Interior Resonance Research Articles

Analytic Method on Vibrate Induced Noise of Engine Block

Numerical prediction of the acoustic radiation of vibrating engine block is conventionally based on the boundary element method (BEM). This approach yields good predictions, provided that the dynamic behavior of engine block is well established. However, interior resonances may lead to singular matrices and nonunique solution. To avoid the problems with interior resonances and singular matrices, this paper introduced the combined Helmholtz integral equation formulation method (CHIEF METHOD). The numerical experiment shows the performance of CHIEF METHOD and then we use this method to estimate the acoustic radiation of engine block. Based on spatially mean square velocity and acoustic radiation efficiency of vibrating engine block, a simplified analytic method is presented in this paper, by which the vibrate induced noise can be predicted in the early design process of engine development.

Numerical Simulation of Transient Gauss pulse Coupling through Small Apertures

Transient electromagnetic pulse (EMP) can easily couple into equipments through small apertures in its shells. To study the coupling effects of transient Gauss pulse to a cubic cavity with openings, coupling course is simulated using sub-gridding finite difference in time domain (FDTD) algorithm in this paper. A new grid partition approach is provided to simulate each kind of apertures with complex shapes. With this approach, the whole calculation space is modeled, and six kinds of aperture with different shapes are simulated. Coupling course is simulate in the whole time domain using sub-gridding FDTD approach. Selecting apertures with dimension of several millimeters to research, coupled electric field waveform, power density and coupling coefficient are calculated. The affect on coupling effects by varied incident angle and varied pulse width are also analyzed. The main conclusion includes interior resonance phenomenon, increase effect around rectangle aperture and several distributing rules of coupled electric field in the cavity. The correctness of these results is validated by comparing with other scholars' results. These numerical results can help us to understand coupling mechanism of the transient Gauss pulse.

Cloaking a sensor via transformation optics

Ideal transformation optics cloaking at positive frequency, besides rendering the cloaked region invisible to detection by scattering of incident waves, also shields the region from those same waves. In contrast, we demonstrate that approximate cloaking permits a strong coupling between the cloaked and uncloaked regions; careful choice of parameters allows this coupling to be amplified, leading to effective cloaks with degraded shielding. The sensor modes we describe are close to but distinct from interior resonances, which destroy cloaking. As one application, we describe how to use transformation optics to hide sensors in the cloaked region and yet enable the sensors to efficiently measure incident waves on the exterior of the cloak, an effect similar to the plasmon-based approach of Alù and Engheta.

Efficient Fast Frequency Sweep Without Nonphysical Resonances

Single-point methods of moment-matching type provide a most effective means for realizing fast frequency sweeps. The impedance formulation of the finite element method is very well-suited for such techniques, but it is susceptible to interior resonances, especially in the case of lossless microwave structures. This deficiency can be resolved by imposing transparent boundary conditions at the waveguide ports, but the resulting matrices exhibit nonpolynomial frequency-dependency unless all waveguide modes are of the transverse electric and magnetic type. In consequence, single-point methods cannot be applied without approximations. This article proposes a fast frequency sweep technique that handles not only transverse electric and magnetic modes but also transverse electric and transverse magnetic modes without any approximations. Moreover, it retains the efficiency of the impedance formulation but prevents interior resonances from occurring. The theoretical background of and numerical evidence for the failure of the impedance formulation is given, and a mathematical proof for the validity of the new method is presented. Computational results demonstrate the reliability and low error of the suggested approach.

Stabilizing the E-Field Integral Equation at the internal resonances through the computation of its numerical null space

In this paper a technique to protect the Method of Moments solution of the E-Field Integral Equation (EFIE) for the scattering by conducting body from effects of its internal resonances, is presented. This method is based on the evaluation of the numerical null space K of the discretized EFIE operator. In order to verify its effectiveness, results for the plane wave scattering from a perfect conducting cube are presented. It is shown that the proposed method is able to give the correct value of the RCS even in presence of the interior resonances.

Well‐posedness of integral equations for modeling electromagnetic scattering from cavities

Integral equations for modeling electromagnetic scattering from indented screens or cavity structures are proposed using a symmetric equivalent current model, which is based on image theory with a perfect electrically conducting (PEC) ground plane. The well posedness of the proposed integral equations is theoretically established in this paper. It is known that all resonance modes for cavity problems either due to the presence of a perfect magnetically conducting (PMC) open aperture or a PEC open aperture. In this paper, it is proved that both kinds of resonance modes cannot exist for the proposed integral equations without excitation. To numerically examine this well posedness, the condition numbers of resultant matrices from the generalized network formulation and proposed integral equations are analyzed as functions of frequency and the depth of cavity. The numerical analysis of the condition numbers obtained also shows that the integral equations proposed in this paper are immune to the problem of the interior resonance for three‐dimensional cavity problems. Moreover, simulation of scattering from standard cavity structures verifies the validity of the proposed integral equations.

Numerical difficulties with boundary element solutions of interior acoustic problems

Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported low-frequency breakdowns and artificial damping due to discretization errors. In this paper, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping, which may potentially lead to erroneous predictions when boundary element methods are used to evaluate the performance of damping materials. This difficulty can be alleviated by first computing an impedance or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then causes nonexistence and nonuniqueness difficulties at the interior resonance frequencies for hard-wall and pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half the acoustic wavelength. Aside from producing much more accurate predictions, the admittance matrix is also much easier to interpolate at low frequencies due to the absence of interior resonances. For the example problem considered, only the formulation using the reactive component of the admittance matrix produces accurate solutions as long as the surface element discretization satisfies the standard six-element-per-wavelength rule.

Reduction of fan noise emission by enclosure modification

In the present study, a structural modification approach to the reduction of sound power radiated by an axial fan mounted to an enclosure is described. An axial fan operating in free-space exhibits a dipole-like radiation pattern at low frequencies when the source is acoustically compact. When a fan is mounted to an enclosure, however, the sound radiation pattern becomes monopole-like since the interaction between the sound fields on the two sides of the fan is prevented by the enclosure: thus the fan can radiate sound more efficiently than when operating in free-space. In addition, it is possible for the source level to be amplified by coupling with interior resonances of the enclosure. The radiation enhancement caused by both effects can be suppressed by introducing an acoustical path that allows the sound fields on both sides of the fan to interact with each other while also changing the resonance frequencies of, and providing damping, for the acoustical cavity, thus reducing the radiation efficiency of the fan towards its free-space value. The effect of the latter type of modification is demonstrated here by using a consumer electronics device equipped with two fans. The sound fields corresponding to various enclosure modifications were visualized by using near-field acoustical holography, and the modifications' effects are quantified in terms of the radiated sound power.

Normally-Integrated Magnetic Field Integral Equations for Electromagnetic Scattering

Modified magnetic field integral equations (MFIEs) are proposed incorporating a testing integration in the direction normal to the boundary of the target surface. The result involves a simpler kernel than the conventional MFIE, facilitating accurate computations. By employing a complex exponential testing function, the equations can be made immune to the interior resonance problem. The influence of the limits of integration, and the overall accuracy, is investigated using high order basis functions.

Scattered-Field Time Domain Boundary Element Method and Its Application to Transient Electromagnetic Field Simulation in Particle Accelerator Physics

Authors have been working in particle accelerator wake field analysis by using the Time Domain Boundary Element Method (TDBEM). A stable TDBEM scheme was presented and good agreements with conventional wake field analysis of the FDTD method were obtained. On the other hand, the TDBEM scheme still contains difficulty of initial value setting on interior region problems for infinitely long accelerator beam pipe. To avoid this initial value setting, we adopted a numerical model of beam pipes with finite length and wall thickness on open scattering problems. But the use of such finite beam pipe models causes another problem of unwanted scattering fields at the beam pipe edge, and leads to the involvement of interior resonant solutions. This paper presents a modified TDBEM scheme, Scattered-field Time Domain Boundary Element Method (S-TDBEM) to treat the infinitely long beam pipe on interior region problems. It is shown that the S-TDBEM is able to avoid the excitation of the edge scattering fields and the involvement of numerical instabilities caused by interior resonance, which occur in the conventional TDBEM.

Numerical difficulties with boundary element solutions of interior acoustic problems

Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported artificial damping due to acoustic radiation and low‐frequency breakdowns. In this presentation, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping due to sound radiation to the farfield. This difficulty can be alleviated by first computing an impedance matrix or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then cause nonexistence/nonuniqueness difficulties at the interior resonance frequencies for hard‐wall or pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half of the acoustic wavelength.

BROADBAND MULTILEVEL FAST MULTIPOLE ALGORITHM FOR ACOUSTIC SCATTERING PROBLEMS

A broadband multilevel fast multipole algorithm (MLFMA) for the acoustic scattering from a sound-hard obstacle is presented. The formulation is based on the Burton–Miller boundary integral equation and Galerkin's method, avoiding any hypersingular integral operators. The resulting matrix equation has good iterative properties for all frequencies and avoids the interior resonance problem. The main novel feature is the use of a broadband MLFMA to accelerate the iterative generalized minimal residual (GMRES) solver. The algorithm is based on a combination of Rokhlin's translation formula for large division cubes and the spectral representation of the Green's function for cubes smaller than one half wavelength, thereby avoiding the sub-wavelength breakdown of the high-frequency MLFMA.

Alternative implementation of combined‐field integral equation using Rao–Wilton–Glisson basis functions for conducting scatterers

AbstractThis paper presents an alternative combined‐field integral equation (CFIE) for conducting scatterers to surmount the nonuniqueness problem around the interior resonance frequencies. The implementation of an alternative CFIE is simplified by transforming the integrals of electric‐field (EFIE) and magnetic‐field (MFIE) integral equations to be of the same type. Through the method of moments (MoM), the commonly used Rao–Wilton–Glisson (RWG) basis and testing functions are exploited in the calculation of the alternative CFIE impedance matrix. The numerical results are illustrated and compared with those of the MFIE. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 753–756, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21465

Iterative solution of high-order boundary element method for acoustic impedance boundary value problems

A high-order boundary element method for time-harmonic acoustic impedance boundary value problems is presented. The method is based on the Galerkin-type formulation of the Burton–Miller integral equation (BMIE) with high-order polynomial basis and testing functions. This formulation has several important advantages: It is free of the interior resonance problem, the hypersingular integral operator of the traditional BMIE formulation can be avoided, it leads to faster convergence in terms of the number of unknowns than the low-order methods and it shows a good performance with iterative solvers. To avoid the numerical difficulties associated to the implementation of the singular surface integral equations, the singularity extraction technique is applied to evaluate the integrals in the singular and near-singular cases. The resulting matrix equation is solved iteratively with the generalized minimal residual method (GMRES) and a simple preconditioner based on the incomplete LU factorization is applied to expedite the convergence. Numerical results indicate that the BMIE formulation with Galerkin method and high-order basis functions has good convergence properties for various geometries and boundary conditions on a wide frequency range when the GMRES method is applied to solve the matrix equation iteratively. This, in turn, indicates that the formulation is well suited for an efficient application of fast solution procedures, such as the fast multipole method.

Stabilized FEM‐BEM Coupling for Helmholtz Transmission Problems

We consider time‐harmonic acoustic scattering at a nonsmooth penetrable object and coupled boundary element finite element schemes for its numerical simulation. Straightforward coupling approaches are haunted by instabilities at wave numbers related to interior resonances, the so‐called spurious resonances. A remedy is offered by adopting the idea underlying the widely used combined field integral equations. We apply it in the form of modified trace operators. These will also feature regularizing operators to offset the lack of compactness of the double‐layer potential integral operators on nonsmooth surfaces. Calderón projectors can be defined based on the modified trace operators. Thus, Costabel’s approach to the symmetric coupling of domain variational formulations and boundary integral equations carries over. The modified traces guarantee uniqueness of solutions of the coupled problem, whereas regularization ensures coercivity. From this we immediately concludeasymptotic quasi‐optimality of a combined finite element and boundary element Galerkin discretization for all frequencies.

Effect of radiation on dust particles in orbital resonances

The effect of electromagnetic radiation on the dynamics of arbitrarily shaped cosmic dust particles is investigated. The paper concentrates on the motion of dust grains near commensurability resonances with a planet—mean-motion resonances—and possible capture of the grains in the resonances. A particle is in resonance with a planet when the ratio of the mean motions of the two objects is a ratio of two small integers. The most fundamental properties of the orbital evolution of spherical dust particles in the mean-motion resonances are shortly rederived: the solar wind effect is also included and the existing result is improved. The results for spherical particles are compared with the detailed numerical calculations for nonspherical particles. It is shown that the fundamental results valid for spherical grains do not hold, in general, for nonspherical particles. While spherical particles are always characterized by the secular decrease of the semi-major axes near mean-motion resonances, this may not be true for nonspherical particles. Nonspherical grains may exhibit an increase of the semi-major axes before capturing in the mean-motion resonances. This is caused by the effect of electromagnetic radiation on nonspherical dust grains. The eccentricities of spherical particles in the exterior resonances approach a limiting value, but nonspherical grains may not follow this behaviour. The interior resonances are characterized by a systematic decrease of eccentricity for spheres, but various behaviours exist in the case of irregularly shaped particles. The motion of a nonspherical dust particle under the action of electromagnetic radiation may be characterized by a small change of the semi-major axis during a long-time interval, but the particle is not captured in any mean-motion resonance. This kind of motion does not exist for spherical grains.

Protecting the method of auxiliary sources (MAS) solutions from the interior resonance problem

The method of auxiliary sources (MAS) has been used for the numerical solution of a variety of electromagnetic problems successfully. Since MAS can be considered as a modified or generalized version of the standard surface integral equation technique (SIE), it has the interior resonance problem, but the remedies to this problem have not been investigated yet. In this letter, it is shown that the interior resonance problem in applying MAS can be completely removed using combined-source solution (CSS).

Regularized Combined Field Integral Equations

Many boundary integral equations for exterior boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. However, if the boundary Γ is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an L2(Γ)-coercive variational formulation. This foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes.This article presents new combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical trace spaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. This permits us to apply the classical convergence theory of conforming Galerkin methods.

On the Variational Formulation of Hybrid Finite Element-Boundary Integral Techniques for Electromagnetic Analysis

This paper addresses the time-harmonic, electromagnetic analysis of a three-dimensional inhomogeneous radiator/scatterer in free-space. Such analysis can be carried out by combining the finite element method (FEM) with the method of moments (MoM), which yields finite element-boundary integral (FE-BI) formulations. A general framework is presented, within which stationary FE-BI formulations can be established (variational boundary-value problems), which relate to equivalent, underlying variational principles (stationary functionals). The formulations are shown to be accurate, robust and computationally efficient. They avoid the problem of interior resonances without resorting to the combined field integral equation and they result in symmetric system matrices, which preserve reciprocity explicitly. Thus, the stationary FE-BI framework combines the FEM and MoM on the continuous level, as opposed to the usual approach of hybridization after discretization, which generally leads to asymmetric matrices. The stationary FE-BI framework allows one to solve either for the electric and magnetic fields on the volume of the problem domain, or for one volume and one exterior surface field quantity, with only marginal differences in computational cost. The volume-surface formulations have the same storage requirements as previous FE-BI formulations and can be more efficiently solved. The volume-volume formulations provide simultaneous solutions of the electric and magnetic fields, which could for instance be used to construct error estimators directly based on Maxwell's equations.

Implicit methods of solution to integral formulations in boundary element method based nearfield acoustic holography

Two integral equation methods are considered which can be inverted to provide the surface velocity and/or pressure given a measurement of the pressure on an imaginary surface in the nearfield of a vibrating or scattering body. This problem is central to nearfield acoustical holography (NAH). The integral equations are discretized using boundary element methods (BEMs). The integral equation methods considered are (1) an indirect formulation method based on the single layer integral equation and (2) a direct formulation method based on a system of equations derived from the Helmholtz–Kirchhoff integral equation. The formation of integral equations from the mentioned methods will not involve the explicit inversion of matrices, but instead will require this inversion to be done implicitly. Since these methods back-track the sound field from the measurement surface to the surface of the source/vibrator they are ill-posed in nature and Tikhonov regularization are used to stabilize the reconstruction. Problems are noted with the use of higher order shape functions in the BEM discretization which are detrimental to the inversion, and it is shown that the breakdown of the method at the interior resonance frequencies can be mitigated successfully. These two approaches are compared with the conventional direct method using measured experimental data from a point-driven rectangular plate.