A new DRBEM model for wave refraction and diffraction

A numerical model using the dual reciprocity boundary element method (DRBEM) is developed to study the combined refraction and diffraction of water waves propagating around islands or solid offshore structures over a seabed with a variable depth. Based on the well-known mild-slope equation, the model has been validated by comparison with both analytical solutions and standard numerical solutions available in the literature. The results show that a considerable improvement in terms of numerical efficiency has been achieved with the adoption of the DRBEM and the model has a great potential to be used in engineering practice to solve wave refraction and diffraction problems.     

A general DRBEM model for wave refraction and diffraction

A numerical model based on the dual reciprocity boundary element method (DRBEM) is presented here for the study of combined wave diffraction and refraction. The model is more general than that presented by Zhu [Zhu S-P. Engng Anal Boundary Elements 1993;12:261–274] in the sense that areas or coastlines where water depth is zero can be dealt with as well. Our comparative study show that the new model is very accurate for long waves (tsunami waves). It is numerically very efficient in comparison with models based on finite elements too. Using the new model, the interaction between the diffraction and refraction effects is examined. It is shown that the diffraction effect is significantly enhanced when there is a combined diffraction and refraction than when there is just diffraction alone.     

Numerical modelling of wave-induced currents in a breakwater situation

A numerical model system is developed to describe waves and wave-induced currents in a breakwater situation, where refraction, diffraction, reflection and wave–current interaction may all play important roles in the transformation of waves. A turbulence transport model is employed to estimate the eddy viscosity, and the friction factors associated with the major variables are clearly defined using a high level closure of combined flow bed friction. The model system is tested against a laboratory situation, and the effects of reflection are demonstrated on the wave-induced currents as well as on the wave distribution.     

A coupled boundary element-finite difference solution of the elliptic modified mild slope equation

The modified mild slope equation of [5] is solved using a combination of the boundary element method (BEM) and the finite difference method (FDM). The exterior domain of constant depth and infinite horizontal extent is solved by a BEM using linear or quadratic elements. The interior domain with variable depth is solved by a flexible order of accuracy FDM in boundary-fitted curvilinear coordinates. The two solutions are matched along the common boundary of two methods (the BEM boundary) to ensure continuity of value and normal flux. Convergence of the individual methods is shown and the combined solution is tested against several test cases. Results for refraction and diffraction of waves from submerged bottom mounted obstacles compare well with experimental measurements and other computed results from the literature.     

An improved partition of unity finite element model for diffraction problems

The partition of unity finite element method (PUFEM) is explored and improved to deal with practical diffraction problems efficiently. The use of plane waves as an external function space allows an efficient implementation of an approximate exterior non‐reflective boundary condition, improving the original proposed by Higdon for general diffraction problems. A ‘virtually’ analytical integration procedure is introduced for multi‐dimensional high‐frequency problems which exhibits a dramatic decrease in the number of operations for a given error compared with standard integration methods. Suitable conjugate gradient type solvers for the whole range of wavenumbers are used, including such cases in which PUFEM can produce nearly singular matrices caused by ‘round‐off’ limits. Copyright © 2001 John Wiley & Sons, Ltd.     

半圆柱型沉积盆地对SH波散射的数值分析

该文实现了一种半无限域SH波散射问题的数值分析方法。采用传递矩阵法得到SH波斜入射时的自由场,将其作为输入;采用集中质量显式有限元方法计算区域内节点的位移;采用透射人工边界计算人工边界点的位移;通过编写的FORTRAN程序实现计算过程。运用该方法对均匀半空间内半圆柱型沉积盆地在SH波入射下的散射进行了分析,与Trifunac M D的解析解进行了对比,验证了该文方法的有效性,分析了不同入射角对地表位移和位移谱放大系数的影响。最后,对成层半空间内半圆柱型沉积盆地在SH波入射下的散射进行了分析。相对于解析方法而言,该方法可以考虑更为复杂地形情况。     

波浪与弹性板作用的数值模拟

该文采用高阶有限元和边界元联合的方法求解波浪与弹性板的相互作用。其中流场采用边界元法求解,结构弹性响应方程采用基于Mindlin板理论的有限元方法求解,通过模态叠加技术实现了弹性板变形与流场相互作用的解耦。通过对一矩形板的计算,验证了该文方法与他人试验结果和数值模拟结果都吻合良好。利用这一模型进一步分析了波浪与弹性圆形板的作用问题,并对圆形板运动响应的收敛性进行了分析。     

Calculation of the Rayleigh-Sommerfeld diffraction integral by exact integration of the fast oscillating factor

We describe a numerical method that can be used to calculate the propagation of light in a medium of constant (possibly complex) index of refraction n. The method integrates the Rayleigh-Sommerfeld diffraction integral numerically. After an appropriate change of integration variables, the integrand of the diffraction integral is split into a slowly varying and an (often fast) oscillating quadratic factor. The slowly varying factor is approximated by a spline fit, and the resulting Fresnel integrals are subsequently integrated exactly. Although the method is not as fast as methods involving a fast Fourier transform, such as plane-wave propagation or Fresnel approximation, it is accurate over a greater range than these methods.     

The spectral element method for elastic wave equations—application to 2‐D and 3‐D seismic problems

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two‐ and three‐dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor‐product of a Gauss–Lobatto–Legendre 1‐D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy‐momentum conserving scheme that can be put into a classical explicit‐implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two‐dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three‐dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three‐dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.     

FEM/BEM for simulation of LSAW devices

This paper presents a modeling of the propagation of surface acoustic, leaky acoustic, and surface skimming bulk waves in piezoelectrics with a finite array of metallic electrodes over their surface. A combined method of matrix Green's function and the finite element method for computation of all acoustic wave fields is an effective tool for simulation of the propagation of acoustic waves in such structures. The proposed method is optimized in the speed of computation of all matrix Green's function components originally obtained. The Fourier transformations of Green's function from kappa-space domain to real space domain are performed by combined trapezoidal and Filon's integration methods for rapidly oscillating functions. The trapezoidal integration method is used on a distance from a point source from zero to a few wavelengths long, but the other has the advantage for a distance from some wavelength to infinity. That allows one, by selectively condensing computation grids around branch and singular points of the sharp behavior of Green's function, to maximize speed and accuracy of computation of integrals. FEM is used, modified originally to achieve acceleration without loss accuracy. Because of the simple geometry of the electrodes, unknown elastic fields are presented as a series of known eigenfunctions with unknown coefficients over the whole region of electrodes. All unknown coefficients are determined by applying the Galerkin method. There is good agreement between numerical and experimental conductances of acoustic wave transducers on materials such as lithium niobate and lithium tantalate.     

Some properties of a hybrid element method for water waves

Two properties of hybrid element method for diffraction and radiation of water waves are examined. For long waves in shallow water the method is shown to give a unique solution for all frequencies. Thus, unlike several other known methods, there are no irregular frequencies for which the approximating matrix equation is singluar. For a sea of arbitrary depth, it is shown that all known global identities such as reciprocity and energy theorems are preserved by the discrete solution. Thus, satisfaction of these identities by the numerical solution is only a necessary but by no means sufficient condition for accuracy.     

The Diffraction Efficiency of Double-grating Holographic Fan-out Elements

Abstract

An analytical expression is derived for the diffraction efficiency of a holographic fan-out element containing two superimposed gratings separated by a small angle. In addition to the usual zero and first orders an infinite set of the significant spurious waves is considered. In deriving the expression all these spurious waves are assumed to satisfy the Bragg condition exactly. The results are compared with a direct numerical solution and give very good agreement for angles up to 1°. Furthermore the analysis provides useful information for angles up to 5°. An analogy is drawn with a hybrid hologram consisting of a thick ‘carrier’ grating and a thin ‘modulation’ grating. Using this model the diffraction efficiencies can be calculated using the standard grating formulae. For fan-out applications the spurious waves around the first orders are considered the most important. From this analysis it can be seen that they can be made negligibly small but at the cost of a reduction in total diffraction efficiency.     

Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error

It is known that the reduction in the finite element space discretization error for elastodynamics problems is related to the reduction in numerical dispersion of finite elements. In the paper, we extend the modified integration rule technique for the mass and stiffness matrices to the dispersion reduction of linear finite elements for linear elastodynamics. The analytical study of numerical dispersion for the modified integration rule technique and for the averaged mass matrix technique is carried out in the 1-D, 2-D and 3-D cases for harmonic plane waves. In the general case of loading, the numerical study of the effectiveness of the dispersion reduction techniques includes the filtering technique (developed in our previous papers) that identifies and removes spurious high-frequency oscillations. 1-D, 2-D and 3-D impact problems for which all frequencies of the semi-discrete system are excited are solved with the standard approach and with the new dispersion reduction technique. Numerical results show that compared with the standard mass and stiffness matrices, the simple dispersion reduction techniques lead to a considerable decrease in the number of degrees of freedom and computation time at the same accuracy, especially for multi-dimensional problems. A simple quantitative estimation of the effectiveness of the finite element formulations with reduced numerical dispersion compared with the formulation based on the standard mass and stiffness matrices is suggested.     

Scalar and electromagnetic diffraction point-spread functions

We describe an accurate technique for computing the diffraction point-spread function for optical systems. The approach is based on the combined method of ray tracing and diffraction, which implies that the computation is accomplished in a two-step procedure. First, ray tracing is employed to compute the wave-front error in a reference plane on the image side of the system and to determine the shape of the vignetted pupil. Next the Rayleigh-Sommerfeld diffraction theory, combined with the Kirchhoff approximation and the Stamnes-Spjelkavik-Pedersen method for numerical integration, is applied to compute the field in the region of the image. The method does not rely on small-angle approximations and works well for a pupil of general shape. Both scalar and electromagnetic computations are discussed and numerical results are presented.     

Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves

A novel method for simulating field propagation is presented. The method, based on the angular spectrum of plane waves and coordinate rotation in the Fourier domain, removes geometric limitations posed by conventional propagation calculation and enables us to calculate complex amplitudes of diffracted waves on a plane not parallel to the aperture. This method can be implemented by using the fast Fourier transformation twice and a spectrum interpolation. It features computation time that is comparable with that of standard calculation methods for diffraction or propagation between parallel planes. To demonstrate the method, numerical results as well as a general formulation are reported for a single-axis rotation.     

Improvement of PUFEM for the numerical solution of high‐frequency elastic wave scattering on unstructured triangular mesh grids

The purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77 :1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries. The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

A new application of an improved DRBEM model for water wave propagation over a frictional uneven bottom

This paper presents a numerical scheme to approximate water wave diffraction, refraction and frictional dissipation over an axi-symmetric pit. Based on an improved extended mild-slope equation (EMSE) including bottom friction effect, as the elliptic governing differential equation, dual reciprocity boundary element method (DRBEM) is employed to model water wave propagation over an axi-symmetric pit. To the authors' knowledge, this is the first application of DRBEM for water wave scattering over a pit. In order to promote accuracy of the model, not only effects of the bottom curvature and the slope-squared terms which are neglected in the mild-slope equation (MSE), are considered, but also effect of the bottom friction is measured by the improved EMSE. Numerical results are compared with existing analytical or numerical solutions or with experimental data by several examples. Through these numerical experiments reliability and efficiency of present DRBEM model for determining the total wave field over an uneven bottom is approved.     

时域径向积分边界元法在平面单相凝固问题中的应用

该文将时域精细积分边界元方法与界面追踪法相结合,给出平面单相凝固热传导问题的一个有效数值分析方法。首先,利用稳态热传导问题的基本解和径向积分法给出瞬态传热问题的边界积分方程,并采用精细积分方法求解离散的微分方程组,获得相变界面的热流密度。然后应用相变界面上的能量守恒方程,采用界面追踪法来预测相变边界的移动位置,从而给出相关问题数值模拟的结果。最后,为验证该文方法的有效性,给出两个数值算例并与解析解进行了对比。结果表明,该文方法具有较高的求解精度,是求解相变热传导问题的一种有效数值方法。     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical analysis of Lamb waves using the finite and spectral cell methods

An accurate and efficient simulation of wave propagation phenomena plays an important role in different engineering disciplines. In structural health monitoring, for example, ultrasonic guided waves are used to detect and localize damage and to assess the structural integrity of the component part under consideration. Because of the complexity of real structures, the numerical simulation of structural health monitoring systems is a computationally demanding task. Therefore, to facilitate the analysis of wave propagation phenomena, the authors propose to combine the finite cell method with the spectral element method. The ensuing novel method is referred to as the spectral cell method. Because it does not rely on body‐fitted meshes, the resulting approach eliminates all discretization difficulties encountered in conventional finite element methods. Moreover, with the aid of mass lumping, it paves the way for the use of explicit time‐integration algorithms. In the first part of the paper, we show that using a lumped mass matrix instead of the consistent one has no detrimental effect on the accuracy of the spectral element method. We introduce the spectral cell method in the second part, showing that, when applied to wave propagation analysis, the spectral cell method yields results comparable with other standard higher order finite element approaches.Copyright © 2014 John Wiley & Sons, Ltd.