Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering

Classical finite-element and boundary-element formulations for the Helmholtz equation are presented, and their limitations with respect to the number of variables needed to model a wavelength are explained. A new type of approximation for the potential is described in which the usual finite-element and boundary-element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions evenly distributed on the unit sphere. Compared with standard piecewise polynomial approximation, the plane-wave basis is shown to give considerable reduction in computational complexity. In practical terms, it is concluded that the frequency for which accurate results can be obtained, using these new techniques, can be up to 60 times higher than that of the conventional finite-element method, and 10 to 15 times higher than that of the conventional boundary-element method.     

Transient analysis of three-dimensional wave propagation using the boundary element method

In the past the time domain solution of the wave equation has been limited to simplified problems. This was due to the limitations of analytical methods and the capacity of computers to manipulate and store ‘large’ blocks of spatial information. With the advent of ‘super computers’ the ability to solve such problems has significantly increased. This paper outlines a method for transient analysis of wave propagation in arbitrary domains using a boundary element method. The technique presented will allow the definition of a domain, the input of impedance conditions on the domain's surface, the specification of inputs on the surface, and the specification of initial conditions within the domain. It will produce a complete solution of the wave equation inside the domain. The techniques are demonstrated using a program with a boundary element formulation of Kirchhoff's equation. The elements used are triangular and compatible.     

Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems

A new adaptive fast multipole boundary element method (BEM) for solving 3-D half-space acoustic wave problems is presented in this paper. The half-space Green's function is employed explicitly in the boundary integral equation (BIE) formulation so that a tree structure of the boundary elements only for the boundaries of the real domain need to be applied, instead of using a tree structure that contains both the real domain and its mirror image. This procedure simplifies the implementation of the adaptive fast multipole BEM and reduces the CPU time and memory storage by about a half for large-scale half-space problems. An improved adaptive fast multipole BEM is presented for the half-space acoustic wave problems, based on the one developed recently for the full-space problems. This new fast multipole BEM is validated using several simple half-space models first, and then applied to model 3-D sound barriers and a large-scale windmill model with five turbines. The largest BEM model with 557470 elements was solved in about an hour on a desktop PC. The accuracy and efficiency of the BEM results clearly show the potential of the adaptive fast multipole BEM for solving large-scale half-space acoustic wave problems that are of practical significance.     

On the application of Biot's theory to acoustic wave propagation in snow

A fluid-saturated, elastic, porous media model is used to describe acoustic wave propagation in snow. This model predicts the existence of two dilatational waves and a shear wave. In homogeneous, isotropic snow the two dilatational waves are uncoupled from one another but involve coupled motion between the interstitial air and ice skeleton. Dilatational waves of the first kind and shear waves are slightly dispersive and attenuated with distance. Dilatational waves of the second kind are strongly dispersive and highly attenuated. The model also predicts that the wave impedance for snow is close to that of air and that snow strongly absorbs acoustic wave energy.Available experimental phase velocity, impedance and attenuation data support the calculated results. Phase velocity measurements indicate three identifiable categories: fast dilatational waves (phase velocity ? 500 m/s), slow dilatational waves (phase velocity < 500 m/s) and shear waves. Wave impedance and attenuation measurements illustrate the low impedance, highly absorbing characteristics of snow. Additional impedance, attenuation and phase velocity data are required to further test and improve the model.     

Use of discontinuous boundary elements for fracture mechanics analysis

The boundary element method (BEM) has long been considered a suitable technique for the analysis of fracture mechanics problems. While research is being published showing how advanced BEM formulations can reduce the data preparation time and produce accurate fracture results, it is still very simple to use more conventional direct boundary element methods to find accurate stress intensity factors very quickly and easily. This is made possible with discontinuous elements. This paper describes modeling and post-processing techniques which can assist the designer in this process. The result are compared against the charts published by Rooke & Cartwright, and show an excellent agreement. The method offers a very quick and accurate method of predicting fracture properties of real-life, non-standard geometries.     

Simulation of acoustic wave propagation in dispersive media with relaxation losses by using FDTD method with PML absorbing boundary condition

We present a method to incorporate the relaxation dominated attenuation into the finite-difference time-domain (FDTD) simulation of acoustic wave propagation in complex media. A dispersive perfectly matched layer (DPML) boundary condition, which is suitable for boundary matching to such a dispersive media whole space, is also proposed to truncate the FDTD simulation domain. The numerical simulation of a Ricker wavelet propagating in a dispersive medium, described by second-order Debye model, shows that the Ricker wavelet is attenuated in amplitude and expanded in time in its course of propagation, as required by Kramers-Kronig relations. The numerical results also are compared to exact solution showing that the dispersive FDTD method is accurate and that the DPML boundary condition effectively dampens reflective waves. The method presented here is applicable to the simulation of ultrasonic instrumentation for medical imaging and other nondestructive testing problems with frequency dependent, attenuating media.     

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Medium‐frequency regime and multi‐scale wave propagation problems have been a subject of active research in computational acoustics recently. New techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion. One such technique, the discontinuous enrichment method, incorporates features of the governing partial differential equation in the approximation, in particular, the solutions of the homogeneous form of the equation. Here, based on this concept and by extension of a conventional space–time finite element method, a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this contribution, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Results for two‐dimensional and three‐dimensional problems, in both low‐frequency and medium‐frequency regimes, show that the proposed DGM outperforms the conventional space–time finite element method. Copyright © 2014 John Wiley & Sons, Ltd.     

Simulation of wave propagation in three-dimensional random media

Quantitative error analyses for the simulation of wave propagation in three-dimensional random media, when narrow angular scattering is assumed, are presented for plane-wave and spherical-wave geometry. This includes the errors that result from finite grid size, finite simulation dimensions, and the separation of the two-dimensional screens along the propagation direction. Simple error scalings are determined for power-law spectra of the random refractive indices of the media. The effects of a finite inner scale are also considered. The spatial spectra of the intensity errors are calculated and compared with the spatial spectra of intensity. The numerical requirements for a simulation of given accuracy are determined for realizations of the field. The numerical requirements for accurate estimation of higher moments of the field are less stringent.     

A new hybrid velocity integration method applied to elastic wave propagation

We present a novel space–time Galerkin method for solutions of second‐order time‐dependent problems. By introducing the displacement–velocity relationship implicitly, the governing set of equations is reformulated into a first‐order single field problem with the unknowns in the velocity field. The resulting equation is in turn solved by a time‐discontinuous Galerkin approach (Int. J. Numer. Anal. Meth. Geomech. 2006; 30 :1113–1134), in which the continuity between time intervals is weakly enforced by a special upwind flux treatment. After solving the equation for the unknown velocities, the displacement field quantities are computed a posteriori in a post‐processing step. Various numerical examples demonstrate the efficiency and reliability of the proposed method. Convergence studies with respect to the h‐ and p‐refinement and different discretization techniques are given. Copyright © 2007 John Wiley & Sons, Ltd.     

A method for computation of discontinuous wave propagation in heterogeneous solids: basic algorithm description and application to one‐dimensional problems

An explicit integration algorithm for computations of discontinuous wave propagation in heterogeneous solids is presented, which is aimed at minimizing spurious oscillations when the wave fronts pass through several zones of different wave speeds. The essence of the present method is a combination of two wave capturing characteristics: a new integration formula that is obtained by pushforward–pullback operations in time designed to filter post‐shock oscillations, and the central difference method that intrinsically filters front‐shock oscillations. It is shown that a judicious combination of these two characteristics substantially reduces both spurious front‐shock and post‐shock oscillations. The performance of the new method is demonstrated as applied to wave propagation through a uniform bar with varying courant numbers, then to heterogeneous bars. Copyright © 2012 John Wiley & Sons, Ltd.     

A 3-D,symmetric, finite element formulation of the Biot equations with application to acoustic wave propagation through an elastic porous medium

A weak solution of the coupled, acoustic-elastic, wave propagation problem for a flexible porous material is proposed for a 3-D continuum. Symmetry in the matrix equations; with respect to both volume, i.e. ‘porous frame’–‘pore fluid’, and surface, i.e. ‘porous frame/pore fluid’–‘non-porous media’, fluid–structure interaction; is ensured with only five unknowns per node; fluid pore pressure, fluid-displacement potential and three Cartesian components of the porous frame displacement field. Taking Biot's general theory as starting point, the discretized form of the equations is derived from a weighted residual statement, using a standard Galerkin approximation and iso-parametric interpolation of the dependent variables. The coupling integrals appearing along the boundary of the porous medium are derived for a number of different surface conditions. The primary application of the proposed symmetric 3-D finite element formulation is modelling of noise transmission in typical transportation vehicles, such as aircraft, cars, etc., where porous materials are used for both temperature and noise insulation purposes. As an example of an application of the implemented finite elements, the noise transmission through a double panel with porous filling and different boundary conditions at the two panel boundaries are analysed. © 1998 John Wiley & Sons, Ltd.     

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W ₂(y) and W ₃(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.     

Non‐reflecting boundary conditions for acoustic propagation in ducts with acoustic treatment and mean flow

We consider a time‐harmonic acoustic scattering problem in a 2D infinite waveguide with walls covered with an absorbing material, in the presence of a mean flow assumed uniform far from the source. To make this problem suitable for a finite element analysis, the infinite domain is truncated. This paper concerns the derivation of a non‐reflecting boundary condition on the artificial boundary by means of a Dirichlet‐to‐Neumann (DtN) map based on a modal decomposition. Compared with the hard‐walled guide case, several difficulties are raised by the presence of both the liner and the mean flow. In particular, acoustic modes are no longer orthogonal and behave asymptotically like the modes of a soft‐walled guide. However, an accurate approximation of the DtN map can be derived using some bi‐orthogonality relations, valid asymptotically for high‐order modes. Numerical validations show the efficiency of the method. The influence of the liner with or without mean flow is illustrated. Copyright © 2011 John Wiley & Sons, Ltd.     

The application of three-dimensional finite elements to a cylinder-cylinder intersection

Recent advances in finite element techniques have enabled full three-dimensional stress analyses to be undertaken. Economy and accuracy can only be achieved simultaneously if the characteristics of the element are understood. Preparatory work is described in which suitable pitching and disposition of brick-type elements are established for the analysis of cylinders. The information derived is put to use in the analysis of a cylinder–cylinder intersection. A favourable comparison is made with an independently computed solution.     

A new boundary element formulation for wave load analysis

A new boundary element (BEM) formulation is proposed for wave load analysis of submerged or floating bodies. The presented formulation, through establishing an impedance relation, permits the evaluation of the hydrodynamic coefficients (added mass and damping coefficients) and the coefficients of wave exciting forces systematically in terms of system matrices of BEM without solving any special problem, such as, unit velocity or unit excitation problem. It also eliminates the need for scattering analysis in the evaluation of wave exciting forces. The imaginary and real parts of impedance matrix give, respectively, added mass and damping matrices whose elements describe the fluid resistance against the motion of the body. The formulation is explained through the use of a simple fluid-solid system under wave excitations, which involves a uniform fluid layer containing a solid cylindrical body. In the formulation, the solid body is taken first as deformable, then, it is specialized when it is rigid. The validity of the proposed method is verified by comparing its result with those available in literature for rigid submerged or floating bodies.     

Seismic crack propagation analysis of concrete structures using boundary elements

The time-domain boundary element method is applied to study the propagation of discrete cracks under dynamic loadings. The principles of linear elastic fracture mechanics are employed to accurately represent the stress singularity in front of the crack tip. A fracture criterion suitable for brittle materials is employed to determine the time and direction of the propagating crack. Discrete crack closure under dynamic loadings is modelled and the response of a 2-D cracked body is obtained by direct integration of the equations of motion. Results are presented for three test problems wherein the accuracy and usefulness of the proposed formulation is established through comparisons with available data. 'It is expected that this formulation can be used to study the seismic cracking of mass concrete structures such as gravity dams.     

Influence of material parameters on acoustic wave propagation modes in ZnO/Si bi-layered structures

The influences of material properties on acoustic wave propagation modes in ZnO/Si bi-layered structures are studied. The transfer matrix method is used to calculate dispersion relations, wave field distributions, and electromechanical coupling coefficients of acoustic wave propagation modes in ZnO/Si bi-layered systems, in which the thickness of the substrate is of the same order of magnitude as the wavelength of the propagating wave modes. The influences of the thin film parameters on the acoustic wave propagation modes and their electromechanical coupling coefficients of the wave modes also are obtained. In addition, some experimental results for characterizing the wave propagation modes and their frequencies have also been obtained, which agree well with the theoretical predictions.     

Characterization of surface acoustic wave propagation in multi-layered structures using extended FEM/SDA software

This paper describes the characterization of SAW propagation in layered substrate and overlayered structures. The software based on the finite element method and spectral domain analysis was newly developed and applied to the characterization of SAW propagation under an infinitely-long Al interdigital transducer on a rotated Y-cut LiTaO₃/sapphire substrate. Because of the finite LiTaO₃ thickness, a series of spurious resonances appears. It is shown that the excitation strength of the spurious resonances changes with frequency as well as the rotation angle, which reflects the frequency and rotation angle dependence of the energy leakage. Next, the analysis was carried out for SAWs propagating in a SiO₂ layer/Al IDT/42°YX-LiTaO₃ structure. It is shown that the influence of the SiO₂ layer is significantly dependent on the location where the SiO₂ layer is deposited. In particular, it is shown that when the SiO₂ layer is deposited only on top of the electrodes, the SAW reflectivity increases compared with when the SiO₂ layer is deposited between and on top of electrodes.     

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.