On the comparison of the solutions obtained by using two different transform methods for the second problem of Stokes for Newtonian fluids

The unsteady flow over a plane wall which is initially at rest and the plate begins suddenly to oscillate in own plane is considered. The solution subject to the boundary and initial conditions is obtained by applying to the governing equation the Laplace transform method or Fourier transform method. A comparison of the solutions obtained by two transform methods for flow considered is given. It is shown that the solution obtained by the Laplace transform method or Fourier transform method is the sum of the steady-state and the transient parts. The transient parts are found in terms of definite integrands whose integrals are oscillatory functions. Therefore, the transient parts are expressed in terms of the tabulated functions.     

无穷域势流问题的有限元／差分线法混合求解

提出了一种将有限元和差分线法相结合求解无穷域势流问题的算法。用两同心圆将求解域划分为存在重叠的有限和无限两个区域,在有限和无限域上分别用有限元和差分线法求解Laplace方程边值问题。用差分线法推导出的关系式修正有限元方程,求解该方程组从而得到原问题的解。本算法将求解无穷域问题转化为代数特征值问题和有限域内线性方程组的...     

Dynamic analysis of functionally graded plates using the hybrid Fourier-Laplace transform under thermomechanical loading

In this study, the analytical solution is presented for dynamic response of a simply supported functionally graded rectangular plate subjected to a lateral thermomechanical loading. The first-order and third-order shear deformation theories and the hybrid Fourier-Laplace transform method are used. The material properties of the plate, except Poisson’s ratio, are assumed to be graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The plate is subjected to a heat flux on the bottom surface and convection on the upper surface. A third-order polynomial temperature profile is considered across the plate thickness with unknown constants. The constants are obtained by substituting the profile into the energy equation and applying the Galerkin method. The obtained temperature profile is considered along with the equations of motion. The governing partial differential equations are solved using the finite Fourier transformation method. Using the Laplace transform, the unknown variables are obtained in the Laplace domain. Applying the analytical Laplace inverse method, the solution in the time domain is derived. The computed results for static, free vibration, and dynamic problems are presented for different power law indices for a plate with simply supported boundary conditions. The results are validated with the known data reported in the literature. Furthermore, the results calculated by the analytical Laplace inversion method are compared with those obtained by the numerical Newmark method.     

Time-fractional radial diffusion in a sphere

The radial diffusion in a sphere of radius R is described using time-fractional diffusion equation. The Caputo fractional derivative of the order 0<α<2 is used. The Laplace and finite sin-Fourier transforms are employed. The solution is written in terms of the Mittag–Leffler functions. For the first and second time-derivative terms, the obtained solutions reduce to the solutions of the ordinary diffusion and wave equations. Several examples of signaling, source and Cauchy problems are presented. Numerical results are illustrated graphically.     

Laplace transform-boundary element coupling method for viscous fluid-structure impact analysis

Based on linearized 2-D Navier-Stokes equation, a Laplace transform-boundary element coupling method for viscous fluid-structure impact analysis is proposed. Under assumption of incompressibility for the fluid, the corresponding equivalent boundary integral equation in terms of the potential function and stream function is first established by Lamb's transform in the Laplace transform domain. It enables us to solve impact water problems in frequency domain by the boundary element method, in which the effect of viscous flow on the dynamic response can be taken into account. Then a complete solution of the problem under consideration in time domain is obtained by means of Durbin's formulas for the numerical inversion of the Laplace transform. Finally, a practical example is given to confirm the validity of the present method. Project supported by the National Defence Foundation of Science & Technology of China (No. J14. 8. 1. JW0515).     

弹性动力学轴对称问题的理论解

本文给出了弹性动力学轴对称问题基本方程的一种理论解。它由满足非齐次边界条件的准静态解和满足齐次边界条件的动态解的叠加构成。在求得准静态解后,代入基本方程,得到动态解所需满足的非齐次方程。由相应的齐次方程的特征值问题,定义了有限Hankel变换。通过这种变换及Laplace变换,求得动态解,从而得到了一个完整的理论解。文中通过对一个实例求解,表明该方法求解过程简便,实用,求解结果精确。     

A theoretical solution for axially symmetric problems in elastodynamics

This paper presents a theoretical solution for the basic equation of axisymmetric problems in elastodynamics. The solution is composed of a quasi-static solution which satisfies inhomogeneous boundary conditions and a dynamic solution which satisfies homogeneous boundary conditions. After the quasi-static solution has been obtained an inhomogeneous equation for dynamic solution is found from the basic equation. By making use of eigenvalue problem of a corresponding homogeneous equation, a finite Hankel transform is defined. A dynamic solution satisfying homogeneous boundary conditions is obtained by means of the finite Hankel transform and Laplace transform. Thus, an exact solution is obtained. Through an example of hollow cylinders under dynamic load, it is seen that the method, and the process of computing are simple, effective and accurate.     

Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain

A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors.     

A new analytical solution to axisymmetric Blot's consolidation of a finite soil layer

A new analytical method is presented to study the axisymmetric Blot's consolidation of a finite soil layer. Starting from the governing equations of axisymmetric Blot's consolidation, and based on the property of Laplace transform, the relation of basic variables for a point of a finite soil layer is established between the ground surface (z= 0) and the depth z in the Laplace and Hankel transform domains. Combined with the boundary conditions of the finite soil layer, the analytical solution of any point in the transform domain can be obtained. The actual solution in the physical domain can be obtained by inverse Laplace and Hankel transforms. A numerical analysis for the axisymmetric consolidation of a finite soil layer is carried out.     

A series solution for the effective properties of incompressible viscoelastic media

This paper presents a series solution for the homogenization problem of a linear viscoelastic periodic incompressible composite. The method uses the Laplace transform and the correspondence principle which are combined with the classical expansion along Neumann series of the solution of the periodic elasticity problem in Fourier space. The terms of the Neumann series appear as decoupled, containing geometry dependent terms and viscoelastic properties dependent terms which are polynomial fractions whose inverse Laplace transforms are provided explicitly.     

求解动荷载作用下多层粘弹性半空间轴对称问题的精确刚度矩阵法

利用Laplace—Hankel联合积分变换，推导出了单层粘弹性半空间轴对称问题在动荷载作用下，层间完全接触情况的刚度矩阵，然后按传统的有限元方法组成总体刚度矩阵。通过求解由总体刚度矩阵所构成的代数方程就可解出动荷载作用下多层粘弹性半空间轴对称问题的矩阵。由于刚度矩阵的元素中只含有负指数项，计算时不会出现溢出的现象。本文还成功地应用了Durbin的Laplace逆变换的数值方法，求解出了多层粘弹性体的时域解。最后，文中还给出了路面动弯沉的计算结果与实测结果的对比来证明推导结果的准确性。     

Thermal stresses in a viscoelastic three-phase composite cylinder

A thermoviscoelastic analysis of a three-phase composite cylinder is presented in this work. The solutions of the heat conduction and thermoelastic problem for three dissimilar media are derived based on the method of analytic continuation associated with the alternation technique. A rapidly convergent series solution for both the temperature and stress field, which is expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. According to the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real time solution can then be solved numerically by taking inverse Laplace transform. Finally, some typical examples of interfacial stresses are discussed.     

自然单元法研究进展

自然单元法是一种基于Voronoi图和Delaunay三角化几何结构,以自然邻点插值为试函数的一种新型数值方法.其既具有无网格方法和经典有限元方法的优点,又克服了两者的一些缺陷,是一种发展前景广阔的求解微分方程的数值方法.自然单元法的形函数满足插值性质,可以像有限元法一样直接施加本质边界条件,不存在基于移动最小二乘拟合的无网格方法不能直接施加本质边界条件的难题.由于自然单元法是无网格方法,可以方便处理有限元方法较难处理的一些问题,例如移动边界和大变形等问题.自然单元法与其他数值方法的最根本区别于其插值格式的不同.将自然邻点插值用于Galerkin过程,就得到基于Voronoi结构的自然单元Galerkin法.自然邻点插值有自然邻点Sibson插值和Laplace插值(非Sibson插值)两种.Laplace插值比Sibson插值在计算上要简单的多,并且不论对凸的或非凸的区域都能精确施加本质边界条件.以Laplace插值为试函数的自然单元法在数值实施上比以Sibson插值为试函数的自然单元法简单.本文对基于Voronoi结构的自然邻点插值和自然单元法的基本思想作了介绍,综述了国内外关于自然单元法的研究成果,总结了自然单元法的优点和尚需解决的问题.      

High-order boundary conditions for the problems of Laplace equation in infinite region and their application

The high-order boundary conditions for the problems of Laplace equation in infiniteregion have been developed.The improvement in accuracy for numerical solution isachieved by imposing the high-order boundary conditions on the exterior boundary of areduced finite region in which the numerical method is used.So both the computing effortsand the required storage in computer are reduced.The numerical examples show that thelst-order boundary condition approaches to the exact boundary condition and it is clearlysuperior to the traditional boundary condition and the2nd-order boundary condition.     

Computer-aided fringe-pattern analyzer—A case of photoelastic fringe

A real-time “Computer-Aided FRinge-pattern AN-alyzer” (CAFRAN) system for the analysis of photoelastic fringe is developed. The pattern-analyzing process which utilizes image analysis or pattern-recognition technique and discretized-computation process in terms of finite-element method play an important role in this system. The former makes it possible to readout the fringe position and the fringe order of both isochromatic and isoclinic informations, and the latter analyzes the sum of the principal stresses which constitutes the Laplace field. Both results are combined to give the distributions of stress components over the domain under consideration. As far as the results of classical problems solved by this technique are concerned, the CAFRAN is promising in reading the fractional fringe order and encouraging the use of this technique to such problems as stress concentration.     

A finite element analysis of a free surface drainage problem of two immiscible fluids

A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two-dimensional model and solved numerically. This problem is a transient two-phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time-stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.     

A new analytical solution to axisymmetric Biot＇s consolidation of a finite soil layer

A new analytical method is presented to study the axisymmetric Biot＇s consolidation of a finite soil layer. Starting from the governing equations of axisymmetric Blot＇s consolidation, and based on the property of Laplace transform, the relation of basic variables for a point of a finite soil layer is established between the ground surface （z= 0） and the depth z in the Laplace and Hankel transform domains. Combined with the boundary conditions of the finite soil layer, the analytical solution of any point in the transform domain can be obtained. The actual solution in the physical domain can be obtained by inverse Laplace and Hankel transforms. A numerical analysis for the axisymmetric consolidation of a finite soil layer is carried out.     

Higher order Boussinesq-type equations for water waves on uneven bottom

Higher order Boussinesq-type equations for wave propagation over variable bathymetry were derived. The time dependent free surface boundary conditions were used to compute the change of the free surface in time domain. The free surface velocities and the bottom velocities were connected by the exact solution of the Laplace equation. Taking the velocities on half relative water depth as the fundamental unknowns, terms relating to the gradient of the water depth were retained in the inverse series expansion of the exact solution, with which the problem was closed. With enhancements of the finite order Taylor expansion for the velocity field, the application range of the present model was extended to the slope bottom which is not so mild. For linear properties, some validation computations of linear shoaling and Booij' s tests were carried out. The problems of wave-current interactions were also studied numerically to test the performance of the enhanced Boussinesq equations associated with the effect of currents. All these computational results confirm perfectly to the theoretical solution as well as other numerical solutions of the full potential problem available.     

THE MIXED STATE HAMILTONIAN DYNAMIC ELEMENT AND A SEMI-ANALYTICAL SOLUTION FOR THE ANALYSIS OF THICK LAMINATED COMPOSITE PLATES

Through introducing the Laplace transformation in the time direction,the mixed state Hamilton canonical equation and a semi-analytical solution arepresented for analyzing the dynamic response of laminated composite plates.Thismethod accounts for the separation of variables,the finite element discretization can beemployed in the plane of laminar,and the exact solution in the thickness direction isderived by the state space control method.To apply the transfer matrix method,therelational expression at the top and bottom surface is established.So the generalsolution in transformation space is deduced by the spot method.By the application ofinversion of Laplace transformation,the transient displacements and stresses can bederived.     

Viscoelastic layer under the action of a moving load

Recently, problems concerning the dynamic behavior of imperfect continuous media under various types of actions have been widely investigated. The method of Laplace transformation is very convenient for describing physical processes concerning unsteady phenomena. In viscoelastic media two complications are added: the representation of the properties of a medium depending on time, and the inversion of the obtained solutions containing this additional complication. Certain approximate methods of inversion in the analysis of viscoelastic stresses are discussed in [1]. In [2, 3] a discussion is given for an effective method of constructing the solution of unsteady problems for finite and for infinite imperfect media using auxiliary functions, and a solution is presented for a half-space. Making use of the idea of the inversion of transforms, discussed in [4], in [5] a solution is obtained and a complete picture is presented for the dynamics of the variation of the stress field in a viscoelastic half-space. In the present study we consider the action of a normal moving load that is suddenly applied to the free surface of a viscoelastic layer. By Laplace and Fourier integral transformations we obtain a solution in the form of a uniformly converging series based on longitudinal and transverse waves reflected in the layer. By means of inverting the transforms by the method discussed in [4, 5], we obtain an exact solution for the stress field in the medium under investigation. We consider the special case of a viscoelastic medium of Boltzmann type, for which we obtain a numerical realization of the solution on a digital computer.