An explicit time-domain finite element method for room acoustics simulations: Comparison of the performance with implicit methods

This paper presents the applicability of an explicit time-domain finite element method (TD-FEM) using a dispersion reduction technique called modified integration rules (MIR) on room acoustics simulations with a frequency-independent finite impedance boundary. First, a dispersion error analysis and a stability analysis are performed to derive the dispersion relation and the stability condition of the present explicit TD-FEM for three-dimensional room acoustics simulations with an infinite impedance boundary. Secondly, the accuracy and efficiency of the explicit TD-FEM are presented by comparing with implicit TD-FEM using MIR through room acoustics simulations in a rectangular room with infinite impedance boundaries. Thirdly, the stability condition of the explicit TD-FEM is investigated numerically in the case with finite impedance boundaries. Finally, the performance of the explicit TD-FEM in room acoustics simulations with finite impedance boundaries is demonstrated in a comparison with the implicit TD-FEM. Although the stability of the present explicit TD-FEM is dependent on the impedance values given at boundaries, the explicit TD-FEM is computationally more efficient than the implicit method from the perspective of computational time for acoustics simulations of a room with larger impedance values at boundaries.     

室内声学时域有限差分模拟中的边界条件

给出了时域有限差分法用于室内声学问题模拟中的边界条件,结合声波方程的基本差分格式,模拟并分析了高斯脉冲在4m×4m房间中的波动过程和脉冲响应;模拟了一9m×6m×4m房间的简正频率,并与经典理论计算值进行了对比;模拟了一12m×5m×4m水平地面房间中的坐席吸声低谷效应,并与Joe LoVtri的模拟结果进行了对比;模拟并实际测量了一10.6m×5.8m×3.4m房间在几个受声点的脉冲响应和早期衰变时间EDT,将模拟结果与实际测量结果进行了对比分析,计算程序是用Metlab语言编写的。模拟与经典理论、相关研究、实际测量几方面的对比分析,验证了本边界条件的可靠性。     

Fundamental accuracy of time domain finite element method for sound-field analysis of rooms

This paper presents an assessment of the accuracy and applicability of a time domain finite element method (TDFEM) for sound-field analysis in architectural space. This TDFEM incorporates several techniques: (1) a hexahedral 27-node isoparametric acoustic element using a spline function; (2) a lumped acoustic dissipation matrix; and (3) Newmark time integration method with an absolute diagonal scaled COCG iterative solver. Sound fields in an irregularly shaped reverberation room of 166 m³ are computed using TDFEM. The computed values and measured values for 125-500 Hz are compared, revealing that the fine structure of the computed band-limited impulse responses agree with measured ones up to 0.1 s, with a cross-correlation coefficient greater than 0.93. The cross-correlation coefficient decreases gradually over time, and more rapidly for higher frequencies. Moreover, the computed decay curves, and the reverberation times, agree well with the respective measured ones, and with a better fit the higher the frequency (up to 500 Hz).     

Source excitation strategies for obtaining impulse responses in finite difference time domain room acoustics simulation

This paper considers source excitation strategies in finite difference time domain room acoustics simulations for auralization purposes. We demonstrate that FDTD simulations can be conducted to obtain impulse responses based on unit impulse excitation, this being the shortest, simplest and most efficiently implemented signal that might be applied. Single, rather than double, precision accuracy simulations might be implemented where memory use is critical but the consequence is a remarkably increased noise floor. Hard source excitation introduces a discontinuity in the simulated acoustic field resulting in a shift of resonant modes from expected values. Additive sources do not introduce such discontinuities, but instead result in a broadband offset across the frequency spectrum. Transparent sources address both of these issues and with unit impulse excitation the calculation of the compensation filters required to implement transparency is also simplified. However, both transparent and additive source excitation demonstrate solution growth problems for a bounded space. Any of these approaches might be used if the consequences are understood and compensated for, however, for room acoustics simulation the hard source is the least favorable due to the fundamental changes it imparts on the underlying geometry. These methods are further tested through the implementation of a directional sound source based on multiple omnidirectional point sources.     

A comparison of measured room acoustics metrics using a spherical microphone array and conventional methods

The traditional microphone configuration used to measure room impulse responses (IRs) according to ISO 3382:2009 is an omnidirectional and figure-8 microphone pair. IRs measurements were taken in a 2500-seat auditorium to determine how the results from a spherical microphone array (an mh acoustics Eigenmike-em32) compare to those from the traditional microphone setup (a Brüel & Kjær Type-4192 omnidirectional microphone and a Sennheiser MKH30 figure-8 microphone). Measurements were obtained at six receiver locations, with three repetitions each in order to first evaluate repeatability. The metrics considered in this study were: reverberation time (T30), early decay time (EDT), clarity index (C80), strength (G), lateral energy fraction (J_LF) and late lateral energy level (L_J). Before calculating these quantities, the IRs were filtered to equalize the frequency response of the microphones and sound source. For the spherical array measurements, the omnidirectional (monopole) and figure-8 (dipole) patterns were extracted using beamforming. In terms of repeatability, the average standard deviation of the three measurements at each receiver location averaged across all metrics, receivers, and octave bands was found to be 0.01 just noticeable differences (JNDs). The analysis comparing the measurements from the two microphone configurations yielded differences which were less than 1 JND for the majority of metrics, with a few exceptions of EDT and C80 slightly above 1 JND. Based on this case study, these results indicate that spherical microphone arrays can be used to obtain valid room IR measurements, which will allow for the development of new metrics utilizing the higher spatial resolution made possible with spherical arrays.     

Determination of the sound energy level of a gunshot and its applications in room acoustics

In some cases an impulsive noise source such as a gunshot can be a preferred alternative when investigating building acoustics, including sound insulation measurements, when compared to conventional steady state noise sources. A gun equipped with blank cartridges is an impulsive noise source that is lightweight and small enough to be easily transported. The differences in the noise characteristics between individual cartridges for the same gun are usually small, so the impulsive source can be replicated to a high degree. This paper is focused on the practical application of the sound exposure levels produced by a gunshot with a known sound energy level in the rooms under investigation. In this way, the equipment and methods required by the conventional method are simplified significantly. Furthermore, reverberation times need not be measured, since the equivalent absorption area can be directly obtained from the measured sound exposure levels. Using Green’s theorem, the roles of the sound source and measuring microphone were exchanged, which simplified the determination of sound insulation as it was easier to change the position of the gun than the microphone. The results obtained using the impulsive noise source were in good agreement with those obtained using the conventional method. Above 100 Hz, their difference in any frequency band of interest was less than 1 dB.     

时域有限差分法在建筑声学中的应用及前景

介绍了声波方程的基本差分格式及稳定条件、数值色散、吸收边界条件等数值计算理论，例举了前人用时域有限差分法对噪声传播过程的模拟和室内声学中座椅吸声低谷效应模拟的模拟结果。本文指出，由于时域有限差分法的特点使其具有在模拟脉冲响应方而的特别优势，因而，应用这一技术研究厅堂的声学特性，尤其是低频特性，将有重要意义。     

A hybrid MLS technique for room impulse response estimation

The measurement of room impulse response (RIR) when there are high background noise levels frequently means one must deal with very low signal-to-noise ratios (SNR). If such is the case, the measurement might yield unreliable results, even when synchronous averaging techniques are used. Furthermore, if there are non-linearities in the apparatus or system time variances, the final SNR can be severely degraded. The test signals used in RIR measurement are often disturbed by non-stationary ambient noise components. A novel approach based on the energy analysis of ambient noise - both in the time and in frequency - was considered. A modified maximum length sequence (MLS) measurement technique, referred to herein as the hybrid MLS technique, was developed for use in room acoustics. The technique consists of reducing the noise energy of the captured sequences before applying the averaging technique in order to improve the overall SNRs and frequency response accuracy. Experiments were conducted under real conditions with different types of underlying ambient noises. Results are shown and discussed. Advantages and disadvantages of the hybrid MLS technique over standard MLS technique are evaluated and discussed. Our findings show that the new technique leads to a significant increase in the overall SNR.     

An alternative method for plotting dispersion curves

Solving the frequency equation and plotting the dispersion curves in problems of wave propagation in cylinders and plates, particularly when the material is anisotropic, are complicated tasks. The traditional numerical methods are usually based on determination of the zeros of the frequency equation by using an iterative find-root algorithm. In this paper, an alternative method is proposed which extracts the solution of the frequency equation in the form of dispersion curves from the three-dimensional illustration of the frequency equation. For this purpose, a three-dimensional representation of the real roots of the frequency equation is first plotted. The dispersion curves, which are the numerical solutions of the frequency equation, are then obtained by a suitable cut in the velocity-frequency plane. The advantages of this method include simplicity, high speed, low possibility of numerical error, and presentation of the results in a graphical form that promotes ease of interpretation. This method is not directly applicable to problems which incorporate high damping or leaky waves. However, if the damping is not very high, it could be a good estimate of the true dispersion curves.     

Microstructure holey fibers as wideband dispersion compensating media for high speed transmission system

This paper presents a dispersion compensating microstructure holey fiber for wideband transmission system. The finite element method with perfectly matched absorbing layers boundary condition is used to investigate the guiding properties. According to simulation, negative dispersion coefficient of −1455 ps/(nm km) and a relative dispersion slope (RDS) close to that of single mode fiber of about 0.0036 nm⁻¹ is obtained at 1.55 μm. The variation of structural parameters is also studied to evaluate the tolerance of the fabrication. The proposed module can be used in 40 Gb/s dense wavelength division multiplexing (DWDM) systems in optical fiber communication networks.     

Guided waves dispersion equations for orthotropic multilayered pipes solved using standard finite elements code

The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0, n), torsional T(0, n) and flexural F(m, n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method.     

Analytical derivation of a formula for the reduction of computation time by the voxel crossing technique used in room acoustical simulation

Computation times of room acoustical simulation algorithms still suffer from the time consuming search for ray-wall-intersections. Spatial subdivision may speed up ray tracing considerably. For room acoustics, where the number of surface polygons (walls) is not so high, the voxel technique appears suitable. The voxel crossing algorithm is very fast. However, its performance was not yet investigated up to now. Voxels are small cubes by which the space is subdivided periodically. The advantage: Only in the rare case a voxel intersects a wall the intersection point needs to be computed. In this paper, by estimating the probabilities of such intersections, an analytical formula is derived, by which the optimum degree of spatial subdivision and the factor of acceleration of the algorithm can be forecasted. It turns out that the computation time increases only with instead of with K₀ (the number of polygons of the room). Thus, on a modern PC, computation time for a full room acoustical simulation even for highly complicated rooms may be reduced by a factor in the order of 100, i.e. to a few seconds.     

Combined wave and ray based room acoustic simulations of audio systems in car passenger compartments,Part I: Boundary and source data

The present series of papers summarizes the results of a three-year research project on the realistic simulation of car audio sound in car passenger compartments using a combined Finite Element (FE) and Geometrical Acoustics (GA) approach. The simulations are conducted for the whole audible frequency range with the loudspeakers of the car audio system as the sound sources. The challenges faced during the project relate to fundamental questions regarding the realistic sound field simulation in small enclosures with strong modal and diffraction effects.The paper denoted here as Part I focuses on boundary and source representations in the FE and GA domain and suggests guidelines for a best-possible acquisition of the required data. Since a straight-forward determination of the boundary and source characteristics is mostly hampered by the immense complexity and inhomogeneity of the materials and loudspeaker configurations inside a car compartment, different measurement and calculation methods have been applied to determine the required data and quantify the corresponding uncertainty. The paper clearly points out the strength and weaknesses of the applied methods depending on the considered frequency range and material characteristics. In order to keep the complexity of the FE simulations at a manageable level, all passive boundaries were considered as locally reacting with impedance conditions.Part II of the study applies the obtained data in combined FE-GA room acoustic simulations and compares the simulated room impulse responses (RIR) with corresponding measurement results. In a final step the observed differences in the RIRs are related to the uncertainty and inherent errors in the boundary and source representation.     

Combined wave and ray based room acoustic simulations of audio systems in car passenger compartments,Part II: Comparison of simulations and measurements

The present series of papers summarizes the results of a three-year research project on the realistic simulation of sound fields in car passenger compartments using a combined Finite Element (FE) and Geometrical Acoustics (GA) approach. The simulations are conducted for the whole audible frequency range with the loudspeakers of the car audio system as the sound sources. The challenges faced during the project relate to fundamental questions regarding the realistic sound field simulation in small enclosures with strong modal and diffraction effects. While Part I of this series of papers focusses on the determination of the boundary and source conditions for the simulation model of the car compartment, the present paper, denoted here as Part II, presents extensive objective and subjective comparisons of the corresponding room acoustic measurement and simulation results.By applying the FE method to the low frequency part of the room transfer function (RTF) the study aims at the quantification of potential objective and subjective benefits with regard to the simulation quality in small rooms, when compared to a purely geometrical acoustics approach. The main challenges and limitations in the simulation domain are due to the very small volume, the difficult to determine source and boundary conditions and the considerable diffraction effects (especially at the seats) in the car passenger compartments. In order to keep the complexity of the FE simulations at a manageable level, all boundary conditions were described by acoustic surface impedances and no fluid-structural coupling was considered in the FE simulation model.While the results of the study reveal that an overall good agreement regarding the energy distribution in time and frequency domain is generally possible even in such complex enclosures, the results also clearly show the limitations of the impedance boundary approach in the FE domain as well as the strong sensitivity of the simulation results with regard to the uncertainty in the boundary and source conditions in both simulation domains. It can thus be concluded, that possible fields of application of the FE extension in room acoustic simulations lie in the prediction of the modally dominated low frequency part of the RTF of well defined rooms and in the prediction of sound fields that are strongly affected by near-field or diffraction effects as in the car passenger compartment. However, due to the considerable problems in the determination of realistic boundary conditions for the FE model, improved measurement techniques are urgently needed to further improve the overall simulation quality.     

Analysis of mode shapes of a rigidly backed cylindrical foam using three-dimensional finite element acoustical analysis

In this study, the three-dimensional finite element frequency domain acoustical analysis is used to determine the modal shapes of cylindrical foam with a rigid backing and subjected to a unit normal incidence impulsive sound pressure loading while placed in the impedance tube. The acoustic results predicted for the foam are validated by data from the two-microphone acoustic measurements, and good agreement between the measured and predicted acoustic results is observed. The mode shapes of the incident face of the foam at a low frequency, resonant and anti-resonant frequencies as well as the frequency that occurring the peak loss modulus are illustrated. It is found that the modal behaviors of the cylindrical foam are dominated by the fluid, although the acoustic properties of the cylindrical foam are also influenced by the circumferential edge constraints and the modal movements of the solid skeleton.     

Design and characterization of single mode circular photonic crystal fiber for broadband dispersion compensation

In this paper, we present a single mode circular photonic crystal fiber (C-PCF) for broadband dispersion compensation covering 1400 to 1610 nm wavelength band over the telecommunication windows. Investigations of guiding properties are carried out using finite element method (FEM) with circular perfectly matched layer boundary condition. Numerical study reveals that a negative dispersion coefficient of about −386.57 to −971.44 ps/(nm km) is possible to obtain over the wavelength ranging from 1400 to 1610 nm with a relative dispersion slope (RDS) of about 0.0036 nm⁻¹ at 1550 nm wavelength. In addition, the single mode behaviour of C-PCF is demonstrated by employing V parameter. According to simulation, it is found that the proposed C-PCF acts as a single mode fiber within 1340 to 1640 nm wavelength. Moreover, effective dispersion, relative dispersion slope, birefringence and confinement loss are also presented and discussed.     

A note on the proper frequency resolution for the room transfer function in the phased beam tracing method

In the simulation of acoustic response of a room, one cannot satisfactorily estimate the transfer function or the impulse response with a sparse frequency resolution. We have studied about the frequency resolution for a satisfactory estimation of room transfer function or the reverberation time using the phased beam tracing method. In order to investigate what happens with the frequency resolution, a number of frequency resolution (0.01-1 Hz), which are usually employed in the numerical analysis, was tested for a room model. It was found that the total accumulated phase in a transfer function and the late part of an impulse response of a room are influenced by the frequency resolution. Main reason for the difference is the correct detection of non-minimum phase zeros, depending on the frequency resolution. A criterion for a proper frequency resolution was suggested by considering the modal density of three-dimensional space. When a sparse dataset was initially given, we showed that cubic spline interpolation can be used to enhance the precision of detection for non-minimum phase zeros.     

Time domain finite volume method for three-dimensional structural–acoustic coupling analysis

This work extends the application of finite volume method (FVM) to structural–acoustic problems. A three-dimensional time domain FVM (TDFVM) is proposed to predict the transient response and natural characteristics of structural–acoustic coupling systems. Acoustic wave equation in heterogeneous medium and structural dynamic equation are solved in fluid and solid sub-domains respectively. The structural–acoustic coupling is implemented according to normal components of particle acceleration continuity condition and normal traction equilibrium condition at the interface. The computational domain is discretized with four-node tetrahedral grid which is generated easily and has strong adaptability to complicated geometries. Numerical experiments are carried out to examine the accuracy of the method in both time domain and frequency domain. The results show good agreement with analytical solutions and numerical results. For structural–acoustic problem, TDFVM has the capability to consider the heterogeneity of both fluid and solid.     

A non-conforming monolithic finite element method for problems of coupled mechanics

In this study, a Lagrange multiplier technique is developed to solve problems of coupled mechanics and is applied to the case of a Newtonian fluid coupled to a quasi-static hyperelastic solid. Based on theoretical developments in [57], an additional Lagrange multiplier is used to weakly impose displacement/velocity continuity as well as equal, but opposite, force. Through this approach, both mesh conformity and kinematic variable interpolation may be selected independently within each mechanical body, allowing for the selection of grid size and interpolation most appropriate for the underlying physics. In addition, the transfer of mechanical energy in the coupled system is proven to be conserved. The fidelity of the technique for coupled fluid–solid mechanics is demonstrated through a series of numerical experiments which examine the construction of the Lagrange multiplier space, stability of the scheme, and show optimal convergence rates. The benefits of non-conformity in multi-physics problems is also highlighted. Finally, the method is applied to a simplified elliptical model of the cardiac left ventricle.     

The simulation of acoustic radiation from a moving line source with variable speed

The purpose of this work is to simulate and investigate the sound field generated by a moving line source with finite length and variable speed. Expect for the variation of the acoustic pressure at the specific field point, the distribution of the surface pressure along the surface of the line source was also considered. For achieving this purpose, a numerical method which combines the Time Domain Boundary Element Method (TDBEM) and moving sound source theory was developed in the present work. After comparing the results with the constant and the variable speed case, it showed that the effect of the variable speed not only influenced the variation rate of the frequency modulation, i.e., Doppler effect, but also the time about the maximum acoustic pressure being observed. In addition, the simulation results also presented that the difference as to the amplitude variation of the acoustic pressure still existed between the moving case and the stationary case even if the length of the line source is very long.