三维动态裂纹问题的超奇异积分方程法

基于弹性材料的动态基本方程，结合广义Betti-Rayleigh互易等式与时域下的边界积分方程，推导得到时域下的超奇异积分方程组。引入Laplace域下的动态基本解，将经过主部分析的积分核函数分解为静态和动态部分，其中动态积分核不具有奇异性。在裂纹前沿附近单元，采用与理论分析一致的平方根位移模型。结合Lubich时间卷积实现拉氏变换，采用配置点法计算超奇异积分，获得问题的数值解。并针对椭圆裂纹算例编写Fortran程序，得到冲击荷载作用下张开型裂纹的动态应力强度因子变化规律，数值结果稳定且收敛速度快。     

时域边界元法分析撞水响应

基于势流理论，考虑流场的可压缩性，首先利用积分变换导得了势流问题的一个动力学倒易定理，在此基础上，进而求得问题对应的时空边界积分方程，然后通过对边界和时间轴同时离散，建立了一组有递推形式的时间边界元方程最后结合液面条件和物体运动方程耦全求解得到了刚体的撞水响应。     

敷设多孔吸声材料声腔特征值分析的径向积分边界元法

由于Helmholtz方程的基本解是频率的函数,因此传统边界元法在处理声场特征值问题时具有天生的缺陷。本文采用Laplace方程基本解生成积分方程,通过径向积分法将在此过程中产生的域积分项转化为边界积分。此方法克服了传统边界元法系数矩阵对频率的依赖,同时克服了特解积分法对特解的依赖,并通过对表面声导纳的多项式逼近,将敷设多孔吸声材料声腔特征值问题转化为矩阵多项式,从而避免了复杂的非线性求解。通过数值算例验证了算法的有效性。     

基于拉氏变换的弹性动力学插值型重构核粒子法

插值型重构核粒子法的形函数具有离散点插值特性和不低于核函数的高阶光滑性，因而不仅可以直接施加本质边界条件，同时也保证了较高的计算精度．本文将弹性动力学方程作拉氏变换后，在变换域内用插值型重构核粒子法求解，最后再借助Durbin数值反演方法求得时间域的解．针对典型的弹性动力学问题，给出了插值型重构核粒子法的数值算例，并验证了本文方法的有效性．     

Boundary element analysis for dynamic response of vibration foundation

The boundary elament method (BEM) for numerical solution to dynamic response of vibration fundation in plane, elastic domains are presented. The dynamic boundary integral equation is derived from the Laplace integral transform of the elestodynamic differential equation. Numerical solution can then be completed by the discrete boundary element in the transform space. Finally, dynamic responsed in time domain will be inverted back from the transform space with the numerical method. Excited harmonic load responses of dynamic rigid foundation are calculated and discussed for different frequencies, Layer depths and foundation embedments. Again, screening of exciting wave is also studied.The support of the ressarch project part in this work by Dr. O. Tullberg, Goteborg Universities' Computing Centre, Sweden, is gratefully acknwledged.     

High accuracy time and space transform method for advection–diffusion equation in an unbounded domain

A new numerical method called high accuracy time and space transform method (TSTM) is introduced to solve the advection–diffusion equation in an unbounded domain. By a spatial transform, the advection–diffusion equation in the unbounded domain Rⁿ is converted to one on the bounded domain [?1, 1]ⁿ, and the Laplace transform is applied to eliminate time dependency. The consequent boundary value problem is solved by collocation on Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that TSTM has exponential rate in time and space. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary element method for solving dynamical response of viscoelastic thin plate (I)

I.lntroducti0nThedynamicresponseofviscoelasticstructuresisoneofimportantresearchdirectionsinsolidmechanics.BecauseofthecompIexityoftheconstitutiverelations0fviscoeIasticmaterials,theproblemofsolvingthedynamicresponseisverydifflcult.Therearesomeavailablenu…     

Separation of principal stresses in dynamic photoelasticity

A new and effective method used to separate the transient principal stresses for dynamic photoelasticity is proposed. This is a hybrid method combining the optical method of dynamic caustics and the boundary element numerical method. Firstly, a modified Cranz-Schardin spark camera is used to record simultaneously the isochromatic fringe patterns of photoelasticity and the shadow spot patterns in the dynamic process. By means of the isochromatic fringe patterns, the difference between transient principal stresses in the whole domain and the principal stresses along the free boundary can be solved. In addition, the method of caustics is a very powerful technique for measuring the concentrative load. Then, the sum of the principal stresses is calculated by the boundary integral equation obtained from the Laplace integral transform of the wave equation. So, the transient principal stresses can be determined from the experimental and numerical results. As an example, the transient principal stresses in a polycarbonate disk under an impact load are resolved. Concurrently published in the Chinese Edition of Acta Mechanica Sinica, Vol. 26, No. 1, 1994     

A Quasilinear Based Procedure for Saturated/Unsaturated Water Movement in Soils

The quasilinear form of Richards equation for one-dimensional unsaturated flow in soils can be readily solved for a wide variety of conditions. However, it cannot explain saturated/unsaturated flow and the constant diffusivity assumption, used to linearise the transient quasilinear equation, can introduce significant error. This paper presents a quasi-analytical solution to transient saturated/unsaturated flow based on the quasilinear equation, with saturated flow explained by a transformed Darcy's equation. The procedure presented is based on the modified finite analytic method. With this approach, the problem domain is divided into elements, with the element equations being solutions to a constant coefficient form of the governing partial differential equation. While the element equations are based on a constant diffusivity assumption, transient diffusivity behaviour is incorporated by time stepping. Profile heterogeneity can be incorporated into the procedure by allowing flow properties to vary from element to element. Two procedures are presented for the temporal solution; a Laplace transform procedure and a finite difference scheme. An advantage of the Laplace transform procedure is the ability to incorporate transient boundary condition behaviour directly into the analytical solutions. The scheme is shown to work well for two different flow problems, for three soil types. The technique presented can yield results of high accuracy if the spatial discretisation is sufficient, or alternatively can produce approximate solutions with low computational overheads by using large sized elements. Error was shown to be stable, linearly related to element size.     

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.     

A TIME DOMAIN BOUNDARY ELEMENT METHOD FOR WATER-SOLID IMPACT ANALYSIS

A reciprocal theorem of dynamics for potential flow problems is first derived by meansof the Laplace transform in which the compressibility of water is taken into account.Based on this the-orem,the corresponding time-space boundary integral equation is obtained.Then,a set of time do-main boundary element equations with recurrence form is immediately formulated through discretiza-tion in both time and boundary.After having carried out the numerical calculation two solutions arefound in which a rigid semicircular cylinder and a rigid wedge with infinite length suffer normal impacton the surface of a half-space fluid.The results show that the present method is more efficient than theprevious ones.     

Linear viscoelastic analysis of a semi-infinite porous medium

This paper considers the problem of a semi-infinite, isotropic, linear viscoelastic half-plane containing multiple, non-overlapping circular holes. The sizes and the locations of the holes are arbitrary. Constant or time dependent far-field stress acts parallel to the boundary of the half-plane and the boundaries of the holes are subjected to uniform pressure. Three types of loading conditions are assumed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, a force uniformly distributed over the whole boundary of the half-plane. The solution of the problem is based on the use of the correspondence principle. The direct boundary integral method is applied to obtain the governing equation in the Laplace domain. The unknown transformed displacements on the boundaries of the holes are approximated by a truncated complex Fourier series. A system of linear equations is obtained by using a Taylor series expansion. The viscoelastic stresses and displacements at any point of the half-plane are found by using the viscoelastic analogs of Kolosov–Muskhelishvili’s potentials. The solution in the time domain is obtained by the application of the inverse Laplace transform. All the operations of space integration, the Laplace transform and its inversion are performed analytically. The method described in the paper allows one to adopt a variety of viscoelastic models. For the sake of illustration only one model in which the material responds as the standard solid in shear and elastically in bulk is considered. The accuracy and efficiency of the method are demonstrated by the comparison of selected results with the solutions obtained by using finite element software ANSYS.     

Dynamic stress intensity factors around a cylindrical crack in an infinite elastic medium subject to impact load

This paper investigates transient stresses around a cylindrical crack in an infinite elastic medium subject to impact loads. Incoming stress waves resulting from the impact load impinge on the crack in a direction perpendicular to the crack axis. In the Laplace transform domain, by means of the Fourier transform technique, the mixed boundary value equations with respect to stresses and displacements were reduced to two sets of dual integral equations. To solve the equations, the differences in the crack surface displacements were expanded in a series of functions that are zero outside the crack. The boundary conditions for the crack were satisfied by means of the Schmidt method. Stress intensity factors were defined in the Laplace transform domain and were numerically inverted to physical space. Numerical calculations were carried out for the dynamic stress intensity factors corresponding to some typical shapes assumed for the cylindrical crack.     

三维势流场的比例边界有限元求解方法

比例边界有限元法(SBFEM)是线性偏微分方程的一种新的数值求解方法。该方法只对计算域边界利用Galerkin方法进行数值离散,相对于有限元方法(FEM)减少了一个空间坐标的维数,而在减少的空间坐标方向利用解析方法进行求解;相对于边界元法(BEM),比例边界有限元方法不需要基本解,避免了奇异积分的计算,所以它结合了有限元和边界元方法的优点。本文建立了利用比例边界有限元法求解三维Laplace方程的数值模型并用于计算三维物体周围的水流场,将计算结果与解析解和边界元方法进行了对比,结果表明此方法可以很好地模拟水流场,且具有较高的计算精度。     

轴向运动弦线横向振动的频域分析

轴向运动弦线是多种工程系统的模型。为明确轴向运动横向振动的频域特性，及探索频域方法的应用特点．本文用频域方法分析轴向运动弦线的横向振动。基于轴向运动弦线横向振动方程和边界条件．通过Laplace变换导出频率域中的控制方程，并将该控制方程和边界条件用状态变量表示。由状态空间中的控制方程导出特征方程，从而求出固有频率。由轴向运动弦线的矩阵函数计算得到系统的传递函数，然后用留数定理计算传递函数的Laplace逆变换．这样就可以得到时域响应。最后分析了轴向运动弦线的横向共振，若简谐外激励的频率与系统固有频率相同，系统响应将随时间无限增加。     

The fundamental solution of Mindlin plates resting on an elastic foundation in the Laplace domain and its applications

The aim of this study is to investigate the method of fundamental solution (MFS) applied to a shear deformable plate (Reissner/Mindlin’s theories) resting on the elastic foundation under either a static or a dynamic load. The complete expressions for internal point kernels, i.e. fundamental solutions by the boundary element method, for the Mindlin plate theory are derived in the Laplace transform domain for the first time. On employing the MFS the boundary conditions are satisfied at collocation points by applying point forces at source points outside the domain. All variables in the time domain can be obtained by Durbin’s Laplace transform inversion method. Numerical examples are presented to demonstrate the accuracy of the MFS and comparisons are made with other numerical solutions. In addition, the sensitivity and convergence of the method are discussed for a static problem. The proposed MFS is shown to be simple to implement and gives satisfactory results for shear deformable plates under static and dynamic loads.     

Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multi- term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ- ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.     

非规则区域中拉普拉斯方程的有限分析5点格式

建立了非规则区域的有限分析5点格式,增加了有限分析法对不规则边界的适应性。应用所提出的方法对水利工程中常见的有压和无压流动进行了计算,与实验和前人的计算结果相比较,本文的方法都能得到较为满意的结果。本文的计算格式也可以应用到其他非规则区域的计算中。     

Exact solutions of multi-term fractional difusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional difusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial diferential equations are converted into time-fractional ordinary diferential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-difusion problems are given to validate the proposed analytical method.