Narrow Results

An analytical model and a fully coupled finite element/boundary element model are developed for a simplified physical model of a submarine. The submerged body is modeled as a ring-stiffened cylindrical shell with finite rigid end closures, separated by bulkheads into a number of compartments and under axial excitation from the propeller-shafting system. Lumped masses are located at each end to maintain a condition of neutral buoyancy. Excitation of the hull axial modes from the propeller-shafting system causes both axial motion of the end closures and radial motion of the shell, resulting in a high level of radiated noise. In the low frequency range, only the axial modes in breathing motion are examined, which gives rise to an axisymmetric case, since these modes are efficient radiators. An expression for the structurally radiated sound pressure contributed by axial movement of the end plates and radial motion of the shell was obtained using the Helmholtz integral equation. In the computational model, the effects of the various influencing factors (ring stiffeners, bulkheads, realistic end closures, and fluid loading) on the free vibrational characteristics of the thin walled cylinder are examined. For both the analytical and computational models, the frequency responses, axial and radial responses of the cylinder, and the radiated sound pressure are compared.

Algorithms for an alternative integral-formulation method (AIM) are developed for predicting acoustic radiation from an arbitrary source in a free field. The main advantages of these algorithms are that the solution is always unique and the efficiency in numerical computations is very high. The input data to these algorithms consist of the normal and tangential components of the particle velocities that are specified on a hypothetical surface enclosing the source, and output data are the acoustic quantities that include the acoustic pressure, particle velocity, and acoustic intensity on and beyond the enclosure. To speed up the numerical computations, the Dijistra algorithm is adopted that searches automatically the shortest path between two neighboring nodes in carrying out line integral. Experiments in both interior and exterior regions are conducted, and the predicted acoustic pressure is checked against the benchmark value measured at the same location. The efficiency of AIM is examined and compared with that of conventional boundary element method (BEM) based Helmholtz integral formulations.

Thin components, such as baffles, extended inlet/outlet tubes, and internal connecting tubes, are commonly used in reactive mufflers for cancelation of sound at particular frequency peaks. To provide additional absorption effects at higher frequencies, porous sound absorbing materials may be used on the muffler interior wall surface or on any internal thin components. If the sound absorbing material is backed by a rigid surface, it is usually modeled by the local normal impedance approach. The local impedance modeling on the interior wall surface is straightforward and has been extensively used in the boundary element method, in which the boundary surface is just moved forward to the contact surface between the lining and air. On the other hand, the local impedance modeling on any internal thin components is relatively rare. This paper first presents a direct mixed-body boundary element formulation for a thin body covered by local impedance on either side or both sides of the thin body. The local impedance can be from the lining material itself, or from the lining material plus a protective perforated metal cover. Several test cases with experimental comparison are presented in this paper.

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.

Different numerical approaches for the physical phenomena of scattering waves from an obstacle are presented. They are based on different integral formulations. Fluid structure interaction effects are numerically treatable as well. We use the Boundary Element Method (BEM) in different approaches because the inherently satisfied Sommerfeld radiation condition makes sure that no reflecting waves from boundaries at infinity occur. One of the biggest disadvantages of numerical methods like BEM is the fact that they have difficulties with handling the high frequency range. For the high frequency range approximations like the Kirchhoff–Helmholtz integral equation have to be used. With varying assumptions of the reflecting behavior of the structure different approaches for the higher frequency range are obtained, where the explicit solving of a system of equations is not necessary. Another high frequency approach is the plane wave approximation which is compared with the Kirchhoff approach of the first kind.

Additionally a modified Kirchhoff approach is introduced. Because the incident pressure on the scatterer's surface is known the integral is evaluated analytically on triangular patches. The discretization is no longer frequency dependent and the size of the patches only depends on the curvature of the structure. Large planar parts can be discretized with one element only. This leads to a substantial advantage in terms of calculation time over the traditional Kirchhoff approach. Like the traditional approach this procedure is valid under the assumption of high frequency or far field conditions.

A semi-analytic method based on the propagation matrix formulation of indirect boundary element method to compute response of elastic (and acoustic) waves in multi-layered media with irregular interfaces is presented. The method works recursively starting from the top-most free surface at which a stress-free boundary condition is applied, and the displacement-stress boundary conditions are then subsequently applied at each interface. The basic idea behind this method is the matrix formulation of the propagation matrix (PM) or more recently the reflectivity method as wide used in the geophysics community for the computation of synthetic seismograms in stratified media. The reflected and transmitted wave fields between arbitrary shapes of layers can be computed using the indirect boundary element (BEM) method. Like any standard BEM methods, the primary task of the BEM-based propagation matrix method (thereafter called PM–BEM) is the evaluation of element boundary integral of the Green's function, for which there are standard method that can be adapted. In addition, effective absorbing boundary conditions as used in the finite difference numerical method is adapted in our implementation to suppress the spurious arrivals from the artificial boundaries due to limited model space. To our knowledge, such implementation has not appeared in the literature. Several examples are presented in this paper to demonstrate the effectiveness of this proposed PM–BEM method for modeling elastic waves in media with complex structure.

Compared to the traditional boundary element method (BEM), the single level fast multipole boundary element method (SLFMBEM) or the multilevel fast multipole boundary element method (MLFMBEM) reduces the computational complexity of a job from O(n²) to O(n^3/2) or O(n log²n), respectively with n being the number of unknowns; this means a dramatical reduction in terms of CPU-time and storage requirement. Large scale problems, unsolvable with the traditional BEM, can be solved by using the FMBEM. In this paper, the traditional BEM, SLFMBEM, and MLFMBEM are formulated within the framework of the Burton–Miller Collocation BEM for acoustic radiation and scattering from 3D structures. Attention is especially paid to the practical aspects of the method in order to get a reliable and efficient computation code. The performance of the method is tested with practical examples, including one for computing the head-related transfer function (HRTF) between 1000 and 18 000 Hz.

A so-called FuzzBEM methodology for analyzing the influence of uncertain acoustic and structural parameters on the radiated sound field of vibrating structures combining fuzzy arithmetic and fast multipole boundary element method is introduced. Uncertainties in acoustic properties may result from uncertain parameters of the vibrating mechanical structures, e.g. material density or geometry, as well as from uncertainties in the acoustic domain, e.g. sound velocity. The use of the transformation method in the proposed approach allows to employ simulation tools based on the crisp number arithmetic by appropriate preprocessing of the fuzzy numbers modeling the uncertain input parameters and postprocessing of the simulation results to determine the fuzzy numbers for the considered output quantities.

In this contribution, the proposed FuzzBEM procedure is applied to a sound radiating, vibrating stiffened cylindrical shell where the investigated uncertainties include the shell wall thickness and the driving frequency of a monofrequency point load and the air density and sound velocity. As exemplary output quantities of acoustic performance, the acoustic pressure at multiple field points and the radiated sound power are evaluated.

The proposed coupling of fuzzy arithmetic and acoustic boundary elements yields run times two orders of magnitudes or more longer than a single BEM calculation. Nevertheless, the systematic parameterization obtained by the proposed fuzzy analysis has the potential to reveal input–output relationships difficult to identify with individual conventional BEM simulation runs.

The fast multipole boundary element method (FMBEM), which is an efficient BEM that uses the fast multipole method (FMM), is known to suffer from instability at low frequencies when the well-known high-frequency diagonal form is employed. In the present paper, various formulations for a low-frequency FMBEM (LF-FMBEM), which is based on the original multipole expansion theory, are discussed; the LF-FMBEM can be used to prevent the low-frequency instability. Concrete computational procedures for singular, hypersingular, Burton-Miller, indirect (dual BEM), and mixed formulations are described in detail. The computational accuracy and efficiency of the LF-FMBEM are validated by performing numerical experiments and carrying out a formal estimation of the efficiency. Moreover, practically appropriate settings for numerical items such as truncation numbers for multipole/local expansion coefficients and the lowest level of the hierarchical cell structure used in the FMM are investigated; the differences in the efficiency of the LF-FMBEM when different types of formulations are used are also discussed.

2.5D and 3D Green's functions are implemented to simulate wave propagation in the vicinity of two-dimensional wedges. All Green's functions are defined by the image-source technique, which does not account directly for the acoustic penetration of the wedge surfaces. The performance of these Green's functions is compared with solutions based on a normal mode model, which are found not to converge easily for receivers whose distance to the apex is similar to the distance from the source to the apex. The applicability of the image source Green's functions is then demonstrated by means of computational examples for three-dimensional wave propagation. For this purpose, a boundary element formulation in the frequency domain is developed to simulate the wave field produced by a 3D point pressure source inside a two-dimensional fluid channel. The propagating domain may couple different dipping wedges and flat horizontal layers. The full discretization of the boundary surfaces of the channel is avoided since 2.5D Green's functions are used. The BEM is used to couple the different subdomains, discretizing only the vertical interfaces between them.

The present study involves numerical assessment of two types of boundary elements, namely constant and discontinuous quadratic elements based on a hypersingular Burton and Miller boundary integral formulation to tackle spurious frequencies manifesting in exterior problems. Convergence trends of the two types of boundary element with/without the inclusion of hypersingular formulation were studied for various combinations of boundary conditions and over a wide range of frequencies. The results indicate that discontinuous quadratic elements and constant elements give comparable results, with the quadratic elements being computationally more efficient as they take lesser computational time. Nevertheless, the constant element formulation is easier to implement, and it may be used for solving exterior wave propagation problems.

The phenomenon of irregular frequencies or spurious modes when solving the Kirchhoff–Helmholtz integral equation has been extensively studied over the last six or seven decades. A class of common methods to overcome this phenomenon uses the linear combination of the Kirchhoff–Helmholtz integral equation and its normal derivative. When solving the Neumann problem, this method is usually referred to as the Burton and Miller method. This method uses a coupling parameter which, theoretically, should be complex with nonvanishing imaginary part. In practice, it is usually chosen proportional or even equal to $\mathrm{\text{i}}/k$. A literature review of papers about the Burton and Miller method and its implementations revealed that, in some cases, it is better to use $-\mathrm{\text{i}}/k$ as coupling parameter. The better choice depends on the specific formulation, in particular, on the harmonic time dependence and on the fundamental solution or Green’s function, respectively. Surprisingly, an unexpectedly large number of studies is based on the wrong choice of the sign in the coupling parameter. Herein, it is described which sign of the coupling parameter should be used for different configurations. Furthermore, it will be shown that the wrong sign does not just make the solution process inefficient but can lead to completely wrong results in some cases.

This paper reviews the equivalent source method (ESM), an attractive alternative to the standard boundary element method (BEM). The ESM has been developed under different names: method of fundamental solutions, wave superposition method, equivalent source method, etc. However, regardless of the method name, the basic concept is very similar; that is to use auxiliary points called equivalent sources to reconstruct the acoustic pressure for radiation or scattering problems. The strength of the equivalent sources are then determined via various approaches such that the boundary conditions on the boundary surface are satisfied. This paper reviews several frequency-domain and time-domain ESMs. There are several distinct advantages in these types of methods: (1) the method is a meshless approach so that it is easy and simple to implement; (2) it does not have a numerical singularity problem that occurs in the BEM; (3) the number of equivalent sources can be fewer than the number of surface collocation points so that the matrix size is reduced and a fast computation is achieved for large problems. The main issue of the ESM is that there is no rule to find out the optimal number and position of equivalent sources. In addition, the ESM suffers from the numerical instability that is associated with the ill-conditioned matrix. Some guidelines have been suggested in terms of finding the number and position of the sources, and several numerical techniques have been developed to resolve the numerical instability. This paper reviews the common theories, numerical issues and challenges of the ESM, and it summarizes recent developments and applications of the ESM to aircraft noise.

In recent years, boundary element method (BEM) and finite element method (FEM) implementations of acoustics in fluids with viscous and thermal losses have been developed. They are based on the linearized Navier–Stokes equations with no flow. In this paper, such models with acoustic losses are applied to an acoustic metamaterial. Metamaterials are structures formed by smaller, usually periodic, units showing remarkable physical properties when observed as a whole. Acoustic losses are relevant in metamaterials in the millimeter scale. In addition, their geometry is intricate and challenging for numerical implementation. The results are compared with existing measurements.

A plasmonic nanosensor is proposed and investigated its sensitivity using the optical properties of plasmonic nanoparticles. To this end, we first consider the nanosensor consisting of bowtie nanoparticles. Then, the nanosensor performance is examined under different factors such as the refractive index of the background environment, the height, and length of the nanoparticles. The proposed bowtie nanoparticles, in this work, are made of gold. The boundary element method is used to simulate the nanosensor. The extinction cross-section is computed in terms of wavelength and the effect of various factors on the resonance wavelength of localized surface plasmon is investigated. It is shown that the nanosensor investigated in this research has a high sensitivity to the changes in the refractive index of the studied sample. The sensitivity of the nanosensor is obtained as 650nm/RIIU. In addition, the required spectral range can be arbitrarily adjusted by the type of nanoparticles.

In this paper, a BEM-based meshless method is developed for buckling analysis of elastic plates with various boundary conditions that include elastic supports and restraints. The proposed method is based on the concept of the Analog Equation Method (AEM) of Katsikadelis. According to this method, the original eigenvalue problem for a governing differential equation of buckling is replaced by an equivalent plate bending problem subjected to an appropriate fictitious load under the same boundary conditions. The fictitious load is established using a technique based on BEM and approximated by using the radial basis functions. The eigenmodes of the actual problem are obtained from the known integral representation of the solution for the classical plate bending problem, which is derived using the fundamental solution of the biharmonic equation. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The method has all the advantages of the pure BEM. To validate its effectiveness, accuracy as well as applicability of the proposed method, numerical results of various problems are presented.

The fast multipole method (FMM) is an effective approach for accelerating the computation efficiency of the boundary element method (BEM) in solving problems that are computationally intensive. This paper presents two different BEMs, i.e., Kress' and Seydou's methods, for solving two-dimensional (2D) acoustic transmission problems with a multilayered obstacle, along with application of the FMM to solution of the related boundary integral equations. Conventional BEM requires O(MN²) operations to compute the equations for this problem. By using the FMM, both the amount of computation and the memory requirement of the BEM are reduced to order O(MN), where M is the number of layers of the obstacle. The efficiency and accuracy of this approach in dealing with the acoustic transmission problems containing a multilayered obstacle are demonstrated in the numerical examples. It is confirmed that this approach can be applied to solving the acoustic transmission problems for an obstacle with multilayers.

This paper studies the dynamic interaction for two adjacent rigid foundations embedded in a viscoelastic soil layer. The vibrations originate from one of the rigid foundations placed in the soil layer, which are subjected to harmonic loads of translation, rocking, and torsion. The dynamic responses of the rigid surface foundations are solved from the wave equations by taking into account their interaction. The solution is formulated using the frequency domain boundary element method (BEM), in conjunction with Kausel–Peek Green’s function for a layered stratum and the thin layer method (TLM) to account for the interaction between the two foundations. This approach allows us to establish a mathematical model for determining the compliance functions of the two adjacent foundations with regard to their spacing, substratum depth, masses, shapes, embedding, load intensity, and frequencies of excitation. The soil heterogeneity was taken into account for the cases of one or two layers of soils over a rigid bedrock and semi-infinite soil. The analysis of the present study indicates that the effect of several parameters on the dynamic interaction response of two adjacent foundations is nonnegligible. In particular, the dominant influence of some parameters, such as the heterogeneity of the soil, shape of the foundations, and the load intensity, compared to the other ones is clearly revealed.

In this paper, hydroelastic behavior of a pontoon-type very large floating structure (VLFS) subjected to a moving single axle vehicle is computed using a novel numerical approach, in which the boundary element method (BEM) is firstly extended to cooperate with the moving element method (MEM), named the BEM–MEM. By utilizing this paradigm, the plate and fluid are discretized into “moving structural element” and “moving boundary element”, respectively, which are conceptual elements and “travel” with the moving vehicle. Thus, the proposed method can absolutely eliminate the need of keeping track the location of the moving load with respect to the floating structure. Particularly, the surrounding fluid is defined based on the potential flow theory and the motion of a floating plate is governed by the vibration equation of a thin plate. The governing equations of motion, moving element and fluid matrices of boundary element are formulated in a relative coordinate system traveling with the moving vehicle. Several examples are numerically conducted to illustrate the performance and ability of the BEM–MEM. Its obtained results are compared with those of the traditional finite element method for validation. The outcomes reveal that the proposed method is effective for the large-time behavior owing to the fact that it does not require a domain with the length greater than the horizontal displacement of the vehicle. The paper also discusses the effect of the liquid and structural parameters on responses of the vehicle and floating structure.

The aim of this paper is to determine the two constant parameters corresponding to the physical properties of a homogeneous heat conductor, namely, the heat capacity and the thermal conductivity, from heat flux and temperature measurements. An iterative nonlinear least-squares boundary element method is proposed. The inversion is performed for both exact and noisy measurements. Numerically, it is shown that the thermal properties can uniquely and stably be retrieved from two measurements containing at least one heat flux measurement for finite homogeneous heat conductors, whilst theoretically, for the semi-infinite conductor it is shown that one heat flux and one internal temperature measurement are necessary and sufficient.