and for unlimited permissible variations of the geometry and for different modal dampings.