Boundary Element Method Model Research Articles

Modified BEM for scattering analysis by a flaw at interface in an anisotropic multi-layered plate

In this paper, a modified boundary element method (BEM) for time-harmonic scattering computation analysis in two-dimensional (2-D), homogeneous, anisotropic and linear elastic plates is proposed. The aim of modification is to correct the spurious scattering introduced by inevitable model truncation at far-field in the traditional BEM model. The far-field wave displacement fields beyond the truncation points of BEM model are assumed to be the superposition of orthogonal propagating guided wave patterns, and are finally incorporated into BEM equation systems as the modified items to account for the contribution of infinite boundaries traditionally omitted. This method is simple and elegant, which has advantages on dimension reduction and needs no absorption medium or perfectly matched layer to suppress the reflected waves compared to FEM. Also, we can obtain the reflection and transmission coefficients of each mode directly without post-processing. This modified BEM is implemented to solve wave scattering problems due to a cavity-type defect at the interface in an infinite anisotropic multi-layered plate. The formulation is finally validated for several benchmark problems.

Effects of gravity on natural oscillations of sessile drops

Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$ . Three types of $\alpha$ – $Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$ . As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$ . This implies that the ‘walking’ drop instability reported in previous literature does not exist.

Experimental characterization of a shape optimized acoustic lens: Application to compact speakerphone design

This work presents the shape optimization and subsequent experimental validation of an acoustic lens with application to a compact loudspeaker, such as found in commercial speakerphones. The shape optimization framework is based on a combined lumped parameter and boundary element method model using free form deformation geometry parameterization. To test the optimized design, the loudspeaker lens is three-dimensionally printed and experimentally characterized under anechoic conditions on a finite baffle with respect to its off-axis frequency response. The overall tendencies of the frequency responses agree well between measurement and simulations within the optimization frequency range and at low frequencies. The optimization process is applied to a model including acoustic lumped parameter approximations. The shortcomings of the assumptions made in the model are revealed by laser Doppler vibrometer measurements of the loudspeaker driver and modelling of the mechanical vibrations of the lens.

A Nonlinear Fractional BEM Model for Magneto-Thermo-Visco-Elastic Ultrasound Waves in Temperature-Dependent FGA Rotating Granular Plates

The primary goal of this study is to create a nonlinear fractional boundary element method (BEM) model for magneto-thermo-visco-elastic ultrasound wave problems in temperature-dependent functionally graded anisotropic (FGA) rotating granular plates in a constant primary magnetic field. Classical analytical methods are frequently insufficient to solve the governing equation system of such problems due to nonlinearity, fractional order heat conduction, and strong anisotropy of mechanical properties. To address this challenge, a BEM-based coupling scheme that is both reliable and efficient was proposed, with the Cartesian transformation method (CTM) used to compute domain integrals and the generalized modified shift-splitting (GMSS) method was used to solve the BEM-derived linear systems. The calculation results are graphed to show the effects of temperature dependence, anisotropy, graded parameter, and fractional parameter on nonlinear thermal stress in the investigated plates. The numerical results validate the consistency and effectiveness of the developed modeling methodology.

Nonlinear dynamic analysis of aquaculture platforms in irregular waves based on Hilbert–Huang​ transform

This study aimed to examine the nonlinear dynamics of aquaculture platforms using the Hilbert–Huang transform (HHT). First, a numerical model that couples the boundary element method and lumped-mass model was used to simulate the dynamic behavior of a multi-body floating aquaculture platform in irregular waves. Second, a time–frequency analysis method based on the HHT was used to analyze the dynamic responses, including motion and mooring force responses. The results highlight that the surge motion of the net cage is dominated by the low-frequency components, whereas the heave and pitch motions are dominated by the wave-frequency components. The pitch response of the net cage has an obvious energy concentration near the peak frequency region of the incident wave. The surge motion of the net cage in the middle of the platform has a greater response to wave components with lower frequencies than that of the other two cages. The inherent characteristic vibration form of the mooring force response is affected by both the low-frequency characteristic of surge motion and the wave-frequency characteristic of motions in the vertical plane. The amplitudes of the wave-frequency components of the mooring force are much larger than those of the low-frequency components, which is like the characteristics of heave and pitch motions. The gravity periods of the wave-frequency mode components of the mooring force response were very close to the gravity period of the surge motion.

Fractional Dual-Phase-Lag Model for Nonlinear Viscoelastic Soft Tissues

The primary goal of this paper is to create a new fractional boundary element method (BEM) model for bio-thermomechanical problems in functionally graded anisotropic (FGA) nonlinear viscoelastic soft tissues. The governing equations of bio-thermomechanical problems are briefly presented, including the fractional dual-phase-lag (DPL) bioheat model and Biot’s model. The more complex shapes of nonlinear viscoelastic soft tissues can be handled by the boundary element method, which also avoids the need for the interior domain to be discretized. The fractional dual-phase-lag bioheat equation was solved using the general boundary element method (GBEM) based on the local radial basis function collocation method (LRBFCM). The poroelastic fields are then calculated using the convolution quadrature boundary element method (CQBEM) The numerical findings show that our proposed numerical model is valid, efficient, and accurate.

Numerical prediction of underwater noise on a flat hull induced by twin or podded propeller systems

In this article, the numerical prediction of underwater noise induced by multi-propulsor systems is investigated on a flat hull by solving the Ffowcs Williams–Hawkings (FW–H) equation in the form proposed by Farassat. To get the noise source, the hydrodynamic analysis is first performed using twin propellers and contra-rotating propellers via a boundary element method (BEM) in non-cavitating flow. In order to fully consider viscous interactions between neighboring propellers, incoming flow, and non-axisymmetric pod, a BEM/Reynolds-averaged Navier–Stokes (RANS) interactive method is utilized, in which two coaxial propellers are handled by a separate BEM model, and their mutual interactions are considered via a RANS solver. The predicted hydroacoustic and hydrodynamic performance from various configurations is validated with experimental data and results from viscous RANS simulations. Along with the noise spectrum in the flow field, the time history of acoustic pressures on the top wall is investigated to see how the mutual interactions among neighboring propellers and the pod affect the acoustic characteristics. A proper way of considering scattered/reflected noise from underwater geometry in a half-space domain is discussed, and importantly, the limited influence of the linear noise terms of the FW–H equation is pointed out, especially when the flow non-linearities, such as wake turbulence and structure-induced vortices, start to dominate the downstream flow.

An Automatic Finite Element Method-Boundary Element Method Coupling Method for Elastic–Plastic Problems of Multiscale Structures in Electronic Packaging

Abstract We introduce a coupled finite and boundary element method for elastic-plastic analysis over multiscale electronic packaging structures. Based on the finite element-boundary element (FE-BE) coupling algorithm, an automatic implementation procedure for the coupling of the abaqus with a self-written elastic BE code is introduced for elastic problems. In the mixed finite element method (FEM)-boundary element method (BEM) model, the effective stiffness and effective forces at the interfacial boundary are evaluated by the self-written BE code. Then, the obtained effective stiffness and effective forces are assembled to the global FE formulations by using the user subroutine (UEL) in abaqus. Numerical simulation of structures with plastic deformation, stress concentration, etc. is carried out by using FEM theory. The boundary element method is used for linear elastic domains with large-scale structures. The proposed method offers several key improvements compared with current analysis methods available for multiscale electronic packaging structures. The benefits are: (i) the powerful pre- and postprocessing of abaqus; (ii) the higher accuracy of the solution; (iii) the computational cost and time can be reduced by using the scheme; and (iv) solving systems with infinite extension by using the BEM as a supplement. Furthermore, we demonstrate the ability of the proposed approach to handle multiscale structures in electronic packaging problems.

Experimental and numerical investigations on the flow-induced vibration and acoustic radiation of a pump-jet propulsor model in a water tunnel

Experimentally investigating the vibration characteristics of a pump-jet propulsor in water is challenging. In this study, the characteristics of flow-induced vibration and acoustic radiation of a pump-jet operating in a water tunnel were experimentally and numerically investigated. A customised vibration acceleration acquisition instrument was used to measure the vibration response of the rotor. Numerical simulation methods are established based on Computational Fluid Dynamics (CFD), coupled Finite Element Method (FEM), and Boundary Element Method (BEM). The CFD model, verified by the hydrodynamic performance, was used to numerically investigate the fluctuation pressure and unsteady forces of the pump-jet. The coupled FEM model verified by the rotor mode and BEM model was used to calculate the flow-induced vibration response under distributed unsteady forces and acoustic radiation, respectively. The results show that the vibration response of the rotor has prominent peaks at Shaft Frequency (SF) and Blade Passing Frequency (BPF) for the experimental and coupled FEM results. The inadequate cancellations of the spatial distribution and phase distribution of the unsteady rotor forces contributed to the high amplitude at SF, in addition to the rotor–shaft system imbalance. The Sound Pressure Level (SPL) after the superposition was maximal at BPF and had a relatively high amplitude at SF, in agreement with the experimental results. The established simulation method can reasonably obtain the characteristics of flow-induced vibration and acoustic radiation of the pump-jet and demonstrates a similar variational trend as the experimental results.

3D structure-soil-structure interaction in an arbitrary layered half-space

The aim of the study is to evaluate the 3D dynamic response of a finite geological region containing two structures and rested on a semi-infinite elastic layered half-space with a dynamic source radiating transient waves. The hybrid modelling approach is applied. It is based on the decomposition of the whole domain under consideration into two sub-regions: a finite-sized near-field elastic isotropic zone with two containment structures and the open semi-infinite far-field layered region. The far-field zone is a semi-infinite elastic isotropic arbitrary layered medium where the near-field finite geological region is located on. The 3D hybrid computational tool is based on the boundary element method (BEM) for the far-field layered zone and the finite element method (FEM) for the finite near-field domain. The model for the semi-infinite layered zone is further extended by the incorporation of a new condensation algorithm which makes it possible to handle 3D wave propagation through arbitrary layered half-space. The condensation algorithm is developed to avoid high computational memory cost while retaining the compatibility with the hybrid FEM-hosted procedure which facilitates the useful solution for the practical three-dimensional engineering problems. The BEM model of the dynamically active far-field zone is inserted as a macro-finite element (MFE) in the FEM commercial program ABAQUS. The accuracy and convergence study of the hybrid numerical scheme is presented. Numerical simulations convincingly illustrate that the dynamic response of structure-soil-structure system depends on different key factors and their mutual interplay. These factors are arbitrary layering of the far-field geological zone, the characteristics of the dynamic source, the site effects phenomena, the structure-soil-structure dynamic interaction, the type and geometrical disposition of foundations and structures and the 3D features of the dynamic motion.

Non-cuboid Iterative room optimizer

In past years, various iterative optimization programs emerged to separately determine the optimal room ratios, sources and listening positions of perfectly reflective cuboid rooms, using the image-source model. Despite its fast computation times, this approach does not account for scattering, phase change at the boundary and cannot be extended to non-cuboid rooms. This presentation describes the current status of a program called NIRO, that offers a solution to those issues, by using the Boundary Element Method (BEM) to compute the frequency response from 20–200 Hz, considering the effects of the boundary’s complex admittance and all acoustical elements inside the room. With BEM as its engine, a room optimization genetic algorithm was developed to optimize source and receiver positions simultaneously with the room geometry, aiming to present the best possible acoustic environment given imposed restraints. To control the room's temporal decay, low-frequency acoustic treatments were added to the BEM model. By using transfer matrix models, the acoustical behavior of different multilayered treatments can be modeled and inserted into the BEM simulation to evaluate the change in the room’s acoustic field and in the frequency response at the receiving positions. 3D waterfall plots illustrate the temporal decay following optimization. Examples will be presented.

An efficient frequency domain analysis method for bridge structure-borne noise prediction under train load and its application in noise reduction

Development of an efficient method for solving the structure-borne noise of the bridge in the frequency domain is of concern. Based on the stochastic theory, the vehicle-track coupling model is established in the frequency domain, and its frequency domain response and the force transmitted to the bridge are analyzed by the pseudo excitation method-symplectic method (PEM-SM). Thereafter, the field vibration and noise test of a 30 m U-shaped concrete subway girder are carried out to compare the cases before and after installing damping fasteners. In the frequency domain, after the input of the force transmitted to the bridge, the finite element–boundary element method (FE-BEM) model of the girder is developed to predict the vibration and structure-borne noise. There is a rationally good agreement between the results of numerical simulations and those of experiment, suggesting an innovative idea for the rapid prediction of structural vibration and noise. Finally, the noise reduction effect and mechanism of the acoustic board and steel spring floating plate beneath the track are discussed. The obtained results reveal that reducing the input of the track system to the bridge is better than preventing the vibration and noise radiation of bridge plates. The above results provide a theoretical and experimental basis for calculating and controlling of vibration and noise in rail transit.

Modelling of the fatigue cracking resistance of grid reinforced asphalt concrete by coupling fast BEM and FEM

We present a computational modelling approach aimed at investigating the effect of fibre grid reinforcement on crack opening displacement and fatigue crack propagation. Grid reinforcements are modelled using elastic membrane finite elements, while the cracked concrete is treated using a symmetric boundary element method (BEM), which in particular allows easy geometrical modelling and meshing of cracks. The BEM is accelerated by the fast multipole method, allowing the handling of potentially large BEM models entailed by three-dimensional configurations hosting multiple cracks. Fatigue crack growth is modelled using the Paris law. The proposed computational approach is first verified on a reinforced cracked beam, and then applied to a three-dimensional configuration featuring a grid-reinforced asphalt pavement.

Energy-maximising experimental control synthesis via impedance-matching for a multi degree-of-freedom wave energy converter

We present, in this paper, an experimental framework for design and synthesis of impedance-matching-based (IM) controllers capable of maximising energy extraction in inherently multi degree-of-freedom wave energy converter (WEC) systems, and its subsequent application to the Intertial Sea Wave Energy Converter (ISWEC) device, by incorporating recent advances in IM-based theory. In particular, we consider a 1/20th scale prototype of the ISWEC system, tested as part of a larger experimental campaign conducted within the tank facilities available at Università degli Studi di Napoli Federico II, subject to a variety of wave conditions. We adopt two different control structures to realise an approximation of the IM principle, fully tuned based upon interpolation of a particular (experimentally obtained) non-parametric empirical transfer function estimate, which defines the optimal frequency-domain input-output response for energy-maximising behaviour. Furthermore, a performance comparison between controller tuning based upon traditional linear boundary element method models, and the presented experimental approach, is also offered, showing that the latter can consistently outperform the former in realistic scenarios, for the set of analysed sea-states.

Numerical analysis of wave\u2013structure interaction of regular waves with surface-piercing inclined plates

The Mocean wave energy converter consists of two sections, hinged at a central location, allowing the device to convert energy from the relative pitching motion of the sections. In a simplified form, the scattering problem for the device can be modelled as monochromatic waves incident upon a thin, inclined, surface-piercing plate of length L' in a finite depth d' of water. In this paper, the flow past such a plate is solved using a Boundary Element Method (BEM) and Computational Fluid Dynamics (CFD). While the BEM solution is based on linear potential flow theory, CFD directly solves the Navier–Stokes equations. Problems of this type are known to exhibit near-perfect reflection (indicated by a reflection coefficient |R|approx 1) of waves at specific wavenumbers k'. In this paper, we show that the resonant motion of the fluid induces large hydrodynamic forces on the plate. Furthermore, we argue that this low-frequency resonance resembles Helmholtz resonance, and that Mocean’s device being able to tune to these low frequencies does not act like an attenuator. For the case where the water is deep (d'>lambda '/2, where lambda '=2pi /k'), we find excellent agreement between our simulations and previous semi-analytical studies on the value of the resonant wave periods in deep water. We also find excellent agreement between the excitation forces on the plate computed using the BEM model, analytical results, and CFD for large inclination angles (alpha > 45^circ ). For alpha le 15^circ , both methods show the same trend, but the CFD predicts a significantly smaller peak in the excitation force compared with BEM, which we attribute to non-linear effects such as the non-linear Froude–Krylov force

Boundary element modeling of fractional nonlinear generalized photothermal stress wave propagation problems in FG anisotropic smart semiconductors

The main aim of this article is to develop an efficient boundary element method (BEM) modeling of the fractional nonlinear generalized photo thermal stress wave propagation problems in the context of functionally graded (FG) anisotropic smart semiconductors. Due to nonlinearity, fractional order heat conduction and strongly anisotropy of mechanical properties, the governing equations system of such problems is often very difficult to solve using classical analytical methods. Therefore, a reliable and efficient coupling scheme based on BEM was proposed to address this challenge, where, the Cartesian transformation method (CTM) has been implemented to calculate the domain integrals, and the generalized modified shift-splitting (GMSS) has been implemented for solving the linear systems arising from BEM. The calculation findings are depicted in graphical forms to display the impacts of temperature-dependent, anisotropy, piezoelectric, graded parameter and fractional parameter on the nonlinear photo thermal stress wave propagation in the considered structure. The numerical findings confirm the consistency and efficacy of the developed modeling methodology.

Numerical and experimental modelling of wave interaction with fixed and floating porous cylinders

We consider wave forces on fixed porous cylinders with and without a solid inner cylinder and wave-induced motions of floating cylinder with and without a porous outer cylinder. Comparisons between experimental measurements and numerical predictions from an iterative boundary element method (BEM) model are presented. The BEM model assumes that pressure drop across porous surface is proportional to the square of the velocity through the surface. It is shown that the BEM model is able to accurately predict the nonlinear variation of the forces with wave amplitude or motion amplitude. It is demonstrated that adding a porous outer cylinder to a solid vertical cylinder leads to increased excitation force on the combined structure. For floating cylinders adding a porous outer cylinder also leads to a corresponding increase in excitation force. However, the porous outer cylinder provides a larger increase in the damping, resulting in reduced motion response. Further numerical simulations indicate that placing the porous cylinder lower in the water column can lead to increased damping without the corresponding increase in excitation forces. It is shown that for low Keulegan Carpenter numbers, the damping coefficient for a porous cylinder is significantly higher than the viscous damping on a solid cylinder. The results suggest that porous materials could be beneficial for motion damping of floating structures.

Adhesive Boundary Element Method Using Virtual Crack Closure Technique

In this study, a new adhesive contact model is built upon a boundary element method (BEM) model developed by Pohrt and Popov (2015). The strain energy release rate (SERR) on the edge of the bonding interface is evaluated using Virtual Crack Closure Technique (VCCT) which is shown to have better accuracy and weaker mesh-size dependency than the closed-form SERR formula derived by Pohrt and Popov. A composite delamination criterion is proposed for crack nucleation and propagation. Numerical results predicted by the present model are in good agreement with the analytical solutions of two classic problems, namely, the axisymmetric parabolic contact and the sinusoidal waviness contact in the plane strain condition. The model of Pohrt and Popov can achieve a similar accuracy for the axisymmetric parabolic contact where the mesh grid is non-conforming to the crack front. Once the conforming mesh grid is used, the accuracy of their model is significantly deteriorated, especially at high work of adhesion and high mesh density. In both BEM models, however, the crack nucleation is found to be mesh-dependent which may be solved by introducing an upper limit for the tensile normal traction.

Band structure analysis of two-dimensional photonic crystals using the wavelet-based boundary element method

By adopting the scaling functions of the B-spline wavelet on the interval (BSWI) as interpolation functions, a wavelet-based boundary element method (BEM) model is constructed to compute the band structures of two-dimensional photonic crystals (2D PCs), which are composed of square or triangular lattices with arbitrarily shaped inclusions. The boundary integral equations of both the matrix and inclusion are developed in a unit cell because of the structural periodicity. In order to make the curve boundary be compatible well, geometric boundaries are interpolated by employing second order BSWI scaling functions while arbitrary order scaling functions are used to approximate boundary variables. For any given angular frequency, an effective technique is provided to generate matrix values related to the boundary shape. Moreover, singular integral problem involved in the presented wavelet-based BEM is considered. Then, combining the Bloch theorem and the interface conditions, a linear eigenvalue equation related to the Bloch wave vector is obtained. Numerical examples are given to verify the performance of the wavelet-based BEM developed herein compared with the conventional BEM.

Numerical simulation with a macroscopic CFD method and experimental analysis of wave interaction with fixed porous cylinder structures

Porous structures have been widely applied in the coastal and ocean engineering due to their wave energy dissipation mechanism. The macroscopic computational fluid dynamics (CFD) approach where the quadratic pressure drop condition of porous surface is introduced to model the wave interaction with porous cylinders. A series of CFD simulations of waves interacting with a single porous cylinder and the combined structure of a porous cylinder with a concentric inner solid column are performed, with corresponding tank tests conducted. The CFD method is compared with experiments, linear potential model, and the quadratic BEM (boundary element method) model. The effects of porosity and porous cylinder radius on wave force and wave heights inside porous cylinder are analyzed to evaluate the performance of porous shell reducing wave loads and wave surface elevation, and the wave force variation with incident wave amplitudes are also investigated. The results demonstrate that the established CFD model is reliable for engineering analysis and thereby being of great significance for reference purpose in the CFD simulations of waves interacting with porous structures.