Frequency Domain Boundary Element Method Research Articles

Optimization Analysis of the Shape and Position of a Submerged Breakwater for Improving Floating Body Stability

Submerged breakwaters can be installed underneath floating structures to reduce the external wave loads acting on the structure. The objective of this study was to establish an optimization analysis framework to determine the corresponding shape and position of the submerged breakwater that can minimize or maximize the external forces acting on the floating structure. A two-dimensional frequency-domain boundary element method (FD-BEM) based on the linear potential theory was developed to perform the hydrodynamic analysis. A metaheuristic algorithm, the advanced particle swarm optimization, was newly coupled to the FD-BEM to perform the optimization analysis. The optimization analysis process was performed by calling FD-BEM for each generation, performing a numerical analysis of the design variables of each particle, and updating the design variables using the collected results. The results of the optimization analysis showed that the height of the submerged breakwater has a significant effect on the surface piercing body and that there is a specific area and position with an optimal value. In this study, the optimal values of the shape and position of a single submerged breakwater were determined and analyzed so that the external force acting on a surface piercing body was minimum or maximum.

Influence of Viscosity and Non-Linearities in Predicting Motions of a Wind Energy Offshore Platform in Regular Waves

AbstractMotion predictions of floating bodies in extreme waves represent a challenging problem in naval hydrodynamics. The solution of the seakeeping problem involves the study of complex non-linear wave-body interactions that require large computational costs. For this reason, over the years, many seakeeping models have been formulated in order to predict ship motions using simplified flow theories, usually based on potential flow theories. Neglecting viscous effects in the wave-induced forces might largely underestimate the energy dissipated by the system. This problem is particularly relevant for unconventional floating bodies at resonance. In these operating conditions, the linear assumption is no longer valid, and conventional boundary element methods, based on potential flow, might predict unrealistic large responses if not corrected with empirical viscous damping coefficients. The application considered in this study is an offshore platform to be operated in a wind farm requiring operability even in extreme meteorological conditions. In this paper, we compare heave and pitch response amplitude operators predicted for an offshore platform using three different seakeeping models of increasing complexity, namely, a frequency-domain boundary element method (BEM), a partly nonlinear time domain BEM, and a non-linear viscous model based on the solution of the unsteady Reynolds-averaged Navier–Stokes (URANS) equations. Results are critically compared in terms of accuracy, applicability, and computational costs.

A polynomial mathematical tool for foundation-soil-foundation interaction

This paper studies the dynamic foundation-soil-foundation interaction for two square rigid foundations embedded in a viscoelastic soil layer. The vibrations come from only one rigid foundation placed in the soil layer and subjected to harmonic loads of translation, rocking, and torsion. The required dynamic response of rigid surface foundations constitutes the solution of the wave equations obtained by taking account of the conditions of interaction. The solution is formulated using the frequency domain Boundary Element Method (BEM) in conjunction with the Kausel-Peek Green's s function for a layered stratum, with the aid of the Thin Layer Method (TLM), to study the dynamic interaction between adjacent foundations. This approach allows the establishment of a mathematical model that enables us to determine the dynamic displacements amplitude of adjacent foundations according to their different separations, the depth of the substratum, foundations masss, foundations embedded, and the frequencies of excitation. This paper attempts to introduce an approach based on a polynomial mathematical tool conducted from several results of numerical methods (BEM-TLM) so that practicing civil engineers can evaluation the dynamic foundations displacements more easy.

Hydrodynamic aspects of moving vehicle with sloshing tanks

Freight vehicles with partially filled liquid tanks will be affected by liquid sloshing during transient motion. The sloshing induced by vehicle motion will lead to extra force on the vehicle and sometimes is a potential threat to safety. Previous studies on this problem usually use a mass-spring analogy to represent the sloshing effect of the liquid tank. Its main disadvantage is that CFD analysis or experimental study has to be performed beforehand, so that an equivalent mass-spring model can be constructed by curve fitting. In this paper, frequency domain boundary element method (BEM) and analytical solution to the sloshing problems are used to derive the modal equation and hydrodynamic parameters of the sloshing fluid. The accuracy of the results will be examined by comparison with available CFD results in the literature. The paper then evaluates the accuracy of equivalent mass-spring model and explores the possibility to approximate the sloshing effects inside cylindrical tanks by using analytical solution to the sloshing inside equivalent rectangular tanks.

Analysis of Cracks in Functionally Graded Piezoelectric Materials by a Frequency-Domain BEM

Time-harmonic crack analysis in two-dimensional piezoelectric functionally graded materials (FGMs) is presented in this paper. A frequency-domain boundary element method (BEM) is developed for this purpose. Since fundamental solutions for piezoelectric FGMs are not available, a boundary-domain integral formulation is derived. This requires only the frequency-domain fundamental solutions for homogeneous piezoelectric materials. The radial integration method is adopted to compute the resulting domain integrals. The collocation method is used for the spatial discretization of the frequency-domain boundary integral equations. Adjacent the crack-tips square-root elements are implemented to capture the local square-root-behavior of the generalized crack-opening-displacements properly. Special regularization techniques based on a suitable change of variables are used to deal with the singular boundary integrals. Numerical examples will be presented and discussed to show the influences of the material gradation and the dynamic loading on the intensity factors.

Numerical study on sub-harmonic generation due to interior and surface breaking cracks with contact boundary conditions using time-domain boundary element method

Recently, the ultrasonic testing (UT) based on the contact acoustic nonlinearity (CAN) has attracted notice as a new technique for detection of closed cracks which cannot be detected by using the linear UT (LUT). In the nonlinear UT (NLUT), detection and size measurement of flaws are conducted via the frequency spectrum analysis of nonlinear ultrasonic waves which consist of higher- and sub-harmonic waves. Although the mechanism of higher-harmonic generation due to CAN has been understood mostly from theoretical and experimental points of view, that of sub-harmonic generation has not been revealed yet. In addition, there are few numerical and theoretical studies on the phenomenon of sub-harmonic generation due to the CAN. In this paper, the boundary integral equation for two-dimensional (2-D) elastic wave scattering by cracks is formulated and numerically solved to investigate the behavior of the sub-harmonic waves. The clapping motion and dynamic friction on crack faces are modeled by considering contact boundary conditions, and their interaction with the characteristic of frequency response in the corresponding linear system is investigated. In this study, time-domain numerical simulations are implemented for both an interior crack in an unbounded elastic solid and a surface breaking crack in an elastic half-space. Some frequency response analyses for a linear system corresponding to the time-domain nonlinear simulations are also performed by using the frequency-domain boundary element method. From obtained numerical results, the causes of sub-harmonic generation are discussed.

Horn effect prediction based on the time domain boundary element method

A time domain boundary element method (TBEM) is applied to predict the horn effect. As the time response calculated by the time domain boundary integral equation contains the resonance components, when transformed to the frequency domain, the result will corrupt at the characteristic frequencies. To overcome this problem, a Burton–Miller-type combined time domain integral equation in half-space is applied. The resonance components are excluded in the time domain calculation, thus the corruptions are avoided in the frequency domain. As a result, the horn effect can be predicted very well at all frequencies. Compared to the frequency domain boundary element method for predicting the horn effect, the TBEM is more efficient due to the lower cost of forming coefficient matrices and solving equations. A numerical simulation is carried out to demonstrate the efficiency of the TBEM, and two experiments are conducted to validate the proposed method in predicting the horn effect. Both numerical and experimental results indicate that the proposed method is reliable and efficient in predicting the horn effect.

Boundary element method simulation of 3D heat diffusion in defective layered media for IRT building applications

This paper presents a numerical model to simulate heat transfer by conduction in defective layered media, looking to contribute to the interpretation of thermographic results obtained in active Infrared Thermography (IRT) tests performed on building elements. The proposed model is developed to simulate 3D heat diffusion in a multilayered system containing a 3D thin defect (considered to be a null thickness inclusion). The proposed model uses the frequency domain Boundary Element Method (BEM) formulation written in terms of normal derivative integral equation formulation, referred to here as TBEM, in order to handle the null thickness of the defect. 3D Green's functions for multilayered media are used to avoid the discretization of the interfaces between layers. These 3D Green's functions are expressed as Bessel integrals and written as the sum of source terms and terms which are defined by imposing the required boundary conditions at each interface. When using this procedure, only the defect's surface is discretized. The proposed solution is verified against a previously developed 3D TBEM formulation for unbounded media using a mirror image source technique. To illustrate the applicability of the proposed methodology in active IRT, time-domain and phase contrast results are simulated and presented.

Computing head related impulse responses and transfer functions using time domain equivalent sources

In the past, head-related impulse responses (HRIR) and head-related transfer functions (HRTF) have primarily been computed using frequency domain boundary element methods or finite-difference time domain methods. The possibility of computing HRIRs and HRTFs using transient equivalent sources is examined using a lumped parameter technique for enforcing the specified boundary condition. It is demonstrated that performing the computations in the time domain is advantageous because only a few thousand time steps are need to fully define the HRIRs and nonuniform meshes can be used to reduce the number of acoustic variables drastically without significantly degrading the solution accuracy. It is also shown that the computations adapt well to parallel processing environments and the times associated with the equivalent source calculations are proportional to the number of processors.

Acoustic viscoelastic modeling by frequency-domain boundary element method

Earth medium is not completely elastic, with its viscosity resulting in attenuation and dispersion of seismic waves. Most viscoelastic numerical simulations are based on the finite-difference and finite-element methods. Targeted at viscoelastic numerical modeling for multilayered media, the constant-Q acoustic wave equation is transformed into the corresponding wave integral representation with its Green’s function accounting for viscoelastic coefficients. An efficient alternative for full-waveform solution to the integral equation is proposed in this article by extending conventional frequency-domain boundary element methods to viscoelastic media. The viscoelastic boundary element method enjoys a distinct characteristic of the explicit use of boundary continuity conditions of displacement and traction, leading to a semi-analytical solution with sufficient accuracy for simulating the viscoelastic effect across irregular interfaces. Numerical experiments to study the viscoelastic absorption of different Q values demonstrate the accuracy and applicability of the method.

Dynamic Interaction of Two Adjacent Foundations Embedded in a Viscoelastic Soil

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

異方性を有する飽和多孔質体中の波動散乱解析のための周波数領域境界要素法の開発

異方性を有する飽和多孔質体中の波動散乱解析のための周波数領域境界要素法の開発

Comparative study of time and frequency domain BEM approaches in frictional contact problem for antiplane crack under harmonic loading

Abstract Two different boundary element methods (BEM) for crack analysis in two dimensional (2-D) antiplane, homogeneous, isotropic and linear elastic solids by considering frictional contact of the crack edges are presented. Hypersingular boundary integral equations (BIE) in time-domain (TD) and frequency domain (FD), with corresponding elastodynamic fundamental solutions are applied for this purpose. For evaluation of the hypersingular integrals involved in BIEs a special regularization process that converts the hypersingular integrals to regular integrals is applied. Simple regular formulas for their calculation are presented. For the problems solution while considering frictional contact of the crack edges a special iterative algorithm of Udzava's type is elaborated and used. Numerical results for crack opening, frictional contact forces and dynamic stress intensity factors (SIFs) are presented and discussed for a finite III-mode crack in an infinite domain subjected to a harmonic crack-face loading and considering crack edges frictional contact interaction using the TD and FD approaches.

Efficiency improvement of the frequency-domain BEM for rapid transient elastodynamic analysis

The frequency-domain fast boundary element method (BEM) combined with the exponential window technique leads to an efficient yet simple method for elastodynamic analysis. In this paper, the efficiency of this method is further enhanced by three strategies. Firstly, we propose to use exponential window with large damping parameter to improve the conditioning of the BEM matrices. Secondly, the frequency domain windowing technique is introduced to alleviate the severe Gibbs oscillations in time-domain responses caused by large damping parameters. Thirdly, a solution extrapolation scheme is applied to obtain better initial guesses for solving the sequential linear systems in the frequency domain. Numerical results of three typical examples with the problem size up to 0.7 million unknowns clearly show that the first and third strategies can significantly reduce the computational time. The second strategy can effectively eliminate the Gibbs oscillations and result in accurate time-domain responses.

Experimental validation of a frequency domain BEM model to study 2D and 3D heat transfer by conduction

Abstract This article presents an experimental validation of 2D and 2.5D boundary element method (BEM) solutions for transient heat conduction in systems containing heterogeneities. The problem is formulated in the frequency domain. The responses in the time domain are obtained by means of an inverse Fourier transform. Complex frequencies with a small imaginary part are introduced to cope with aliasing. Their effect is taken into account by rescaling the results in the time domain.For validation purposes, the solutions provided by the proposed BEM formulation were first verified against analytical solutions and then compared with experimental results. In the laboratory tests a steel inclusion was embedded in a confined host medium and unsteady temperatures were applied to its boundary. Two host media were tested: molded expanded polystyrene and medium-density fiberboard. The systems were subjected to plane and point heat sources. The thermal properties of these materials have been previously defined experimentally. The results show that the BEM solutions agree well with the experimental results.

Second-Harmonic Generation Imaging of Metal Nano-Objects with Cylindrical Vector Beams

We introduce an imaging technique based on second-harmonic generation with cylindrical vector beams that is extremely sensitive to three-dimensional orientation and nanoscale morphology of metal nano-objects. Our experiments and second-harmonic field calculations based on frequency-domain boundary element method are in very good agreement. The technique provides contrast for structural features that cannot be resolved by linear techniques or conventional states of polarization and shows great potential for simple and cost-effective far-field optical imaging in plasmonics.

Stabilization of time domain acoustic boundary element method for the interior problem with impedance boundary conditions

The time domain boundary element method (BEM) is associated with numerical instability that typically stems from the time marching scheme. In this work, a formulation of time domain BEM is derived to deal with all types of boundary conditions adopting a multi-input, multi-output, infinite impulse response structure. The fitted frequency domain impedance data are converted into a time domain expression as a form of an infinite impulse response filter, which can also invoke a modeling error. In the calculation, the response at each time step is projected onto the wave vector space of natural radiation modes, which can be obtained from the eigensolutions of the single iterative matrix. To stabilize the computation, unstable oscillatory modes are nullified, and the same decay rate is used for two nonoscillatory modes. As a test example, a transient sound field within a partially lined, parallelepiped box is used, within which a point source is excited by an octave band impulse. In comparison with the results of the inverse Fourier transform of a frequency domain BEM, the average of relative difference norm in the stabilized time response is found to be 4.4%.

시간 영역 음향 경계요소법에서의 비유일성 문제 해결을 위한 방법에 관하여

Kirchhoff 적분식을 이용하여 외부 음향 문제의 시간 영역 응답을 계산하는 경우, 주파수영역 해석과 마찬가지로 가상적인 내부 음향 모드에 기인한 비유일성 문제가 발생한다. 이를 해결하는 방법들 중의 하나로서 CHIEF(Combined Helmholtz Integral Equation Formulation) 방법이 쓰이는데, 이는 몇몇 내부 수음점의 응답을 0으로 추가하여 구속하는 조건을 부가하는 기법이다. 이 기법은 주파수 영역 경계요소법에서는 간편한 수식 때문에 많이 사용되고 있지만, 시간 영역에서는 사용된 예가 없다. 본 연구에서는 대상체 내부의 가상 수음점과 경계 표면의 절점들간의 최소 거리에 대한 지연시간을 고려하여, 계산하고자 하는 미지수인 현재 시간의 경계 표면 음장을 구속함으로써, 시간 영역 해석에 적합하도록 CHIEF 방법을 수식화하였다. 예제로서, 반지름 방향으로 진동하는 구의 음향 방사 문제를 다루었다. CHIEF 방법을 적용함에 따라 저차의 내부 음향 모드에 기인한 비유일성 문제를 해결할 수 있었고, 비요동 모드에 의한 수치적 불안정성을 피할 수 있었다. 그러나, 유효주파수 밖에 남은 내부 음향의 고차모드들에 의한 수치적 불안정성은 증가하였다. The time-domain solution from the Kirchhoff integral equation for an exterior problem is not unique at certain eigen-frequencies associated with the fictitious internal modes as happening in frequency-domain analysis. One of the solution methods is the CHIEF (Combined Helmholtz Integral Equation Formulation) approach, which is based on employing additional zero-pressure constraints at some interior points inside the body. Although this method has been widely used in frequency-domain boundary element method due to its simplicity, it was not used in time-domain analysis. In this work, the CHIEF approach is formulated appropriately for time-domain acoustic boundary element method by constraining the unknown surface pressure distribution at the current time, which was obtained by setting the pressure at the interior point to be zero considering the shortest retarded time between boundary nodes and interior point. Sound radiation of a pulsating sphere was used as a test example. By applying the CHIEF method, the low-order fictitious modes could be damped down satisfactorily, thus solving the non-uniqueness problem. However, it was observed that the instability due to high-order fictitious modes, which were beyond the effective frequency, was increased.

Study on the scattering of 2-D Rayleigh waves by a cavity based on BEM simulation

A frequency domain boundary element method is introduced in this paper to simulate the scattering of two-dimensional Rayleigh waves by a cavity on the surface of an elastic half-space. This numerical method allows the Rayleigh wave to escape the truncated boundary without producing any spurious reflections. The numerical results of the displacements and stresses of a 2-D Rayleigh wave which travels along a flat surface of an elastic half-infinite medium are in good agreement when compared with theoretical ones. The problem of a Rayleigh wave scattering by a cavity on the surface of an elastic half-infinite medium is simulated by the proposed numerical method. The result is taken as the benchmark to introduce an approximate analytical method for such problems of scattering by a cavity on the surface. This numerical method is formulated for a two-dimensional homogeneous, isotropic, linearly elastic half-space and its implementation in a frequency domain boundary scheme is discussed in detail.

Comparisons of internal solitary wave and surface wave actions on marine structures and their responses

In this paper, a time-domain numerical model is established for computing the action of internal solitary wave on marine structures and structure motion responses. For a cylindrical structure, its side and bottom are discretized by pole and surface elements, respectively. The drag and inertial forces in the perpendicular direction of the structure are computed by the Morison equation from the pole elements, and the Froude–Krylov force in the axial direction of the structure due to internal wave motion is computed by integration of the dynamic pressure over the surface elements. The catenary theory is used to analyze the reaction force due to mooring lines, and the motion equation of the marine structure is solved by the fourth-order Runge–Kutta method in the time domain. The model is used to calculate the interaction of the internal solitary wave with a Spar platform with mooring system, and the surface wave action with the platform has also been computed by a frequency-domain boundary element method for comparison. Through the comparison based on a practical internal wave and surface wave states, it can be concluded that the internal wave force on the structure is only 9% of the one due to surface waves. However, the motion response due to the internal wave is much greater than the one due to the surface waves. It shows that the low-frequency effect of internal solitary waves is a great threat to the safety of marine structures.