基于无网格稳定化方案求解非稳态强对流问题的自适应节点加密技术

无网格Taylor最小二乘(MFLS)稳定化方案可有效地消除无网格Galerkin方法求解对流占优问题时产生的数值伪振荡,但当对流作用很强或纯对流时,它的求解效果不尽人意.因此,本文基于MFLS稳定化方案给出了一种自适应节点加密技术.该技术将无网格方法中背景积分单元作为自适应节点加密时物理量梯度指标的控制单元,并计算该控制单元上的物理量梯度指标;然后将其与给定的物理量梯度指标限进行比较,标识出大梯度区域从而进行自适应节点加密.数值实验表明,当求解对流作用很强的问题或纯对流问题时,这种基于MFLS稳定化方案的自适应节点加密技术不仅能有效地标示出数值振荡区域,而且可以彻底地消除数值伪振荡.     

Influence of local cubic anisotropy on the transition towards an equipartition regime in a 3D texture-less random elastic medium

At long lapse times in randomly fluctuating media with macroscopic isotropy (texture-less media), the energy of elastic waves is equipartitioned between compressional (P) and shear (S) waves. This property is independent of the local isotropy or anisotropy of the heterogeneous constitutive tensor and of the type of source. However the local symmetry of the constitutive tensor does influence the rate of convergence to equipartition and this paper discusses the precise influence of local anisotropy on the time required to reach equipartition. More particularly, a randomly-fluctuating medium is considered, whose behavior is statistically isotropic, and locally cubic. After calculating all the differential and total scattering cross-sections in that case, an analytical formula is derived for the rate of convergence to the equipartition regime, function of the second-order statistics of the mechanical parameter fields (bulk and shear moduli and anisotropy parameter). The local anisotropy is shown to influence strongly that transition rate, with a faster transition when the fluctuations of the anisotropy parameter are positively correlated to those of the shear modulus. A numerical model is constructed to illustrate numerically these results. Since the asymptotic regime of equipartition cannot be simulated directly because it would require too large a computational domain, boundaries are introduced and mechanical properties are chosen so as to minimize their influence on equipartition.     

Numerical Analysis of Coupled Stokes/Darcy Flows in Industrial Filtrations

Free flow channel confined by porous walls is a feature of many of the natural and industrial settings. Viscous flows adjacent to saturated porous medium occur in cross-flow and dead-end filtrations employed primarily in pharmaceutical and chemical industries for solid–liquid or gas–solid separations. Various mathematical models have been put forward to describe the conjugate flow dynamics based on theoretical grounds and experimental evidence. Despite this fact, there still exists a wide scope for extensive research in numerical solutions of these coupled models when applied to problems with industrial relevance. The present work aims towards the numerical analysis of coupled free/porous flow dynamics in the context of industrial filtration systems. The free flow dynamics has been expressed by the Stokes equations for the creeping, laminar flow regime whereas the flow behaviour in very low permeability porous media has been represented by the conventional Darcy equation. The combined free/porous fluid dynamical behaviour has been simulated using a mixed finite element formulation based on the standard Galerkin technique. A nodal replacement technique has been developed for the direct linking of Stokes and Darcy flow regimes which alleviates specification of any additional constraint at the free/porous interface. The simulated flow and pressure fields have been found for flow domains with different geometries which represent prototypes of actual industrial filtration equipment. Results have been obtained for varying values of permeability of the porous medium for generalised Newtonian fluids obeying the power law model. A series of numerical experiments has been performed in order to validate the coupled flow model. The developed model has been examined for its flexibility in dealing with complex geometrical domains and found to be generic in delivering convergent, stable and theoretically consistent results. The validity and accuracy of the simulated results has been affirmed by comparing with available experimental data.     

A cracked beam or plate transversely loaded by a stamp

In this paper the problem of an infinite elastic beam or a plate containing a crack is considered. The medium is loaded transversely through a stamp which may be rigid or elastic. The problem is a coupled crack-contact problem which cannot be solved by treating the crack and contact problems separately and by using a superposition technique. First the Green's functions for the general case are obtained. Then the integral equations for a cracked infinite strip loaded by a frictionless stamp are obtained. With the question of fracture in mind, the primary interest in the paper has been in calculating the stress intensity factors. The results are given for a rigid flat stamp with sharp edges and for an elastic curved stamp. The effect of friction at the supports on the stress intensity factors is also studied and a numerical example is given.     

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针对连续Bernoulli-Euler和Timoshenko梁单元的动态刚度矩阵,分析了在使用连续梁单元 进行结构动态特性分析中的数值问题. 基于连续梁单元的运动方程,导出了连续 Bernoulli-Euler和Timoshenko梁单元的动态刚度矩阵. 分析了影响动态刚度矩阵中双曲函 数自变量的各个独立变量及其产生的影响,并给出了初估连续梁单元合理长度的方法. 使用 单一连续Bernoulli-Euler和Timoshenko梁单元的动态刚度矩阵分别进行了悬臂梁频响曲线 的数值求解. 研究表明,在合理选择连续梁单元的长度时,大多数工程结构的动态特性分析 中都不会产生数值问题.     

Two-dimensional interactions due to moving load in generalized thermoelastic solid with diffusion

The present paper is concerned with the investigation of disturbances in＇a homogeneous, isotropic elastic medium with generalized thermoelastic diffusion, when a moving source is acting along one of the co-ordinate axis on the boundary of the medium. Eigen value approach is applied to study the disturbance in Laplace-Fourier transform domain for a two dimensional problem. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using a numerical technique for the inversion of Laplace transform based on Fourier expansion techniques. These expressions are calculated numerically for a copper like material and depicted graphically. As special cases, the results in generalized thermoelastic and elastic media are obtained. Effect of presence of diffusion is analyzed theoretically and numerically.     

非均匀介质中流动和传热问题的有限分析法

数值求解非均匀介质中的输运问题广泛应用于科学计算和工程领域．介质的强非均匀性给相关问题的准确求解带来极大的困难．近年来，本课题组将有限分析法拓展到该领域，建立了非均匀介质中输运问题的有限分析法．该算法基于网格奇点邻域内类拉普拉斯方程局部解析解构建，算法具有很高的精度，且不依赖于介质的非均匀性强度．不管相邻网格传导率差异如何，仅需对原始网格进行很少地细分就可以获得非常准确的计算结果，因此与其他传统数值算法相比，可以大幅提高计算精度和效率．该算法可广泛应用于求解非均匀多孔介质中的渗流、复合材料中的热传导及电场分布等问题．     

Transient analysis of a three-dimensional plate by the ray grouping technique

The propagation of transient waves in an elastic plate excited by an arbitrary loading is investigated. Exact three-dimensional transform solutions are derived for a plate suddenly loaded on one of its bounding surfaces, with the inversion effected by Cagniard's technique. The solution is based on three canonical problems for finding the n-th reflected waves from only the information on the (n − 1)-th reflected waves. The technique automatically groups rays which arrive simultaneously at one point, thus simplifying the computations needed by the ray tracing technique. A nonaxially symmetric sample loading is selected for demonstrating the technique, and numerical results are presented. The Ray Grouping Technique can be extended to the layered medium case.     

A numerical study of rays in random media

Statistics of electromagnetic rays in a random medium are studied numerically by the Monte Carlo method. Two dimensional random surfaces with prescribed correlation functions are used to simulate the random media. Rays are then traced in these sample media. Statistics of the ray properties such as the ray positions and directions are computed. Histograms showing the distributions of the ray positions and directions at different points along the ray path as well as at given points in space are given. The numerical experiment is repeated for different cases corresponding to weakly and strongly random media with isotropic and anisotropic irregularities. Results are compared with those derived from theoretical investigations whenever possible.     

Effects of Medium Permeability Anisotropy on Chemical-Dissolution Front Instability in Fluid-Saturated Porous Media

This paper deals with the theoretical aspects of chemical-dissolution front instability problems in two-dimensional fluid-saturated porous media including medium anisotropic effects. Since a general anisotropic medium can be described as an orthotropic medium in the corresponding principal directions, a two-dimensional orthotropic porous medium is considered to derive the analytical solution for the critical condition, which is used to judge whether or not the chemical dissolution front can become unstable during its propagation. In the case of the mineral dissolution ratio (that is defined as the ratio of the dissolved-mineral equilibrium concentration in the pore-fluid to the molar concentration of the dissolvable mineral in the solid matrix of the fluid-saturated porous medium) approaching zero, the corresponding critical condition has been mathematically derived when medium permeability anisotropic effects are considered. As a complementary tool, the computational simulation method is used to simulate the morphological evolution of chemical dissolution fronts in two-dimensional fluid-saturated porous media including medium anisotropic effects. The related theoretical and numerical results demonstrated that: (1) a decrease in the medium anisotropic permeability factor (or ratio), which is defined as the ratio of the principal permeability in the transversal direction to that in the longitudinal direction parallel to the pore-fluid inflow direction, can stabilize the chemical dissolution front so that it becomes more difficult for a planar chemical-dissolution front to evolve into different morphologies in the chemical dissolution system; (2) the medium anisotropic permeability ratio can have significant effects on the morphological evolution of the chemical dissolution front. When the Zhao number of the chemical dissolution system is greater than its critical value, the greater the medium anisotropic permeability ratio, the faster the irregular chemical-dissolution front grows.     

A novel statistical linearization solution for randomly excited coupled bending-torsional beams resting on non-linear supports

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique.

     

Galerkin technique based on beam functions in application to the parametric instability of thermal convection in a vertical slot

A Fourier–Galerkin spectral technique for solving coupled higher‐order initial‐boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so‐called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived. The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth‐order algebraic for both linear and nonlinear problems. Though algebraic, the fifth‐order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems. An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement. The new technique is applied to the G‐jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters. Copyright © 2008 John Wiley & Sons, Ltd.     

CONSERVATION LAW AND APPLICATION OF J-INTEGRAL IN MULTI-MATERIALS

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衰变热源作用下饱和多孔介质热固结问题的扩展精细积分法

热源作用下饱和多孔介质热固结效应是土木及能源工程领域的一个重要课题.由于问题的复杂性,已有的研究大多将介质假定为均匀各向同性,且将热源假定为恒定强度.实际工程中,天然饱和多孔介质常表现出明显的分层特性,热源强度也存在衰变性,为此本工作采用扩展精细积分法对衰变热源作用下层状饱和多孔介质的热固结问题进行研究.借助于积分变换,将饱和多孔介质热固结问题的偏微分方程转化为变换域内的常微分方程;然后对饱和多孔介质微层元进行合并消元,并结合边界条件,推导出衰变热源作用下层状饱和多孔介质热固结问题在积分变换域内的扩展精细积分解;对所得解答进行相应的数值积分逆变换,可获得所求温度、超静孔压及竖向位移在物理域内的解答.基于上述求解过程,编制相应的计算程序进行数值计算,通过与已有文献对比,验证本文扩展精细积分法在求解层状饱和多孔介质热固结问题中的适应性和正确性;最后通过几组算例,分析热源衰变周期、热源埋深及介质的成层性对热固结效应的影响.结果表明:热源衰变周期对温度和超静孔压的峰值、以及达到峰值的时间均有明显影响,衰变周期越长,二者峰值均越大,且达到峰值所需时间越长;热源埋深对超静孔压及竖向位移变化影响显著,深埋热源作用时热源两侧竖向位移呈对称分布,而浅埋热源两侧则无此现象;饱和多孔介质的分层特性对热固结效应影响明显.     

Application of a contact model due to L. A. galin to problems of contact interaction between stringers and massive deformable bodies

L. A. Galin’s contact model for a narrow beam bending on an elastic half-space and Melan’s contact model for a stringer are used to consider two problems of contact interaction between one or two identical symmetrically loaded stringers with small rectangular cross-sections and an elastic half-space. The basic characteristics of these problems are expressed by explicit formulas, and the results of their numerical analysis are given as well.     

Theoretical Analyses of the Effects of Solute Dispersion on Chemical-Dissolution Front Instability in Fluid-Saturated Porous Media

This article deals with the theoretical aspects of chemical-dissolution front instability problems in two-dimensional fluid-saturated porous media including solute dispersion effects. Since the solute equilibrium concentration is much smaller than the molar density of the dissolvable mineral in a mineral dissolution system, a limit case, in which the ratio of the solute equilibrium concentration (in the pore fluid) to the molar density of the dissolvable mineral (in the solid matrix of the porous medium) approaches zero, is considered in the theoretical analysis. Under this assumption, the critical condition under which a planar chemical-dissolution front becomes unstable has been mathematically derived when solute dispersion effects are considered. The present theoretical results clearly demonstrated that: (1) the propagation speed of a planar chemical-dissolution front in the case of considering solute dispersion effects is the same as that when solute dispersion effects are neglected. This indicates that solute dispersion does not affect the propagation speed of the planar chemical-dissolution front in a fluid-saturated porous medium. (2) The consideration of solute dispersion can cause a significant increase in the critical Zhao number, which is used to judge whether or not a planar chemical-dissolution front may become unstable in the fluid-saturated porous medium. This means that the consideration of solute dispersion can stabilize a planar chemical-dissolution front, because an increase in the critical Zhao number reduces the likelihood of the planar chemical-dissolution front instability in a fluid-saturated porous medium. In addition, the present results can be used as benchmark solutions for verifying numerical methods employed to simulate detailed morphological evolution processes of chemical dissolution fronts in two-dimensional fluid-saturated porous media.     

Experimental study of plane elastic contact problems by the pseudocaustics method

The optical method of pseudocaustics can be used for the experimental solution of plane elasticity, smooth contact problems for finite or infinite media in contact of arbitrary shape. This technique constitutes an alternative to the various numerical and experimental techniques for the approximate solution of plane elasticity contact problems. The success in the application of the method of pseudocaustics to plane elasticity contact problems is due to the possibility inherent in this method of the direct determination of the derivative Φ'(z) of the complex potential Φ(z) of N.I. Muskhelishvili along the boundaries of the media in contact. Then, the Muskhelishvili complex potentials, Φ(z) and Ψ(z), completely characterizing the state of stress and strain in a plane elastic medium, can easily be determined at any point of the media in contact after simple algebraic calculations. Two applications of the proposed method to contact problems of practical interest are also made.     

一维流体饱和粘弹性多孔介质层的动力响应

本文研究了不可压流体饱和粘弹性多孔介质层的一维动力响应问题。基于粘弹性理论和多孔介质理论,在流相和固相微观不可压、固相骨架服从粘弹性积分型本构关系和小变形的假定下,建立了不可压流体饱和粘弹性多孔介质层一维动力响应的数学模型,利用Laplace变换,求得了原初边值问题在变换空间中的解析解,并利用Laplace逆变换的Crump数值反演方法,得到原动力响应问题的数值解。数值研究了饱和标准线性粘弹性多孔介质层的动力响应,分析了固相位移、渗流速度、孔隙压力及固相有效应力等的响应特征。结果表明,与不可压流体饱和弹性多孔介质相同,不可压流体饱和粘弹性多孔介质中亦只存在一个纵波,并且固相骨架的粘性对动力行为有显著的影响。     

非均匀介质散射问题的体积分方程数值解法

将非均匀介质视为某一均匀背景介质中的扰动，可建立用均匀背景介质格林函数作基本解的体积分方程．给出了配置法求解体积分方程的数值方法，首先解得扰动域内各点以速度扰动为权的波场函数，然后回代计算得到观测面上各接收点的散射波场．与边界元法和Ｂｏｒｎ近似法计算结果比较表明该方法具有很高的精度，可得到穿过非