Toru Otsuru

NUMERICAL ANALYSIS OF LARGE-SCALE SOUND FIELDS USING ITERATIVE METHODS PART II: APPLICATION OF KRYLOV SUBSPACE METHODS TO FINITE ELEMENT ANALYSIS ∗

The convergence behavior of the Krylov subspace iterative solvers towards the systems with the 3D acoustical BEM is investigated through numerical experiments. The fast multipole BEM, which is an efficient BEM based on the fast multipole method, is used for solving problems with up to about 100,000 DOF. It is verified that the convergence behavior of solvers is much affected by the formulation of the BEM (singular, hypersingular, and Burton-Miller formulation), the complexity of the shape of the problem, and the sound absorption property of the boundaries. In BiCG-like solvers, GPBiCG and BiCGStab2 have more stable convergence than others, and these solvers are useful when solving interior problems in basic singular formulation. When solving exterior problems with greatly complex shape in Burton-Miller formulation, all solvers hardly converge without preconditioning, whereas the convergence behavior is much improved with ILU-type preconditioning. In these cases GMRes is the fastest, whereas CGS is one of the good choices, when taken into account the difficulty of determining the timing of restart for GMRes. As for calculation for rigid thin objects in hypersingular formulation, much more rapid convergence is observed than ordinary interior/exterior problems, especially using BiCG-like solvers.

Application of modified integration rule to time-domain finite-element acoustic simulation of rooms

The applicability of the modified integration rule for time-domain finite-element analysis is tested in sound field analysis of rooms involving rectangular elements, distorted elements, and finite impedance boundary conditions. Dispersion error analysis in three dimensions is conducted to evaluate the dispersion error in time-domain finite-element analysis using eight-node hexahedral elements. The results of analysis confirmed that fourth-order accuracy with respect to dispersion error is obtainable using the Fox-Goodwin method (FG) with a modified integration rule, even for rectangular elements. The stability condition in three-dimensional analysis using the modified integration rule is also presented. Numerical experiments demonstrate that FG with a modified integration rule performs much better than FG with the conventional integration rule for problems with rectangular elements, distorted elements, and with finite impedance boundary conditions. Further, as another advantage, numerical results revealed that the use of modified integration rule engenders faster convergence of the iterative solver than a conventional rule for problems with the same degrees of freedom.

Different Types of Finite Elements

Our contribution concentrates on the numerical solution of acoustic wave problems applying the Finite Element Method (FEM). After a short introduction, we provide a detailed discussion about the concepts of FEM and apply it to the acoustic wave equation. We introduce the standard (Lagrangian) FE basis functions for approximating the continuous acoustic pressure as well as an alternative approach utilizing spline functions. In a second part we focus on two methods to improve the accuracy of the FE solution: (i) increasing the order of the FE basis functions using higher order elements (p–FEM) and (ii) applying spectral FEM to a mixed formulation of the wave equation.

Direct and modal frequency response analysis of sound fields in small rooms by finite element method

In this paper, sound pressures are computed by two techniques using finite element method. One is a technique by solving the system of linear equations directly (direct analysis) and the other is a technique by modal superposition (modal analysis). To confirm the accuracy of the direct analysis, sound pressures obtained by the technique are compared with those obtained by the modal analysis in a room with the volume of 10 m3. Then, as in the modal analysis, two methods are employed: one is a simplified method based on a real eigenvalue problem assuming that the damping matrix, [C], has orthogonality; and another is the method based on complex eigenvalue problem. Those obtained by the direct analysis are in good agreement with those obtained by the two kinds of modal analyses regardless of absorption conditions, even if the analysis is carried out at the frequency close to an eigen frequency. Next, diffuseness of sound field below 315 Hz in a room, which is used in the measurement of ISO140‐3, is investiga...

Application of in-situ method for measuring sound absorption coefficient of direct piercing carved wood panel with daun sireh motif

An application of In-Situ method of measuring sound absorption coefficient on the surface of direct piercing carved wood panel using the concept of ensemble averaged is discussed. The method offer an easier way to measure the absorption performance for each individual aperture of carved wood panel with floral pattern which was replicated from one of the oldest mosque, namely Masjid Abidin located in Terengganu, Malaysia. Two pieces of 20 mm thick of cengal wood (Neobalanocarpus heimii) with 30% and 40% perforation ratio were respectively measured in a reverberation room in order to determine the value of . At lower frequencies (0.1 kHz-1.5 kHz), the measured values of for both direct piercing carved wood panel with floral pattern (Daun Sireh motif) are shown that the sound absoption for both direct piercing carved wood panel are in perform level. From the measurements, clearly, the installation of the direct piercing carved wood panel with floral pattern (Daun Sireh motif) in the Masjid Abidin can provide better air circulation and additional natural sunlight, as well as better sound intelligibility inside the building.

A synthesized boundary condition for wave-based room acoustics simulations using ensemble averaged impedance measured in-situ

The importance of wave-based room acoustics simulation has been increasing in various fields of researches and design-stages. The problem remained, however, is how to model the rooms’ absorptive boundary conditions. It is known that traditional absorption indices like sound absorption coefficient and normal impedance are not always suit for such a fine method like finite element method. Then, the concept and measurement technique of ensemble averaged impedance were presented. In our previous studies, a series of in-situ measurement of ensemble averaged impedances were successfully performed to result excellent reproducibility and repeatability. Herein, the outline of the concept of ensemble averaged impedance including the measurement technique is summarized, first. Second, several example results of practical measurements are given to exhibit that their uncertainties stay within the range for wave-based simulations to keep their resulting uncertainties less than just noticeable difference. Then, a mathem...

Finite element sound field analysis for investigation on measurement mechanism of reverberation time in non-diffuse sound field

As well known, the reverberation time is essential to evaluate room acoustics; however, it is not so easy to predict in a practical room where the assumption of diffuse sound field is not satisfied. On the other hand, the numerical analyses based on the wave equation have been intensively used to explore many kinds of acoustic problems. In this study, 15 kinds of rooms with different absorption surfaces are analyzed by time domain finite element method. This study shows effectiveness of the analysis for investigation on measurement mechanism of the reverberation time of sound field in rooms where the assumption of diffuse sound field is not satisfied. To evaluate a sound field in a room for the measurement, the ratio of incident sound energy to a part of boundary in those to all boundary of a sound field in a room is calculated from results of the time domain finite element analysis. At first, the calculation method of the ratio of incident sound energy to a part of boundary in those to all boundary of a ...

Applications of large-scale finite element sound field analysis onto a music hall using ensemble averaged surface normal impedance

To analyze the sound field in a practical room with complicated boundaries, the authors have developed large-scale finite element sound field analysis in both frequency and time domains. Although the surface normal impedance values of boundaries are required in the modeling process of the analysis, insufficient amount of the impedances are available to date. Then, to provide rather practical boundary conditions for numerical simulations on room acoustics, the authors have also proposed the concept and theoretical background of ensemble averaged surface normal impedance including the fundamental measurement technique. Herein, a brief summary of the finite element sound field analysis is given first. Next, the concept of the ensemble averaged surface normal impedance is explained. Then, several application analyses of a music halls sound frequencies are conducted to show the resulting accuracy of the Large-scale finite element sound field analyisis.

Absolute Diagonal Scaling Preconditioner of COCG Method for Symmetric Complex Matrix

The iterative method, i.e., Conjugate Orthogonal Conjugate Gradient (COCG) method included in the Krylov Subspace methods, can solve a linear system of equations Ax = b. This COCG method is well known to solve by the system with large sparse matrix arises for realistic problems by FEM. The characteristic of the matrix is that the diagonal entries only are complex besides real nondiagonal entries. In this paper, we propose new preconditioner for a matrix with complex diagonal entries only and real nondiagonal entries. We refer to as an absolute diagonal scaling (A-D-s) technique. This method can reduce computation times and amount of memory compared with nondiagonal scaling method.

Comparison of sound fields in rooms computed by finite‐element method and by cone beam method

Sound fields in realistic rooms were simulated by different kinds of computational methods, and comparisons were conducted. One is based on the wave acoustics and the other is based on the geometrical acoustics, i.e., the finite‐element method (FEM) and the conical beam method (CBM). As for the software applied here, the authors have developed their own FEM code, namely LsFE‐SFA, to conduct FEM computations, and they employed commercial software, i.e. raynoise, to conduct CBM computations. First, theoretical basis of LsFE‐SFA was given briefly. Next, sound‐pressure distributions in a reverberation room with the volume of 165 cubic meters were computed by both methods to compare the agreement with measured sound‐pressure levels. The boundary conditions due to the sound absorption were changed to examine the relation between the absorbent condition and the agreements. The agreement between FEM and measurement were found to be better throughout the examination. Next, both FEM and CBM were applied onto the an...