Boundary Element Calculations Research Articles

The influence of reflections from the train body and the ground on the sound radiation from a railway rail

PurposeThe vibration of the rails is a significant source of railway rolling noise, often forming the dominant component of noise in the important frequency region between 400 and 2000 Hz. The purpose of the paper is to investigate the influence of the ground profile and the presence of the train body on the sound radiation from the rail.Design/methodology/approachTwo-dimensional boundary element calculations are used, in which the rail vibration is the source. The ground profile and various different shapes of train body are introduced in the model, and results are observed in terms of sound power and sound pressure. Comparisons are also made with vibro-acoustic measurements performed with and without a train present.FindingsThe sound radiated by the rail in the absence of the train body is strongly attenuated by shielding due to the ballast shoulder. When the train body is present, the sound from the vertical rail motion is reflected back down toward the track where it is partly absorbed by the ballast. Nevertheless, the sound pressure at the trackside is increased by typically 0–5 dB. For the lateral vibration of the rail, the effects are much smaller. Once the sound power is known, the sound pressure with the train present can be approximated reasonably well with simple line source directivities.Originality/valueNumerical models used to predict the sound radiation from railway rails have generally neglected the influence of the ground profile and reflections from the underside of the train body on the sound power and directivity of the rail. These effects are studied in a systematic way including comparisons with measurements.

Modeling of Sound Radiation from a Loaded Rolling Tire

In recent work it has been possible to model the surface vibration of a loaded rolling tire, including the influence of the acoustic mode that forms within the tire cavity. For example, it has been found that the acoustic cavity mode splits owing to the deformation of the tire when it is loaded, and that the split widens owing to Doppler effects when the tire is rolling. It has also been demonstrated that the force at the hub that results from the acoustic cavity mode is strongly dependent on interaction with circumferential structural modes of the treadband. The finite element models developed to-date, thus allow a prediction of the tire's surface vibration. To predict the sound radiation from the deformed, rolling tire, the surface vibration data has been used as the input to boundary element calculations; the effect of the ground surface is modeled by creating an "image" tire. The calculations have been used to illustrate the nature of the exterior sound radiation that results from the acoustic cavity mode, which can be relatively efficient due to the sonic phase speed of the mode within the tire cavity, as opposed to the subsonic structural wave propagation along the treadband.

Coseismic fault model of the 2017 MW 6.5 Jiuzhaigou earthquake and implications for the regional fault slip pattern

On August 8, 2017, an MW 6.5 earthquake occurred in Jiuzhaigou County, Sichuan Province, China, on the eastern margin of the Qinghai–Tibet Plateau. This study investigates the coseismic deformation field and fault model with ascending and descending Sentinel-1 synthetic aperture radar (SAR) images, aftershock distribution, and elastic half-space dislocation model. The regional fault slip pattern is then quantitatively examined using the boundary element method. The results show that the ascending and descending interferometric synthetic aperture radar (InSAR) coseismic deformation fields display an overall NNW–SSE trend, with more significant deformation on the southwest side of the fault. The coseismic fault geometry is divided into NW and SE sub-faults with strikes of 162.1° and 149.3°, respectively. The coseismic fault slip is dominated by a left-lateral strike-slip movement with an average rake of −2.31°, mainly occurring at a depth of 0–13.04 km with a shape of an approximately inverted triangle. The fault slip features two peak slip zones, with a maximum of 1.39 m. The total seismic moment is 6.34 × 1018 N·m (MW 6.47). The boundary element calculation quantitatively indicates that the regional fault slip pattern may be mainly attributable to the changing strike and dip. The strike changes from NNW–SSE to nearly NS direction, and the dip gradually decreases from the Jiuzhaigou earthquake fault in the north to the Huya fault in the south. With these characteristics, the Huya and the Jiuzhaigou earthquake faults form the eastern boundary of the Minshan uplift zone and accommodate the accumulated deformation.

Hydrodynamic interactions between a point force and a slender filament

Motivated by the transport of cargo inside cells, we study the hydrodynamic interactions between a point-force and a rigid, slender filament at a separation distance between the filament radius and its length. Using asymptotic calculations and boundary element computations, we show that the leading-order interactions are captured by a new resistive-force theory with modified drag coefficients.

Damage separation model: A replaceable particle method based on strain energy field.

We present a realistic model for simulating particle fragmentation in granular assemblies, the damage separation model (DSM), that addresses the limitations of previous methods by replacing the particle with smaller ones after fragmentation. The method is based on the calculation of the strain energy field inside the particle, and it solves the two major issues of the existing replaceable particle methods: the oversimplification of particle stress, and the unrealistic geometrical constraints needed in postbreakage replacements. Our model is formulated with three modules: (i) a boundary element calculation of stress and strain fields inside the spheropolygons that represent individual particles; (ii) a strain-energy-based theoretical framework to determine the onset of fragmentation; and (iii) an advanced geometrical algorithm, the subset separation method (SSM), to handle the postbreakage replacements in the discrete element simulations. Especially, the SSM effectively calculates the fragments required by the replacement with no geometrical limitation on the number, location, and orientation of the fracture planes. A uniaxial compression test based on laboratory setups is used to validate the method. A comparison is further conducted to study the performance of four different replaceable irregular particle methods. Results indicate that our method overcomes most of the existing issues, including stability, accuracy, and artificial constraints on the number and shape of fragments. The DSM has great potential for capturing the morphological changes of particle breakage and comminution with an unprecedented numerical resolution.

Improved accuracy for radiation damping in coupled finite element/equivalent source computations.

In coupled structural-acoustic computations, radiation damping is due to the resistive component of the surface pressure created by structural vibrations. Equivalent sources using tripole sources as basis functions can be used to compute the surface pressure forces for exterior radiation problems. This technique is similar to the Burton and Miller method for eliminating numerical difficulties due to interior acoustic resonances in boundary element computations and has been proven to yield unique solutions. However, numerical computations presented here will show that for the specific equivalent source formulation under investigation, tripole sources overpredict the resistive component of the surface impedance, especially in the mid-to-high frequency range. It will also be shown that for frequency domain calculations, an accurate representation for the resistive component of the pressure forces can be derived from an analytical representation for the source radiation resistance. Unfortunately, this technique is not applicable to time domain computations. It is also shown that more accurate results can be obtained by allowing both the simple and dipole source amplitudes to be independent variables and enforcing boundary conditions in both the exterior and interior directions simultaneously to reduce the magnitude of the interior acoustic field.

Boundary Element Calculations for Normal Contact of Soft Materials With Tensed Surface Membrane

This work considers the non-adhesive frictionless contact problem of soft materials with surface being tensed by equi-biaxial tension. The boundary element method based on Fast Fourier Transform and conjugate gradient algorithm is extended to deal with this problem. By comparing with existing analytical solutions for the axisymmetric contact between a rigid parabolic indenter and an elastic half space with constant membrane/surface tension, our numerical simulations are validated having great accuracy and efficiency. Then, with this numerical tool, we further calculate the contact responses of a soft substrate indented by a smooth indenter with general quadric profile and a rough indenter with self-affine fractal surface, respectively. Some nontrivial contact behavior resulted from the presence of membrane tension are demonstrated and discussed.

Semi-discrete Green's function for solution of anisotropic thermal/electrostatic Boussinesq and Mindlin problems: Application to two-dimensional material systems

Green's function (GF) for steady state Laplace/Poisson equation is derived for an anisotropic, finite two-dimensional (2D) composite material by solving a combined Boussinesq–Mindlin problem. A semi-discrete model of the material is developed in which only one Cartesian coordinate axis is discretized, while the other is treated as a continuous variable. The Fourier integral for the continuous coordinate is obtained analytically. Thus, a 2D problem needs only a 1D discretization. An approximate analytical estimate shows that the numerical convergence of our model is at least an order of magnitude better than fully discretized models. Numerical results are reported for the GF for a phosphorene composite containing an array of metallic inclusions. The GF is useful as a starting solution in boundary element calculations. It can be used for deriving the full solution of the Laplace/Poisson equation for an arbitrary distribution of sources and boundary values, used for modeling heat flow and electrostatic potential distribution in a 2D composite. These material systems are of strong topical interest because of their potential application in revolutionary new solid-state devices for energy conversion and quantum computing. This paper is another step towards developing GF based characterization techniques for modern 2D materials.

Fracture Analysis for Torsion Problems of a Deep Sea Spar Platform Main Body

Attention should be paid to variations of crack effects on the stability of offshore structure during building and using. The body of the deep-water Spar platform is the important support of the whole structure, once produced edge cracks in certain parts, the overall structure of the platform would be severely affected. By using boundary integral equations which applied to the edge cracks of the cylinder, this paper discussed singularity of the crack tip; then used the boundary element numerical calculation method to divide the crack boundary into many units, used different interpolation functions to calculate the stress intensity factor of crack tip; Finally, calculated torsional fracture of spar platform with edge cracks under different wind loads, got the allowed maximum crack length of Spar’s body with cracks under different wind scale. It was concluded that used the boundary element calculation method was practical in ocean engineering.

Overcoming of One More Pitfall in Boundary Element Calculations with Computer Simulations of Ion-Selective Electrode Response.

Computer simulations of ion-selective membrane electrodes using diffusion layer models based on finite-differences principle for calculating diffusion processes in both phases and taking into account the local ion exchange equilibrium at the interface are successfully used for clarifying and even predicting the influence of different diffusion factors on several time-dependent characteristics of electrodes. It is shown here that a well-established approach based on the assumption of the constant concentration of the interfering ion in the sample solution fails for solutions containing strongly interfering ions where the concentration of the interfering ion in the boundary layer of the solution can be far lower in comparison with its concentration in the bulk. The limitation is demonstrated by a drastic discrepancy between experimental and calculated curves for the dependence of potential on time. This limitation can be overcome by taking into account the change of the interfering ion concentration in the boundary layer in accordance with the electroneutrality condition. A good agreement between simulation results and experimental data is demonstrated.

Sound field reconstruction using multipole equivalent source model with un-fixed source locations.

The equivalent source methods (ESM) that have been developed to this point can generally be classified into two categories: one in which a relatively large number of lower order sources are fixed at various locations, and one in which a series of higher order sources are fixed at one location. The present work started with a model in the latter category, but the individual sources were then allowed to move separately to locations which were determined by using a nonlinear optimization procedure based on the measured sound field data. To test this approach, experiments were conducted using a small loudspeaker cabinet; measurements were made using an array of microphones on all sides of the loudspeaker. It was found that by allowing the source components to move, the sound field representation in both the near and far fields was improved, particularly at high frequencies, compared to the model with fixed source locations. By comparison with results obtained from boundary element calculations based on laser vibrometer measurements of the loudspeaker's diaphragm and tweeter velocities, it was found that the proposed ESM can also be used to accurately predict the sound pressure distribution on the source surface.

Two-dimensional pulse dynamics and the formation of bound states on electrified falling films

The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling waves on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model, which is valid in the presence of weak inertia, and second using the Stokes equations. For obtuse angles, gravity is destabilising and solitary pulses exist even in the absence of an electric field. For acute angles, spatially non-uniform solutions exist only beyond a critical value of the electric field strength; moreover, solitary-pulse solutions are present only at sufficiently high supercritical electric-field strengths. The electric field increases the amplitude of the pulses, can generate recirculation zones in the humps and alters the far-field decay of the pulse tails from exponential to algebraic with a significant impact on pulse interactions. A weak-interaction theory predicts an infinite sequence of bound-state solutions for non-electrified flow, and a finite set for electrified flow. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. In addition, the electric field is shown to trigger a switch from absolute to convective instability, thereby regularising the dynamics, and this is confirmed by time-dependent simulations of the long-wave model.

Human sperm swimming in a high viscosity mucus analogue

Remarkably, mammalian sperm maintain a substantive proportion of their progressive swimming speed within highly viscous fluids, including those of the female reproductive tract. Here, we analyse the digital microscopy of a human sperm swimming in a highly viscous, weakly elastic mucus analogue. We exploit principal component analysis to simplify its flagellar beat pattern, from which boundary element calculations are used to determine the time-dependent flow field around the sperm cell. The sperm flow field is further approximated in terms of regularised point forces, and estimates of the mechanical power consumption are determined, for comparison with analogous low viscosity media studies. This highlights extensive differences in the structure of the flows surrounding human sperm in different media, indicating how the cell-cell and cell-boundary hydrodynamic interactions significantly differ with the physical microenvironment. The regularised point force decomposition also provides cell-level information that may ultimately be incorporated into sperm population models. We further observe indications that the core feature in explaining the effectiveness of sperm swimming in high viscosity media is the loss of cell yawing, which is related with a greater density of regularised point force singularities along the axis of symmetry of the flagellar beat to represent the flow field. In turn this implicates a reduction of the wavelength of the distal beat pattern — and hence dynamical wavelength selection of the flagellar beat — as the dominant feature governing the effectiveness of sperm swimming in highly viscous media.

FE-BE computation of electromagnetic noise of a permanent-magnetic excited synchronous ma-chine considering dynamic rotor eccentricity

Electromagnetic noise in Electrical Machines (EMs) occurs due to vibrations caused by magnetic forces acting onto rotor and stator surface. This is the dominant source for the considered permanent-magnetic excited synchronous machine in this paper. The radiated electromagnetic noise is sequentially calculated by a Finite Element (FE) and Boundary Element (BE) computation. An electromagnetic FE model is created to determine magnetic forces. Structure-borne sound and rotor dynamics are calculated using a structural dynamic FE model for the EM housing and the rotor. In order to predict resonance frequencies and amplitudes as reliable as possible, it is important to know the direction-dependent stiffness of the laminated rotor stacks and mechanical joints as well as their structural damping. Thereby, the properties of the laminated stack can be determined experimentally by a shear and dilatation test. Mechanical joint properties can be modelled by Thin-Layer Elements (TLEs) and the overall damping by the model of constant hysteretic damping. The radiated sound power is determined by a direct BE computation. The influence of dynamic rotor eccentricity on radiated sound power is examined for a run-up of the EM. All FE models are verified by data from experimental modal analysis.

Boundary Element Calculation of Near-Boundary Solutions in 3D Generally Anisotropic Solids by the Self-Regularization Scheme

AbstractThis research targets investigation of the internal elastic field near the boundaries of 3D anisotropic bodies by the boundary element method (BEM). This analysis appears to be most important, especially for the interest in the internal solutions near corners or crack tips, where structural failure is most likely to occur. Although the BEM is very efficient for analyzing the problem, nearly singular integration will arise when the internal point of interest stays near the boundary. The present work is to show how the self-regularized boundary integral equation (BIE) can be applied to the interior analysis for 3D generally anisotropic bodies. So far, to the authors' best knowledge, no implementation work has been reported in the literature for successfully treating the near boundary solutions in 3D generally anisotropic solids. In the end, a few benchmark examples are presented to demonstrate the veracity of the present approach for the interior BEM analysis of 3D anisotropic elasticity.

Emission of Train-Induced Ground Vibration - Prediction of Axle-Load Spectra and its Experimental Verification

Train passages induce forces on the track, train-induced vibrations propagate through the soil and excite neighbouring buildings. The emission, which is the first part of the prediction of vibrations near railway lines, is presented by focusing on the dynamic axle loads. The calculation of the axle loads is based on the vehicle-track-soil interaction. This interaction calculus utilises the dynamic stiffness of the vehicle (the inertia of the wheelset) and the dynamic stiffness of the track-soil system. Based on various time consuming finite-element boundary-element calculations, an approximate track-soil model has been established. The vehicle-track-soil analysis yields several transfer functions between the various geometric or stiffness irregularities and the axle loads of the train. Geometric irregularities of the vehicle (the wheels) and the track (rail surface and track alignment) are the simplest components. Geometric irregularities of the subsoil (trackbed irregularities) have to be transferred to effective irregularities at rail level. The bending stiffness of the track is filtering out the short-wavelength contribution. Stiffness irregularities occur due to random variations in the ballast or the subsoil, which must also be transferred to effective track irregularities, and due to the discrete rail support on sleepers. All necessary transfer functions for the prediction of axle-load spectra are presented as general formula and as specific graphs for differing vehicle and track parameters. The prediction method is applied to a ballast track and a slab track and compared with corresponding axle-box measurements. Moreover, ground vibration measurements at numerous sites are exploited for the axle-load spectra and the validation of the prediction method. All theoretical and experimental results confirm that the dynamic axle-load spectra have an approximate value of 1 kN per third of octave and increase with train speed, track stiffness and around the vehicle-track resonance.

About Some Boundary Integral Operators on the Unit Disk Related to the Laplace Equation

We introduce four integral operators related to the Laplace equation in three dimensions on the circular unit disk. Two of them are related to the weakly singular operator and the other two are related to the hypersingular operator. We provide series expressions for their kernels using proposed bases for the Sobolev trace spaces involved in the symmetric Dirichlet and antisymmetric Neumann Laplace screen problems on the disk. We then provide explicit and closed variational forms suitable for boundary element computations. We develop numerical computation schemes for the associated Galerkin matrices and test their use as preconditioners for the matrices arising from the integral equations associated with the solution of the mentioned screen problems.

Flow noise induced by small gaps in low-Mach-number turbulent boundary layers

The flow-noise induced by small gaps underneath low-Mach-number turbulent boundary layers at Reθ = 4755 is studied using large-eddy simulation and Lighthill's theory. The gap leading-edge height is 13% of the boundary-layer thickness, and the gap width and trailing-edge height are varied to investigate their effects on surface-pressure fluctuations and sound generation. The maximum surface pressure fluctuations, which increase with gap width and trailing-edge height, occur at the trailing edge or near the reattachment point if there is separation from the trailing edge. The downstream recovery towards an equilibrium boundary layer is significantly faster for gap flows compared to step flows, and the recovery distance scales with the reattachment length for gaps with trailing-edge separation. The acoustic field is dominated by the forward-facing step in the gap and resembles forward-step sound for wide gaps and/or asymmetric gaps with trailing edge higher than leading edge. In these cases, the dominant acoustic source mechanisms are the impingement of the separated shear layer from the leading edge onto the trailing edge and the unsteady separation from the trailing edge, coupled with edge diffraction. For narrow and symmetric gaps, the destructive interference of sound from the leading and trailing edges causes a significant decline in low-frequency sound and thereby creates a broad spectral peak in the mid-frequency range. The effects of gap acoustic non-compactness and free-stream convection are investigated by comparing solutions based on a compact gap Green's function with those from a boundary-element calculation. They are found to be negligible at the typical hydroacoustc Mach number of 0.01, but become significant at Mach numbers as low as 0.1 and moderately high frequencies.

Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. For hypersingular integral equations in 2D with a positive-order Sobolev space, we analyse the mathematical relation between the (h − h/2)-error estimator from [S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008), pp. 135–162], the two-level error estimator from [M. Maischak, P. Mund, and E. Stephan, Adaptive multilevel BEM for acoustic scattering, 585 Comput. Methods Appl. Mech. Eng. 150 (1997), pp. 351–367], and the averaging error estimator from [C. Carstensen and D. Praetorius, Averaging techniques for the a posteriori bem error control for a hypersingular integral equation in two dimensions, SIAM J. Sci. Comput. 29 (2007), pp. 782–810]. All of these a posteriori error estimators are simple in the following sense: first, the numerical analysis can be done within the same mathematical framework, namely localization techniques for the energy norm. Second, there is almost no implementational overhead for the realization.

Computing Floquet–Bloch modes in biperiodic slabs with boundary elements

A new variational formulation for the boundary element computation of Floquet–Bloch solutions to the Helmholtz equation is derived. The method is applicable to geometries that are layered in the vertical and biperiodic in two horizontal directions. The discretization leads to a nonlinear Hermitian eigenvalue problem which is solved using a homotopy approach. Numerical examples demonstrate that with the boundary element approach a high accuracy can be achieved with a small number of degrees of freedom in the discretization.