Modified perturbation method for eigenvalues of structure with interval parameters

In overcoming the drawbacks of traditional interval perturbation method due to the unpredictable effect of ignoring higher order terms,a modified parameter perturbation method is presented to predict the eigenvalue intervals of the uncertain structures with interval parameters.In the proposed method,interval variables are used to quantitatively describe all the uncertain parameters.Different order perturbations in both eigenvalues and eigenvectors are fully considered.By retaining higher order terms,the original dynamic eigenvalue equations are transformed into interval linear equations based on the orthogonality and regularization conditions of eigenvectors.The eigenvalue ranges and corresponding eigenvectors can be approximately predicted by the parameter combinatorial approach.Compared with the Monte Carlo method,two numerical examples are given to demonstrate the accuracy and efficiency of the proposed algorithm to solve both the real eigenvalue problem and complex eigenvalue problem.     

The impact of water column variability on horizontal wave number estimation and mode based geoacoustic inversion results

The influence of water column variability on low-frequency, shallow water geoacoustic inversion results is considered. The data are estimates of modal eigenvalues obtained from measurements of a point source acoustic field using a horizontal aperture array in the water column. The inversion algorithm is based on perturbations to a background waveguide model with seabed properties consistent with the measured eigenvalues. Water column properties in the background model are assumed to be known, as would be obtained from conductivity, temperature, and depth measurements. The scope of this work in addressing the impact of water column variability on inversion is twofold. Range-dependent propagation effects as they pertain to eigenvalue estimation are first considered. It is shown that mode coupling is important even for weak internal waves and can enhance modal eigenvalue estimates. Second, the effect of the choice of background sound speed profile in the water column is considered for its impact on the estimated bottom acoustic properties. It is shown that a range-averaged sound velocity profile yields the best geoacoustic parameter estimates.     

等截面消声管道传递损失计算的简化方法

该文提出了等截面消声管道传递损失计算的简化方法,方法利用消声管道截面形式的特点,将三维声学计算问题简化为二维问题,消声管道的传递损失可以表示为与轴向波数有关的表达式,轴向波数可以通过计算截面的特征值得到。对于规则截面结构,使用传递矩阵法计算特征值;复杂非规则截面的特征值使用二维有限元方法得到,进而可以计算消声管道的传递损失。仿真结果与文献中的数值方法及实验值在较宽的频率范围内吻合较好,说明了方法的正确性,此外,该方法可以考虑均匀流对消声管道声学性能的影响。方法的计算效率高,对消声管道的前期优化设计具有实际意义。     

Numerical solutions of the acoustic eigenvalue equation in the rectangular room with arbitrary (uniform) wall impedances

Two numerical procedures for finding the acoustic eigenvalues in the rectangular room with arbitrary (uniform) wall impedances are developed. One numerical procedure applies Newton's method. Here, starting with soft walls, the eigenvalues are found by increasing the impedances of each wall pair in small increments up to the terminal impedances. Another procedure poses the eigenvalue problem as one of homotopic continuation from a non-physical reference configuration in which all eigenvalues are known and obvious. The continuation is performed by the numerical integration of two differential equations. The latter procedure was found to be faster and finds all possible solutions. The set of eigenvalues allowed the room modal natural frequencies and damping constants to be obtained. From sound decays measured in a hard-walled rectangular room, and from the collective-modal-decay curve, the impedances of the hard walls are estimated. These are then used to find the reverberation times of the modes in the room with the floor lined with sound absorbing material of known acoustic impedance. It was found that a single reverberation time, for all modes, is only supported in the rectangular room with hard walls and at the higher frequency bands, consistent with Sabine's theory, which assumes a diffuse sound field. In the rectangular room with hard walls and at the lower frequency bands, and in the rectangular room with the floor lined with sound absorbing material and for all frequency bands, modes with rather distinctive reverberation times may produce sound decays not always consistent with Sabine's prediction.     

电子结构方法中带结构特征值问题的统一视角

在(相对论)电子结构方法中,四元数矩阵特征值问题和计算激发能的线性响应(Bethe-Salpeter)特征值问题是两个经常出现的结构特征值问题. 尽管前一个问题已被十分仔细地研究,后一个问题在一般形式下,即不假设电子Hessian正定性的复矩阵情况,并没有得到完全的理解. 鉴于它们非常相似的数学结构,本文从一个统一的角度研究了这两个问题,揭示了它们特征向量的“李群”结构,为将来设计对角化算法和数值优化方法提供了一个统一的框架. 利用和处理四元数矩阵特征值问题相同的归约算法,本文给出了表征线性响应问题特征值(实数、纯虚、或复数)的充分必要条件. 这一结果可以看作是实矩阵情况下已知条件的自然推广.     

A theorem on maximum variance

The square variance function of a finite-dimensional Hamiltonian H obeys a maximum principle that leads to the determination of its maximum and minimum eigenvalues. A systematic algorithm is presented that generates a sequence of monotonically increasing values for the square variance. It is shown that the method converges to the exact two-dimensional eigenvalue problem determined by the lowest and highest eigenvalues. Preliminary numerical results are briefly outlined.     

Comparison of measured broadband noise attenuation spectra from circular flow ducts and from lined engine intakes with predictions

Sound propagation in lined circular ducts is investigated in the presence of uniform and sheared flow. The modal solutions are obtained by solving an eigenvalue equation which, in the case of sheared flow, is derived by using finite differences and by matching the pressure and the radial component of the particle velocity at the interface of the regions of uniform and sheared flow. For the uniform flow region, standard Bessel function solutions are used. The attenuation of acoustic energy at a given frequency and for a given liner length is computed on the assumption that at the inlet to the lined duct, the acoustic energy is equally distributed among the propagating modes. The total number of propagating modes is determined from the hard wall “cut off” condition. The failure to find some of the modal solutions on the attenuation computed in this way is discussed. It is shown that the reliability of this method of computing liner attenuation depends on the ability to successfully compute most of the modal solutions over a large range of frequencies, flow conditions and duct wall impedance values. A numerical technique is developed which uses a fraction of the total number of solutions to compute the total attenuations without appreciable loss of accuracy. Measured attenuation spectra from a flow duct facility and from lined intake ducts of the RB.211 engine are compared with predictions. In general very good agreement between predictions and measurements is obtained.     

Shape optimization towards stability in constrained hydrodynamic systems

Hydrodynamic stability plays a crucial role for many applications. Existing approaches focus on the dependence of the stability properties on control parameters such as the Reynolds or the Rayleigh number. In this paper we propose a numerical method which aims at solving shape optimization problems in the context of hydrodynamic stability. The considered approach allows to guarantee hydrodynamic stability by modifying parts of the underlying geometry within a certain flow regime. This leads to a formulation of a shape optimization problem with constraints on the eigenvalues related to the linearized Navier–Stokes equations. In that context the eigenvalue problem is generally non-symmetric and may involve complex eigenvalues. To validate the proposed numerical approach we consider the flow around a body in a channel. The shape of the body is parameterized and can be changed by means of a discrete number of design variables. It is our aim to find a design which minimizes the drag force and ensures at the same time hydrodynamic stability while keeping the volume of the body constant. The numerical results show that a transition from an unstable design to a stable one is attainable by considering an adequate change of the body shape. The resulting bodies are long and flat which corresponds to common intuition.     

Krein Signatures for the Faddeev-Takhtajan Eigenvalue Problem

One of the difficulties in analyzing eigenvalue problems that arise in connection with integrable systems is that they are frequently non-self-adjoint, making it difficult to determine where the spectrum lies. In this paper, we consider the problem of locating and counting the discrete eigenvalues associated with the Faddeev-Takhtajan eigenvalue problem, for which the sine-Gordon equation is the isospectral flow. In particular we show that for potentials having either zero topological charge or topological charge ± 1, and satisfying certain monotonicity conditions, the point spectrum lies on the unit circle and is simple. Furthermore, we give an exact count of the number of eigenvalues. This result is an analog of that of Klaus and Shaw for the Zakharov-Shabat eigenvalue problem. We also relate our results, as well as those of Klaus and Shaw, to the Krein stability theory for J-unitary matrices. In particular we show that the eigenvalue problem associated to the sine-Gordon equation has a J-unitary structure, and under the above conditions the point eigenvalues have a definite Krein signature, and are thus simple and lie on the unit circle.     

On the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods

Recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators. The standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed. They are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds. Further, the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues. A new method is presented which combines multigrid techniques with the Lanczos process. The resulting scheme requires O(mn) arithmetic operations and O(n) storage requirement, where n is the number of unknowns and m, the number of needed eigenvalues. The discretization of the considered differential operators is realized by means of p-finite elements and is applicable on general geometries. Numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods, at least on current workstations. The ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a PC shows the potential of this method. Practical applications are found, e.g. in the numerical simulation of quantum billiards.     

Fast time-evolution method for dynamical systems

A fast time-evolution method is developed for systems for which the dynamical behavior can be reduced to the eigenvector/eigenvalue problem. The method does not use the eigenvectors/eigenvalues themselves and is based on a polynominal expansion of the formal operator solution in the eigenfrequency domain. It is complementary to the standard time-integration approaches and allows one to calculate or simulate the state of a system at arbitrary times. The time evolution of, e.g., classical harmonic atomic systems and quantum systems described by linear Hamiltonians can be treated by this method.     

求解α本征值的k-α迭代方法误差估计

分析求解α本征值的k-α迭代方法,指出迭代法中参数的选择影响使用标准差的平均值估计出来的α误差.如果参数选取不当,实际估计误差可能和真实估计误差相差很大.MCNP4C程序中使用的参数给出的误差估计通常和真实估计误差有一定的偏差.本文给出一个新的参数选择办法,能够让实际估计误差几乎等于真实估计误差,并用数值实验进行验证.     

流管实验装置中声传播计算的模态方法

流管实验装置是测量有流动情况下航空发动机消声短舱内声衬声阻抗的主要装置。本文发展了一种解析的模态匹配方法进行在平均流有声衬条件下矩形流管中声传播的计算。用同伦方法求解特征值问题,并与用环绕积分求解的结果进行比较。声场通过轴向阻抗间断面的声压和声质点速度积分相等计算。第一个算例是无流动、硬壁、有限长、考虑端口反射的情况,并与北航流管实验台测量数据进行了对比;第二个算例为有流动情况下有限长声衬管道不考虑端口反射的声场计算,它与文献中NASA流管实验结果和CAA计算结果符合得很好。     

On the effect of viscosity and thermal conductivity on sound propagation in ducts: A re-visit to the classical theory with extensions for higher order modes and presence of mean flow

The dispersion equation for the axisymmetric modes of viscothermal acoustic wave propagation in uniform hard-walled circular ducts containing a quiescent perfect gas is classical. This has been extended to cover the non-axisymmetric modes and real fluids in contemporary studies. The fundamental axisymmetric mode has been the subject of a large number of studies proposing approximate solutions and the characteristics of the propagation constants for narrow and wide ducts with or without mean flow is well understood. In contrast, there are only few publications on the higher order modes and the current knowledge about their propagation characteristics is rather poor. On the other hand, there is a void of papers in the literature on the effect of the mean flow on the quiescent modes of propagation. The present paper aims to contribute to the filling of these gaps to some extent. The classical theory is re-considered with a view to cover all modes of acoustic propagation in circular ducts carrying a real fluid moving axially with a uniform subsonic velocity. The analysis reveals a new branch of propagation constants for the axisymmetric modes, which appears to have escaped attention hitherto. The solution of the governing wave equation is expressed in a modal transfer matrix form in frequency domain and numerical results are presented to show the effects over wide ranges of frequency, viscosity and mean flow parameters on the propagation constants. The theoretical formulation allows for the duct walls to have finite impedance, but no numerical results are presented for lined ducts or ducts carrying a sheared mean flow.     

Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics

The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a time-independent single-frequency sound source. Its solution consists of a set of discrete modes radiating into the upper atmosphere, usually related to the continuous spectrum. In this article, we present two spectral methods, the Chebyshev-Tau and Chebyshev-Collocation methods, to solve for the atmospheric acoustic normal modes, and corresponding programs are developed. The two spectral methods successfully transform the problem of searching for the modal wavenumbers in the complex plane into a simple dense matrix eigenvalue problem by projecting the governing equation onto a set of orthogonal bases, which can be easily solved through linear algebra methods. After the eigenvalues and eigenvectors are obtained, the horizontal wavenumbers and their corresponding modes can be obtained with simple processing. Numerical experiments were examined for both downwind and upwind conditions to verify the effectiveness of the methods. The running time data indicated that both spectral methods proposed in this article are faster than the Legendre-Galerkin spectral method proposed previously.     

Dynamics of two-dimensional composites of elastic and viscoelastic layers

Models of frequency response, acoustic transmission, and transient wave propagation are presented for a two-dimensional composite of elastic and viscoelastic layers, simply supported at the two boundaries. The three models adopt transfer matrices to relate state variables over the two faces of a layer. In the frequency domain, a viscoelastic constitutive law is derived by nonlinear fitting a Padé series to measured data of complex shear modulus. For an elastic material, the eigenproblem admits positive real eigenvalues and their negatives. For a viscoelastic material, it admits positive complex eigenvalues and their negative conjugates. The imaginary part of the eigenvalue acts as a velocity-dependent viscous damper. Modal analysis solving transient response utilizes the complex eigenquantities and the static-dynamic superposition method.     

三自由度含间隙碰撞振动系统Neimark-Sacker分岔的反控制

分岔反控制作为传统分岔控制的逆问题, 其目的是在预先指定的系统参数点通过控制主动设计出具有所期望特性的分岔解. 以一类三自由度含间隙双面碰撞振动系统为研究对象, 在不改变原系统平衡解结构的前提下, 考虑到在碰撞振动系统反控制过程中由Poincaré映射的隐式特点和传统的映射Neimark-Sacker分岔临界准则带来的困难, 通过对原系统施加线性反馈控制器并利用不直接依赖于特征值计算的Neimark-Sacker分岔显式临界准则研究了此系统的分岔反控制问题. 首先对原系统施加线性反馈控制, 建立闭环控制系统的六维Poincaré映射. 由于六维映射的雅克比矩阵的特征值没有解析的表达式, 利用高维映射Neimark-Sacker分岔的显式临界准则, 获得了系统出现拟周期碰撞振动运动的控制参数区域. 然后采用中心流形-正则形方法分析了拟周期分岔解的稳定性. 数值仿真结果表明本文方法可以在指定的系统参数点通过控制设计出稳定的拟周期碰撞运动.     

A derivation of the Christoffel equation with damping

This paper provides a derivation of the Christoffel eigenvalue equation for acoustic wave propagation in an acoustically damped piezoelectric medium. The damping tensor is shown to couple into both the stress and displacement constitutive equations. Application of the quasi-static approximation leads to an additional term in the Christoffel equation that generates a complex k-vector, due both to introduction of a complex term and to breaking of symmetry in the left-hand side of the eigenvalue equation, subsequently resulting in damping and a phase shift for a plane wave solution. Shown are the effects of damping on the eigenvalues of the piezoelectrically stiffened Christoffel equation for plane wave propagation in unconstrained quartz over a 1 MHz to 1 GHz frequency range.     

Constructive proof of the existence of bound state in one dimension

A proof is given for the theorem that at least one bound state exists in a one-dimensional attractive potential however weak it may be. The proof is constructive in that it provides a method to explicitly solve the eigenvalue problem for the eigenvalues as well as the eigenfunctions. The method is well suited to precise numerical calculations.     

Application of spectral representations to the nonrelativistic and the relativistic Bethe-Salpeter equation

The eigenvalue problem of the scalar Bethe-Salpeter equation is solved by application of the vertical Dyson representation. The method of solution is developed in complete analogy to the solution of Schrödinger's equation by a Stieltjes representation in the case of a Yukawa potential. The eigenvalues are zeros of a characteristic determinant, which can be understood as a generalization of the nonrelativistic Jost function.