Application of boundary integral equations for analyzing the dynamics of elastic,viscoelastic, and poroelastic bodies

Two approaches (classical and nonclassical) of the boundary integral equation method for solving three-dimensional dynamical boundary value problems of elasticity, viscoelasticity, and poroelasticity are considered. The boundary integral equation model is used for porous materials. The Kelvin–Voigt model and the weakly singular hereditary Abel kernel are used to describe the viscoelastic properties. Both approaches permit solving the dynamic problems exactly not only in the isotropic but also in the anisotropic case. The boundary integral equation solution scheme is constructed on the basis of the boundary element technique. The numerical results obtained by the classical and nonclassical approaches are compared.     

To the solution of the thermoelasticity problem for conical shells

We consider the stress-strain state of thin conical shells in the case of arbitary distribution of the temperature field over the shell. We obtain equations of the general theory based on the classical Kirchhoff-Love hypotheses alone. But since these equations are very complicated, attempts to construct exact solutions by analytic methods encounter considerable or insurmountable difficulties. Therefore, the present paper deals with boundary value problems posed for simplified differential equations. The total stress-strain state is constructed by “gluing” together the solutions of these equations. Such an approach (the asymptotic synthesis method) turns out to be efficient in studying not only shells of positive and zero curvature [1, 2] and cylindrical shells [3] but also conical shells [4, 5]. Here we illustrate it by an example of an arbitrary temperature field, and the problem is reduced to solving differential equations with polynomial coefficients and with right-hand side containing the Heaviside function, the delta function, and their derivatives.     

An exact axisymmetric solution for a simply supported piezoelectric cylindrical shell

Summary Three-dimensional axisymmetric solution is presented for a simply supported piezoelectric cylindrical shell. The variables are expanded in Fourier series to satisfy the boundary conditions at the ends. The solution of the governing differential equations with variable coefficients is constructed as a product of an exponential function and a power series. The coefficients of terms of all degrees in the governing equations are set to zero, yielding a characteristic equation for the exponent and recursive relations for the coefficients of the power series. Results are presented illustrating the effect of thickness parameter of the shell. An inverse problem of inferring the applied temperature from the measured potential difference has been solved. Accepted for publication 26 July 1996     

On the Solution of Mixed Problems in Linear Anti-plane Piezoelectricity

In the present paper we consider interior and exterior mixed boundary value problems of anti-plane shear in the static theory of linear piezoelectricity. Using the boundary integral equation method we reduce the problems to systems of singular integral equations with discontinuous coefficients to which the classical Nöether’s theorems on existence of the solution can be applied. This allows us to establish well-posedness results and to obtain integral solutions of the corresponding mixed boundary value problems for a rather general class of piezoelectric materials. Mathematics Subject Classifications (2000) 45E05, 45F15, 74F15.     

Dimension Reduction in the Context of Structured Deformations

In the present paper the basic boundary value problems (BVPs) of the full coupled linear theory of elasticity for triple porosity materials are investigated by means of the potential method (boundary integral equation method) and some basic results of the classical theory of elasticity are generalized. In particular, the Green’s identities and the formula of Somigliana type integral representation of regular vector and regular (classical) solutions are presented. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The uniqueness theorems for classical solutions of the internal and external BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method and the theory of singular integral equations.     

A class of variational difference schemes for a singular perturbation problem

In this paper, a singularly perturbed boundary value problem for second order self-adjoint ordinary differential equation is discussed. A class of variational difference schemes is constructed by the finite element method. Uniform convergence about small parameter is proved under a weaker smooth condition with respect to the coefficients of the equation. The schemes studied in refs. [1], [3], [4] and [5] belong to the class.     

Hermite梯度重构核近似配点法及其在功能梯度材料板的静力分析中的应用

由于直接配点法在求解边值问题时边界上的求解精度较低,本文提出了Hermite梯度重构核近似配点法（HGCM）来改进边界求解精度。重构核近似是无网格法中一种常用的近似函数,但是其在求解高阶导数时格式复杂且非常耗时。HGCM采用梯度重构核近似构建形函数的任意高阶导数,提高了计算效率;通过Hermite配点法构建离散方程,提高了边界求解精度。这种方法在求解对应变系数四阶偏微分方程的功能梯度材料板的静力问题时精度高,计算效率高,并可进一步推广应用于高阶偏微分方程描述的边值问题。     

求解Helmholtz方程基于核重构思想的最小二乘配点法

基于核重构思想构造近似函数，将配点法和最小二乘原理相结合对微分方程进行离散， 建立了Helmholtz方程的最小二乘配点格式，并分别研究了Helmholtz方程的波传播问题和 边界层问题. 通过数值算例可以发现，给出的数值计算结果非常接近于精确解，计算精度明显高于SPH 法的数值结果，且随着节点数目的增加，其精确度越来越高，具有良好的收敛性.     

The second initial-boundary value problem for a quasilinear parabolic equations with nonlinear boundary conditions

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆａｓｔｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｏｆｄｉｖｅｒｇｅｎｃｅｆｏｒｍａｓｕｔ =(ａ(ｕ)ｕｘ) ｘ ｂ(ｕ) ｘ ｃ(ｕ)　　 (ａ( 0 ) = ∞ ) ( 1 )ｈａｓｉｍｐｏｒｔａｎｔｐｈｙｓｉｃａｌｂａｃｋｇｒｏｕｎｄ ,ｓｕｃｈａｓ [1 ] .Ｉｎｒｅｃｅｎｔｙｅａｒｓ ,ｓｏｍｅｒｅｓｕｌｔｓａｂｏｕｔ ( 1 )ｈａｖｅｂｅｅｎｏｂｔａｉｎｅｄ .Ｆｏｒｅｘａｍｐｌｅ ,[2 ] ,[3]ｒｅｓｐｅｃｔｉｖｅｌｙｄｉｓｃｕｓｓｅｄｔｈｅＣａｕｃｈｙｐｒｏｂｌｅｍｓｆｏｒｅｑｕａｔｉｏｎ( 1 )ａｎｄｕｔ=(…     

两种滑动轴承油膜压力计算方法的比较

对常用的两种计算滑动轴承油膜力的方法进行了比较。一种是在迭代求解雷诺方程过程中,如果出现角压力,则用零压力来替代的方法;另一种是根据质量守恒理论（ＪＦ０理论）,由Ｅｌｒｏｄ等人提出的空穴算法。指出用负压力充零的方法代替Ｅｌｒｏｌｄ空穴算法,误差不大,但却可以克服Ｅｌｒｏｄ空穴算法不稳定和振荡的缺点,大大节约计算时间。     

Analytic Solution of an Inhomogeneous Kinetic Equation in the Problem of Second-Order Thermal Creep

An analytic solution of the problem of second-order thermal creep is obtained. A method for solving the half-space boundary value problem for an inhomogeneous linearized kinetic BGK equation forms the basis of the solution. The general solution of the input equation is constructed in the form of an expansion of the corresponding characteristic equation in terms of the eigenfunctions. Substitution of the solution in the boundary conditions leads to a Riemann boundary value problem. The unknown thermal creep velocity is found from the condition of solvability of the boundary value problem. The numerical analysis performed confirms the existence of negative thermophoresis (in the direction of the temperature gradient) for high-conductivity aerosol particles at low Knudsen numbers.     

On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

Generalized form of boundary value problems method for material modeled as micro-polar media subjecting to the thermo-mechanical interaction

In this paper, the periodic structure material is modeled as the continuum homogeneous micro-polar media subjecting to thermo-mechanical interaction. Meanwhile, a series of equivalent quantities such as the equivalent stress, couple stress, displacement gradient and torsion tensor were defined by the integral forms of the boundary values of the external surface force, moment, displacement and the angular displacement, and were proved to satisfy the equivalence conditions of virtual work. Based on above works, the displacement boundary value problem was established to deduce the equivalent constitutive equation. Assume the representative volume element is composed of the spatial cross-framework, and applying the boundary value problem of displacement on frame structures, the equivalent elastic coefficients, temperature coefficients of equivalent stress and the temperature gradient coefficients of equivalent couple stress are deduced. In addition, themethod can also be extended to the stress boundary value problem to deduce the equivalent constitutive equation. The calculations indicate that the equivalent result can be obtained from the two kinds of boundary value problems.     

Asymptotic Behavior of the Principal Eigenvalue for Cooperative Elliptic Systems and Applications

The asymptotic behavior of the principal eigenvalue for general linear cooperative elliptic systems with small diffusion rates is determined. As an application, we show that if a cooperative system of ordinary differential equations has a unique positive equilibrium which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann boundary condition or the Robin boundary condition also has a unique positive steady state which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. Moreover, as the diffusion coefficients approach zero, the positive steady state of the reaction-diffusion system converges uniformly to the equilibrium of the corresponding kinetic system.     

On constructing a Green’s function for a semi-infinite beam with boundary damping

The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained.     

EFFICIENT ANALYTIC SERIES SOLUTIONS FOR TWO–DIMENSIONAL POTENTIAL FLOW PROBLEMS

The solution of Laplace's equation for a wide range of spatial domains and boundary conditions is a valuable asset in the study of potential theory. Recently, classical analytic series techniques based on separation of variables have been modified to solve Laplace's equation with both irregular and free boundaries. Computationally the free boundary problem is reduced to an iterative sequence of curve-fitting exercises. At each iteration the series coefficients for a known boundary problem are evaluated numerically. In this paper a new interpolation approach is presented for the estimation of the series coefficients. It has the advantages of providing a conceptually simpler view of the series technique and of estimating the series coefficients significantly faster than alternative approaches. Owing to the choice of basis functions in the truncated series solution, rigorous estimates of the error in the approximation are immediately available. A free boundary problem from steady hillside seepage with irregular boundaries will be used to illustrate the new technique.     

三维位势问题的梯度边界积分方程的新解法

三维位势问题的边界元分析中,关于坐标变量的边界位势梯度的计算是一个困难的问题. 已有一些方法着手解决这个问题,然而,这些方法需要复杂的理论推导和大量的数值计算. 本文提出求解一般边界位势梯度边界积分方程的辅助边值问题法. 该方法构造了与原边界值问题具有相同解域的辅助边值问题,该辅助边值问题具有已知解,因此通过求解此辅助边值问题,可获得梯度边界积分方程对应的系统矩阵,然后将此系统矩阵应用于求解原边值问题,求解过程非常简单,只需求解一个线性系统即可获得原边值问题的解. 值得注意的是,在求解原边值问题时,不再需要重新计算系统矩阵,因此辅助边值问题法的效率并不很差. 辅助边值问题法避免了强奇异积分的计算,具有数学理论简单、程序设计容易、计算精度高等优点,为坐标变量梯度边界积分方程的求解提供了一个新的途径. 3个标准的数值算例验证了方法的有效性.      

Equation of longitudinal vibrations of a beam with coefficients periodically depending on the coordinate

The equation of small longitudinal vibrations of a beam is considered for the case in which the coefficients are periodic and have “gaps” (sharp decreases in the value) at some point in the space inside the periodicity interval. We suggest to describe the process locally in the domain of minimum of the coefficients by an equation of the boundary layer approximation. Attention is mainly paid to the analysis of this equation. A time hierarchy of simplified models for describing this process is established.     

Existence of bounded solutions on the real line for Liénard system

The existence of monotone and non-monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Liénard system and the system given by symmetric transformation or quasi-symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi-stable, the existence of infinitely many monotone solusion is obtained.     

Mathematical model of heat transfer in a fluid with account for its relaxation properties

Using the terms that take account for the temporal and spatial nonlocality (time variation of the heat flux and the temperature gradient) in the formula of Fourier’s law for the heat flux a differential equation for a fluid in motion is derived that contains the second time derivative and themixed derivative with respect to the spatial and temporal variables. Numerical solution of the problem of heat transfer in the laminar fluid flow in a plane channel demonstrates that, in view of the lag in the time variation of the heat flux from zero to a certain maximum value, the boundary condition of the first kind (thermal shock) cannot be instantaneously realized. The process of its stabilization on the wall is characterized by a certain time interval, whose duration is determined by the relaxation properties of the fluid. At large values of the dimensionless coefficients of the heat flux relaxation and the temperature gradient the boundary condition of the first kind can be realized only as the steady state is attainted, as Fo→∞. In this case, the flow does not contain temperature jumps and negative temperature values.