Transient analysis of wave propagation in non‐homogeneous elastic unbounded domains by using the scaled boundary finite‐element method

The scaled boundary finite‐element method has been developed for the dynamic analysis of unbounded domains. In this method only the boundary is discretized resulting in a reduction of the spatial dimension by one. Like the finite‐element method no fundamental solution is required. This paper extends the scaled boundary finite‐element method to simulate the transient response of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. To reduce the number of degrees of freedom and the computational cost, the technique of reduced set of base functions is applied. The scaled boundary finite‐element equation for an unbounded domain is reformulated in generalized coordinates. The resulting acceleration unit‐impulse response matrix is obtained and assembled with the equation of motion of standard finite elements. Numerical examples of non‐homogeneous isotropic and transversely isotropic unbounded domains demonstrate the accuracy of the scaled boundary finite‐element method. Copyright © 2006 John Wiley & Sons, Ltd.     

Response of unbounded soil in scaled boundary finite‐element method

The scaled boundary finite‐element method is a powerful semi‐analytical computational procedure to calculate the dynamic stiffness of the unbounded soil at the structure–soil interface. This permits the analysis of dynamic soil–structure interaction using the substructure method. The response in the neighbouring soil can also be determined analytically. The method is extended to calculate numerically the response throughout the unbounded soil including the far field. The three‐dimensional vector‐wave equation of elasto‐dynamics is addressed. The radiation condition at infinity is satisfied exactly. By solving an eigenvalue problem, the high‐frequency limit of the dynamic stiffness is constructed to be positive definite. However, a direct determination using impedances is also possible. Solving two first‐order ordinary differential equations numerically permits the radiation condition and the boundary condition of the structure–soil interface to be satisfied sequentially, leading to the displacements in the unbounded soil. A generalization to viscoelastic material using the correspondence principle is straightforward. Alternatively, the displacements can also be calculated analytically in the far field. Good agreement of displacements along the free surface and below a prism foundation embedded in a half‐space with the results of the boundary‐element method is observed. Copyright © 2001 John Wiley & Sons, Ltd.     

A semi-analytical artificial boundary for time-dependent elastic wave propagation in two-dimensional homogeneous half space

This paper presents a time-dependent semi-analytical artificial boundary for numerically simulating elastic wave propagation problems in a two-dimensional homogeneous half space. A polygonal boundary is considered in the half space to truncate the semi-infinite domain, with an appropriate boundary condition imposed. Using the concept of the scaled boundary finite element method, the wave equation of the truncated semi-infinite domain is represented by the partial differential equation of non-constant coefficients. The resulting partial differential equation has only one spatial coordinate variable and time variable. Through introducing a few auxiliary functions at the truncated boundary, the resulting partial differential equations are further transformed into linear time-dependent equations. This allows an artificial boundary to be derived from the time-dependent equations. The proposed artificial boundary is local in time, global at the truncated boundary and semi-analytical in the finite element sense. Compared with the scaled boundary finite element method, the main advantage in using the proposed artificial boundary is that the requirement for solving a matrix form of Lyapunov equation to obtain the unit-impulse response matrix is avoided, so that computer efforts are significantly reduced. The related numerical results from some typical examples have demonstrated that the proposed artificial boundary is of high accuracy in dealing with time-dependent elastic wave propagation in two-dimensional homogeneous semi-infinite domains.     

Free‐vibration analysis of a three‐dimensional soil–structure system

A procedure which involves a non‐linear eigenvalue problem and is based on the substructure method is proposed for the free‐vibration analysis of a soil–structure system. In this procedure, the structure is modelled by the standard finite element method, while the unbounded soil is modelled by the scaled boundary finite element method. The fundamental frequency, and the corresponding radiation damping ratio as well as the modal shape are obtained by using inverse iteration. The free vibration of a dam–foundation system, a hemispherical cavity and a hemispherical deposit are analysed in detail. The numerical results are compared with available results and are also verified by the Fourier transform of the impulsive response calculated in the time domain by the three‐dimensional soil–structure–wave interaction analysis procedure proposed in our previous paper. The fundamental frequency obtained by the present procedure is very close to that obtained by Touhei and Ohmachi, but the damping ratio and the imaginary part of modal shape are significantly different due to the different definition of damping ratio. This study shows that although the classical mode‐superposition method is not applicable to a soil–structure system due to the frequency dependence of the radiation damping, it is still of interest in earthquake engineering to evaluate the fundamental frequency and the corresponding radiation damping ratio of the soil–structure system. Copyright © 2001 John Wiley & Sons, Ltd.     

Time‐domain analysis of unbounded media using mixed‐variable formulations

Formulation of a matrix‐valued force–displacement relationship which can take radiation damping into account is of major importance when modelling unbounded domains. This can be done by means of fundamental solutions in space and time in connection with convolution integrals or by means of a frequency dependent boundary element representation, but for discrete frequencies Ω only. In this paper a method for interpolating discrete values of dynamic stiffness matrices by a continuous matrix valued rational function is proposed. The coupling between interface degrees of freedom is fully preserved. Another crucial point in soil–structure interaction analysis is how to implement an approximation in the spectral domain into a time‐domain analysis. Well‐known approaches for the scalar case are based on the partial‐fraction expansion of a scalar rational function. Here, a more general procedure, applicable to MDOF‐systems, for the transformation of spectral rational approximations into the time‐domain is introduced. Evaluation of the partial‐fraction expansion is avoided by using the so‐called mixed variables. Thus, unknowns in the time‐domain are displacements as well as forces. Copyright © 2001 John Wiley & Sons, Ltd.     

Implementation of a second‐order paraxial boundary condition for a water‐saturated layered half‐space in plane strain

A half‐space finite element and a transmitting boundary are developed for a water‐saturated layered half‐space using a paraxial boundary condition. The exact dynamic stiffness of a half‐space in plane strain is derived and a second‐order paraxial approximation of the stiffness is obtained. A half‐space finite element and a transmitting boundary are then formulated. The development is verified by comparison of the dynamic stiffness of impermeable and permeable rigid strip foundations with other published results. The advantage of using the paraxial boundary condition in comparison with the rigid boundary condition is examined. It is shown that the paraxial boundary condition offers significant gain and the resulting half‐space finite element and transmitting boundary can represent the effects of a water‐saturated layered half‐space with good accuracy and efficiency. In addition, the numerical method described herein maintains the strengths and advantages of the finite element method and can be easily applied to demanding problems of soil–structure interaction in a water‐saturated layered half‐space. Copyright © 2010 John Wiley & Sons, Ltd.     

2.5D direct‐current resistivity forward modelling and inversion by finite‐element–infinite‐element coupled method

To reduce the numerical errors arising from the improper enforcement of the artificial boundary conditions on the distant surface that encloses the underground part of the subsurface, we present a finite‐element–infinite‐element coupled method to significantly reduce the computation time and memory cost in the 2.5D direct‐current resistivity inversion. We first present the boundary value problem of the secondary potential. Then, a new type of infinite element is analysed and applied to replace the conventionally used mixed boundary condition on the distant boundary. In the internal domain, a standard finite‐element method is used to derive the final system of linear equations. With a novel shape function for infinite elements at the subsurface boundary, the final system matrix is sparse, symmetric, and independent of source electrodes. Through lower upper decomposition, the multi‐pole potentials can be swiftly obtained by simple back‐substitutions. We embed the newly developed forward solution to the inversion procedure. To compute the sensitivity matrix, we adopt the efficient adjoint equation approach to further reduce the computation cost. Finally, several synthetic examples are tested to show the efficiency of inversion.     

Scaling and self‐similarity in one‐dimensional unsteady open channel flow

The conditions under which the Saint Venant equations system for unsteady open channel flow, as an initial–boundary value problem, becomes self‐similar are investigated by utilizing one‐parameter Lie group of point scaling transformations. One of the advantages of this methodology is that the self‐similarity conditions due to the initial and boundary conditions can also be investigated thoroughly in addition to the conditions due to the governing equation. The obtained self‐similarity conditions are compared with the scaling relationships that are derived through the Froude similitude. It is shown that the initial–boundary value problem of a one‐dimensional unsteady open channel flow process in a prototype domain can be self‐similar with that of several different scaled domains. However, the values of all the flow variables (at specified time and space) under different scaled domains can be upscaled to the same values in the prototype domain (at the corresponding time and space), as shown in this study. Distortion in scales of different space dimensions has been implemented extensively in physical hydraulic modelling, mainly because of cost, space and time limitations. Unlike the traditional approach, the distinction is made between the longitudinal–horizontal and transverse–horizontal length scales in this study. The scaled domain obtained by the proposed approach, when scaling ratios of channel width and water depth are equal, is particularly important for the similarity of flow characteristics in a cross‐section because the width‐to‐depth ratio and the inclination angles of the banks are conserved in a cross‐section. It is also shown that the scaling ratio of the roughness coefficient under distorted channel conditions depends on that of hydraulic radius and longitudinal length. The proposed scaling relations obtained by the Lie group scaling approach may provide additional spatial, temporal and economical flexibility in setting up physical hydraulic models. Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamic impedances of pile groups with embedded caps in homogeneous elastic soils using CIFECM

Significant research has been reported on the dynamic analysis of pile groups. However, in most of the cases, the effect of pile cap is neglected despite the fact that there may be additional interactions due to the presence of the cap. This paper presents the dynamic impedances for the pile groups with caps embedded in isotropic homogeneous elastic soils. A general three-dimensional finite element procedure is developed. The system is sub-structured into bounded near-field and an unbounded far-field. The pile-soil system of the near-field is modeled using solid finite elements, and the unbounded elastic soil system of the far-field is modeled using the consistent infinitesimal finite element cell method (CIFECM) in the frequency domain.     

Systematic lumped‐parameter models for foundations based on polynomial‐fraction approximation

Based on the approximation by polynomial‐fraction, a series of systematic lumped‐parameter models are developed in this paper for efficiently representing the dynamic behaviour of unbounded soil. Concise formulation is first employed to represent the dynamic flexibility function of foundation with a ratio of two polynomials. By defining an appropriate quadratic error function, the optimal coefficients of the polynomials can be directly solved from a system of linear equations. Through performing partial‐fraction expansion on this polynomial‐fraction and designing two basic discrete‐element models corresponding to the partial fractions, systematic lumped‐parameter models can be conveniently established by connecting these basic units in series. Since the systematic lumped‐parameter models are configured without introducing any mass, the foundation input motion can be directly applied to these models for their applications to the analysis of seismic excitation. The effectiveness of these new models is strictly validated by successfully simulating a semi‐infinite bar on an elastic foundation. Subsequently, these models are applied for representing the dynamic stiffness functions for different types of foundation. Comparison of the new models with the other existing lumped‐parameter models is also made to illustrate their advantages in requiring fewer parameters and featuring a more systematic expansion. Copyright © 2002 John Wiley & Sons, Ltd.     

Vertical earthquake response of megawatt‐sized wind turbine with soil‐structure interaction effects

The dynamic response of a wind turbine on monopile is studied under horizontal and vertical earthquake excitations. The analyses are carried out using the finite element program SAP2000. The finite element model of the structure is verified against the results of shake table tests, and the earthquake response of the soil model is verified against analytical solutions of the steady‐state response of homogeneous strata. The focus of the analyses in this paper is the vertical earthquake response of wind turbines including the soil‐structure interaction effects. The analyses are carried out for both a non‐homogeneous stratum and a deep soil using the three‐step method. In addition, a procedure is implemented which allows one to perform coupled soil‐structure interaction analyses by properly tuning the damping in the tower structure. The analyses show amplification of the ground surface acceleration to the top of the tower by a factor of two. These accelerations are capable of causing damage in the turbine and the tower structure, or malfunctioning of the turbine after the earthquake; therefore, vertical earthquake excitation is considered a potential critical loading in design of wind turbines even in low‐to‐moderate seismic areas. Copyright © 2015 John Wiley & Sons, Ltd.     

Analytical layer-element solution to axisymmetric dynamic response of transversely isotropic multilayered half-space

Starting with the governing equations of motion and the constitutive equations of transversely isotropic elastic body, and based on the corresponding algebraic operations and the Hankel transform, the analytical layer-elements of a finite layer and a half-space are obtained in the transformed domain. According to the continuity conditions between adjacent layers, the global stiffness matrix equation is obtained by assembling the analytical layer-element of each single layer. The solutions in the transformed domain are acquired by introducing the boundary conditions into the global stiffness matrix equation, and thus, the corresponding solutions in frequency domain are achieved by taking the inversion of Hankel transform. Finally, some numerical examples are given to illustrate the accuracy of the proposed method, and to study the influence of properties and the frequency of excitation on the dynamic response of the medium.     

Seismic modelling for geological fractures

Based on knowledge of a commutative group calculation of the rock stiffness and on some geophysical assumptions, the simplest fractured medium may be regarded as a fracture embedded in an isotropic background medium, and the fracture interface can be simulated as a linear slip interface that satisfies non‐welded contact boundary conditions: the kinematic displacements are discontinuous across the interface, whereas the dynamic stresses are continuous across the interface. The finite‐difference method with boundary conditions explicitly imposed is advantageous for modelling wave propagation in fractured discontinuous media that are described by the elastic equation of motion and non‐welded contact boundary conditions. In this paper, finite‐difference schemes for horizontally, vertically, and orthogonally fractured media are derived when the fracture interfaces are aligned with the boundaries of the finite‐difference grid. The new finite‐difference schemes explicitly have an additional part that is different from the conventional second‐order finite‐difference scheme and that directly describes the contributions of the fracture to the wave equation of motion in the fractured medium. The numerical seismograms presented, to first order, show that the new finite‐difference scheme is accurate and stable and agrees well with the results of previously published finite‐difference schemes (the Coates and Schoenberg method). The results of the new finite‐difference schemes show how the amplitude of the reflection produced by the fracture varies with the fracture compliances. Later, comparisons with the reflection coefficients indicate that the reflection coefficients of the fracture are frequency dependent, whereas the reflection coefficients of the impedance contrast interface are frequency independent. In addition, the numerical seismograms show that the reflections of the fractured medium are equal to the reflections of the background medium plus the reflections of the fracture in the elastic fractured medium.     

Dynamic stiffness of deep foundations with inclined piles

The influence of inclined piles on the dynamic response of deep foundations and superstructures is still not well understood and needs further research. For this reason, impedance functions of deep foundations with inclined piles, obtained numerically from a boundary element–finite element coupling model, are provided in this paper. More precisely, vertical, horizontal, rocking and horizontal–rocking crossed dynamic stiffness and damping functions of single inclined piles and 2 × 2 and 3 × 3 pile groups with battered elements are presented in a set of plots. The soil is assumed to be a homogeneous viscoelastic isotropic half‐space and the piles are modeled as elastic compressible Euler–Bernoulli beams. The results for different pile group configurations, pile–soil stiffness ratios and rake angles are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Representation of radiation damping by fractional time derivatives

When modelling unbounded domains, formulation of a matrix‐valued force–displacement relationship which can take radiation damping into account is of major importance. In this paper, a method to describe the dynamic stiffness by a system of fractional differential equations in the time‐domain is presented. Here, a doubly asymptotic rational approximation of the low‐frequency force–displacement relationship is used, whereas a direct interpretation of the asymptotic part as a fractional derivative is possible. The numerical solution of the corresponding system of fractional differential equations is demonstrated using the infinite beam on elastic foundation as an example. Copyright © 2003 John Wiley & Sons, Ltd.     

基于比例边界有限元方法的拱坝子结构分区形式研究

比例边界有限元方法是一种半解析的数值计算方法,具有降维、网格灵活、严格模拟无限域和无需基本解等特点。比例边界有限元方法的基本理论是在整体坐标与局部坐标的比例边界转换基础之上建立的,相似中心的选取是否合理对分析计算具有重要的影响,导致在模拟拱坝这种不规则的空间壳体结构时,具有一定的局限性。采用子结构方法,将坝体分为若干满足相似性要求的区域可解决上述问题,以某拱坝为例给出了合理的坝体子结构分区形式,验证了子结构方法的精确性,为建立基于比例边界有限元方法的坝体-库水-地基系统的计算模型奠定了基础。     

A hybrid absorbing boundary condition for elastic staggered‐grid modelling

We recently proposed an efficient hybrid scheme to absorb boundary reflections for acoustic wave modelling that could attain nearly perfect absorptions. This scheme uses weighted averaging of wavefields in a transition area, between the inner area and the model boundaries. In this paper we report on the extension of this scheme to 2D elastic wave modelling with displacement‐stress formulations on staggered grids using explicit finite‐difference, pseudo‐implicit finite‐difference and pseudo‐spectral methods. Numerical modelling results of elastic wave equations with hybrid absorbing boundary conditions show great improvement for modelling stability and significant absorption for boundary reflections, compared with the conventional Higdon absorbing boundary conditions, demonstrating the effectiveness of this scheme for elastic wave modelling. The modelling results also show that the hybrid scheme works well in 2D rotated staggered‐grid modelling for isotropic medium, 2D staggered‐grid modelling for vertically transversely isotropic medium and 2D rotated staggered‐grid modelling for tilted transversely isotropic medium.     

Problems encountered from the use (or misuse) of Rayleigh damping

Rayleigh damping is commonly used to provide a source of energy dissipation in analyses of structures responding to dynamic loads such as earthquake ground motions. In a finite element model, the Rayleigh damping matrix consists of a mass‐proportional part and a stiffness‐proportional part; the latter typically uses the initial linear stiffness matrix of the structure. Under certain conditions, for example, a non‐linear analysis with softening non‐linearity, the damping forces generated by such a matrix can become unrealistically large compared to the restoring forces, resulting in an analysis being unconservative. Potential problems are demonstrated in this paper through a series of examples. A remedy to these problems is proposed in which bounds are imposed on the damping forces. Copyright © 2005 John Wiley & Sons, Ltd.     

Pseudo‐spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media

Staggering grid is a very effective way to reduce the Nyquist errors and to suppress the non‐causal ringing artefacts in the pseudo‐spectral solution of first‐order elastic wave equations. However, the straightforward use of a staggered‐grid pseudo‐spectral method is problematic for simulating wave propagation when the anisotropy level is greater than orthorhombic or when the anisotropic symmetries are not aligned with the computational grids. Inspired by the idea of rotated staggered‐grid finite‐difference method, we propose a modified pseudo‐spectral method for wave propagation in arbitrary anisotropic media. Compared with an existing remedy of staggered‐grid pseudo‐spectral method based on stiffness matrix decomposition and a possible alternative using the Lebedev grids, the rotated staggered‐grid‐based pseudo‐spectral method possesses the best balance between the mitigation of artefacts and efficiency. A 2D example on a transversely isotropic model with tilted symmetry axis verifies its effectiveness to suppress the ringing artefacts. Two 3D examples of increasing anisotropy levels demonstrate that the rotated staggered‐grid‐based pseudo‐spectral method can successfully simulate complex wavefields in such anisotropic formations.