Numerical study of dynamic processes in a continuous medium with a crack initiated by a near-surface disturbance by means of the grid-characteristic method

The purpose of this work is to study the problem of the near-surface disturbance propagation in a massive rock containing various heterogeneities, i.e., empty or filled cracks. Numerical solutions have been obtained for problems of wave propagation in such highly heterogeneous media, including those taking into account the plastic properties of the rock that can be manifested in the vicinity of a seismic gap or a well bore. All kinds of elastic and elastoplastic waves are analyzed resulting from the propagation of the initial disturbance and the waves arising from the reflection from the cracks and from the boundaries of the integration domain. An investigation was carried out of wave identification by means of seismograms obtained at the receiver located near the ground surface. In this study, the grid-characteristic method is employed using computational grids with triangular meshes and boundary conditions formulated at the interface between the rock and the crack, and on free surfaces in an explicit form. The proposed numerical method is extremely general and is suitable for investigations of the processes of seismic waves’ interaction with heterogeneous inclusions because it ensures the construction of the most correct computational algorithms at the boundaries of the integration domain and at the medium’s interface.     

超声波在各向同性固体中传播的数值模拟

介绍以应用动量守恒和应力-应变关系为各向同性弹性介质进行微变形建立超声波传播方程,并利用一种有限元方法和差分法,分析超声波(P波)在各向问性介质中传播,及在裂缝上的散射结果.并且通过计算机程序模拟二维超声波的传播,散射的数值模拟与几何理论结果是相符合的.因此超声检测仿真软件是可以预测超卢检测过程中的波形.超声探头的建模发展趋势是利用FDM和FEM方法研究超声检测的仿真软件,分析结果将广泛应用到检测过程中.     

Numerical study of the anisotropy of wave responses from a fractured reservoir using the grid-characteristic method

This work is aimed at studying the problem of subsurface disturbance propagation in a massive rock containing various inhomogeneities: empty or filled cracks. Numerical solutions to the problems of wave propagation in such substantially inhomogeneous media have been obtained. The dependence that the parameters of the response from a fractured reservoir have on the problem parameters is studied. The latter parameters are the density of the crack location, the fractured reservoir extent, amount of cracks, the initial disturbance location, crack inclination, and disturbance frequency. The concept of response anisotropy is introduced and the dependence that the anisotropy has on the abovementioned parameters is studied.     

基于分层模型的复合材料耦合Lamb波频率—波数域特性研究

针对复合材料层合板中耦合Lamb波的传播问题,基于分层模型提出解析建模与有限元数值模拟相结合的方法对其进行预测和评估。利用Legendre正交多项式展开法推导多层各向异性复合材料层合板中耦合Lamb波的控制方程,并对频率-波数域频散特性曲线实现数值求解。基于平面壳单元构建复合材料层合板的有限元模型,采用波结构加载法生成单一Lamb波基本模态,设计复合材料层合板的不同纤维取向、边界和界面约束条件,并经二维傅里叶变换获得有限元模拟数据的频率-波数域频散特性曲线。通过对比验证,结果表明两种方法均有较好的吻合性。     

Design of maintenance schedules for fatigue-prone metallic components using reliability-based optimization

This contribution focuses on the design of optimal maintenance schedules for metallic structures prone to develop fatigue cracks. The crack propagation phenomenon is addressed using a fracture mechanics approach. The problem of maintenance scheduling is addressed within the framework of reliability-based optimization (RBO). Thus, it is possible to minimize the costs associated with maintenance and eventual failure while explicitly considering uncertainties in the crack propagation phenomenon and inspection activities. The underlying RBO problem is solved using an efficient method recently developed by the authors. A numerical example demonstrating the application of the proposed approach is presented.     

Influence of natural disasters on ground facilities

The purpose of this article is to study the problem of the propagation of waves that result in earthquakes in different geological media: homogeneous, multilayer, gradient, with fractured layer, and karst cavern. The authors pose the problem of analyzing the impact of waves on ground structures: buildings and dams. Numerical solutions of problems of wave propagation in heterogeneous media are obtained. On the basis of the analysis of wave patterns, the types of waves propagated from the focus of the earthquake are qualified. The comparison of the impact of elastic waves on the day surface for the cases of different geological media is done. Synthetic seismograms for these media are obtained. The influence of elastic waves on the stability of ground structures is qualitatively examined. The grid-characteristic method for triangle meshes with the formulation of boundary conditions on interfaces of rock-crack, building-rock, rock-water, and dam-water, as well as free surfaces in an explicit form, is used in this paper.     

基于TDR的多孔介质中含水/水合物饱和度测量方法仿真研究

为了优化设计时域反射技术（TDR）中的探针结构,建立基于TDR响应的含水/水合物测量模型,利用有限元数值模拟方法建立了TDR测量过程数值仿真模型。分别以空气、不同浓度氯化钠溶液以及不同水合物含量的石英砂为被测介质验证了模型的正确性、研究了被测介质电导率和介电常数对反射波形的影响规律。通过改变被测介质的介电常数来模拟含水/水合物饱和度不同的石英砂,随着石英砂中水合物含量的增加,含水量逐渐降低,表观介电常数随之减小,电磁波传播速度随之以非线性形式增加;与理论值相比较,仿真计算所得到的传播速度最大误差处于5%以内。下一步需要在数值模型中的被测区域中填充各相异性材料来更加真实地模拟含水合物沉积物被测介质。     

The simulation of 3D elastic scattering produced by thin rigid inclusions using the traction boundary element method

A mixed formulation that uses both the traction boundary element method (TBEM) and the boundary element method (BEM) is proposed to compute the three-dimensional (3D) propagation of elastic waves scattered by two-dimensional (2D) thin rigid inclusions. Although the conventional direct BEM has limitations when dealing with thin-body problems, this model overcomes that difficulty. It is formulated in the frequency domain and, taking into account the 2-1/2D configuration of the problem, can be expressed in terms of waves with varying wavenumbers in the zdirection, k_z. The elastic medium is homogeneous and unbounded and it should be noted that no restrictions are imposed on the geometry and orientation of the internal crack.     

Domain decomposition for wave propagation problems

The problem posed by domain decomposition methods is to find the correct modeling of physical phenomena across the interfaces separating the subdomains. The technique described here for wave propagation problems is based on physical grounds since it relies on the fact that the wave equation can be decomposed into incoming and outgoing wave modes at the boundaries of the subdomains. The inward propagating waves depend on the solution exterior to the subdomains and therefore are computed from the appropriate boundary conditions, while the behavior of the outward propagating waves is determined by the solution inside the subdomains. The technique is applied to the anisotropic-viscoelastic wave equation, which practically includes all the possible rheologies of one-phase media.     

An Inverse Procedure for Crack Detection in Anisotropic Laminated Plates Using Elastic Waves

. This paper presents a computational inverse technique to detect the location and length of cracks in anisotropic laminated plates. The scattered elastic harmonic wave fields in the laminated plates with horizontal or vertical crack are calculated using the strip element method, whereby the anisotropic laminated plate is discretized into strip elements in the thickness direction. By applying the principle of virtual work, the governing differential equations of the wave propagation are derived for the field variables. These differential equations are solved analytically together with the vertical boundary conditions. The crack length and its location are then identified by minimizing an error function, which is defined as the difference between the scattered wave fields in plates with actual and searched parameters. A uniform micro-genetic algorithm is employed to search for the correct parameters that minimize the error function. Numerical examples are given to demonstrate the efficiency of the procedure in the detection of the location and the length of both horizontal and vertical cracks in composite laminates.     

Seismic scalar wave equation with variable coefficients modeling by a new convolutional differentiator

Studying seismic wavefields in the Earth's interior requires an accurate calculation of wave propagation using accurate and efficient numerical techniques. In this paper, we present an alternative method for accurately and efficiently modeling seismic wavefields using a convolutional generalized orthogonal polynomial differentiator. Our approach uses optimization and truncation to form a localized operator. This preserves the fine structure of the wavefield in complex media and avoids non-causal interaction when parameter discontinuities are present in the medium. We demonstrate this approach for scalar wavefield modeling in heterogeneous media and conclude that the method could be readily extended to elastic wavefield calculations. Our numerical results indicate that this method can suppress numerical dispersion and allow for the study of wavefields in heterogeneous structures. The results hold promise not only for future seismic studies, but also for any field that requires high-precision numerical solution of partial differential equation with variable coefficients.     

Propagation and transformation of periodic nonlinear shallow-water waves in basins with selected breakwater systems

This paper describes the development of a Boussinesq three-equation model for simulating propagation and transformation of periodic nonlinear waves (cnoidal waves) in an arbitrary shallow-water basin. The Boussinesq equations in terms of depth-averaged horizontal velocities and free-surface elevation are solved numerically in a curvilinear coordinate system. An Euler’s predictor-corrector finite-difference algorithm is applied for numerical computation. The effects of irregular boundary, non-uniform water depth and coastal structures inside a basin are all included in the model simulation. A second-order cnoidal wave solution for the Boussinesq equations is used as an incident wave condition. A set of open boundary conditions is also applied to effectively transmit waves out of the computational domain. Model tests were conducted by simulating waves propagating past an isolated breakwater. The effect of variable depth was examined with modeling waves over an uneven bottom with convex ramp topography. The overall evolution of wave propagation, diffraction and reflection in coupled harbors with various layouts of inner and outer breakwaters was also studied. Data comparisons reveal that the simulated wave heights agree reasonably well with laboratory measurements, especially in the region of inner basin.     

Wave propagation in delaminated beam

The wave propagation model investigated herein is based on the known fact that material discontinuities affect the propagation of elastic waves in solids. The change in certain material characteristics, such as a local change in stiffness or inertia caused by a crack or the presence of material damage, will affect the propagation of transmitted elastic waves and will modify the received signal. Wave frequencies associated with the highest detection sensitivity depend, among other things, on the type of structure, the type of material, and the type of damage. This paper presents a method of wave propagation, which can be further used to detect small delaminations in beam-like structures. The considered beam is modelled by spectral finite elements.     

Adjoint design sensitivity analysis of dynamic crack propagation using peridynamic theory

Based on the peridynamics of the reformulated continuum theory, an adjoint design sensitivity analysis (DSA) method is developed for the solution of dynamic crack propagation problems using the explicit scheme of time integration. Non-shape DSA problems are considered for the dynamic crack propagation including the successive branching of cracks. The adjoint variable method is generally suitable for path-independent problems but employed in this bond-based peridynamics since its path is readily available. Since both original and adjoint systems possess time-reversal symmetry, the trajectories of systems are symmetric about the u-axis. We take advantage of the time-reversal symmetry for the efficient and concurrent computation of original and adjoint systems. Also, to improve the numerical efficiency of large scale problems, a parallel computation scheme is employed using a binary space decomposition method. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It is demonstrated that the peridynamic adjoint sensitivity involving history-dependent variables can be accurate only if the path of the adjoint response analysis is identical to that of the original response.     

A spectrally formulated plate element for wave propagation analysis in anisotropic material

A new spectrally formulated plate element is developed to study wave propagation in composite structures. The element is based on the classical lamination plate theory. Recently developed method based on singular value decomposition (SVD) is used in the element formulation. Along with this, a new strategy based on the method of solving polynomial eigenvalue problem (PEP) is proposed in this paper, which significantly reduces human intervention (and thus human error), in the element formulation. The developed element has an exact dynamic stiffness matrix, as it uses the exact solution of the governing elastodynamic equation of plate in frequency–wavenumber domain as the interpolating functions. Due to this, the mass distribution is modeled exactly, and as a result, a single element captures the exact frequency response of a regular structure, and it suffices to model a plate of any dimension. Thus, the cost of computation is dramatically reduced compared to the cost of conventional finite element analysis. The fast Fourier transform (FFT) and Fourier series are used for inversion to time–space domain. This element is used to model plate with ply drops and to capture the propagation of Lamb waves.     

1D wave propagation in saturated viscous geomaterials: Improvement and validation of a fractional step Taylor–Galerkin finite element algorithm

This paper tackles the numerical simulation of 1D wave propagation in saturated viscous porous media, and especially in soil-like geomaterials. For this purpose, an improved fractional step Taylor–Galerkin algorithm is first formulated and then validated on the basis of a new analytical solution.The algorithm, based on a stress–velocity–pressure formulation of the hydro-mechanical problem, combines an explicit Taylor–Galerkin method with a fractional time-stepping, while an accurate Runge–Kutta-type integrator is introduced to treat the viscosity of the porous skeleton. The overall algorithm results in an efficient stabilized scheme allowing for linear equal interpolation of field variables, even when the so-called “undrained incompressible limit” is approached.The accuracy and stability of the method are verified with reference to a 1D benchmark problem, concerning the propagation of P waves along a saturated viscoelastic soil stratum. For this problem, a frequency-domain analytical solution is derived, assuming incompressible interstitial fluid and soil grains. The assumption of Maxwell viscoelastic soil skeleton is analytically convenient to preserve the linearity of the problem, while the same rheology of a more realistic elasto-viscoplastic non-linear behaviour is maintained.The performance of the fractional step Taylor–Galerkin algorithm is explored simulating the dynamic response of the stratum to harmonic, impulsive and seismic input excitations. In particular, parametric analyses are performed to confirm the effectiveness of the method in reproducing fully undrained responses, as well as in dealing with weakly viscous materials.     

Parallel implementation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation

Numerical simulation of wave propagation in poroelastic media demands significantly more computational capability compared to elastic media simulation. Use of serial codes in a single scientific workstation limits the size of problem. To overcome this difficulty, a parallel velocity-stress staggered-grid finite-difference method is developed for efficient simulation of wave propagation in 2-D poroelastic media. The finite difference formulation of Biot's theory has the properties of fourth order accuracy in space and second order accuracy in time. The model is decomposed into small subdomains for each processor. After each processor updates wavefields within its domain, the processors exchange the wavefields via message passing interface (MPI). The parallel implementation reduces the computational time and also allows one to study larger problems. From our numerical experiment, consistent with other 1-D experiments, it is found that the presence of heterogeneity of porous medium can produce significant P-wave attenuation in the seismic frequency range.     

Modeling crack in viscoelastic media using the extended finite element method

In this paper,the problem of modeling crack in 2D viscoelastic media is studied using the extended finite element method.The paper focuses on the definition of enrichment functions suitable for cracks assessment in viscoelastic media and the generalized domain integrals used in the determination of crack tip parameters.The opening mode and mixed mode solutions of crack tip fracture problems in viscoelastic media are also undertaken.The results obtained by the proposed method show good agreement with the ana...     

Lagrangian finite element modelling of dam–fluid interaction: Accurate absorbing boundary conditions

The dynamic dam–fluid interaction is considered via a Lagrangian approach, based on a fluid finite element (FE) model under the assumption of small displacement and inviscid fluid. The fluid domain is discretized by enhanced displacement-based finite elements, which can be considered an evolution of those derived from the pioneering works of Bathe and Hahn [Bathe KJ, Hahn WF. On transient analysis of fluid–structure system. Comp Struct 1979;10:383–93] and of Wilson and Khalvati [Wilson EL, Khalvati M. Finite element for the dynamic analysis of fluid–solid system. Int J Numer Methods Eng 1983;19:1657–68]. The irrotational condition for inviscid fluids is imposed by the penalty method and consequentially leads to a type of micropolar media. The model is implemented using a FE code, and the numerical results of a rectangular bidimensional basin (subjected to horizontal sinusoidal acceleration) are compared with the analytical solution. It is demonstrated that the Lagrangian model is able to perform pressure and gravity wave propagation analysis, even if the gravity (or surface) waves are dispersive. The dispersion nature of surface waves indicates that the wave propagation velocity is dependent on the wave frequency.For the practical analysis of the coupled dam–fluid problem the analysed region of the basin must be reduced and the use of suitable asymptotic boundary conditions must be investigated. The classical Sommerfeld condition is implemented by means of a boundary layer of dampers and the analysis results are shown for the cases of sinusoidal forcing.The classical Sommerfeld condition is highly efficient for pressure-based FE modelling, but may not be considered fully adequate for the displacement-based FE approach. In the present paper a high-order boundary condition proposed by Higdom [Higdom RL. Radiation boundary condition for dispersive waves. SIAM J Numer Anal 1994;31:64–100] is considered. Its implementation requires the resolution of a multifreedom constraint problem, defined in terms of incremental displacements, in the ambit of dynamic time integration problems. The first- and second-order Higdon conditions are developed and implemented. The results are compared with the Sommerfeld condition results, and with the analytical unbounded problem results.Finally, a number of finite element results are presented and their related features are discussed and critically compared.