变系数瞬态热传导问题边界元格式的特征正交分解降阶方法

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法,重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程,建立了变系数瞬态热传导问题边界元格式的POD降阶模型,并用常数边界条件下建立的瞬态热传导问题的POD降阶模态,对光滑时变边界条件瞬态热传导问题进行降阶分析.首先,对一个变系数瞬态热传导问题,建立其边界域积分方程,并将域积分转换成边界积分;其次,离散并重组积分方程,获得可用于降阶分析的矩阵形式的时间微分方程组;最后,用POD模态矩阵对该时间微分方程组进行降阶处理,建立降阶模型并对其求解.数值算例验证了本文方法的正确性和有效性.研究表明:1)常数边界条件下建立的低阶POD模态矩阵,能够用来准确预测复杂光滑时变边界条件下的温度场结果;2)低阶模型的建立,解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.     

Closed form solutions for the acoustical impulse response over a masslike or an absorbing plane

The transient sound field caused by a Dirac delta impulse function above an infinite locally reacting plane can be calculated by applying the inverse Fourier transform of the corresponding half-space Green's function in frequency domain. As a starting point, the representation given by Ochmann [J. Acoust. Soc. Am. 116(6), 3304-3311 (2004)] is used, which consists of discrete and continuous superposition of point sources. For a locally reacting plane with masslike character and also with pure absorbing behavior, it is possible to express the resulting impulse response in closed form. Such a result is surprising, because corresponding formulations in the frequency domain are not available yet. Hence, the first main result is the closed form solution Eq. (22) for an impulse response over an infinite plane with a pure imaginary impedance. The second main result is the closed form solution Eq. (53) for an impulse response over an infinite plane with a pure real impedance. As a particular application of both main results, a convolution technique is used for deriving formulas for point sources with a general time dependency. For special signals like an exponentially decaying time signal or a triangular shaped impulse, the resulting sound field can be presented in terms of elementary functions.     

Assessing the effect of a barrier between two rooms subjected to low frequency sound using the boundary element method

The boundary element method (BEM) has been used to compute the acoustic wave propagation through a single vertical panel, which separates two rooms, made of concrete, when one of the rooms is excited by a steady-state, spatially sinusoidal, harmonic line load pressure at low frequencies. This work focuses on how the connection of the panel to the ceiling affects the acoustic insulation provided by the wall. Perfect double-fixed partitions and acoustic barrier-type structures with differently-sized gaps between the ceiling and the barrier are studied. The BEM model is formulated in the frequency domain and takes the air-solid interaction fully into account. Insulation dips are localised in the frequency domain and identified with dips associated with both the wall's natural dynamic vibration modes and with those associated with the air in the rooms. The influence of the wall's thickness on acoustic insulation is also analysed. The computed results obtained with the acoustic barrier type structure are compared with those obtained by a rigid model. The importance of the rooms' surface conditions is assessed, modelling the rooms with cork.     

Vibrations of oblique shear-deformable plates

The proposed numerical analysis of moderately thick plates subject to rather general boundary conditions is based on the direct boundary element method (BEM) in the frequency domain. First order shear-deformation theory of the Reissner-Mindlin-type is considered. A step forward in efficiency is obtained when the force and double force with moment Green's functions of the rectangular simply supported base plate of the same stiffness are applied. The time-reduced equations of hard-hinged polygonal plates correspond to those of a background Kirchhoff plate having frequency-dependent effective parameters like mass, lateral and in-plane load, and is further forced by imposed fictitious curvatures. This analogy holds even for the quasi-static shear forces and bending moments, i.e., when inertia effects become negligible. Furthermore, it can be shown that, in the static case, these stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting from the Kirchhoff theory in the background. Since this analogy is restricted to hard-hinged supports of straight edges, it is necessary to apply, e.g., the direct BEM of analysis to the plate of general planform and boundary conditions. The main effort is thus to study the properties and effective representations of the Green's dyadics and their singularities, in view of their proper integration. Similarly as for Kirchhoff plates, the strong singularity of the infinite domain is identified for the rectangular plate and subject to indirect integration. The resulting direct BEM proves to be efficient, robust and, in connection with proper pre- and post-processors, becomes an effective tool of engineering analyses just within the frequency limits given by the first two of the three spectral branches.     

Evaluation of mutual radiation impedances between circular pistons by impulse response and asymptotic methods

A general approach is presented to evaluate the mutual radiation impedance between circular pistons of arbitrary size and spacing in an infinite rigid planar baffle. The impedance is expressed as a Fourier transform of a generalized impulse response which is defined by an integral relationship. Although the integral must, in general, be numerically evaluated, several special cases of interest can readily be evaluated by using asymptotic techniques. Several asymptotic expressions for the mutual radiation impedance are developed and their limitations are noted. Numerical results are then presented for the generalized impulse response and mutual radiation impedance corresponding to pistons of equal size and arbitrary spacing. The time domain characteristics of the generalized impulse responses as a function of spacing are noted and related to the frequency characteristics of the mutual radiation impedances as a function of spacing. In addition, the accuracy of a simple closed form expression for the mutual radiation impedance is presented as a function of normalized frequency and spacing.     

直接模拟中不同边界条件的实施及对沉降规律的影响

在牛顿流体中, 对颗粒在4种不同边界的垂直通道中的沉降运动进行了直接数值模拟. 计算结果表明:通过计算区域随颗粒运动而移动构建的无限长通道能准确模拟颗粒自由下落到稳定沉降的发展过程; 周期性边界条件由于流场变化, 对颗粒沉降产生了影响, 不能模拟颗粒的自由沉降过程; 底部封闭边界适合模拟封闭容器内颗粒与固壁的相互作用过程, 若颗粒达到稳定沉降, 也能模拟无限长通道内的沉降过程; 流化边界适合模拟流化床内气固两相流动. 计算结果有助于更好地理解和使用不同边界条件. 关键词： 直接数值模拟 边界条件 沉降 任意拉格朗日-欧拉方法     

Time domain identification of standing wave parameters in gas piping systems

A method for experimentally determining the natural frequencies and modal pressures of an air or gas piping system is presented. Such information is of interest in installations where pressure pulsations caused by pumps or compressors are of importance. In the method a time domain based technique is used which was originally developed as an alternative to frequency response methods for determining the vibration parameters (natural frequencies, modes, damping factors) of structures, to avoid difficulties often encountered in interpreting complex and non-conclusive frequency response data such as arises from systems having numerous modes, some of which may be highly damped or closely spaced in frequency. In this application, a straight steel pipe with a sound source at one end and closed at the other end was used. Two microphones were used to measure the pressure at two locations in the pipe. The free pressure response following a rapidly swept sinewave input was recorded, digitized and then used in a computational procedure based on a lumped parameter representation of the system. The natural frequencies and the corresponding modal pressure ratios at the two stations, thus obtained, are compared with mention here that although in the experiment reported here an external frequency sweep excitation was used, the technique works as well with free decay response after a system shut-off, impulse response or random responses from normal system operation.     

On the instability of time-domain acoustic boundary element method due to the static mode in interior problems

In the analysis of interior acoustic problems, the time domain boundary element method (TBEM) suffers the monotonically increasing instability when using the direct Kirchhoff integral. This instability is related to the non-oscillatory static acoustic mode describing the constant spatial response in an enclosure. In this work, nonphysical natures of non-oscillatory static mode influencing the instability of TBEM calculation are investigated, and a method for stabilization is studied. In TBEM calculation, the static mode is represented by two non-oscillatory eigenmodes with different eigenvalues. The monotonically increasing instability is caused by the unstable poles of non-oscillatory eigenmodes as well as very small, very low frequency noise of an input signal. Interior problems with impedance boundary condition also exhibit the monotonically increasing instability stemming from its pseudo non-oscillatory static mode due to the lack of dissipation at very low frequencies. Calculation of transient sound fields within rigid and lined boxes provides numerical evidences. It is noted that the stabilization effort by modifying the coefficient matrix based on the spectral decomposition can be used only for correcting the unstable pole. The filtering method based on the eigen-analysis must be additionally used to avoid the remaining instability caused by very low frequency noise of input signal.     

Fan interaction noise reduction using a wake generator: experiments and computational aeroacoustics

A control grid (wake generator) aimed at reducing rotor-stator interaction modes in fan engines when mounted upstream of the rotor has been studied here. This device complements other active noise control systems currently proposed. The compressor model of the instrumented ONERA CERF-rig is used to simulate suitable conditions. The design of the grid is drafted out using semi-empirical models for wake and potential flow, and experimentally achieved. Cylindrical rods are able to generate a spinning mode of the same order and similar level as the interaction mode. Mounting the rods on a rotating ring allows for adjusting the phase of the control mode so that an 8 dB sound pressure level (SPL) reduction at the blade passing frequency is achieved when the two modes are out of phase. Experimental results are assessed by a numerical approach using computational fluid dynamics (CFD). A Reynolds averaged Navier-Stokes 2-D solver, developed at ONERA, is used to provide the unsteady force components on blades and vanes required for acoustics. The loading noise source term of the Ffowcs Williams and Hawkings equation is used to model the interaction noise between the sources, and an original coupling to a boundary element method (BEM) code is realized to take account of the inlet geometry effects on acoustic in-duct propagation. Calculations using the classical analytical the Green function of an infinite annular duct are also addressed. Simple formulations written in the frequency domain and expanded into modes are addressed and used to compute an in-duct interaction mode and to compare with the noise reduction obtained during the tests. A fairly good agreement between predicted and measured SPL is found when the inlet geometry effects are part of the solution (by coupling with the BEM). Furthermore, computed aerodynamic penalties due to the rods are found to be negligible. These results partly validate the computation chain and highlight the potential of the wake generator system proposed.     

Band gap structures for viscoelastic phononic crystals based on numerical and experimental investigation

Many materials used as phononic crystals (PCs) are viscoelastic one. It is believed that viscosity results in damping to attenuate wave propagation, which may help to tune the defect modes or band gaps of viscoelastic phononic crystals. To investigate above phenomenon, firstly, we have extended the application of boundary element method (BEM) to the study of viscoelastic phononic crystals with and without a point defect. A new developed BEM within the framework of Bloch theory can easily deal with viscoelastic phononic crystals with arbitrary shapes of the scatterers. Experimental methods have been put forward based on the self-made viscoelastic phononic crystals. Verified by the experimental results, systematic comprehensive parametric studies on the band structure of viscoelastic phononic crystals with varying factors (final–initial value ratio, relaxation time, volume fraction of scatterers, shapes of scatterers) have been discussed by the numerical simulation. To further address the possibility to change the defect modes, the band structure of viscoelastic phononic crystals with a point defect has been studied based on the numerical and experimental methods. From present research work, it can be found that by adjusting the two viscous parameters combined with considering the effect of volume fraction and shapes, a wider and lower initial forbidden frequency or lower and higher quality factor resonant frequency can be obtained.     

A direct mixed-body boundary element method for packed silencers

Bulk-reacting sound absorbing materials are often used in packed silencers to reduce broadband noise. A bulk-reacting material is characterized by a complex mean density and a complex speed of sound. These two material properties can be measured by the two-cavity method or calculated by empirical formulas. Modeling the entire silencer domain with a bulk-reacting lining will involve two different acoustic media, air and the bulk-reacting material. Traditionally, the interior silencer domain is divided into different zones and a multi-domain boundary element method (BEM) may be applied to solve the problem. However, defining different zones and matching the elements along each interface is tedious, especially when the zones are intricately connected. In this paper, a direct mixed-body boundary element method is used to model a packed silencer without subdividing it into different zones. This is achieved by summing up all the integral equations in different zones and then adding the hypersingular integral equations at interfaces. Several test cases, including a packed expansion chamber with and without an absorbing center bullet, and a parallel baffle silencer, are studied. Numerical results for the prediction of transmission loss (TL) are compared to experimental data.     

Computational simulation of wave propagation problems in infinite domains

This paper deals with the computational simulation of both scalar wave and vector wave propagation problems in infinite domains. Due to its advantages in simulating complicated geometry and complex material properties, the finite element method is used to simulate the near field of a wave propagation problem involving an infinite domain. To avoid wave reflection and refraction at the common boundary between the near field and the far field of an infinite domain, we have to use some special treatments to this boundary. For a wave radiation problem, a wave absorbing boundary can be applied to the common boundary between the near field and the far field of an infinite domain, while for a wave scattering problem, the dynamic infinite element can be used to propagate the incident wave from the near field to the far field of the infinite domain. For the sake of illustrating how these two different approaches are used to simulate the effect of the far field, a mathematical expression for a wave absorbing boundary of high-order accuracy is derived from a two-dimensional scalar wave radiation problem in an infinite domain, while the detailed mathematical formulation of the dynamic infinite element is derived from a two-dimensional vector wave scattering problem in an infinite domain. Finally, the coupled method of finite elements and dynamic infinite elements is used to investigate the effects of topographical conditions on the free field motion along the surface of a canyon.     

Sound propagation around rigid barriers laterally confined by tall buildings

This paper focuses on the propagation of sound waves in the presence of acoustic barriers placed close to very tall buildings. The boundary element method (BEM) is used to model the acoustic barrier, while the presence of the tall buildings is taken into account by using the image source method. Different geometries are analyzed, representing the cases of a single building, two buildings forming a corner and three buildings defining a laterally confined space. The acoustic barrier is assumed to be non-absorbing, and all the buildings and the ground are modeled as infinite rigid plane surfaces. Calculations are performed in the frequency domain and time signals are then obtained by means of Inverse Fourier Transforms. The sound pressure loss provided by the acoustic barrier is computed, illustrating the importance of the lateral confinements.     

Dispersive elastodynamics of 1D banded materials and structures: analysis

The elastodynamics of 1D periodic materials and finite structures comprising these materials are studied with particular emphasis on correlating their frequency-dependent characteristics and on elucidating their pass-band and stop-band behaviors. Dispersion relations are derived for periodic materials and are employed in a novel manner for computing both pass-band and stop-band complex mode shapes. Through simulations of harmonically induced wave motion within a finite number of unit cells, conformity of the frequency band structure between infinite and finite periodic systems is shown. In particular, only one or two unit cells of a periodic material could be sufficient for “frequency bandedness” to carry over from the infinite periodic case, and only three to four unit cells are necessary for the decay in normalized transmission within a stop band to practically saturate with an increase in the number of cells. Dominant speeds in the scattered wave field within the same finite set of unit cells are observed to match those of phase and group velocities of the infinite periodic material within the most active pass band. Dynamic response due to impulse excitation also is shown to capture the infinite periodic material dynamical characteristics. Finally, steady-state vibration analyses are conducted on a finite fully periodic structure revealing a conformity in the natural frequency spread to the frequency band layout of the infinite periodic material. The steady-state forced response is observed to exhibit mode localization patterns that resemble those of the infinite periodic medium, and it is shown that the maximum localized response under stop-band conditions could be significantly less than in an equivalent homogenous structure and the converse is true for pass-band conditions.     

Prediction of sound level at high-frequency bands by means of a simplified boundary element method

A simplified boundary element method (BEM) for dealing with high-frequency sound is proposed. The boundary integral equation is modified into a quadratic form to enable the prediction of sound levels in the one-third octave band analysis. Monopole and dipole source terms in the conventional BEM are transformed into the auto- and cross-spectra of two vibrating sources, in which the cross-spectra are eventually neglected by assuming that the correlation coefficients involved are negligible. The present method is compared with the Rayleigh integral for calculating the sound pressure radiated from a baffled panel, in terms of the application limit. The characteristic length of the boundary element and the applicable frequency range can be determined by the lower limit value of the correlation coefficient. As a test example, the field pressure radiated from a partially vibrating sphere is predicted and the resultant trend is in good agreement with the analytic solution as far as the related correlation coefficient satisfies the assumption. The overdetermination process for overcoming nonuniqueness in exterior radiation problems is unnecessary in the present method because phase information can be ignored. The results of the calculation show that the proposed method is acceptable for solving the exterior radiation problem at a high-frequency range in a timely manner.     

边界元素法在集成电路CAD中的应用

本文采用边界元素法(BEM)计算任意多边形的二维电阻和复杂结构的二维电容。同一计算程序既能计算电阻又能计算电容,而只需作微小不同的后处理。对于边界元素法,由于仅需计算边界上的积分方程,其离散边界上的网格点数大大少于有限差分法及有限元素法所需的网格点数,使CPU执行时间显著减少,并简化了网格划分工作。此外,计算结果还指出边界元素法具有精度高和处理复杂边界能力强的优点。     

Boundary element method for calculation of elastic wave transmission in two-dimensional phononic crystals

A boundary element method (BEM) is presented to compute the transmission spectra of two-dimensional (2-D) phononic crystals of a square lattice which are finite along the x-direction and infinite along the y-direction. The cross sections of the scatterers may be circular or square. For a periodic cell, the boundary integral equations of the matrix and the scatterers are formulated. Substituting the periodic boundary conditions and the interface continuity conditions, a linear equation set is formed, from which the elastic wave transmission can be obtained. From the transmission spectra, the band gaps can be identified, which are compared with the band structures of the corresponding infinite systems. It is shown that generally the transmission spectra completely correspond to the band structures. In addition, the accuracy and the efficiency of the boundary element method are analyzed and discussed.     

Topological phase boundary in a generalized Kitaev model

We study the effects of the next-nearest-neighbor hopping and nearest-neighbor interactions on topological phases in a one-dimensional generalized Kitaev model. In the noninteracting case, we define a topological number and calculate exactly the phase diagram of the system. With addition of the next-nearest-neighbor hopping, the change of phase boundary between the topological and trivial regions can be described by an effective shift of the chemical potential. In the interacting case, we obtain the entanglement spectrum, the degeneracies of which correspond to the topological edge modes, by using the infinite time-evolving block decimation method. The results show that the interactions change the phase boundary as adding an effective chemical potential which can be explained by the change of the average number of particles.     

A new formulation and error analysis for vibrating dam-reservoir systems with upstream transmitting boundary conditions

This paper proposes and validates a new formulation to investigate the dynamic response of dam-reservoir systems with upstream transmitting boundary conditions (TBCs). The mathematical derivations are provided for the new formulation as well as for exact and various approximate TBCs. The developed analytical equations can be solved numerically to assess the accuracy of a given TBC and determine the associated error independently of FEM or BEM modeling of the reservoir. The method is first validated in the case of semi-infinite reservoirs and an excellent agreement is obtained against classical techniques. The paper presents a fundamental understanding of the behavior of various TBCs and a systematic identification of their influence on the system's dynamic response, considering: (i) dam flexibility, (ii) water compressibility, (iii) reservoir bottom wave absorption, (iv) reservoir truncation length, and (v) excitation frequency. The new method is used to obtain exact error estimators to evaluate the effects of various TBCs on the dam-reservoir first resonant frequency and hydrodynamic forces acting on the dam upstream face. The proposed formulation can be programmed easily and used efficiently for rigorous assessment of classical or newly developed TBCs for vibrating dam-reservoir systems or similar fluid-structure problems.     

A BEM approach to validate a model for predicting sound propagation over non-flat terrain

A two-dimensional boundary element model for sound propagation in a homogeneous atmosphere above non-flat terrain has been constructed. An infinite impedance plane is taken into account in the Green's function in the underlying integral equation, so that only the non-flat parts of the terrain need to be discretised in the boundary element model. This Green's function is undefined for points below the impedance plane, and therefore valleys and hollows are taken into account by coupling the exterior domain above the ground with one or several interior domains below the ground, as suggested in a recent paper [J. Sound Vibrat. 223 (1999) 355]. The resulting BEM model, which can handle arbitrary combinations of barriers and hollows, has been used for validating a ray model for various difficult configurations, including combinations of valleys and barriers.