Fictitious frequency revisited

The nonexistence and nonuniqueness problems associated with integral equation methods for exterior acoustics are revisited. The Fredholm alternative theorem in conjunction with the singular value decomposition updating technique is used to simultaneously determine the fictitious frequencies and corresponding modes in exterior acoustics. After selecting the combined Helmholtz interior integral equation formulation (CHIEF) points, the influence row vectors are obtained. A criterion in selecting the minimum number of CHIEF points and their positions is proposed by testing the orthogonal condition between the influence row vector and the right unitary vector. It is proved in the discrete system for arbitrary-shape problems that the source of numerical instability of irregular frequency originates from the zero divided by zero using the generalized coordinates of unitary vectors. The mathematical structures of the four influence matrices in the dual boundary element method (BEM) are examined by using the left and right unitary matrices. Extracting the true eigenvalue and filtering out the fictitious frequency can be unified by using the updating term and updating document, respectively. Radiation problems of a cylinder and a square rod are demonstrated to see the validity of the present formulation.     

Subharmonic resonance of a trapped wave near a vertical cylinder by narrow-banded random incident waves

It is well-known that near an infinite linear array of periodically spaced cylinders trapped waves of certain eigenfrequencies can exist. If there are only a finite number of cylinders in an infinite sea, trapping is imperfect. Simple harmonic incident waves can excite a nearly trapped wave at one of the eigen frequencies through a linear mechanism. However, the maximum amplification ratio increases monotonically with the number of the cylinders; hence the solution is singular in the limit of infinitely many cylinders. Recently, a nonlinear theory of subharmonic resonance of perfectly trapped waves has been completed. In this article the theory is further extended to random incident waves with a narrow spectrum centered near twice the natural frequency of the trapped wave. The effects of detuning and bandwidth of the spectrum are examined. Dedicated to Professor J. N. Newman on his 70th birthday. We wish to express our profound admiration for Professor Newman’s scientific contributions and leadership in the ship-hydrodynamics discipline. The relation between this article and an early work of his reflects in part his impact on us.     

On the improvement of the solution accuracy for exterior acoustic problems with BEM and FMBEM

The boundary element formulations suffer from the non-uniqueness of the solution. Consequently, fictitious resonances degrade the exterior field prediction. This paper investigates the benefits of using different approaches to mitigate fictitious resonances to improve the solution accuracy in real industrial cases. For direct BEM simulations, over-determination points are added to the interior of the cavity as suggested by the CHIEF method. For indirect BEM and Fast Multipole BEM simulations, the impedance condition is put over the interior boundary and two different values of absorption are applied to observe the effects on the response.The different BEM methods are applied to simulate the exterior sound radiation of three different gearbox housings. The numerical results are compared with high quality measurements enabling the benefits and the improvements on the solution accuracy of each method to be evaluated.     

Dual boundary element method for three-dimensional thermoelastic crack problems

This paper describes the formulation and numerical implementation of the three-dimensional dual boundary element method (DBEM) for the thermoelastic analysis of mixed-mode crack problems in linear elastic fracture mechanics. The DBEM incorporates two pairs of independent boundary integral equations; namely the temperature and displacement, and the flux and traction equations. In this technique, one pair is applied on one of the crack faces and the other pair on the opposite one. On non-crack boundaries, the temperature and displacement equations are applied. This revised version was published online in July 2006 with corrections to the Cover Date.     

A comparison of techniques for overcoming non‐uniqueness of boundary integral equations for the collocation partition of unity method in two‐dimensional acoustic scattering

The Partition of Unity Method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements. The non‐uniqueness of solution obtained from the conventional boundary integral equation (CBIE) is well known. The CBIE derived through Green's identities suffers from a problem of non‐uniqueness at certain characteristic frequencies. Two of the standard methods of overcoming this problem are the so‐called Combined Helmholtz Integral Equation Formulation (CHIEF) method and that of Burton and Miller. The latter method introduces a hypersingular integral, which may be treated in various ways. In this paper, we present the collocation partition of unity boundary element method (PUBEM) for the Helmholtz problem and compare the performance of CHIEF against a Burton–Miller formulation regularised using the approach of Li and Huang. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary element method solutions for elastic wave scattering in 3D

Time-harmonic elastic wave scattering problems such as those encountered in ultrasonic non destructive evaluation are solved by the boundary element method (BEM). Selected results for spherical and spheroidal shaped voids and inclusions are compared with analytical and other numerical solutions. Results for ellipsoids, which require a full three-dimensional formulation, are provided as a benchmark for comparison when other numerical methods would be developed for this problem class in the future. The modelling of cracklike defects with this formulation is discussed. Recent theoretical findings regarding the fictitious eigenfrequency difficulty (FED) are confirmed by a numerical study.     

Dual boundary‐element method: Simple error estimator and adaptivity

This paper is concerned with the effective numerical implementation of the adaptive dual boundary‐element method (DBEM), for two‐dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single‐region analysis. Taking advantage on the use of non‐conforming parametric boundary‐elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM. Copyright © 2011 John Wiley & Sons, Ltd.     

The trapping of surface waves by multiple submerged horizontal cylinders

The existence of edge waves, or trapped modes, travelling above a single long horizontal submerged cylinder is well established in the linearised theory of water waves. In the present paper, the possibility of wave-trapping by multiple submerged horizontal circular cylinders is considered. The trapped mode solutions are constructed by means of a multipole approach combined with an addition formula for Bessel functions and requires finding the non-trivial solutions of a real infinite system of algebraic equations. The case of a single submerged cylinder is returned to briefly, where results for symmetric trapped modes are reproduced and new numerical results for antisymmetric modes are presented. A large range of results are also presented for multiple cylinders.     

Unified formulation of radiation conditions for the wave equation

A family of radiation boundary conditions for the wave equation is derived by truncating a rational function approximation of the corresponding plane wave representation, and it is demonstrated how these boundary conditions can be formulated in terms of fictitious surface densities, governed by second‐order wave equations on the radiating surface. Several well‐established radiation boundary conditions appear as special cases, corresponding to different choices of the coefficients in the rational approximation. The relation between these choices is established, and an explicit formulation in terms of selected directions with ideal transmission is presented. A mechanical interpretation of the fictitious surface densities enables identification of suitable conditions at corners and boundaries of the radiating surface. Numerical examples illustrate excellent results with one or two fictitious layers with suitable corner and boundary conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

MHD convective heat transfer of nanofluids through a flexible tube with buoyancy: A study of nano-particle shape effects

This paper presents an analytical study of magnetohydrodynamics and convective heat transfer of nanofluids synthesized by three different shaped (brick, platelet and cylinder) silver (Ag) nanoparticles in water. A two-phase nanoscale formulation is adopted which is more appropriate for biophysical systems. The flow is induced by metachronal beating of cilia and the flow geometry is considered as a cylindrical tube. The analysis is carried out under the low Reynolds number and long wavelength approximations and the fluid and cilia dynamics is of the creeping type. A Lorentzian magnetic body force model is employed and magnetic induction effects are neglected. Solutions to the transformed boundary value problem are obtained via numerical integration. The influence of cilia length parameter, Hartmann (magnetic) number, heat absorption parameter, Grashof number (free convection), solid nanoparticle volume fraction, and cilia eccentricity parameter on the flow and heat transfer characteristics (including effective thermal conductivity of the nanofluid) are examined in detail. Furthermore a comparative study for different nanoparticle geometries (i.e. bricks, platelets and cylinders) is conducted. The computations show that pressure increases with enhancing the heat absorption, buoyancy force (i.e. Grashof number) and nanoparticle fraction however it reduces with increasing the magnetic field. The computations also reveal that pressure enhancement is a maximum for the platelet nano-particle case compared with the brick and cylinder nanoparticle cases. Furthermore the quantity of trapped streamlines for cylinder type nanoparticles exceeds substantially that computed for brick and platelet nanoparticles, whereas the bolus magnitude (trapped zone) for brick nanoparticles is demonstrably greater than that obtained for cylinder and platelet nanoparticles. The present model is applicable in biological and biomimetic transport phenomena exploiting magnetic nanofluids and ciliated inner tube surfaces.     

波动谱元模拟中透射边界稳定性分析

无限域波动数值模拟中,人工边界的稳定性是获得可靠模拟结果的前提。具有高阶精度的谱元法和透射边界两者结合的数值模拟方案显示出较好的模拟精度和数值稳定性,然而,仍然存在数值失稳现象,其失稳机理和稳定条件尚不明确,相应的理论分析极为欠缺。该文针对透射边界在高阶谱元法中的稳定性,依据高阶谱单元中非等间距节点的周期延拓特点,通过构建内域和边界数值格式的向量形式来分析人工边界反射系数。进而保证边界对谱元法中存在的真实模态和虚假模态的反射系数均小于等于1,从而得到透射边界的稳定条件,其表现为无量纲边界参数和谱元参数之间的关系,其含义为透射边界人工波速与介质物理波速的比值限定在一定范围内。同时揭示了透射边界引发高频失稳的机理,即边界对谱元法中虚假模态的反复反射放大所致。最后采用数值实验验证了透射边界稳定条件。     

Evaluation of boundary integrals for plate bending

The alternative to quadrature, as a procedure for dealing with the integrations required in the direct boundary element method (DBEM), is to carry out the integration analytically and code the results directly. The potential benefits are efficient computer programs; the avoidance of numerical instability; and generally, better accuracy. The technique is developed in this paper. Serious problems arise when Gauss quadrature is employed for the integration of functions which contain, or are close to singularities. A numerical integration approach may fail at the first stage of the analysis, that is, during the assembly of the discrete equations; or it may fail at the subsequent stage of computing domain points near the boundary. The severity of the problem is dependent both on the strength of the singularity, and on geometry. These points are illustrated with examples.     

Crack identification in two-dimensional unilateral contact mechanics with the boundary element method

The identification of a unilateral frictionless crack is performed in nonlinear elastostatics by using boundary measurements for given static loadings. The procedure proposed takes into account the possibility of a partial or total closure of the crack during the identification process; that makes the present formulation more complex than others referred to permanently open cracks. The Linear Complementarity Problem (LCP), which provides at each step contact tractions and relative displacements along the crack, is discretised by means of the Dual Boundary Element Method (DBEM) and solved explicitly by Lemke's algorithm. The identification procedure is based on a first-order nonlinear optimisation technique in which the gradients of the cost function are obtained by solving again a LCP with a considerably reduced number of variables. Some numerical examples show the applicability of the method. Received 23 November 1998     

Improving the stability of time domain dual boundary element method for three dimensional fracture problems: A time weighting approach

This paper deals with the stability of time domain dual boundary element method (DBEM). A time-weighted time domain DBEM is presented in this study and used for the first time in order to improve the stability of the standard time domain dual boundary element method. In this research a time weighting function with a prediction algorithm based on constant velocity algorithm has been utilized. The present approach was tested for three-dimensional fracture problems. The computer cost for the time of the presented approach is very close to the standard form. The results of numerical experiments carried out within this study indicate that the time weighting method, which is suggested for time domain DBEM, has more stability in comparison with the conventional method.     

The dual boundary element formulation for elastoplastic fracture mechanics

In this paper the extension of the dual boundary element method (DBEM) to the analysis of elastoplastic fracture mechanics (EPFM) problems is presented. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. In order to avoid collocation at crack tips, crack kinks and crack-edge corners, both crack surfaces are discretized with discontinuous quadratic boundary elements. The elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the strain tensor on the region expected to yield. This region is discretized with internal quadratic, quadrilateral and/or triangular cells. This formulation was implemented for two-dimensional domains only, although there is no theoretical or numerical limitation to its application to three-dimensional ones. A centre-cracked plate and a slant edge-cracked plate subjected to tensile load are analysed and the results are compared with others available in the literature. J-type integrals are calculated.     

Low‐Reynolds‐number flow around an oscillating circular cylinder using a cell viscousboundary element method

Flow fields from transversely oscillating circular cylinders in water at rest are studied by numerical solutions of the two‐dimensional unsteady incompressible Navier–Stokes equations adopting a primitive‐variable formulation. These findings are successfully compared with experimental observations. The cell viscous boundary element scheme developed is first validated to examine convergence of solution and the influence of discretization within the numerical scheme of study before the comparisons are undertaken. A hybrid approach utilising boundary element and finite element methods is adopted in the cell viscous boundary element method. That is, cell equations are generated using the principles of a boundary element method with global equations derived following the procedures of finite element methods. The influence of key parameters, i.e. Reynolds number Re, Keulegan–Carpenter number KC and Stokes' number β, on overall flow characteristics and vortex shedding mechanisms are investigated through comparisons with experimental findings and theoretical predictions. The latter extends the study into assessment of the values of the drag coefficient, added mass or inertia coefficient with key parameters and the variation of lift and in‐line force results with time derived from the Morison's equation. The cell viscous boundary element method as described herein is shown to produce solutions which agree very favourably with experimental observations, measurements and other theoretical findings. Copyright © 2001 John Wiley & Sons, Ltd.     

Study on harbor resonance and focusing by using the null-field BIEM

In this paper, the resonance of a circular harbor is studied by using the semi-analytical approach. The method is based on the null-field boundary integral equation method in conjunction with degenerate kernels and the Fourier series. The problem is decomposed into two regions by employing the concept of taking free body. One is a circular harbor, and the other is a problem of half-open sea with a coastline subject to the impermeable (Neumann) boundary condition. It is interesting to find that the SH wave impinging on the hill can be formulated by the same mathematical model. After finding the analogy between the harbor resonance and hill scattering, focusing of the water wave inside the harbor as well as focusing in the hill scattering are also examined. Finally, two numerical examples, circular harbor problems of 60° and 180° opening entrance, are both used to verify the validity of the present formulation.     

Instability and treatments of the coupled discrete element and lattice Boltzmann method by the immersed moving boundary scheme

The immersed moving boundary (IMB) scheme has been extensively used to couple the discrete element method (DEM) with the lattice Boltzmann method (LBM). In the literature, only the formulation of IMB for lattice nodal cells covered by a single-solid particle was given. The treatment of situations where a nodal cell is covered by two or more solid particles is seldom discussed. It is found that some numerical instability can occur for such situations due to an inappropriate computation of the weighting function in the IMB formulation. This work presents an enhanced treatment that can resolve the issue and validates it using some benchmark tests. Furthermore, to avoid the extra costs associated with the treatment and simplify the complicated procedure introduced, a simplified IMB scheme is proposed. The accuracy of both enhanced and simplified IMB schemes are validated by test cases including single-particle sedimentation, two-particle drafting-kissing-tumbling phenomenon, and multiple-particle sedimentation. Then, the robustness of both schemes is examined and discussed using a specially designed flow past cylinders test. The simplified IMB scheme is proved to be robust and sufficiently accurate and simpler and more effective than the enhanced scheme.     

Acoustic scattering of spherical waves incident on a long fluid-saturated poroelastic cylinder

Acoustic scattering of spherical waves generated by a monopole point source in a perfect (inviscid and ideal) compressible fluid by a fluid-saturated porous cylinder of infinite length is studied theoretically in the present study. The formulation utilizes the Biot theory of dynamic poroelasticity along with the appropriate wave-field expansions, the translational addition theorem for spherical wave functions, and the pertinent boundary conditions to obtain a closed-form solution in the form of infinite series. The analytical results are illustrated with a numerical example in which a monopole point source within water is located near a porous cylinder with a water-saturated Ridgefield sandstone formation. The numerical results reveal the effects of source excitation frequency, the cylinder interface permeability condition, and the location of the point source and the field point on the backscattered pressure magnitudes. Limiting cases are considered, and the obtained numerical results are validated by already well-known solutions.     

Scattering from cylinders using the two-dimensional vector plane wave spectrum

The two-dimensional vector plane wave spectrum (VPWS) is scattered from parallel circular cylinders using a boundary value solution with the T-matrix formalism. The VPWS allows us to define the incident, two-dimensional electromagnetic field with an arbitrary distribution and polarization, including both radiative and evanescent components. Using the fast Fourier transform, we can quickly compute the multiple scattering of fields that have any particular functional or numerical form. We perform numerical simulations to investigate a grating of cylinders that is capable of converting an evanescent field into a set of propagating beams. The direction of propagation of each beam is directly related to a spatial frequency component of the incident evanescent field.