Complex Impedance Boundary Research Articles

Approximate acoustic boundary conditions in the time-domain using volume penalization.

This paper presents an immersed boundary method for modeling complex impedance boundary conditions in wave-based finite-difference time-domain simulations. The fully parallelizable and physically motivated Brinkman method allows for the representation of complex geometries on simple Cartesian grids as porous material by introducing a friction term and an effective volume. The parameters are specified using blending functions, enabling impedance boundary conditions without the need for grid fitting or special boundary treatment. Representative acoustic configurations are analyzed to assess the method. In detail, acoustic materials on and in front of a rigid wall, a reacting surface as well as fully reflecting walls are examined. Comparison with analytical solutions shows satisfactory agreement of the resulting impedances in the range from 20 Hz up to 4 kHz. The method is derived for the (non-)linear Euler equations and the acoustic wave equation. An extensive stability analysis is carried out.

On Computing Impulse Responses from Frequency-Domain Finite Element Solutions

In acoustics, knowledge of the impulse response of a system is often important. While impulse responses may be measured, they may also be predicted using numerical methods. This work considers the generation of impulse responses from frequency-domain finite element solutions. It is shown that these impulse responses, obtained by inverse Fourier transformation, are noncausal. Through error analysis, it is demonstrated that the noncausality can be reduced by increasing the duration of a source signal used to excite a simulated system. It is found that increasing the source signal duration increases the dispersion-related phase error present in the simulated impulse responses. The findings of this study are used to simulate an impulse response for a system with a nonuniform, frequency-dependent, complex impedance boundary condition.

An accurate treatment of non-homogeneous boundary conditions for development of the BEM

This paper proposes an enhancement of the treatment of non-homogeneous boundary conditions to improve the boundary element method (BEM) formulation. The standard formulation is modified by introducing the boundary conditions in the integral kernels. The boundary conditions are implicitly defined through known parameters depending on the geometry, rather than by prescribing nodal values as is done in the standard formulation. The main advantage of this procedure is that the right-hand side of the system of equations is integrated taking the exact distribution of loads into account. This approach is implemented in the Bézier–Bernstein space to yield a geometry-independent field approximation. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The application of the proposed method covers the resolution of complex boundary value problems as optimization with uncertain data, material modelling with graded impedance, and the definition of general boundary constraints. The performance of the proposed method is shown by solving the Helmholtz equation in two dimensions. The proposed method is numerically compared to the standard BEM formulation in two benchmark problems. Finally, the application of complex impedance boundary conditions is analysed in a numerical example.

Numerical Investigation of Second-Mode Attenuation over Carbon/Carbon Porous Surfaces

Axisymmetric direct numerical simulations of a spatially developing hypersonic boundary layer over a sharp 7 deg half-angle cone at at Reynolds numbers , , and have been carried out. The streamwise extent of the simulated domain is 0.9 m. The boundary layer is excited with the one-time application of a broadband pulse that is advected downstream over an impermeable wall and a complex impedance boundary condition (IBC) modeling the acoustic response of carbon/carbon (C/C) ultrasonically absorptive porous surfaces. The complex IBCs are derived as an output of a pore-scale aeroacoustic analysis: the inverse Helmholtz solver (IHS) and algebraic low-order models. The IHS estimate of the C/C impedance is compared to other methods for porous acoustic absorbers, revealing uncertainties in the applicability of analytical models for porous absorbers. The introduction of the IBCs in all cases leads to a significant attenuation of the instability waves, up to one order of magnitude reduction in pressure perturbation amplitude for the case.

Numerical and experimental study on the reduction of refrigerant pressure pulsation within compressor pipes

The pulsation of compressor pipe refrigerant pressure that occurs during the suction and discharge processes of a compressor is a major cause of vibration and noise. This makes it important to be able to predict the pressure pulsation within the pipe using numerical analysis, leading to a design that can reduce it. In this study, a method is presented to obtain the complex propagation speed and impedance boundary conditions of the refrigerant, which are necessary for the numerical analysis, by using the least square estimation and experimental data. Experimental verification was carried out by installing a total of 14 pressure sensors at 100 mm intervals on the compressor suction pipe connected to the refrigerant supply equipment. To verify the numerical analysis model the pressure pulsation was measured by varying the compressor operating frequency. Also, a pressure-structure interaction analysis was used to investigate whether an elastic structure installed within the pipe would affect any reduction in the pressure pulsation. The results showed that installation of the elastic structure within the pipe resulted in a significant pulsation reduction effect for the first operating frequency, while a slight reduction effect was observed for the second operating frequency. It was found that the pressure pulsation reduction was most significant when the longitudinal natural frequency of the installed elastic structure matches the operating frequency of the compressor.

Numerical Method to Calculate Low-Frequency Sound Field in Non-anechoic Pools

A numerical method of low-frequency sound field calculation in non-anechoic pools with impedance boundary based on finite element software ACTRAN is proposed in this paper. A virtual Kundt’s tube is established in ACTRAN to obtain the normal incidence surface impedance of non-anechoic pools’ walls. The normal incidence surface impedance is then added to the surface of fluid domain to simulate the sound field boundary of non-anechoic pools. The numerical results are obtained by computing the model established in ACTRAN. The sound field of a spherical sound source has been tested in a glass tank to verify the accuracy of the proposed numerical method. As a comparison, the sound field in the glass tank with absolutely soft boundary is also calculated. It is concluded that the proposed method can effectively predict the low-frequency sound field in non-anechoic pools with arbitrary material impedance boundary and will provide a reference analysis method for the study of the low-frequency sound field characteristic in non-anechoic pools with complex impedance boundary.

Formant frequencies and bandwidths of the vocal tract transfer function are affected by the mechanical impedance of the vocal tract wall.

The acoustical properties of the vocal tract, the air-filled cavity between the vocal folds and the mouth opening, are determined by its individual geometry, the physical properties of the air and of its boundaries. In this article, we address the necessity of complex impedance boundary conditions at the mouth opening and at the border of the acoustical domain inside the human vocal tract. Using finite element models based on MRI data for spoken and sung vowels /a/, /i/ and // and comparison of the transfer characteristics by analysis of acoustical data using an inverse filtering method, the global wall impedance showed a frequency-dependent behaviour and depends on the produced vowel and therefore on the individual vocal tract geometry. The values of the normalised inertial component (represented by the imaginary part of the impedance) ranged from 250,hbox {g}/hbox {m}^{2} at frequencies higher than about 3 kHz up to about 2.5times 10^{5},hbox {g}/hbox {m}^{2} in the mid-frequency range around 1.5–3 kHz. In contrast, the normalised dissipation (represented by the real part of the impedance) ranged from 65 to 4.5times 10^{5},hbox {Ns}/hbox {m}^{3}. These results indicate that structures enclosing the vocal tract (e.g. oral and pharyngeal mucosa and muscle tissues), especially their mechanical properties, influence the transfer of the acoustical energy and the position and bandwidth of the formant frequencies. It implies that the timbre characteristics of vowel sounds are likely to be tuned by specific control of relaxation and strain of the surrounding structures of the vocal tract.

HIFU scattering by the ribs: constrained optimisation with a complex surface impedance boundary condition

High intensity focused ultrasound (HIFU) enables highly localised, non-invasive tissue ablation and its efficacy has been demonstrated in the treatment of a range of cancers, including those of the kidney, prostate and breast. HIFU offers the ability to treat deep-seated tumours locally, and potentially bears fewer side effects than more established treatment modalities such as resection, chemotherapy and ionising radiation. There remains however a number of significant challenges which currently hinder its widespread clinical application. One of these challenges is the need to transmit sufficient energy through the ribcage to ablate tissue at the required foci whilst minimising the formation of side lobes and sparing healthy tissue. Ribs both absorb and reflect ultrasound strongly. This sometimes results in overheating of bone and overlying tissue during treatment, leading to skin burns. Successful treatment of a patient with tumours in the upper abdomen therefore requires a thorough understanding of the way acoustic and thermal energy is deposited. Previously, a boundary element (BE) approach based on a Generalised Minimal Residual (GMRES) implementation of the Burton-Miller formulation was developed to predict the field of a multi-element HIFU array scattered by human ribs, the topology of which was obtained from CT scan data [1]. Dissipative mechanisms inside the propagating medium have since been implemented, together with a complex surface impedance condition at the surface of the ribs. A reformulation of the boundary element equations as a constrained optimisation problem was carried out to determine the complex surface velocities of a multi-element HIFU array which generated the acoustic pressure field that best fitted a required acoustic pressure distribution in a least-squares sense. This was done whilst ensuring that an acoustic dose rate parameter at the surface of the ribs was kept below a specified threshold. The methodology was tested at an excitation frequency of 1 MHz on a spherical multi-element array in the presence of anatomical ribs.

Scattering of high-intensity focused ultrasound by the ribs: Constrained optimization with a complex surface impedance boundary condition

One of the challenges of trans-rib high-intensity focused ultrasound (HIFU) treatment is the need to transmit sufficient energy through the ribcage to ablate tissue whilst minimizing the formation of side lobes, and sparing healthy tissue. Ribs strongly absorb and reflect ultrasound. This may result in overheating of bone and overlying tissue during treatment, leading to skin burns. Successful treatment of a patient with tumors of the liver or the pancreas therefore requires a thorough understanding of the way acoustic and thermal energy is deposited. A boundary element approach was developed to predict the field of a multi-element HIFU array scattered by human ribs in 3D. This forward model has been reformulated as a constrained least squares optimization problem where the velocity of each individual element on the array is an optimization variable. A locally reacting complex surface impedance condition at the surface of the ribs has been implemented. The methodology has been tested at an excitation frequency of 1 MHz on a spherical section multi-element array and a total of six array-rib configurations have been investigated. The results were compared against other focusing and rib-sparing approaches, demonstrating the efficacy and flexibility of the constrained optimization approach.

Dielectric characteristics of Si-added and Si-doped TbMnO3

Polycrystalline Si-added TbMnSixO3 (x=0.05, 0.1) and Si-doped Tb1−xMnSixO3, TbMn1−xSixO3 (x=0.05, 0.1) samples were fabricated by a conventional solid-state reaction. The dielectric properties of ceramics have been investigated in a temperature range of 77–350K and a frequency range of 0.1−200kHz. Especially, an extraordinarily high low-frequency dielectric constant (~105) at room temperature was observed in TbMnSi0.1O3, and the values of ε′(T) at other frequencies can reach around 104. It turns out that the addition of element Si changes dielectric properties of TbMnO3 remarkably. Through the measured data of complex impedance, bulk contribution and grain boundary effects to electrical response are identified by the analysis of complex plane diagrams (Nyquist diagram).

연소시스템의 열음향 불안정 예측을 위한 Helmholtz Solver 개발

본 연구에서는 실제 로켓엔진 및 가스터빈용 연소기 내부의 열음향 불안정을 효과적으로 예측하기 위하여, 헬름홀츠 방정식과 시간지연모델을 이용한 3차원 유한요소법 해석코드를 개발하였다. 연소응답항에 의해 수치적으로 야기되는 비선형성은 반복법으로 선형화 하였으며, Arnoldi 방법을 사용하여 대용량 고유치 문제를 해석하였다. 해석결과인 복소각주파수와 음향 압력장을 통해 각 음향모드의 공진주파수, 진폭의 증폭/감쇠 여부 그리고 모드 형태를 예측할 수 있다. 이론해가 존재하는 두 가지 문제를 통해 출구 임피던스와 예혼합 화염이 종 방향 음향장에 미치는 영향에 대한 예측 정확도를 평가하였으며, 배플 유무에 따른 횡 방향 음향 모드의 주파수 변이를 상온 음향시험 결과와 비교/검증하였다. In order to effectively predict thermo-acoustic instability within real combustors of rocket engines and gas turbines, in the present study, the Helmholtz equation in conjunction with the time lag hypothesis is discretized by the finite element method on three-dimensional hybrid unstructured mesh. Numerical nonlinearity caused by the combustion response term is linearized by an iterative method, and the large-scale eigenvalue problem is solved by the Arnoldi method available in the ARPACK. As a consequence, the final solution of complex valued eigenfrequency and acoustic pressure field can be interpreted as resonant frequency, growth rate, and modal shape for acoustic modes of interest. The predictive capabilities of the present method have been validated against two academic problems with complex impedance boundary and premixed flame, as well as an ambient acoustic test for liquid rocket combustion chamber with/without baffle.

A new hybrid passive/active cell to realize a complex impedance boundary condition

Hybrid passive/active cells previously developed at the LMFA have proved their efficiency for global noise reduction. A hybrid cell combines passive absorbent properties of a porous layer and active control at its rear face. Active control is mainly used to increase absorption at low frequencies by cancelling the imaginary part of the surface impedance presented by the absorber. A specific decentralized control algorithm allowed realizing large panels to reduce multi-tone noise radiated by a flow duct. Generally, the design of such absorbers starts by determining the optimal impedance for the targeted application, defined as that which, for each frequency, produces the highest reduction. However, in many cases results show that this optimal impedance is complex and that both real and imaginary parts are frequency dependent. This paper investigates the potential of a new hybrid absorber intended to realize any of impedance. The new cell uses one microphone on each side of a resistive cloth. Normal velocity can then be deduced by a simple pressure difference, which allows an easy synthesis of the error signal. First experiments carried out in a standing wave tube are described and different control strategies are examined, allowing us to validate this promising concept.

Benchmark cases for outdoor sound propagation models

The computational tools available for prediction of sound propagation through the atmosphere have increased dramatically during the past decade. The numerical techniques include analytical solutions for selected index of refraction profiles, ray trace techniques which include interaction with a complex impedance boundary, a Gaussian beam ray trace algorithm, and more sophisticated approximate solutions to the full wave equation; the fast field program (FFP) and the parabolic equation (PE) solutions. This large array of computational approaches raises questions concerning under what conditions the various approaches are reliable and concerns about possible errors in specific implementations. This paper presents comparisons of predictions from the several models assuming a complex impedance ground and four atmospheric conditions. For the cases studied, it was found that the FFP and PE algorithms agree to within numerical accuracy over the full range of conditions and agree with the analytical solutions where available. Comparisons to ray solutions define regimes where ray approaches can be used. There is no attempt to compare calculated transmission losses to measurements.

Status of numerical methods and the experimental database

Propagation of sound in the atmosphere is governed by a number of interacting physical mechanisms including geometrical spreading, molecular absorption, reflection from a complex impedance boundary, refraction, diffraction, and scattering. Accurate predictions of sound transmission loss from a source to a receiver must somehow account for all of these phenomena simultaneously. Although this goal is beyond current capabilities, the application of numerical techniques to solve the full wave equation has led to significant advances. Current implementations agree well with experiments for low turbulence and flat ground. At moderate ranges, even the effects of turbulence can be accounted for. Members of NATO Panel 3 Research Study Group 11 began an effort in 1990 to compare predictions of the various models and different implementations of the same model with the goal of identifying any differences and sources of those differences and to provide benchmark cases which future researchers can use to check new models. Further, the amount of well-documented experimental data to which the numerical solutions can be compared is very limited. During the summer of 1991, a Joint Acoustic Propagation Experiment (JAPE) was organized in order to extend the database. This paper will summarize the benchmark effort, review the available experimental data and comparison to theory prior to the JAPE, and finally, summarize the JAPE.

Scattering of sound by atmospheric turbulence: A numerical simulation above a complex impedance boundary

A computer simulation of the effect of small-scale turbulence on atmospheric sound propagation over a complex impedance boundary is developed. The atmospheric turbulence is modeled as an ensemble of spherically symmetric eddies with Gaussian refractive profiles. Single scatter is assumed and a closed form of the first Born approximation for scattering is obtained, giving each eddy’s contribution to the total fluctuation of the sound pressure at a receiver downrange. The numerical simulation is accomplished by repeated calculations for many ‘‘realizations,’’ or snapshots of the turbulent medium. Each eddy’s scatter contribution is added up coherently for a particular configuration of eddies, yielding that realization’s total sound pressure. The eddies are then given a random change in their coordinates. The total sound pressure is calculated for this realization, and the process repeated. Better agreement with the analytical results of Karavainikov [Akust. Zh. 3, 175–186 (1957)] for an unbounded medium is obtained at the higher frequencies with the use of the first Rytov approximation to scattering. A complex impedance boundary is added and the predictions of the standard deviations of the amplitude fluctuations, amplitude probability distributions, and structure functions are then tested against experimental data. Good agreement is found whenever the average sound pressure level was well above the background noise level.

Scattering of sound by atmospheric turbulence

A computer simulation of the effect of small scale turbulence on atmospheric sound propagation over a complex impedance boundary is developed. The atmosphere is broken up into spherically symmetric eddies characterized by a Gaussian profile. Single scatter is assumed and a closed form of the first Born approximation for scattering is obtained, giving each eddie's contribution to the total fluctuation of the sound pressure at a receiver downrange. The numerical simulation was accomplished with the concept of a “realization,” or snapshot of the turbulent medium. Each eddie's scatter contribution was added up for a particular configuration of eddies, giving that realization's total sound pressure fluctuation. The eddies were then given a random change in their coordinates. The total sound pressure was calculated for this realization, and the process repeated. A complex impedance boundary was added and the predictions of the standard deviations of the amplitude fluctuations, amplitude probability distributions, and structure functions were then tested against experimental data. Good agreement was found whenever the average intensity of the fluctuations was well above the background noise level. [Work supported by U. S. Army Construction Engineering Research Laboratory.]

Impedance formulation of the fast field program for acoustic wave propagation in the atmosphere

In an earlier paper, the present authors’ work in adapting the fast field program (FFP) formulation to atmospheric propagation above a complex impedance boundary was described. It was found that numerical overflow problems for high frequencies and multiple layers limited the utility of the FFP in solving atmospheric problems. In this paper is a description of a new formulation which eliminates the overflow problems inherent in the earlier formulation. The results of these two formulations are compared for a test case and the superiority of the new formulation is demonstrated. Results of the FFP2 for a simple atmospheric profile are compared with field measurements and the applicability discussed.

Approximate methods to describe the reflections from cylinders and spheres with complex impedance

We have studied the scattering of plane waves from cylinders and spheres with complex impedance boundary conditions using the Kirchhoff method and the Luneburg–Kline method. A stationary phase approximation is made in the Kirchhoff derivation to eliminate spurious contributions from the boundary of the insonified region. Terms of order up to (ka)−2 are retained. Dependence of the cross section on the complex impedance, the incident wave frequency, as well as on the azimuthal angle is illustrated. The curves are found to obey several near symmetries. Further, the Luneburg–Kline method is found to lead to predictions which are often qualitatively different from those of the Kirchhoff method, and the former is observed to be more reliable. Convergence of the Luneburg–Kline terms is studied and is found to be quite satisfactory.

Propagation of a spherical wave near a plane boundary with a complex impedance

An analysis, using complex angles of incidence as the variable in a steepest descents integration, is made of the propagation of sound from a point source over a locally reacting complex impedance boundary. The integration process contains more terms than are normally used in the analysis, and the solution is valid over a large range of horizontal distances. Conditions for the existence of surface waves are discussed in some detail. A comparison is made of measured and calculated values of pressure on the boundary. Subject Classification: [43]28.40; [43]20.15, [43]20.30.

Sound propagation over a complex impedance boundary

An expression is developed for the propagation of an acoustical wave over a boundary of complex impedance, e.g., the ground. The total field produced may be considered to be the sum of the direct wave from the source, a specularly reflected component from a perfect reflector, a modification produced by the departure of the boundary from infinite impedance, and a possible surface wave. The analysis involves splitting the incident spherical wave into a set of cylindrical waves with various angles of incidence and integrating these waves together with their reflected counterparts over the complex angle plane. The encirclement of a pole determines whether a surface wave exists. Calculations of the total acoustical field for different frequencies are made ewer a wide horizontal range, and its dependence upon height above the boundary of both the source and the receiver, and the impedance of the boundary, is discussed. Height above the boundary and the impedance of the boundary also affect the existence of the surface wave. Comparison with experimental results is made.