Mixed unsplit-field perfectly matched layers for transient simulations of scalar waves in heterogeneous domains

We discuss a new formulation for transient scalar wave simulations in heterogeneous semi-infinite domains. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries and adopt perfectly matched layers (PMLs) as the boundary wave absorbers. Within this framework, we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs. We use, as typically done, complex-coordinate stretching transformations in the frequency domain, and recover the governing partial differential equations in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which, numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error.     

Expanded superposition method for impedance functions of inclined‐pile groups

This paper presents a superposition method expanded for computing impedance functions (IFs) of inclined‐pile groups. Closed‐form solutions for obtaining horizontal, vertical, and rocking IFs, estimated by using pile‐to‐pile interaction factors, are proposed. IFs of solitary inclined piles, crossed IFs, and explicit incorporation of compatibility conditions for pile‐head movements are also appropriately taken into consideration. All of these factors should be known in advance and will be computed and shown for the most relevant cases. The accuracy of the proposed closed‐form solutions is verified for 2 × 2 and 3 × 3 square inclined‐pile groups embedded in an isotropic viscoelastic homogeneous half‐space soil medium, with hysteretic damping. The pile‐to‐pile interaction factors are computed by means of a three‐dimensional time‐harmonic boundary elements–finite elements coupling formulation. The results indicate that the IFs obtained from the proposed method are in good agreement with those obtained from the coupling formulation. Furthermore, crossed vertical‐rocking IFs of solitary piles need to be appropriately considered for obtaining rocking IFs when the number of piles is small. Copyright © 2015 John Wiley & Sons, Ltd.     

Simple superposition approach for dynamic analysis of piled embedded footings

The effectiveness and accuracy of the superposition method in assessing the dynamic stiffness and damping coefficients (impedance functions) of embedded footings supported by vertical piles in homogeneous viscoelastic soil is addressed. To this end, the impedances of piled embedded footings are compared to those obtained by superposing the impedance functions of the corresponding pile groups and embedded footings treated separately, with the magnitude of the relative average differences being around 10–30%. The results are presented in a set of dimensionless graphs and simple expressions that can be used to estimate the dynamic stiffness and damping of piled embedded footings, provided that the impedance functions of the two individual components are known. This is precisely the reason why the superposition approach studied here is appealing, because such impedance functions for both embedded footings and pile groups are available for a wide range of cases. How to estimate the kinematic response functions of the system when those of the individual components are known is also discussed. To address the problem, parametric analyses performed using a 3D frequency‐domain elastodynamic BEM‐FEM formulation are presented for different pile–soil stiffness contrasts, embedment depths, pile‐to‐pile separations and excitation frequencies. Vertical, horizontal, rocking, and cross‐coupled horizontal‐rocking impedance functions, together with translational and rotational kinematic response functions, are discussed. The results suggest that the superposition concept, in conjunction with a correction strategy as that presented herein, can be employed in geotechnical design. For kinematic effects, the response functions of the embedded footing are found to provide reasonable estimates of the system's behaviour. Copyright © 2011 John Wiley & Sons, Ltd.     

Impedances of rigid cylindrical foundations embedded in transversely isotropic soils

A complete formulation and implementation for assessment of the response to dynamic loads of cylindrical rigid structures embedded in transversely isotropic elastic half‐spaces is presented. The analysis is performed in the frequency domain and the steady‐state structure response is obtained. The method is based on a non‐singular version of the indirect boundary element method which uses influence functions, instead of Green's functions, as fundamental solutions. These influence functions are the response of an elastic half‐space to distributed, internally applied loads. The proposed method imposes full bonding contact between the foundation and the surrounding soil. Numerical results for displacement (vertical and horizontal) and rotation (twisting and rocking) impedances, showing the influence of the soil anisotropy, are presented. Results for the soil–structure interface tractions and for the displacement field throughout the half‐space are also shown. Copyright © 2006 John Wiley & Sons, Ltd.     

饱和土半空间中圆柱形孔洞对平面P波的散射

在 Biot饱和多孔介质动力学理论的基础上,首次建立了求解饱和土半空间中圆柱形孔洞对平面 P 波散射问题的波函数展开法。首先。分析了具有圆柱形孔洞的饱和土半空间场地在平面 P 波入射下产生散射波系,并将入射波和散射波的波函数在圆柱坐标下展开。然后,引入边界条件,求出散射波函数的待定系数,从而,得到饱和土半空间中圆柱形孔洞对平面P波的散射问题的解析解。根据所得的波函数的解,可求解区域内的位移、应力的值,同时,分析了入射波频率、入射角对柱面上的应力集中因子的影响。     

A new analytical model to study the influence of weld on the vertical dynamic response of prestressed pipe pile

This investigation is concerned with the mathematical analysis of a viscoelastic prestressed pipe pile embedded in multilayered soil under vertical dynamic excitation. The pile surrounding soil is governed by the plane strain model, and the soil plug is assumed to be an additional mass connected to the pipe pile shaft by applying the distributed Voigt model. Meanwhile, the prestressed pipe pile is assumed to be a vertical, viscoelastic, and hollow cylinder governed by the one‐dimensional wave equation. Then, analytical solutions of the dynamic response of the pipe pile in the frequency domain are derived by means of the Laplace transform and impedance function transfer method. Subsequently, the corresponding quasi‐analytical solution in the time domain for the case of the prestressed pipe pile undergoing a vertical semi‐sinusoidal exciting force applied at the pile top is obtained by employing the inverse Fourier transform. Utilizing these solutions, selected results for the velocity admittance curve and the reflected wave curve are presented for different heights of the soil plug to examine the influence of weld properties on the vertical dynamic response of prestressed pipe pile. The reasonableness of the theoretical model is verified by comparing the calculated results based on the presented solutions with measured results. Copyright © 2017 John Wiley & Sons, Ltd.     

地下管线弹性波探测波场的建模与分析

在地下管线弹性波探测中,由位于地下圆孔边界上有限时长的力源激发形成瞬态场,应用波函数展开法对此波场进行建模。以波函数和待定系数组成的级数序列描述探测波场中的势函数,根据界面处应力和位移连续的边界条件求解待定系数,从而得到波场中弹性波的解析解。据此进行数值研究,分析探测波场中弹性波的传播特征,说明了散射波的类型以及能量对比关系,并总结了反射信号幅值与地下管线物理参数之间由声阻抗匹配程度支配的反射信号强度变化规律,而管线半径对反射信号强度的影响要大于物理参数的作用。最后以实验结果验证上述结论。     

Vertical dynamic response of a pipe pile in saturated soil layer

An analytical solution is developed in this paper to investigate the vertical time-harmonic response of a pipe pile embedded in a viscoelastic saturated soil layer. The wave propagation in the saturated soil is simulated by Biot’s 3D poroelastic theory and that in the pipe pile is simulated by 1D elastodynamic theory. Potential functions are applied to decouple the governing equations of the soil. The analytical solutions of the outer and inner soil in frequency domain are obtained by the method of separation of variables. The vertical response of the pipe pile is then obtained based on the continuity assumption of the displacement and stress between the pipe pile and both the outer and inner soil. The solution is compared with existing solutions to verify the validity. Numerical examples are presented to analyze the vibration characteristics of the pile.     

低应变下变阻抗薄壁管桩动力响应频域解析解

管桩低应变完整性检测时,桩顶一点受到瞬态集中荷载的作用,其动力响应问题是一个三维波动问题。基于三维波动理论,建立了任意段变阻抗薄壁管桩动力响应的计算模型和波动方程,结合初边值条件,采用Laplace变换法求得了该波动方程的频域解析解,采用Fourier逆变换求得了时域响应。将计算结果与三维有限元结果进行了对比分析,解析解的计算结果和三维有限元结果较为接近,两者在入射波峰、缺陷反射峰和桩底反射峰处都较为吻合,两种方法位移响应曲线基本相同。给出了桩顶不同点的动力响应,并探讨了高频干扰问题。对变截面桩和变模量桩的动力响应特性进行了分析。     

二维饱和地基中空沟主动隔振分析

根据饱和土Biot理论,在频域内提出了一种适用于弹性波波动问题的有限元-无限元解法,其中有限单元用于近场介质的模拟,而形函数基于一维解析解答的无限单元则被用于人工边界,它可反映波传递到无限远处的特性;应用该法计算算例,以证明其有效性与正确性。在此基础上,对饱和地基中空沟主动隔振效果进行了详细的数值分析,结果表明,空沟深度是影响隔振效果的主要因素,一般沟深越大,隔振效果越好;当沟深一定时,空沟到振源的距离不宜超过1.5倍饱和土剪切波长;空沟的宽度对隔振效果的影响相对较小。     

The method of fundamental solution for 3‐D wave scattering in a fluid‐saturated poroelastic infinite domain

By using a complete set of poroelastodynamic spherical wave potentials (SWPs) representing a fast compressional wave P_I, a slow compressional wave P_II, and a shear wave S with 3 vectorial potentials (not all are independent), a solution scheme based on the method of fundamental solution (MFS) is devised to solve 3‐D wave scattering and dynamic stress concentration problems due to inhomogeneous inclusions and cavities embedded in an infinite poroelastic domain. The method is verified by comparing the result with the elastic analytical solution, which is a degenerated case, as well as with poroelastic solution obtained using other numerical methods. The accuracy and stability of the SWP‐MFS are also demonstrated. The displacement, hoop stress, and fluid pore pressure around spherical cavity and poroelastic inclusion with permeable and impermeable boundary are investigated for incident plane P_I and SV waves. The scattering characteristics are examined for a range of material properties, such as porosity and shear modulus contrast, over a range of frequency. Compared with other boundary‐based numerical strategy, such as the boundary element method and the indirect boundary integral equation method, the current SWP‐MFS is a meshless method that does not need elements to approximate the geometry and is free from the treatment of singularities. The SWP‐MFS is a highly accurate and efficient solution methodology for wave scattering problems of arbitrary geometry, particularly when a part of the domain extends to infinity.     

二阶声波方程频域PML边界条件及频域变网格步长并行计算

研究了二阶声波方程频域PML边界条件和频域变网格并行计算技术。PML边界是一种较为理想的吸收边界方法,多用在求解时域应力速度方程中,但对于频域声波正演,二阶位移方程更常用。从一阶声波方程PML吸收边界条件导出频域二阶位移方程PML边界条件,模拟算例得到的频率切片、时间切片和地震记录对比都说明该边界条件吸收效果很好。频域单炮正演不同频率间是独立的,据此低频部分采用大网格计算,高频采用小网格,实现变网格步长计算技术,这是较时间域正演的一个优势,在保证模拟质量的同时,减少计算量和内存消耗。     

Soil‐pile interaction in the pile vertical vibration considering true three‐dimensional wave effect of soil

The dynamic response of an end bearing pile embedded in a linear visco‐elastic soil layer with hysteretic type damping is theoretically investigated when the pile is subjected to a time‐harmonic vertical loading at the pile top. The soil is modeled as a three‐dimensional axisymmetric continuum in which both its radial and vertical displacements are taken into account. The pile is assumed to be vertical, elastic and of uniform circular cross section. By using two potential functions to decompose the displacements of the soil layer and utilizing the separation of variables technique, the dynamic equilibrium equation is uncoupled and solved. At the interface of soil‐pile system, the boundary conditions of displacement continuity and force equilibrium are invoked to derive a closed‐form solution of the vertical dynamic response of the pile in frequency domain. The corresponding inverted solutions in time domain for the velocity response of a pile subjected to a semi‐sine excitation force applied at the pile top are obtained by means of inverse Fourier transform and the convolution theorem. A comparison with two other simplified solutions has been performed to verify the more rigorous solutions presented in this paper. Using the developed solutions, a parametric study has also been conducted to investigate the influence of the major parameters of the soil‐pile system on the vertical vibration characteristics of the pile. Copyright © 2013 John Wiley & Sons, Ltd.     

一维问题的等效远置动力人工边界条件

在土-结构动力相互作用分析中,黏性人工边界条件和黏弹性人工边界条件虽具有计算效率和精度较高的优点,但施加边界单元的工作量较大;远置边界虽然具有处理技术简单、精度高等优点,但又存在计算成本高的问题。为克服黏性人工边界和黏弹性人工边界施加边界黏弹性单元工作量大、远置边界计算成本高的缺点,基于波的传播理论和波在介质交界面上的反射、透射原理,通过设置完全透射层并大大降低波在透射层的传播速度的方法,提出了一种等效远置动力人工边界条件并对其进行数值验证分析。研究结果表明,所提出的等效远置动力人工边界条件仅需通过合理选取透射层的参数,即可实现反射波向无限地基传播的模拟,具有精度和效率较好、不需施加边界单元工作量等优点,可应用到土-结构动力相互作用分析当中。     

动力分析中无限地基两种近似方法的应用研究

在结构-地基的动力相互作用分析所提出的众多方法中,无质量地基和黏性边界地基仍然是应用最为广泛的近似方法。针对当前结构-地基动力相互作用分析中地基模型处理中的一些问题,通过结构-地基动力相互作用分析公式的推导以及各变量物理意义的解释,详细阐明了当前考虑地基影响的动力分析中无质量地基和黏性边界地基的概念和应用范围,对当前部分研究中应用这两种地基中存在的问题进行了分析说明,为简化地基动力分析模型提供了思路。     

裂缝性地层声波测井的联合时频特征

裂缝性油气藏越来越受到人们的重视,如何有效地识别和评价裂缝性地层也成为人们关注的一个焦点。Cohen类时频分析方法具有同时对信号的时间和频率进行分析的优点,同时Hilbert-Huang变换(HHT)中的经验模态分解（EMD）方法能将信号分解为有限个具有单分量特性的固有模态函数（IMF）,可以满足Cohen类时频分析对信号单分量的要求。尝试将这两种方法联合应用于阵列声波信号特性的提取,对地层不同性质裂缝的信号能量的时频分布特征进行分析,结果显示了这种联合时频分析方法在不同的裂缝性质地层中有不同的表现特征,对于实现裂缝性地层分析和评价具有一定的现实意义。     

曲线坐标系下的完全匹配层吸收边界条件

在地震波数值模拟中,需要采用吸收边界条件以吸收人为边界反射。本文针对曲线坐标系下的二阶弹性波方程提出了一种完全匹配层(PML)吸收边界条件。与直角坐标系下的PML吸收边界条件类似,曲线坐标系下的PML吸收边界条件是一种在频率域中给出的人工边界条件,由相应的复坐标变换得到。在变换到时间域后,完全匹配层中将出现复杂的卷积运算。为了避免这些卷积运算,引入了4个中间变量。为了简化自由边界条件,采用正交贴体网格对起伏地表模型进行网格剖分。数值算例表明,该方法可以有效消除人为边界反射。     

基于子结构法的成层场地中沉积河谷的散射分析

土层场地对地震动的影响明显,研究成层场地局部复杂地形的波动场特性,对于抗震设防具有重要意义。基于土-结构相互作用理论,含有局部复杂地形的成层场地,可分解成广义结构（即近场复杂地形及其周围土体）和具有规则开挖边界的成层无限地基（即远场）。远场的格林函数可通过精细积分求解频域-波数域内的对偶波动方程获得,利用傅里叶逆变换得到频域-空间域内的柔度矩阵。近场可采用比例边界有限元进行模拟,通过高性能连分式的传递边界求解动刚度。自由场波动可通过传递矩阵法求得,将动刚度和自由场波动位移代入控制方程即可得到散射场的动力响应。数值算例验证了方法的准确性,并利用提出的方法讨论含有软夹层场地局部复杂场地的波动场特性。     

The study of reflection/transmission phenomena in a corrugated interface between two magnetoelastic transversely isotropic media

In this paper, an integrated approach has been carried out with an intent to study the reflection and transmission phenomena of plane SH-type wave on a corrugated interface separating two magnetoelastic transversely isotropic half-space. Closed form expressions for reflection and transmission coefficients have been derived for both plane and corrugated surface. Rayleigh’s method of approximation have been incorporated to deduce equations for the first- and second-order approximation of corrugation. Analytical solutions for both the half-spaces have been worked out. All possible cases have been apprehended specially for anisotropic and an isotropic medium. The effects of magnetoelastic coupling parameter, angle at which the wave crosses the magnetic field, corrugation amplitude, frequency factor and wave number have been explained by collaborating with graphical analysis.     

Longitudinal vibration of a pile embedded in layered soil considering the transverse inertia effect of pile

The dynamic response of a viscoelastic bearing pile embedded in multilayered soil is theoretically investigated considering the transverse inertia effect of the pile. The soil layers surrounding the pile are modeled as a set of viscoelastic continuous media in three-dimensional axisymmetric space, and a simplified model, i.e., the distributed Voigt model, is proposed to simulate the dynamic interactions of the adjacent soil layers. Meanwhile, the pile is assumed to be a Rayleigh–Love rod with material damping and can be divided into several pile segments allowing for soil layers and pile defects. Both the vertical and radial displacement continuity conditions at the soil–pile interface are taken into account. The potential function decomposition method and the variable separation method are introduced to solve the governing equations of soil vibration in which the vertical and radial displacement components are coupled. On this basis, the impedance function at the top of the pile segment is derived by invoking the force and displacement continuity conditions at the soil–pile interface as well as the bottom of pile segment. The impedance function at the pile head is then obtained by means of the impedance function transfer method. By means of the inverse Fourier transform and convolution theorem, the velocity response in the time domain can also be obtained. The reasonableness of the assumptions of the soil-layer interactions have been verified by comparing the present solutions with two published solutions and a set of well-documented measured pile test data. A parametric analysis is then conducted using the present solutions to investigate the influence of the transverse inertia effect on the dynamic response of an intact pile and a defective pile for different design parameters of the soil–pile system.