Boundary Element Method Results Research Articles

Experimental Prediction Method of Free-Field Sound Emissions Using the Boundary Element Method and Laser Scanning Vibrometry

Acoustic emissions play a major role in the usability of many product categories. Therefore, mitigating the emitted sound directly at the source is paramount to improve usability and customer satisfaction. To reliably predict acoustic emissions, numerical methods such as the boundary element method (BEM) are employed, which allow for predicting, e.g., the acoustic emission into the free field. BEM algorithms need appropriate boundary conditions to couple the sound field with the structural motion of the vibrating body. In this contribution, firstly, an interpolation scheme is presented, which allows for appropriate interpolation of arbitrary velocity data to the computational grid of the BEM. Secondly, the free-field Helmholtz problem is solved with the open-source BEM software framework NiHu. The forward coupling between the device of interest and BEM is based on the surface normal velocities (i.e., a Neumann boundary condition). The BEM simulation results are validated using a previously established aeroacoustic benchmark problem. Furthermore, an application to a medical device (knee prosthesis frame) is presented. Furthermore, the radiated sound power is evaluated and contextualized with other low-cost approximations. Regarding the validation example, very good agreements are achieved between the measurements and BEM results, with a mean effective pressure level error of 0.63 dB averaged across three microphone positions. Applying the workflow to a knee prosthesis frame, the simulation is capable of predicting the acoustic radiation to four microphone positions with a mean effective pressure level error of 1.52 dB.

Heat transfer comparison investigation of the permanent magnet synchronous motor for electric vehicles based on the BEM and the FEM

In the field of heat transfer in permanent magnet synchronous motors (PMSM) for electric vehicles, the boundary element method (BEM) has been applied for the first time to calculate the steady-state temperature of the PMSM with a spiral water-cooled system. In this investigation, the boundary-integration equation for the steady-state heat transfer problem of a water-cooled PMSM is first derived on the basis of thermodynamic theory, and the system of constant coefficient differential equations is obtained by discretizing its boundaries, while the temperature results obtained from the BEM are compared with the finite element method (FEM) results. Furthermore, the temperature distribution and heat transfer characteristics obtained from the FEM and BEM were verified twice using the PMSM prototype and test platform. The results show that the maximum relative error between the temperature calculation results of FEM and BEM is 1.97%, and the maximum relative error between the results of BEM and the test does not exceed 3%, which finally verifies the validity and accuracy of BEM in solving the heat transfer problems of water-cooled PMSM.

Frequency‐domain fundamental solutions and boundary element method for consistent couple stress elastodynamic problems

AbstractIn this article, the elastodynamic problems are investigated within the framework of a consistent couple‐stress theory (CCST). The basic equations for the CCST elastodynamic problems are derived at first. These dynamic formulas are then analyzed in the frequency domain by the Fourier transform. The frequency‐domain fundamental solutions to the transformed elastodynamic problems are subsequently derived for the two‐dimensional (2D) plane‐strain state. Based on the fundamental solutions and the elastodynamic reciprocal theorem, the frequency‐domain boundary integral equations (BIEs) are established. Then, the BIEs are numerically solved by a collocation method in conjunction with analytical and numerical treatments of the arising singular integrals. To obtain the time‐domain dynamic responses, an exponential window method (EWM) is adopted to transform the frequency‐domain results to the time‐domain solutions. By using the developed boundary element method (BEM), several typical couple‐stress elastodynamic problems are numerically investigated and the size‐effects are studied by using different values of the length‐scale parameter in the CCST. Good agreements between the results obtained by the developed BEM for a tiny length‐scale parameter and the corresponding classical BEM results or analytical solutions are achieved, which validates the high accuracy of the present BEM.

Bubble dynamics in a pressure gradient with reentrant jet break through and energy loss

The dynamics of a bubble in a pressure gradient is investigated experimentally and numerically with particular emphasis on the behavior at reentrant jet impact and break through the opposite side of the bubble with corresponding energy loss and vorticity generation. High speed photography observations of a bubble generated by electric spark energy deposit in a low ambient pressure tank are coupled with wavelet based Optical Flow Velocimetry (wOFV) and Boundary Element Method (BEM) numerical analysis to examine the flow field resulting from the reentrant jet formation and break through. We study, as an illustration, the effects of the constant pressure gradient due to gravity on the bubble dynamics. Energy losses between the first and second cycle are measured for the bubbles generated under various conditions characterized by a non-dimensional pressure gradient parameter, and the corresponding measured energy loss is used in the numerical simulations. Good correspondence is seen between the image analysis, the wOFV computations, and the BEM results and insight is gained on the involved physics.

A Study on Plate Bending Analysis Using Boundary Element Method

This study presents a method for level ice-structure interaction analysis to estimate the fatigue damage of arctic structures by applying plate theory to the behavior of level ice. The boundary element method (BEM), which incurs a lower computational cost than the finite element method (FEM), was introduced to solve the plate bending problem. The BEM formulation was performed by applying the BEM to plate theory. Finally, to check the validity of the proposed method, the BEM results and FEM results obtained using the ABAQUS commercial software were compared. The response results of the BEM analysis agreed well with those of the FEM analysis. Based on the results of the analysis, the BEM approach is considered to be very powerful in level ice-structure interaction analysis for estimating level ice-induced fatigue damage. Further work is being conducted to perform level ice fracture analysis based on the stress field calculated using the boundary element method.

Analytical Model of Perturbed EMF in a Transmit-Receive Coil System Due to Thin Flaws

This work proposes a model based on the semianalytical boundary element method (BEM) to predict the perturbation of the voltage induced in an eddy current coil due to narrow opening flaws that are usually encountered in eddy current nondestructive tests. The modeling is performed for rectangular narrow opening flaws that are located on the surface of an aluminum plate of finite thickness. The excitation to the coil varies time harmonically and the BEM results are corroborated by experiments performed in a controlled environment in the laboratory.

Boundary element analysis of static plane problems in size-dependent consistent couple stress elasticity

Based on a consistent couple stress elasticity theory, a two-dimensional (2D) boundary element method (BEM) is developed to solve the boundary value problems in the couple stress elastic materials. The displacement boundary integral equations (BIEs) are applied to compute the mechanical quantities on the boundary. The integral representations for the generalized strains and stresses at internal points are also presented together with the explicit expressions for the high-order fundamental solutions. The collocation method is adopted for the spatial discretization of the BIEs. Some special numerical and semi-analytical techniques are provided for the evaluation of weakly and strongly singular integrals. The sinh transformation is used to deal with the nearly singular integrals for near-boundary internal points or boundary nodes in thin structures. An error-controlled integration scheme is adopted for computing the non-singular integrals to improve the accuracy. Three representative 2D boundary value problems in the consistent couple stress elasticity theory are studied by the developed BEM. The correctness and accuracy of the present BEM results are verified by comparison with the corresponding analytical results. Then, multiple flat elliptical holes in a plate are numerically studied to show the interaction effects on the stress concentration.

Hydrodynamic Analysis of a Multibody Wave Energy Converter in Regular Waves

A performance assessment of wave power absorption characteristics of isolated and multiple wave energy converter (WEC) rotors was presented in this study for various wave-heading angles and wave frequencies. Numerical hydrodynamic analysis of the WEC was carried out using the three-dimensional linear boundary element method (BEM) and nonlinear computational fluid dynamics (CFD). Experimental results were used to validate the adopted numerical models. Influence with and without power take-off (PTO) was estimated on both isolated and multiple WEC rotors. Furthermore, to investigate the interaction effect among WECs, a q-factor was used. Incorporation of viscous and PTO damping into the linear BEM solution shows the maximum reduction focused around peak frequency but demonstrated an insignificant effect elsewhere. The q-factor showed both constructive and destructive interactions with the increase of the wave-heading angle and wave frequencies. Further investigation based on the prototype WEC rotor was carried, and calculated results of the linear BEM and the nonlinear CFD were compared. The pitch response and q-factor of the chosen wave frequencies demonstrated satisfactory consistency between the linear BEM and nonlinear CFD results, except for some wave frequencies. Estimated optimal time-averaged power using linear BEM show that the maximum extracted power close to the zero wave-heading angle around the resonance frequency decreases as the wave-heading angle increases. Overall, the linear BEM on the extracted power is overestimated compared with the nonlinear CFD results.

A new boundary element formulation for modeling and simulation of three‐temperature distributions in carbon nanotube fiber reinforced composites with inclusions

A new boundary element method (BEM) formulation is developed in this paper for modeling and simulation of nonlinear three‐temperature (3T) distributions in carbon nanotube (CNT) fiber‐reinforced composites embedded with many rigid‐line inclusions that can be treated as having constant temperature distributions. The fast multipole method (FMM) has been applied to accelerate the proposed BEM solution of the considered problem of a large number of nano‐inclusions, and therefore, the computational complexity and computational time are reduced. The CPU time and memory requirement are also reduced by implementing generalized modified shift‐splitting (GMSS) iteration method which does not require the entire matrix to be stored in the memory. Then, we introduced a new formula which can play an important role in solving thermal problems arising in CNT nanotechnology. Effective thermal conductivity of fiber‐reinforced composites can be evaluated using the computed temperature and heat flux fields. The fast multipole BEM results are depicted graphically to show the effects of CNT manufacturing methods and thermal conductivities of different CNT composites on the temperature and heat flux distributions. The computational performance of the proposed methodology has been demonstrated. The fast multipole BEM results are compared graphically with the finite element method (FEM) results and with the experimental results reported in the literature. Excellent agreements are found in both cases, which clearly demonstrate the validity, accuracy, and efficiency of our proposed algorithm and modeling and simulation technique. The findings and solutions of this study contribute to the further development of CNT nanotechnology applications and devices.

A pure boundary element analysis of Reissner's shear deformable plates subjected to higher-order polynomial loading

In this paper, a direct Boundary Element Method (BEM) solution without load domain integrals for Reissner's plates subjected to higher-order polynomial loads is derived. To this end, the Multiple Reciprocity Method (MRM), which consists to transform original domain integrals into a series of equivalent boundary integrals using Green's second identity in a recursive way, is used. Moreover, higher-order fundamental solutions of thick plates in bending, which are essential in MRM technique, are also derived and presented in an explicit form for all terms of the series. Despite this present approach deals directly with higher-order polynomial distribution load, this boundary element solution can be also applied to other load distribution functions, since it is possible to expand them in a polynomial series. Numerical results are presented for differing cases of boundary conditions and load functions, such as linear, double quadratic, exponential, and double sinusoidal distributions. The BEM results are compared to analytical solutions showing good agreement among them.

A BEM/RANS Interactive Method Applied to an Axial Tidal Turbine Farm

In this study, an interactive method coupling a boundary element method (BEM) with a viscous flow solver solving the Reynolds-averaged Navier–Stokes (RANS) equations is applied to multiturbine interaction problems. The BEMis first applied to a single turbine problem to predict its performance with/without yaw in noncavitating/cavitating conditions. Improved wake alignment models, the full wake alignment and the unsteady wake alignment, are used to align the blade wake. The former is adequate for steady state with zero yaw, and the latter is used for unsteady predictions in the case of nonzero yaw in the incoming flow. The BEM results are compared with the experimental measurements and the results from full-blown RANS simulations for a range of tip speed ratios. The comparisons show satisfactory agreement between the numerical and experimental approaches. Afterward, the BEM/RANS coupling method is applied to multiturbine interaction problems with different layouts and different turbine-to-turbine offsets in an axial turbine farm. The method is shown to work well in this multiturbine interaction problem because of the capability of using a strictly Cartesian grid in the RANS method, which minimizes the artificial diffusion and improves the numerical accuracy of long-range flow development. Representation of a turbine by the body force/mass source fields in the BEM/RANS coupling approach reduces the number of cells required for 3D full-blown RANS simulations, and therefore reduces the computational cost in an efficient way. Introduction In recent years, interests in tidal energy harvesting have been growing because of the needs and perspectives of clean and renewable energy. Among all different types of tidal stream generators, the axial tidal turbine has been most frequently implemented because of its similarity to wind turbines and the analysis tools which are originally developed for propellers but equally applicable to turbines without extensive modifications. Because of the predominantly bidirectional flow in most tidal current sites, typical tidal turbine farms arrange the turbine in multiple lines that are perpendicular to the flow (Turnock et al. 2011). In such applications, turbine-to-turbine interaction needs to be studied to reduce the energy-shadow effect and improve the overall turbine farm efficiency.

Exploiting the limit of BEM solvers in moonpool type floaters

Solvers based on Boundary Element Method (BEM) are fast and they have proven to provide accurate results in a wide range of applications. On the other hand, computational fluid dynamics (CFD) solvers are high-fidelity tools able to account for viscous effects. However, they are computationally demanding. In the present work, the limitations of BEM solvers are exploited considering the case study of a moonpool type floater in which the viscous effects near the sharp edges of the body (vortex shedding) are not negligible. The BEM results are compared with the results from an unsteady Reynolds averaged Navier-Stokes (URANS) CFD solver and experimental data, while viscous corrections of the BEM method are assessed.

Iterative multipole solution for wave interaction with submerged partially perforated semi-circular breakwater

This study develops an iterative multipole solution for water wave interaction with a submerged partially perforated semi-circular breakwater based on the linear potential theory. Multipole expansion method and separation variable method are used for getting the velocity potential expressions of the flow fields, and the expansion coefficients in velocity potentials are determined by iteratively solving the non-linear pressure loss boundary condition on the partially perforated wall. The calculation results of the present multipole solution are in good agreement with the numerical results of multi-domain BEM (boundary element method) solution. The effects of the radius, porosity and perforated position on the reflection coefficient, transmission coefficient and wave force of breakwater are thoroughly discussed. Some meaningful suggestions are presented for engineering design.

Investigation of Rayleigh wave interaction with surface defects

The current article is concerned with the interaction of Rayleigh waves with surface defects of arbitrary shape in a homogeneous, isotropic, linearly elastic half-space. Using a linear superposition principle, the interaction generates a scattered field which is equivalent to the field radiated from a distribution of horizontal and vertical tractions on the surface of the defect. These tractions are equal in magnitude but opposite in sign to the corresponding tractions obtained from the incident wave. The scattered field is then computed as the superposition of the displacements radiated from the tractions at every point of the defect surface using the reciprocity theorem approach. The far-field vertical displacements are compared with calculations obtained by the boundary element method (BEM) for circular, rectangular, triangular and arbitrary-shaped defects. Comparisons between the theoretical and BEM results, which are graphically displayed, are in excellent agreement. It is also discussed the limitations of the proposed approximate theory.
 Keywords:
 half-space; Rayleigh wave; surface defect; reciprocity theorem; boundary element method (BEM).

Landing gear interwheel tonal noise characterization with the Boundary Element Method

Landing gear noise is mainly broadband in nature, but it can also feature some tonal contributions. Tonal noise significantly increases noise annoyance and can originate from facing cavities in landing gear wheels. The present work aims at using the Boundary Element Method (BEM) with a simple harmonic point source model to characterize the resonance between the two facing cylindrical cavities in the wheels of the generic nose landing gear LAGOON, where a flow independence of the tonal noise emission has been reported experimentally. Three configurations of increasing complexity are successively considered: a single wheel, the two facing wheels linked by an axle, and finally the whole landing gear, including the main strut. In terms of resonant frequencies, the BEM is shown to give results in very good agreement with analytical formulae and CFD/CAA computations from the literature. Some acoustic measurements are also performed for the whole LAGOON configuration and a very good agreement is observed between BEM results and the experimental database. The directivity of the resonant modes is detailed for each case and compared with success to CFD/CAA results, except for convective amplification effects. The amplification of each resonant mode is quantified through a so-called scattering factor and it is shown that the facing cavities present much sharper resonances than the single cavity, and that the presence of the main strut only increases the amplification of the axisymmetric mode. Finally, the modelisation of the interwheel space by an annular duct is discussed and a parametric study is conducted on the axle diameter. The results of this parametric study shed light on the annular duct model domain of validity while demonstrating the ability of the BEM to treat geometries of arbitrary complexity with low computational cost when analytical models are unavailable.

Numerical and Experimental Study of the Solo Duck Wave Energy Converter

The Edinburgh Duck is one of the highly-efficient wave energy converters (WECs). Compared to the spine-connected Duck configuration, the solo Duck will be able to use the point absorber effect to enhance its power capture performance. In this paper, a 3D computational fluid dynamic (CFD) model is developed to predict the hydrodynamic performance of the solo Duck WEC in regular waveswithin a wide range ofwave steepness until the Duck capsizes. A set of experiments was designed to validate the accuracy of the CFD model. Boundary element method (BEM) simulations are also performed for comparison. CFD results agree well with experimental results and the main difference comes from the friction in the mechanical transmission system. CFD results also agree well with BEM results and differences appear at large wave steepness as a result of two hydrodynamic nonlinear factors: the nonlinear waveform and the vortex generation process. The influence of both two nonlinear factors iscombined to be quantitatively represented by the drag torque coefficient.The vortex generation process is found to cause a rapid drop of the pressure force due to the vortexes taking away the kinetic energy from the fluid.

A new solution technique for cathodic protection systems with homogeneous region by the boundary element method

The purpose of this work is to efficiently evaluate the design of cathodic protection (CP) systems of tank bottoms using concentric ring or linear anodes. As customary in current CP systems, the outer surface of the tank bottom is usually in electrical contact with a slender homogeneous layer of conductive concrete (or something similar) which in turn is in direct contact with the homogeneous deep soil region. The boundary element method (BEM) together with a subregion technique has been widely adopted to analyse such CP systems where the domain consists of two (or even more) homogeneous zones. However, the numerical solution of the final matrix system of equations can be quite time-consuming, especially if the slender intermediate layer is to be discretised, requiring a considerable number of elements, due to its somewhat reduced thickness. To overcome this problem, the present work proposes a new methodology in which the slender subregion is indirectly introduced, as a theoretically created polarisation curve, acting as a new boundary condition at the boundary of the soil domain (original common interface). Numerical simulations have been carried out using BEM implementations and results are discussed, including CP studies of practical axisymmetric and three-dimensional engineering problems.

Experimental Validation of Fluid–Structure Interaction Computations of Flexible Composite Propellers in Open Water Conditions Using BEM-FEM and RANS-FEM Methods

In the past several decades, many papers have been published on fluid–structure coupled calculations to analyse the hydro-elastic response of flexible (composite) propellers. The flow is usually modelled either by the Navier–Stokes equations or as a potential flow, by assuming an irrotational flow. Phenomena as separation of the flow, flow transition, boundary layer build-up and vorticity dynamics are not captured in a non-viscous potential flow. Nevertheless, potential flow based methods have been shown to be powerful methods to resolve the hydrodynamics of propellers. With the upcoming interest in flexible (composite) propellers, a valid question is what the consequences of the potential flow simplifications are with regard to the coupled fluid–structure analyses of these types of propellers. This question has been addressed in the following way: calculations and experiments were conducted for uniform flows only, with a propeller geometry that challenges the potential flow model due to its sensitivity to leading edge vortex separation. Calculations were performed on the undeformed propeller geometry with a Reynolds-averaged-Navier–Stokes (RANS) solver and a boundary element method (BEM). These calculations show some typical differences between the RANS and BEM results. The flexible propeller responses were predicted by coupled calculations between BEM and finite element method (FEM) and RANS and FEM. The applied methodologies are briefly described. Results obtained from both calculation methods have been compared to experimental results obtained from blade deformation measurements in a cavitation tunnel. The results show that, even for the extreme cases, promising results have been obtained with the BEM-FEM coupling. The BEM-FEM calculated responses are consistent with the RANS-FEM results.

A theoretical approach to multiple scattering of surface waves by shallow cavities in a half-space

The current theoretical work investigates multiple scattering of surface waves from shallow cavities at the surface of an elastic half-space. The multiple scattered field is shown to be equivalent to the field generated by the application of a distribution of tractions, obtained from the incident wave, on the surfaces of the cavities. Using a self-consistent method, an analytical approach is proposed to derive an explicit set of equations which approximate the multiple scattered field. Its far-field displacement amplitude is then calculated and verified by numerical calculation obtained by the boundary element method (BEM). The analytical and BEM results are graphically presented and show excellent agreement when the depths of cavities are small compared to the wavelength. The improvement of the proposed approach relative to a first order approximation is reported and its limitations are discussed based on the results of comparison.

A new solution technique for cathodic protection systems with homogeneous region by the boundary element method

The purpose of this work is to efficiently evaluate the design of cathodic protection (CP) systems of tank bottoms using concentric ring or linear anodes. As customary in current CP systems, the outer surface of the tank bottom is usually in electrical contact with a slender homogeneous layer of conductive concrete (or something similar) which in turn is in direct contact with the homogeneous deep soil region. The boundary element method (BEM) together with a subregion technique has been widely adopted to analyse such CP systems where the domain consists of two (or even more) homogeneous zones. However, the numerical solution of the final matrix system of equations can be quite time-consuming, especially if the slender intermediate layer is to be discretised, requiring a considerable number of elements, due to its somewhat reduced thickness. To overcome this problem, the present work proposes a new methodology in which the slender subregion is indirectly introduced, as a theoretically created polarisation curve, acting as a new boundary condition at the boundary of the soil domain (original common interface). Numerical simulations have been carried out using BEM implementations and results are discussed, including CP studies of practical axisymmetric and three-dimensional engineering problems.