Three-dimensional Boundary Element Method Research Articles

Hydrodynamic forces in higher modes of a thin cantilever beam resonator

The hydrodynamic force in a thin uniform beam cantilever is essential quantity for designing a resonator. We obtained the hydrodynamic drag forces in a uniform cantilever vibrating at the first four transverse modes using a semi-analytical boundary element method (BEM) and finite element method (FEM) in ANSYS. The present three-dimensional (3D) BEM inertial and damping forces are compared with FEM over a frequency range with less than 9.5% deviation until frequency parameter of 100; thereafter, it deviates numerically due to neglecting the non-linear convective forces. Considering the hydrodynamic damping force, the quality factor is estimated for the first four transverse modes of a uniform cantilever beam. As a result, the present 3D BEM model behaves well at the first bending mode compared to FEM. We also study the effect of the aspect ratio (ratio of width to length) of the beam for all four modes. However, the first mode quality factor sufficiently correlates with the numerical value for all aspect ratios. For uniform beams operating with higher modes, the influence of nonlinear and resonance frequency by the numerical model is more pronounced at smaller lengths. Hence, the 3D BEM approach can estimate the hydrodynamic forces on cantilever-based resonators, for example, atomic force microscopy and biosensors.

Hydrodynamic forces in array of uniform and non-uniform cantilever beams

In this paper, we have implemented the three-dimensional Boundary Element Method (BEM) to solve the hydrodynamic forces in arrays of two, three, and five thin beams oscillating with small amplitudes in a fluid. We solve the unsteady Stokes equation by considering the oscillatory Stokeslet. The non-dimensional hydrodynamic function is validated through numerical solutions obtained using the Finite Element Method (FEM) at various frequency parameters and gap ratios. We analyse the influence of the relative phase and amplitude ratio of a uniform beam and a non-uniform two-beam array in terms of added mass and damping coefficients. Additionally, we compare the solution with experimental results available in the literature as a function of gap ratio. The analysis presented in this paper can be useful for optimizing arrays used in Atomic Force Microscope (AFM), biosensors, and energy harvesters.

A fully nonlinear BEM-beam coupled solver for fluid–structure interactions of flexible ships in waves

A fully nonlinear hydroelastic numerical method is proposed for the fluid–structure interactions of elastic ships. The boundary value problem is solved by the three-dimensional boundary element method. To simulate the instantaneous evolution on the free surface, an effective mesh optimization solution consisting of the local coordinate system and an elastic mesh technique has been developed. The mixed Eulerian–Lagrangian method is implemented to update the fully nonlinear free surface conditions in the local coordinate system. The hull structure is simplified as a non-uniform Timoshenko beam, and the modal superposition approach is adopted to analyse the symmetric responses in the head sea. Convergence of the model is evaluated for mesh resolutions and the time step size, and then comprehensive investigations of vertical motions as well as vertical bending moment are carried out for an advancing ship in a range of wave frequencies. The numerical solutions are compared with the experimental data and a satisfactory agreement is demonstrated. The distribution of high-order harmonics and whipping responses are predicted by the amplitude spectrum, in which the water entry and water exit are considered. The high-frequency vertical bending moments are investigated, in which the effects of higher harmonics and whipping responses are addressed.

An adaptive binary-tree subdivision method for evaluation of nearly singular integrals in 3D BEM

PurposeThe purpose of this paper is to present an adaptive binary-tree element subdivision method (BTSM) for the evaluation of nearly singular integrals in three-dimensional boundary element method, which can facilitate automatic and high-quality patch generation.Design/methodology/approachIn this method, the nearly singular element is split into two sub-elements. Each sub-element is then examined to determine if it is to be subdivided based on a specific subdivision criterion. The specific subdivision ensures that those sub-elements far from the source point are sparse. And then those sub-elements in close proximity to the source point are replaced by regular triangular elements.FindingsWith the proposed method, the sub-elements obtained are automatically refined as they approach the projection point, and they are “good” in shape and size for standard Gaussian quadrature. Thus, the proposed method can be used to evaluate nearly singular integrals accurately for cases of different element shapes and various locations of the source point.Originality/valueNumerical examples for surface elements with various relative locations of the source point are presented. The results demonstrate that the proposed method has much better accuracy and robustness than some other methods.

Improvement to the fictitious pile method for pile-soil interaction problems

Dynamic pile-soil interaction analysis is often a key task in many civil engineering problems and several methods have been developed and applied. To many researchers, the fictitious pile method is a good candidate because of its high computational efficiency compared with the three-dimensional finite element method (FEM) and boundary element method (BEM). However, due to the assumptions made in the method, application of the method to dynamic pile-soil interaction at sufficiently high frequency may be problematic. A study on this issue is performed in this paper by comparing with FEM. It is found that the assumption that the vertical load (termed the patch load) applied by the fictitious pile to the ground is uniformly distributed on the cross-section is an important factor invalidating the method at high frequencies. Upon this founding, the original fictitious pile method is then improved by replacing the uniformly distributed patch load with a series of ring loads which make the vertical displacements at the intermediate radii of the ring loads identical. A set of linear algebraic equations are established to determine the ring loads. Results from both the improved and original fictitious pile methods are compared with those from FEM as benchmark. The comparisons show that the improved fictitious pile method has a much better accuracy (or for a given accuracy, works to higher frequencies) than the original one but the calculation efficiency remains similar. Although for the cases studied in this paper the improved fictitious pile method is much more accurate than the original one, the effectiveness of former may be still unsatisfactory for other cases. Therefore, the effect of parameters, including soil Young's modulus, pile radius and pile length, on the accuracy of the fictitious pile method is also investigated. It turns out that the pile radius and length have adverse effects on the accuracy of the fictitious pile method while stiffer soil makes the method work better. Based on the parameters of the pile and soil, a critical frequency is derived below which the fictitious pile method can be well applied.

Hydrodynamic Forces and Reactions of Two Floating Bodies in Close Proximity

The engineering problem of two floating bodies in close proximity occurs in the case of a floating natural liquefied gas terminal when the gas is transferred from a gas carrier to a gas storage and regasification unit. An integral part of the design of such a terminal is the calculation of loads due to the incident waves. The calculation includes the assessment of first order wave forces as well as hydrodynamic reactions i.e. the added mass and the radiational damping. The calculation is based on potential flow theory with the use of three-dimensional boundary element method. Compared to the case of one floating body, the case of two bodies is more complex because it contains 12 degrees of freedom i.e. 6 degrees of freedom for each body. Therefore, the hydrodynamic coefficients and the incident waves forces must contain the interaction of the floating bodies which are in close proximity. The paper presents the results of calculations, done for two barges, which contain transfer response functions and significant force values for given sea conditions.

Three-Dimensional Boundary Element Strategy for Stress Sensitivity of Fractional-Order Thermo-Elastoplastic Ultrasonic Wave Propagation Problems of Anisotropic Fiber-Reinforced Polymer Composite Material.

A new three-dimensional (3D) boundary element method (BEM) strategy was developed to solve fractional-order thermo-elastoplastic ultrasonic wave propagation problems based on the meshless moving least squares (MLS) method. The temperature problem domain was divided into a number of circular sub-domains. Each node was the center of the circular sub-domain surrounding it. The Laplace transform method was used to solve the temperature problem. A unit test function was used in the local weak-form formulation to generate the local boundary integral equations, and the inverse Laplace transformation method was used to find the transient temperature solutions. Then, the three-dimensional elastoplastic problems could be solved using the boundary element method (BEM). Initial stress and strain formulations are adopted, and their distributions are interpolated using boundary integral equations. The effects of the fractional-order parameter and anisotropy are investigated. The proposed method’s validity and performance are demonstrated for a two-dimensional problem with excellent agreement with other experimental and numerical results.

A three-dimensional boundary element method algorithm for simulations of magnetic fluid droplet dynamics

This paper outlines a numerical algorithm capable of simulating the full three-dimensional dynamics of magnetic fluid droplets in external magnetic fields by solving boundary integral equations. The algorithm works with arbitrary droplet and carrier fluid viscosity ratios. It is validated with known theoretical relationships. It also enables evaluating various approximations often used to describe ellipsoidal droplets by comparing the droplet dynamics calculated from such approximations to the results obtained from first principles using our numerical algorithm. The algorithm may be used to investigate droplet configurations in arbitrary magnetic fields and to indirectly calculate the physical properties of magnetic fluid droplets and predicting the magnetic field thresholds above which the droplet shape can develop instabilities in the form of various spikes.

A 3D boundary element method for analysing the hydrodynamic performance of a land-fixed oscillating water column device

In this work, the theoretical hydrodynamic performance of an onshore Oscillating Water Column (OWC) device, subjected to normal wave incidence, is investigated. Based on linear surface-wave theory, the full solution of the boundary value problem (BVP) is obtained by using a three-dimensional (3D) boundary element method (BEM) with nine-noded quadrilateral elements. The method of subdomains is used to solve the BVP by applying the BEM separately to each of its regions, and the continuity of velocity potential and flux are used to match the problem. The capture width, hydrodynamical coefficients and free surface elevations are analysed for different OWC geometric parameters of the chamber. Numerical results for the hydrodynamic performance were found to be in good agreement with cases reported elsewhere. The findings show that the chamber configuration and the linearized air compressibility have a significant impact on the capture width bandwidth, the resonant frequency and the free surface elevation within the chamber.

Wave scattering by a three-dimensional submerged horizontal rectangular plate in a channel: experiments and numerical computations

Experiments are conducted in a wave tank to investigate wave scattering by a three-dimensional submerged horizontal rectangular plate in a channel. The free-surface elevation around the plate is presented for various water depths and depths of submergence of the plate. The wave forces and moments are obtained using an underwater load measuring system. The numerical simulations are performed with a parallelized three-dimensional boundary element method. The numerical set-up follows closely the set-up of the laboratory experiments. The numerical results are compared with the experimental results for non-breaking waves. In most cases a good agreement is found for the free-surface elevation, the vertical force and the moment. A physical interpretation of the flow around the plate is provided. The pressure distribution on the lower surface of the plate differs from the linear distribution from the leading edge to the trailing edge that is obtained in the equivalent two-dimensional problem. The reflection by the lateral walls is investigated.

A Direct Method for Solving Singular Integrals in Three-Dimensional Time-Domain Boundary Element Method for Elastodynamics

The analytically time integrable time-space domain (ATI-TSD) is discovered based on which the minimum time-space domain is identified for treatment on singularities in the three-dimensional time-domain boundary element method (3D TD-BEM) formulation. A direct method to solve singular integrals in the 3D TD-BEM formulation for elastodynamic problems is proposed. The wavefront singularity can be analytically eliminated in ATI-TSD, while the dual singularity can be treated by the direct method using Kutt’s quadrature in the identified minimum time-space domain. Three benchmark examples are presented to verify the correctness and the applicability of the direct method for solving the singular integrals in 3D TD-BEM.

Three-dimensional preconditioned FM-IBEM solution to broadband-frequency seismic wave scattering in a layered sedimentary basin

A three-dimensional (3-D) fast multipole indirect boundary element method (FM-IBEM) combined with an inner-outer preconditioned generalized minimal residual (GMRES) method algorithm is proposed to solve the broadband-frequency seismic wave scattering problem in complex sites. This method is established in three steps: first, the scattering wave field is developed with the dynamic Green's function; second, the dynamic Green's functions are expanded to generate a traditional FM-IBEM framework; and finally, the preconditioning matrix and GMRES algorithm are introduced to pre-process the dynamic Green's functions. Using this method, the broadband-frequency scattering of normally incident seismic waves in a hemispherical layered sedimentary basin is investigated in the frequency and time domains. The results show that the proposed method is highly accurate and efficient for solving low- and high-frequency seismic wave problems. Compared with the FM-IBEM without preconditioning, the computational time and storage space required for the FM-IBEM with preconditioning are reduced by 84% and 54%, respectively. The layered sedimentary basin has significant wave amplification (of up to 10 times) for dimensionless frequencies, η, in the range of 0.5–3.0; however, for η > 5, the amplification effect becomes weak.

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.

Proposal of numerical models to predict the diffuse field sound absorption of finite sized porous materials – BEM and FEM approaches

The prediction of the sound absorption coefficient in diffuse field conditions is of great interest to an acoustic project. Foams and granular materials like porous concrete have been used in many internal or external applications as sound absorbers. However, many doubts have regularly been discussed in scientific literature. The simple use of experimental impedance tube methods, that consider only the incident acoustic plane waves, is not equivalent to the experimental measurement in a reverberant room, which is necessary for a correct characterization of a given constructive solution. The scientific community has explored several methods to predict and approximate the sound absorption coefficient in diffuse field conditions, using only acoustic plane wave data. However, many of them do not allow a realistic approximation of the reverberant room measured data. Therefore, two numerical models are herein proposed which consider the porous media on the fluid-equivalent theory, first, a two-dimensional (2D) finite element method, and, second, a three-dimensional (3D) boundary element method (BEM). These methods are compared with analytical equations and experimental data from the reverberant room, studying the influence of the porous material panel size in the sound absorption coefficient in diffuse field conditions. To illustrate the proposed method’s practical interest, the BEM 3D model was used to determine the diffuse field’s sound absorption coefficient for finite-sized panels of porous concrete. This test was done for plane panels and for slit-like surfaces to highlight the possibilities of the proposed method.

Solitary-wave loads on a three-dimensional submerged horizontal plate: Numerical computations and comparison with experiments

A parallelized three-dimensional (3D) boundary element method is used to simulate the interaction between an incoming solitary wave and a 3D submerged horizontal plate under the assumption of potential flow. The numerical setup follows closely the setup of laboratory experiments recently performed at Shanghai Jiao Tong University. The numerical results are compared with the experimental results. An overall good agreement is found for the two-dimensional wave elevation, the horizontal force and the vertical force exerted on the plate, and the pitching moment. Even though there are some discrepancies, the comparison shows that a model solving the fully nonlinear potential flow equations with a free surface using a 3D boundary element method can satisfactorily capture the main features of the interaction between nonlinear waves and a submerged horizontal plate.

Surface plasmon resonance characteristics of a graphene nano-disk based on three-dimensional boundary element method

The predecessor of China Optics was China Optics and Applied Optics Digest, which was founded in 1985. At that time, it was the only retrieval journal in the field of optics in China. At the end of 2008, China Optics and Applied Optics Digest was renamed

Field Tests and Three-Dimensional Semi-Analytical Boundary Element Method Analysis of a Row of Holes as Active Barrier in Saturated Soil

Abstract The adverse effects of ground vibrations can be eliminated or prevented by installation of various types of wave barriers, such as a row of holes. The analysis is accomplished with the aid of a series of field experiments and numerical simulation. Three-dimensional semi-analytical boundary element models are established to use a row of holes as active wave barrier to isolate the ground vibrations generated by the machine foundation laid on the surface of saturated soil foundation, and the effects of the model parameters on isolation effectiveness are investigated and discussed in detail. A simple single program multiple data parallel algorithm is developed with the aid of Matlab software to increase calculating efficiency. The results show that a row of holes can isolate the ground vibrations successfully. Increasing the radius and the depth and decreasing the net spacing between two successive holes can all result in an increase in the screening effectiveness. According to the results, it is suggested in the design that the hole radius should take the value of 0.1∼0.15λR (λR is the Rayleigh wave length), and the hole depth and the net spacing should take the value no more than 1.0λR and 0.1λR, respectively. Moreover, the distance between the machine foundation and barrier can affect the screening effectiveness, and the larger the distance is, the poorer the screening effectiveness is. The number of holes in a row has less effect on the screening effectiveness. However, the bigger the number of holes in a row is, the larger the screening zone behind the barriers is.

Viscous fluid flow in a microchannel with hydrodynamic traps

Simulation of creeping flows in complex three-dimensional domains is crucial for microfluidics applications, creating multipurpose microfluidic devices, which are used, for example, to study multistage chemical reactions and as analytical devices in medicine.In this work we study the features of the incompressible viscous fluid flow in a flat microchannel with a nontrivial internal structure (the hydrodynamic traps of various configurations) under the constant pressure drop.The three-dimensional boundary element method, improved by utilization high-efficient fast multipole method and heterogeneous computational workstation, was used in numerical simulation.We studied the flow pattern around C-shaped hydrodynamic traps distributed in a flat microchannel and consisting of five cylindrical elements of the same size. Such geometry is widely used in microfluidic devices for fixing particles in the flow, for example, biological objects during tests in medicine. The influence of the distance between the rows of traps on the flow pattern and distribution of the longitudinal and transverse components of the flow velocity was considered.

Three-dimensional simulation of drop motion in channels of different cross-sections

Three-dimensional Boundary Element Method accelerated using a heterogeneous Fast Multipole Method is employed to study two-phase flow in a microchannel of different cross-sections. The flow of a mixture of two Newtonian liquids with equal viscosities and densities is described by Stokes equations. A comparison of the simulation results of the droplet dynamics in cylindrical capillaries with experimental data available in the literature was carried out and showed the qualitative agreement. The deformable droplet motion is considered in a channel with a circle, square, and hexagon cross-sectional shape. The influence of droplet size on a relative velocity in a channel is studied. The applied approach to numerical simulation can be used to solve a wide range of problems related to emulsion flows in microscales.

A three-dimensional, p-version BEM: High-order refinement leveraged through regularization

Numerical results for polynomial (p-version) refinement in the three-dimensional boundary element method are presented. Results are based on a weakly singular form of the gradient boundary element formulation in potential theory. The weakly singular formulation facilitates direct application of numerical quadrature, completely removing the need for closed-form integration. The new results, which include interpolations as high as tenth order, show the efficiency of the approach while confirming a previous assertion that whole-body regularization in conjunction with only linear or quadratic elements will lead to extremely poor results.