饱和两相介质近场波动问题的一种时域全显式数值计算方法

基于 u -p形式的饱和两相介质弹性波动方程,开展了饱和两相介质近场波动问题时域显式数值计算方法的研究。通过对波动方程中的质量矩阵和孔隙流体压缩矩阵进行对角化处理,消除了方程中的动力耦联,实现了波动方程的解耦。分别应用中心差分法和Newmark常平均加速度法求解固相位移和速度,基于向后差分法求解孔隙流体压力,推导得到了饱和两相介质动力响应的时域显式逐步积分的计算列式,建立了饱和两相介质近场波动问题的一种新的时域全显式数值计算方法。进行了该文方法中矩阵对角化合理性的验证。将该方法的数值解与相应的解析解进行对比,二者符合良好,验证了该方法的正确性。将该文建立的时域数值计算方法与透射人工边界方法相结合,应用于饱和两相介质的近场波动问题,进行了饱和土场地地震响应的计算研究,计算结果符合弹性波动理论的基本规律,表明该方法对于饱和两相介质近场波动问题时域计算求解的适用性。基于该方法中时域递推计算格式的传递矩阵,进行了该方法稳定性特性的研究。该文建立的数值计算方法具有时域全显式算法的基本特征。方法中对动力响应的全部分量均采用递推和迭代的模式进行求解,避免了求解耦联的动力方程组。该方法具有较高的计算效率,是进行饱和两相介质近场波动问题时域计算求解的一种有效的算法。     

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.     

A numerical study on the generation of impulsive noise by complex flows discharging from a muzzle

A numerical study on impulsive noise generation produced by complex flows discharging from a muzzle is achieved and the basic structures generating impulsive noise are analyzed. Complex flow features by a muzzle flow and noise generation mechanisms by several sources of noise are discussed from numerical simulations. Two‐dimensional axisymmetric Euler equations are used for governing equations. High‐order dispersion relation preserving finite difference method and an optimized four‐level marching method are used for spatial discretization and time integration, respectively. In order to show the capability of this method to capture blast waves and to examine the basic generation mechanism of acoustic waves from a muzzle, the interaction between a shock/blast wave and a vortex ring is implemented. From the numerical simulation of the 7.62‐mm NATO rifle G3 with a DM‐41 round in the near field, complex blast waves, jet flow, various vortices and their interaction phenomena are described and noise generation mechanism due to the interaction of complex flow structures is observed. The present results demonstrate that numerical simulation using computational aeroacoustic methods provides not only a reliable way to determine the blast wave dynamics of the muzzle flow but also allows an opportunity to study the physics and detailed mechanisms of the noise generation and propagation due to the interaction of complex flow structures generated from a muzzle system. Copyright © 2008 John Wiley & Sons, Ltd.     

非饱和地基中Love波的传播特性

基于非饱和多孔介质的波动方程,考虑了土中水,气体与土骨架之间的粘性耦合作用,建立了弹性半空间上非饱和土层中Love波的弥散方程。首先分析了饱和度与频率对非饱和孔隙介质中剪切波速的影响。然后运用数值方法得到了不同饱和度下土层中多种Love模态波的弥散特性和位移分布情况,并用图表的形式给出。数值计算结果表明,上覆非饱和土层中Love波的传播速度和衰减系数不仅具有频散性,而且与土层的饱和度有关。在不同饱和度时的高模态(n≥2)的Love波的截止频率值不同。此外,讨论了饱和度对Love波水平位移幅值的影响。     

Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.     

The comparisons of thermo-elastic waves for Lord-Shulman mode and Green-Lindsay mode based on Guo��s eigen theory

The motion equations of anisotropic media, coupled to the heat conduction equations, are studied here based on the L-S model and the G-L model. The complete set of uncoupled elastic and heat wave equations for anisotropic media are deduced. The results show that the L-S model is suitable for elastic materials and the G-L model is more suitable for dissipative materials. Based on these laws, we discuss the propagation behaviors of heat wave and elastic waves for isotropic media.     

Computational simulation of chemical dissolution‐front instability in fluid‐saturated porous media under non‐isothermal conditions

This paper primarily deals with the computational aspects of chemical dissolution‐front instability problems in two‐dimensional fluid‐saturated porous media under non‐isothermal conditions. After the dimensionless governing partial differential equations of the non‐isothermal chemical dissolution‐front instability problem are briefly described, the formulation of a computational procedure, which contains a combination of using the finite difference and finite element method, is derived for simulating the morphological evolution of chemical dissolution fronts in the non‐isothermal chemical dissolution system within two‐dimensional fluid‐saturated porous media. To ensure the correctness and accuracy of the numerical solutions, the proposed computational procedure is verified through comparing the numerical solutions with the analytical solutions for a benchmark problem. As an application example, the verified computational procedure is then used to simulate the morphological evolution of chemical dissolution fronts in the supercritical non‐isothermal chemical dissolution system. The related numerical results have demonstrated the following: (1) the proposed computational procedure can produce accurate numerical solutions for the planar chemical dissolution‐front propagation problem in the non‐isothermal chemical dissolution system consisting of a fluid‐saturated porous medium; (2) the Zhao number has a significant effect not only on the dimensionless propagation speed of the chemical dissolution front but also on the distribution patterns of the dimensionless temperature, dimensionless pore‐fluid pressure, and dimensionless chemical‐species concentration in a non‐isothermal chemical dissolution system; (3) once the finger penetrates the whole computational domain, the dimensionless pore‐fluid pressure decreases drastically in the non‐isothermal chemical dissolution system.     

Discontinuous Galerkin method based on peridynamic theory for linear elasticity

Modeling of discontinuities (shock waves, crack surfaces, etc.) in solid mechanics is one of the major research areas in modeling the mechanical behavior of materials. Among the numerical methods, the discontinuous Galerkin method (DGM) poses some advantages in solving these problems. In this study, a novel formulation for DGM is derived for elastostatics based on the peridynamic theory. Derivation of the proposed formulation is presented. Numerical analyses are performed for different problems, and the numerical results are compared to that of the known exact solutions of the problems. The proposed weak formulation is stable and coercive. Peridynamic discontinuous Galerkin formulation is found to be robust and successful in modeling elastostatic problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Weak,anisotropic symmetric formulations of Biot's equations for vibro‐acoustic modelling of porous elastic materials

In this paper a fully anisotropic symmetric weak formulation of Biot's equations for vibro‐acoustic modelling of porous elastic materials in the frequency domain is proposed. Starting from Biot's equations in their anisotropic form, a mixed displacement–pressure formulation is discussed in terms of Cartesian tensors. The anisotropic equation parameters appearing in the differential equations are derived from material parameters which are possible to determine through experimental testing or micro‐structural simulations of the fluid and the porous skeleton. Solutions are obtained by applying the finite element method to the proposed weak form and the results are verified against a weak displacement‐based formulation for a foam and plate combination. Copyright © 2010 John Wiley & Sons, Ltd.     

The spectral element method for elastic wave equations—application to 2‐D and 3‐D seismic problems

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two‐ and three‐dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor‐product of a Gauss–Lobatto–Legendre 1‐D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy‐momentum conserving scheme that can be put into a classical explicit‐implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two‐dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three‐dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three‐dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.     

The method of fundamental solutions for acoustic wave scattering by a single and a periodic array of poroelastic scatterers

The method of fundamental solutions (MFS) is now a well-established technique that has proved to be reliable for a specific range of wave problems such as the scattering of acoustic and elastic waves by obstacles and inclusions of regular shapes. The goal of this study is to show that the technique can be extended to solve transmission problems whereby an incident acoustic pressure wave impinges on a poroelastic material of finite dimension. For homogeneous and isotropic materials, the wave equations for the fluid phase and solid phase displacements can be decoupled thanks to the Helmholtz decomposition. This allows for a simple and systematic way to construct fundamental solutions for describing the wave displacement field in the material. The efficiency of the technique relies on choosing an appropriate set of fundamental solutions as well as properly imposing the transmission conditions at the air–porous interface. In this paper, we address this issue showing results involving bidimensional scatterers of various shapes. In particular, it is shown that reliable error indicators can be used to assess the quality of the results. Comparisons with results computed using a mixed pressure–displacement finite element formulation illustrate the great advantages of the MFS both in terms of computational resources and mesh preparation. The extension of the method for dealing with the scattering by an infinite array of periodic scatterers is also presented.     

Effects of fiber ellipticity and orientation on dynamic stress concentrations in porous fiber-reinforced composites

Interaction of time harmonic fast longitudinal and shear incident plane waves with an elliptical fiber embedded in a porous elastic matrix is studied. The novel features of Biot dynamic theory of poroelasticity along with the classical method of eigen-function expansion and the pertinent boundary conditions are employed to develop a closed form series solution involving Mathieu and modified Mathieu functions of complex arguments. The complications arising due to the non-orthogonality of angular Mathieu functions corresponding to distinct wave numbers in addition to the problems associated with appearance of additional angular dependent terms in the boundary conditions are all avoided by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. A MATHEMATICA code is developed for computing the Mathieu functions in terms of complex Fourier coefficients which are themselves calculated by numerically solving appropriate sets of eigen-systems. The analytical results are illustrated with numerical examples in which an elastic fiber of elliptic cross section is insonified by a plane fast compressional or shear wave at normal incidence. The effects of fiber cross sectional ellipticity, angle of incidence (fiber two-dimensional orientation), and incident wave polarization (P, SV, SH) on dynamic stress concentrations are studied in a relatively wide frequency range. Limiting cases are considered and fair agreements with well-known solutions are established.     

A framework for coupled deformation–diffusion analysis with application to degradation/healing

This paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation–diffusion in linearized elastic solids. The mathematical model takes into account the effect of the deformation on the diffusion process, and the affect of the transport of an inert chemical species on the deformation of the solid. We then present a robust computational framework for solving the proposed mathematical model, which consists of coupled non‐linear partial differential equations. It should be noted that many popular numerical formulations may produce unphysical negative values for the concentration, particularly, when the diffusion process is anisotropic. The violation of the non‐negative constraint by these numerical formulations is not mere numerical noise. In the proposed computational framework, we employ a novel numerical formulation that will ensure that the concentration of the diffusant be always snon‐negative, which is one of the main contributions of this paper. Representative numerical examples are presented to show the robustness, convergence, and performance of the proposed computational framework. Another contribution of this paper is to systematically study the affect of transport of the diffusant on the deformation of the solid and vice versa, and their implication in modeling degradation/healing of materials. We show that the coupled response is both qualitatively and quantitatively different from the uncoupled response. Copyright © 2011 John Wiley & Sons, Ltd.     

A simple multi‐directional absorbing layer method to simulate elastic wave propagation in unbounded domains

The numerical analysis of elastic wave propagation in unbounded media may be difficult due to spurious waves reflected at the model artificial boundaries. This point is critical for the analysis of wave propagation in heterogeneous or layered solids. Various techniques such as Absorbing Boundary Conditions, infinite elements or Absorbing Boundary Layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this paper, a simple absorbing layer method is proposed: it is based on a Rayleigh/Caughey damping formulation which is often already available in existing Finite Element softwares. The principle of the Caughey Absorbing Layer Method is first presented (including a rheological interpretation). The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types and incidences. A comparison with the PML method is first performed for pure P‐waves and the method is shown to be reliable in a more complex 2D case involving various wave types and incidences. It may thus be used for various types of problems involving elastic waves (e.g. machine vibrations, seismic waves, etc.). Copyright © 2010 John Wiley & Sons, Ltd.     

Acoustic scattering of spherical waves incident on a long fluid-saturated poroelastic cylinder

Acoustic scattering of spherical waves generated by a monopole point source in a perfect (inviscid and ideal) compressible fluid by a fluid-saturated porous cylinder of infinite length is studied theoretically in the present study. The formulation utilizes the Biot theory of dynamic poroelasticity along with the appropriate wave-field expansions, the translational addition theorem for spherical wave functions, and the pertinent boundary conditions to obtain a closed-form solution in the form of infinite series. The analytical results are illustrated with a numerical example in which a monopole point source within water is located near a porous cylinder with a water-saturated Ridgefield sandstone formation. The numerical results reveal the effects of source excitation frequency, the cylinder interface permeability condition, and the location of the point source and the field point on the backscattered pressure magnitudes. Limiting cases are considered, and the obtained numerical results are validated by already well-known solutions.     

Finite element modelling of saturated porous media at finite strains under dynamic conditions with compressible constituents

Two finite element formulations are proposed to analyse the dynamic conditions of saturated porous media at large strains with compressible solid and fluid constituents. Unlike similar works published in the literature, the proposed formulations are based on a recently proposed hyperelastic framework in which the compressibility of the solid and fluid constituents is fully taken into account when geometrical non‐linear effects are relevant on both micro‐ and macroscales. The first formulation leads to a three‐field finite element method (FEM), which is suitable for analysing high‐frequency dynamic problems, whereas the second is a simplification of the first, leading to a two‐field FEM, in which some inertial effects of the pore fluid are disregarded, hence the second formulation is suitable for studying low‐frequency problems. A fully Lagrangian approach is considered, hence all terms are expressed with reference to the material setting; the balance equations for the pore fluid are also expressed in terms of the chemical potential and the mass flux of the pore fluid in order to take the compressibility of the fluid into account. To improve the numerical response in the case of wave propagation, a discontinuous Galerkin FEM in the time domain is applied to the three‐field formulation. The results are compared with analytical and semi‐analytical solutions, highlighting the different effects of the discontinuous Galerkin method on the longitudinal waves of the first and second kind. Copyright © 2010 John Wiley & Sons, Ltd.     

Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media

Weight‐adjusted inner products are easily invertible approximations to weighted L² inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time‐domain method for wave propagation which is low storage, energy stable, and high‐order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight‐adjusted DG methods to the case of matrix‐valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind‐like dissipation incorporated through simple penalty fluxes. A semidiscrete convergence analysis is given, and numerical results confirm the stability and high‐order accuracy of weight‐adjusted DG for several problems in elastic wave propagation.     

一种量测松散介质对应力波衰减效应的实验方法及其在珊瑚砂中的应用

砂土等松散介质对在其中传播的应力波有非常明显的衰减作用,因此,松散介质常常作为爆炸波消波材料被广泛应用于防护工程中。为了准确地量测松散介质对应力波的衰减效应,基于并改进了传统SHPB装置,提出了一种定量研究应力波在砂土介质中衰减规律的新方法。该文方法适用于所有在冲击荷载的应变率范围内（约1~102 s-1）应变率效应不明显的松散介质。方法基于拟合的透射系数T₂,通过杆中的三波（入射波、反射波和透射波）计算得到试件两端真实的峰值应力,还可以计算试件的弹性波速、峰值应力速度、试件端部应力波的前沿升时等关键参数,简单实用,可操作性强。采用提出的方法,对干燥珊瑚砂进行了应力波衰减实验,得出了应力波荷载峰值随传播距离的衰减规律。经对比实验与参数讨论发现,拟合透射系数引起的结果误差不超过2.83%,具有很好的可靠性与实用性。     

Boundary element solution of 3-D wave scatter problems in a poroelastic medium

This study details the development of boundary integral equations suitable for treating problems involving the scatter of a plane harmonic wave by an inclusion embedded in an infinite poroelastic medium. The pore pressure-solid displacement form of the harmonic equations of motion are developed from the form of the equations originally presented by Biot. Fundamental solutions and a generalized reciprocal work relation are developed, and these are used to formulate a solution representation in terms of an integral over the inclusion surface. The corresponding boundary integral equations are developed in a form that is integrable in the usual sense, eliminating the need to evaluate Cauchy principal value integrals. These boundary integral equations are discretized and implemented into a boundary element computer program. The so-called forbidden frequency problem which causes computational difficulties in boundary integral treatments of wave scatter in elastic and acoustic media is shown to be absent in the poroelastic case. Numerical results obtained from the boundary element program are compared with analytical results for some test problems, and the program appears to produce accurate results at moderate frequencies.