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

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.     

Wave propagation in cracked elastic slabs and half-space domains—TBEM and MFS approaches

In this paper, the traction boundary element method (TBEM) and the method of fundamental solutions (MFS), formulated in the frequency domain, are used to evaluate the 3D scattered wave field generated by 2D empty cracks embedded in an elastic slab and a half-space. Both models overcome the thin-body difficulty posed when the classical BEM is applied.The crack exhibits arbitrary cross section geometry and null thickness. In neither model are the horizontal formation surfaces discretized, since appropriate fundamental solutions are used to take them into consideration.The TBEM models the crack as a single line. The singular and hypersingular integrals that arise during the TBEM model's implementation are computed analytically, which overcomes one of the drawbacks of this formulation. The results provided by the proposed TBEM model are verified against responses provided by the classical BEM models derived for the case of an empty cylindrical circular cavity.The MFS solution is approximated in terms of a linear combination of fundamental solutions, generated by a set of virtual sources simulating the scattered field produced by the crack, using a domain decomposition technique. To avoid singularities, these fictitious sources are not placed close to the crack, and the use of an enriched function to model the displacement jumps across the crack is unnecessary.The performances of the proposed models are compared and their limitations are shown by solving the case of a C-shaped crack embedded in an elastic slab and a half-space domain.The applicability of these formulations is illustrated by presenting snapshots from computer animations in the time domain for an elastic slab containing an S-shaped crack, after applying an inverse Fourier transformation to the frequency domain computations.     

Coupling of the BEM with the MFS for the numerical simulation of frequency domain 2-D elastic wave propagation in the presence of elastic inclusions and cracks

This paper proposes a coupling formulation between the boundary element method (BEM displacement and TBEM traction formulations) and the method of fundamental solutions (MFS) for the transient analysis of elastic wave propagation in the presence of multiple elastic inclusions to overcome the specific limitations of each of these methods. The full domain of the original problem is divided into sub-domains, which are handled separately by the BEM or the MFS. The coupling is enforced by imposing the required boundary conditions.The accuracy, efficiency and stability of the proposed algorithms, using different combinations of BEM and MFS, are verified by comparing the solutions against reference solutions. The computational efficiency of the proposed coupling formulation is illustrated by computing the CPU time and the error at high frequencies.The potential of the proposed procedures is illustrated by simulating the propagation of elastic waves in the vicinity of an empty crack, with null thickness placed close to an elastic inclusion.     

Use of constant,linear and quadratic boundary elements in 3D wave diffraction analysis

The performance of the Boundary Element Method (BEM) depends on the size of the elements and the interpolation function used. However, improvements in accuracy and efficiency obtained with both expansion and grid refinement increases demand on the computational effort. This paper evaluates the performance of constant, linear and quadratic elements in the analysis of the three-dimensional scattering caused by a cylindrical cavity buried in an infinite homogeneous elastic medium subjected to a point load. A circular cylindrical cavity for which analytical solutions are known is used in the simulation analysis. First, the dominant BEM errors are identified in the frequency domain and related to the natural vibration modes of the inclusion. Comparisons of BEM errors are then made for different types of boundary elements, maintaining similar computational costs. Finally, the accuracy of the BEM solution is evaluated when the nodal points are moved inside linear and quadratic discontinuous elements.     

Benchmark Solutions for Three-Dimensional Transient Heat Transfer in Two-Dimensional Environments Via the Time Fourier Transform

The evaluation of heat propagation in the time domain generated by transient heat sources placed in the presence of three-dimensional media requires the use of computationally demanding numerical schemes. The implementation of numerical 3D solutions may benefit from the existence of benchmark solutions to test the accuracy of approximate schemes.
With this purpose inmind, this article presents analyticalnumerical solutions to evaluate the heat-field elicited by monopole heat sources in the presence of three different inclusions, namely, a cylindrical circular solid inclusion, a cylindrical circular cavity with null fluxes and a cavity with null temperatures prescribed along its boundary, buried in an unbounded medium. The problem is first subjected to a time and space Fourier Transform, which allows the solution to be obtained in the frequency domain as summation of 2D solutions for different spatial wavenumbers. Then, using the inverse Fourier transforms in the wavenumber and frequency domains, the 3D time responses are synthesized. Complex frequencies are used to avoid the aliasing phenomena.
This methodology is first validated calculating the fundamental time solutions for one, two and three dimensions in an unbounded medium. Simulation analyses of these idealized models are then used to study the patterns of heat propagation in the vicinity of the inclusions.     

3D acoustic scattering from an irregular fluid waveguide via the BEM

The BEM is used to calculate the variation in the pressure field generated by a dilatational point load inside a channel filled with a homogeneous fluid, in the presence of an irregular floor. The Green's functions are defined in the frequency domain and obtained by superposing virtual acoustic sources combined so as to generate the boundary conditions of the free or rigid surfaces of the channel. The responses in the time domain are obtained by means of Fourier transforms, making use of complex frequencies. The main features and spectral representation of the signals scattered by irregular floors are then described and used to elucidate the most important aspect of wave acoustics, which can provide the basis for the development of non-destructive testing and imaging methods.     

Coupling the BEM/TBEM and the MFS for the numerical simulation of acoustic wave propagation

The coupling of the boundary element method (BEM)/the traction boundary element method (TBEM) and the method of fundamental solutions (MFS) is proposed for the transient analysis of acoustic wave propagation problems in the presence of multi-inclusions to overcome the limitations posed by each method. The full domain is divided into sub-domains which are modeled using the BEM/TBEM and the MFS, and the sub-domains are coupled with the imposition of the required boundary conditions. The accuracy of the proposed algorithms, using different combinations of BEM/TBEM and MFS, is verified by comparing the solutions against reference solutions. The applicability of the proposed method is shown by simulating the acoustic behavior of a rigid acoustic screen in the vicinity of a dome and by computing the acoustic attenuation provided by a fluid-filled thin inclusion separating two railway tracks in an underground train station.     

Wave propagation in the presence of empty cracks in an elastic medium

This paper proposes the use of a traction boundary element method (TBEM) to evaluate 3D wave propagation in unbounded elastic media containing cracks whose geometry does not change along one direction. The proposed formulation is developed in the frequency domain and handles the thin-body difficulty presented by the classical boundary element method (BEM). The empty crack may have any geometry and orientation and may even exhibit null thickness. Implementing this model yields hypersingular integrals, which are evaluated here analytically, thereby surmounting one of the drawbacks of this formulation. The TBEM formulation enables the crack to be modelled as a single line, allowing the computation of displacement jumps in the opposing sides of the crack. Furthermore, if this formulation is combined with the classical BEM formulation the displacements in the opposing sides of the crack can be computed by modelling the crack as a closed empty thin body.     

Point force excitation of a thick-walled elastic infinite pipe with an embedded inhomogeneity

The propagation of elastic waves in thick-walled pipe with an embedded inhomogeneity is considered. The pipe is excited by a point force applied on its surface and the time harmonic problem is solved using the null field approach, a method whose main characteristics are surface integral representations and expansions in spherical and cylindrical vector wave functions. Entering in the expression for the scattered field are the transition matrix for the cavity, the reflection matrices for the inner and outer surfaces of the pipe, the transformation functions between the spherical and cylindrical vector wave functions and also the translation for the cylindrical waves. Numerical examples, both in the frequency and time domain, are presented for a spherical cavity and an open circular crack.     

2.5D coupled BEM–FEM used to model fluid and solid scattering wave

We describe a hybrid method to study fluid and solid interaction problems in the frequency domain. The numerical method is based on subdomain decomposition. The BEM is used to model unbounded solid mediums, whereas the confined subdomains, both fluid and solid, are represented by the FEM. The analysis is carried out by superposing two‐and‐a‐half dimension (2.5D) problems for different longitudinal wave numbers. A novel 2.5D FEM formulation for inviscid fluids is proposed, which include the energy lost at the fluid boundary enclosure. The fluid and solid subdomains are coupled, and appropriate boundary conditions are imposed at the interfaces. The proposed technique is verified from analytical solutions. A cylindrical cavity located in an unbounded solid medium excited by a dilatational point source is studied. Computed results are in good agreement with the analytical solution. Later, noise and vibration in a concrete tunnel due to an internal pressure load is analysed with the proposed methodology. Results show that tunnel and soil displacements increase with the load speed, as did the air pressure inside the tunnel, according with the travelling ranges defined by the wave propagation velocities in each medium. Copyright © 2014 John Wiley & Sons, Ltd.     

Wave motion between two fluid-filled boreholes in an elastic medium

The boundary element method (BEM) is used to fully simulate the propagation of waves between two fluid-filled boreholes. The sources are placed in one of the boreholes while the receivers are placed in the other. This model is frequently used in cross-hole seismic prospecting techniques to assess the characteristics of the elastic medium between the two boreholes. This work studies the dependence of the wave propagation patterns on the distance between the source and the receiver, their location and orientation relative to the axis of a circular borehole and type of elastic formation (fast and slow formations). In addition, this BEM model is used to compute the influence of the deformed boreholes whose cross-section is not circular. Both the spectra responses and the time-domain responses are computed to elucidate the main physical features of the problem solved.     

Coupling BEM/TBEM and MFS for the simulation of transient conduction heat transfer

The coupling between the boundary element method (BEM)/the traction boundary element method (TBEM) and the method of fundamental solutions (MFS) is proposed for the transient analysis of conduction heat transfer in the presence of inclusions, thereby overcoming the limitations posed by each method. The full domain is divided into sub‐domains, which are modeled using the BEM/TBEM and the MFS, and the coupling of the sub‐domains is achieved by imposing the required boundary conditions. The accuracy of the proposed algorithms, using different combinations of BEM/TBEM and MFS formulations, is checked by comparing the resulting solutions against referenced solutions. The applicability of the proposed methodology is shown by simulating the thermal behavior of a solid ring incorporating a crack or a thin inclusion in its wall. The crack is assumed to have null thickness and does not allow diffusion of energy; hence, the heat fluxes are null along its boundary. The thin inclusion is modeled as filled with thermal insulating material. Copyright © 2010 John Wiley & Sons, Ltd.     

The scattering of 3D sound sources by rigid barriers in the vicinity of tall buildings

The acoustic scattering of a three-dimensional (3D) sound source by an infinitely long rigid barrier in the vicinity of a tall building is analyzed using the boundary element method (BEM). The acoustic barrier is modeled using boundary elements, and is assumed to be non-absorbing, while the image source method is used to model the tall building as an infinite vertical barrier. A frequency domain BEM formulation is used, and time domain responses are then obtained by applying an inverse Fourier transformation.Since the geometry of the problem does not vary along one direction, the 3D solution can be calculated as the summation of a sequence of 2D problems, each solved for a different spatial wavenumber, k_z. To obtain the 3D solution, a discrete form wavenumber transform is performed by considering an infinite number of virtual point sources equally spaced along the z axis. Complex frequencies are used to minimize the influence of these neighboring fictitious sources.Numerical simulations are performed using barriers of varying sizes, evaluating the attenuation of the sound pressure level in the vicinity of the building façade. The creation of shadow zones by the barriers is analyzed and time responses are presented to better understand the sound propagation around these obstacles.     

Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids

The dynamic responses of fluid-saturated semi-infinite porous continua to transient excitations such as seismic waves or ground vibrations are important in the design of soil-structure systems. Biot's theory of porous media governs the wave propagation in a porous elastic solid infiltrated with fluid. The significant difference to an elastic solid is the appearance of the so-called slow compressional wave. The most powerful methodology to tackle wave propagation in a semi-infinite homogeneous poroelastic domain is the boundary element method (BEM). To model the dynamic behavior of a poroelastic material in the time domain, the time domain fundamental solution is needed. Such solution however does not exist in closed form. The recently developed ‘convolution quadrature method’, proposed by Lubich, utilizes the existing Laplace transformed fundamental solution and makes it possible to work in the time domain. Hence, applying this quadrature formula to the time dependent boundary integral equation, a time-stepping procedure is obtained based only on the Laplace domain fundamental solution and a linear multistep method. Finally, two examples show both the accuracy of the proposed time-stepping procedure and the appearance of the slow compressional wave, additionally to the other waves known from elastodynamics.     

Elastodynamic scattering by fluid-filled nonplanar cracks

A recent modification of the null field approach is adapted to the study of scattering of elastic waves by fluid-filled nonplanar cracks. The fluid-filled crack is modeled as a surface over which friction-free boundary conditions apply. A closed surface is formed by adding a fictitious surface, on which latter surface boundary conditions of welded contact are applied. The surface fields on the closed surface are expanded in vector spherical harmonics in a manner which takes the edge conditions into account. Some numerical results on farfield quantities, such as scattering cross sections and backscattering amplitudes (both in the frequency and time domains), are presented for rotationally symmetric cracks.     

充液环肋圆柱壳耦合振动的波动解

研究了充液环肋圆柱壳结构的耦合振动特性。基于Love壳体理论,考虑壳体内部完全充液,采用波动法建立充液环肋圆柱壳耦合振动的频率特征方程,得到了不同边界条件下的耦合频率值。通过与已有文献数据对比,验证了该文研究方法的有效性和正确性。最后通过算例,分析了充液因素、环肋参数、边界条件、壳体几何参数等对充液环肋圆柱壳耦合振动的影响。     

Defining an accurate MFS solution for 2.5D acoustic and elastic wave propagation

This paper proposes a simple methodology to assess the accuracy of the method of fundamental solutions (MFS) when applied to 2.5D acoustic and elastic wave propagation. The proposed technique is developed in the frequency domain. It copes with the precision uncertainty difficulty presented by the MFS solution through its dependency on the number and position of virtual sources and collocation points.The methodology relies on the correlation between the errors registered along surfaces, where boundary or continuity conditions are known a priori, with those obtained along the system domain. Circular cylindrical domains are modeled to illustrate the efficiency of the proposed methodology, since in this case analytical solutions are available.A numerical example is used to illustrate the application of the methodology to a more complex case. An elastic column exhibiting an embedded curved crack, with null thickness, is used to illustrate the applicability of the proposed technique. Since, there are no known analytical solutions; the results provided by the traction boundary element method (TBEM) are used as reference solutions.     

弹性介质中充液圆柱壳的轴对称振动分析

从理论上分析了各向同性弹性介质中有限长充液圆柱壳的自由振动特性。利用经典FLUGGE壳体运动方程,结合弹性动力学方程,得到了弹性介质中充液圆柱壳耦合系统的特征方程。利用数值方法得到了弹性介质中圆柱壳中空和充液时的实固有频率。计算结果表明弹性介质中圆柱壳的实固有频率比真空中圆柱壳的实固有频率高,而且存在截止轴向模态。弹性介质刚度不仅对充液圆柱壳的实固有频率影响显著,而且影响其截止轴向模态。     

流体饱和半空间中埋置球面P1、P2和SV波源动力格林函数

基于Biot流体饱和孔隙介质理论,采用Hankel积分变换方法,在频域内求解了流体饱和半空间中埋置球面P1、P2和SV波源的动力格林函数。首先由Hankel积分变换将空间域内球面波展开为波数域内柱面波的叠加;然后在半空间表面对称位置虚拟放置一同样大小的球面波源,这样对于球面膨胀波源(P1和P2波源),地表剪应力为零,但存在非零正应力和孔隙水压,对于球面剪切波源(SV波源),地表正应力和孔隙水压为零,但存在非零剪应力;最后叠加球面波源、虚拟波源和残余半空间表面应力产生的动力响应,即可求得流体饱和半空间中埋置球面波源波数域内的动力响应,空间域内埋置球面波源的动力格林影响函数则由Hankel逆变换求得。该文给出的球面波源动力格林函数,为建立以球面P1、P2和SV波动力格林函数为基本解的间接边界元方法,求解饱和多孔介质中三维轴对称弹性波散射问题奠定了基础。     

Coupled axisymmetric vibration of nonlocal fluid-filled closed spherical membrane shell

In this paper, the axisymmetric vibration of a fluid-filled spherical membrane shell is studied based on nonlocal elasticity theory. The membrane shell is considered elastic, homogeneous and isotropic. The shell model is reformulated using the nonlocal differential constitutive relations of Eringen. The membrane shell is completely filled with an inviscid fluid. The motion of the fluid is governed by the wave equation. Nonlocal governing equations of motion for the fluid-filled spherical membrane shell are derived. Along the contact surface between the membrane and the fluid, the compatibility requirement is applied and Legendre polynomials, associated Legendre polynomials and spherical Bessel functions are used to obtain the natural frequencies of the fluid-filled spherical membrane shells. The frequencies for both empty and fluid-filled spherical membrane shell are evaluated, and their comparisons are performed to confirm the validity and accuracy of the proposed method. An excellent agreement is found between the present and previous ones available in the literature. The variations of the natural frequencies with the small-scale parameter, density ratio, wave speed ratio and Poisson’s ratio are also examined. It is observed that the frequencies are affected when the size effect is taken into consideration.