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,...     

A time-domain boundary element formulation for the dynamic analysis of non-linear porous media

This paper presents an original time-domain boundary element formulation for the dynamic analysis of porous media. Integral equations for displacements, stresses and pore-pressures, based on non-transient fundamental solutions are considered. Elastoplastic models are also dealt with by the present methodology, extending the applicability of boundary elements to model complex porodynamic problems. At the end of the paper, a discussion concerning two numerical examples is presented, illustrating the potentialities of the new procedure.     

Simulation of ductile tearing by the BEM with 2D domain discretization and a local damage model

A numerical procedure based on the Boundary Element Method with internal cells and dedicated to the simulation of the ductile tearing of thin metal sheets is presented. Plasticity is handled with an integral formulation based on the initial strain approach involving a discretization of the planar domain. Time integration is performed in an implicit way for the local strain-stress relationships while the global algorithm relies on an explicit formulation. Damage is represented by the scalar parameter of the uncoupled local damage model of Rice and Tracey. Within the scope of our applications, the cracks propagate along paths a priori known. As damage spreads, boundary elements are gradually released. Elastoplastic problems with large yielding zones are solved and compared to reference solutions. At last, the ductile tearing of a specimen is addressed. The calibration of the critical damage parameter leads to numerical results in good agreement with the experimental ones.     

An efficient finite element procedure for analyzing three‐phase porous media based on the relaxed Picard method

Effective simulation of the solid‐liquid‐gas coupling effect in unsaturated porous media is of great significance in many diverse areas. Because of the strongly nonlinear characteristics of the fully coupled formulations for the three‐phase porous media, an effective numerical solution scheme, such as the finite element method with an efficient iterative algorithm, has to be employed. In this paper, an efficient finite element procedure based on the adaptive relaxed Picard method is developed for analyzing the coupled solid‐liquid‐gas interactions in porous media. The coupled model and the finite element analysis procedure are implemented into a computer code PorousH2M, and the proposed procedure is validated through comparing the numerical simulations with the experimental benchmarks. It is shown that the adaptive relaxed Picard method has salient advantage over the traditional one with respect to both the efficiency and the robustness, especially for the case of relatively large time step sizes. Compared with the Newton‐Raphson scheme, the Picard method successfully avoids the unphysical ‘spurious unloading’ phenomenon under the plastic deformation condition, although the latter shows a better convergence rate. The proposed procedure provides an important reference for analyzing the fully coupled problems related to the multi‐phase, multi‐field coupling in porous media. Copyright © 2014 John Wiley & Sons, Ltd.     

Green's First Identity Method for Boundary-Only Solution of Self-Weight in BEM Formulation for Thick Slabs

The present paper develops a new technique for treatment of self-weight for building slabs in the boundary element method (BEM). Due to the use of BEM in the analysis, all defined variables are presented on the slab boundary (mesh is defined only along the slab boundary). Self-weight, however, is usually defined over slab domain, hence domain discretisation is required, which spoils the main advantage of the BEM. In this paper a new method is presented to transform self-weight domain integrals to the boundary for such slabs. The proposed method is based on using the so-called Green's first identity. All new kernels for generalized displacements, stress-resultants, and tractions are derived and listed explicitly. The present formulation is implemented into computer code and several examples are tested. Results are compared against results obtained from other numerical method to prove the accuracy and validity of the present formulation.     

A discontinuous Galerkin method with plane waves for sound‐absorbing materials

Poro‐elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper, we present a discontinuous Galerkin method (DGM) with plane waves for poro‐elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave‐based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave‐based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material. Copyright © 2015 John Wiley & Sons, Ltd.     

A boundary point interpolation method for stress analysis of solids

 A boundary point interpolation method (BPIM) is proposed for solving boundary value problems of solid mechanics. In the BPIM, the boundary of a problem domain is represented by properly scattered nodes. The boundary integral equation (BIE) for 2-D elastostatics has been discretized using point interpolants based only on a group of arbitrarily distributed boundary points. In the present BPIM formulation, the shape functions constructed using polynomial basis function in a curvilinear coordinate possess Dirac delta function property. The boundary conditions can be implemented with ease as in the conventional boundary element method (BEM). The BPIM for 2-D elastostatics has been coded in FORTRAN, and used to obtain numerical results for stress analysis of two-dimensional solids. Received 10 January 2000     

变形多孔介质流固耦合模型及数值模拟研究

假定骨架、固体颗粒和水均是可压缩的,在此基础上采用两相不混溶流体的理论推导了水的连续方程,通过引入气压恒定这一假定进一步简化为水的非饱和渗流连续方程。基于广义Biot理论给出了固体骨架积分形式的平衡方程,结合非饱和渗流连续方程采用加权残值法推导了流固耦合方程组的有限元列式。通过干燥介质吸水的数值模拟来考察非饱和流固耦合模型的预测能力,数值模拟的结果表明耦合模型可以准确地反映吸水过程的规律。将耦合模型应用于水下大断面隧洞开挖的瞬态分析,可以模拟出开挖引起的EDZ区域孔隙水压力急剧升高、有效应力减小、渗透系数动态变化以及排水对洞室稳定性的影响,计算的结果与国外大型原位实验的一般性观测结论相吻合。     

A 3-D boundary element formulation of viscoelastic media with gravity

This paper presents a boundary element formulation for 3-D linear and viscoelastic bodies subjected to the body force of gravity. The Laplace transformation is first used to suppress the time variable, and solutions of displacements and stresses are found in the transformed domain. The time domain solutions are then found by an accurate and efficient numerical inversion method which requires only real calculations for all quantities. Input and output data, and solutions in the transformed and time domains are connected through an Interactive Data Language code written by the authors. While particular solutions of stresses and displacements related to the body force of gravity (which is applied at time t = 0 and is kept constant) are derived, the Green's functions in the Laplace domain are obtained through the correspondence principle. The new formulation has been implemented into an existing 3-D BEM program, and several numerical examples involving 3-D viscoelastic bodies are presented. Although the discussion in this paper focuses on Maxwell viscoelastic and isotropic media, other linear isotropic and even anisotropic viscoelastic models can also be incorporated, without difficulty, into the 3-D viscoelastic BEM program.     

Simulation of 3D flow in porous media by boundary element method

In the present paper problem of natural convection in a cubic porous cavity is studied numerically, using an algorithm based on a combination of single domain and subdomain boundary element method (BEM). The modified Navier–Stokes equations (Brinkman-extended Darcy formulation with inertial term included) were adopted to model fluid flow in porous media, coupled with the energy equation using the Boussinesq approximation. The governing equations are transformed by the velocity–vorticity variables formulation which separates the computation scheme into kinematic and kinetic parts. The kinematics equation, vorticity transport equation and energy equation are solved by the subdomain BEM, while the boundary vorticity values, needed as a boundary conditions for the vorticity transport equation, are calculated by single domain BEM solution of the kinematics equation. Computations are performed for steady state cases, for a range of Darcy numbers from 10^?6 to 10^?1, and porous Rayleigh numbers ranging from 50 to 1000. The heat flux through the cavity and the flow fields are analyzed for different cases of governing parameters and compared to the results in some published studies.     

Analysis of anisotropic Kirchhoff plates using a novel hypersingular BEM

In this article a hypersingular boundary element method (BEM) for bending of thin anisotropic plates is presented. A new complex variable fundamental solution is implemented in the algorithm. For spatial discretization a collocation method with discontinuous quadratic elements is adopted. The domain integrals arising from the transversely applied load are transformed analytically into boundary integrals by means of the radial integration technique. The considered numerical examples prove that the novel BEM formulation presented in this study is much more efficient than previous formulations developed for the analysis of this kind of problems.     

A non-singular boundary integral formula for frequency domain analysis of the dynamic <Emphasis Type="Italic">T</Emphasis>-stress

A non-singular 2-D boundary integral equation (BIE) in the Fourier-space frequency domain for determining the dynamic T-stress (DTS) is presented in this paper. This formulation, based upon the Fourier transform of the asymptotic expansion for the stress field in the vicinity of a crack tip, can be conveniently implemented as a post-processing step in a frequency-domain boundary element analysis of cracks. The proposed BIE is accurate as it can be directly collocated at the crack tip in question. The technique is also computationally effective as it simply requires a similar computing effort as that used in determining the dynamic stress components at an interior point of a domain. Five numerical examples involving both straight and curved cracks are studied to validate the proposed technique. For the frequency domain analysis of the DTS in these examples, the exponential window method is employed to obtain its time history.     

Hybrid‐Trefftz stress element for bounded and unbounded poroelastic media

The equations that govern the dynamic response of saturated porous media are first discretized in time to define the boundary value problem that supports the formulation of the hybrid‐Trefftz stress element. The (total) stress and pore pressure fields are directly approximated under the condition of locally satisfying the domain conditions of the problem. The solid displacement and the outward normal component of the seepage displacement are approximated independently on the boundary of the element. Unbounded domains are modelled using either unbounded elements that locally satisfy the Sommerfeld condition or absorbing boundary elements that enforce that condition in weak form. As the finite element equations are derived from first‐principles, the associated energy statements are recovered and the sufficient conditions for the existence and uniqueness of the solutions are stated. The performance of the element is illustrated with the time domain response of a biphasic unbounded domain to show the quality of the modelling that can be attained for the stress, pressure, displacement and seepage fields using a high‐order, wavelet‐based time integration procedure. Copyright © 2010 John Wiley & Sons, Ltd.     

Non‐local boundary integral formulation for softening damage

A strongly non‐local boundary element method (BEM) for structures with strain‐softening damage treated by an integral‐type operator is developed. A plasticity model with yield limit degradation is implemented in a boundary element program using the initial‐stress boundary element method with iterations in each load increment. Regularized integral representations and boundary integral equations are used to avoid the difficulties associated with numerical computation of singular integrals. A numerical example is solved to verify the physical correctness and efficiency of the proposed formulation. The example consists of a softening strip perforated by a circular hole, subjected to tension. The strain‐softening damage is described by a plasticity model with a negative hardening parameter. The local formulation is shown to exhibit spurious sensitivity to cell mesh refinements, localization of softening damage into a band of single‐cell width, and excessive dependence of energy dissipation on the cell size. By contrast, the results for the non‐local theory are shown to be free of these physically incorrect features. Compared to the classical non‐local finite element approach, an additional advantage is that the internal cells need to be introduced only within the small zone (or band) in which the strain‐softening damage tends to localize within the structure. Copyright © 2003 John Wiley & Sons, Ltd.     

Time dependent axisymmetric thermoelastic boundary element analysis

A boundary element method is developed for problems of quasistatic axisymmetric thermoelasticity. Unlike previous approaches, this new time domain formulation is written exclusively in terms of surface quantities, thereby eliminating the need for volume discretization. Furthermore, since the exact three-dimensional infinite space fundamental solutions are employed, very accurate solutions are obtainable, including the determination of surface stresses. In the integral formulation, the fundamental solutions are separated into steady-state and transient components. The steady-state portion, which contains all of the singularities, is integrated analytically in the circumferential direction, yielding the familiar axisymmetric kernels. The remaining non-singular transient integrands are treated by a combination of analytical and numerical quadrature. The method is implemented in a general purpose boundary element code, which includes multiregion capability along with higher order conforming surface elements. Several numerical examples are provided to illustrate the validity of the formulation and the attractiveness of this approach for practical engineenring analysis.     

Modified flux-vector-based Green element method for problems in steady-state anisotropic media

In this study we present a new numerical technique for solving problems in steady-state heterogeneous anisotropic media, namely the ‘flux-vector-based’ Green element method (‘$\mathit{q}$-based’ GEM) for anisotropic media. This method, which is appropriate for problems where the permeability has either constant or continuous components over the whole domain, is based on the boundary element method (BEM) formulation for direct, steady-state flow problems in anisotropic porous media, which is applied to finite element method (FEM) meshes. For situations involving media discontinuities, an extension of this ‘$\mathit{q}$-based’ GEM formulation is proposed, namely the modified ‘$\mathit{q}$-based’ GEM for anisotropic media. Numerical results are presented for various physical problems that simulate flow in an anisotropic medium with diagonal layers of different permeabilities or around faults and wells, and they show that the new method, with the extensions proposed, is very suitable for steady-state problems in such media.     

Initial conditions contribution in a BEM formulation based on the convolution quadrature method

This work presents a two‐dimensional boundary element method (BEM) formulation for the analysis of scalar wave propagation problems. The formulation is based on the so‐called convolution quadrature method (CQM) by means of which the convolution integral, presented in time‐domain BEM formulations, is numerically substituted by a quadrature formula, whose weights are computed using the Laplace transform of the fundamental solution and a linear multistep method. This BEM formulation was initially developed for scalar wave propagation problems with null initial conditions. In order to overcome this limitation, this work presents a general procedure that enables one to take into account non‐homogeneous initial conditions, after replacing the initial conditions by equivalent pseudo‐forces. The numerical results included in this work show the accuracy of the proposed BEM formulation and its applicability to such kind of analysis. Copyright © 2006 John Wiley & Sons, Ltd.     

Transient dynamic boundary element analysis using Gaussian-based mass matrix

This paper presents a new boundary element formulation for transient dynamic analysis. The formulation is based on the solution of the problem using the static fundamental solution. The inertia term is approximated using particular solutions using radial bases functions. Another collocation scheme is preformed to determine suitable coefficients for the radial bases function approximation. The Gaussian radial bases function is used as it rapidly decays leading to better conditioned and sparse matrices. The necessary kernels are derived and their limiting values at the singular node are given. The formulation is implemented into a computer program that accounts for boundary and internal nodes. Two examples are presented to show the accuracy and the validity of the present formulation. A numerical discussion on using the Gaussian function with compact supported or cut-off radius is also given.     

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.     

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.