Computation of flow of viscoelastic fluids by parameter differentiation

A technique combining the features of parameter differentiation and finite differences is presented to compute the flow of viscoelastic fluids. Two flow problems are considered: (i) three-dimensional flow near a stagnation point and (ii) axisymmetric flow due to stretching of a sheet. Both flows are characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. The exact numerical solutions are obtained using the technique described in the paper. Also, the first-order perturbation solutions (in terms of the viscoelastic fluid parameter) are derived. A comparison of the results shows that the perturbation method is inadequate in predicting some of the vital characteristic features of the flows, which can possibly be revealed only by the exact numerical solution.     

Spectral properties of some problems in mechanics of strongly inhomogeneous media

In the present paper, we derive effective dynamic equations and analyze the spectral properties of strongly inhomogeneous media such as elastic porous materials saturated by a fluid and two-fluid mixtures (suspensions).We also study the structure of the natural vibration spectrum of bounded volumes for several effective models and the convergence rate of solutions of the original boundary value problems for two-phase media to the corresponding solutions of the effective (homogenized) boundary value problems.     

On the forward and backward in time problems in the Kelvin–Voigt thermoviscoelastic materials

This paper studies the uniqueness of solutions to the forward and backward in time boundary value problems associated with the Kelvin–Voigt viscoelastic model of the thermoelastic materials. For thermoviscoelastic materials with a center of symmetry, it is shown the uniqueness of solutions to the forward in time boundary value problems without any assumptions upon the thermoviscoelastic constitutive coefficients other than the symmetry properties and those induced by the dissipation inequality. While for the final boundary value problems two uniqueness theorems are presented: the first one is essentially based on the assumption that the specific heat is of negative definite sign, while the second is established in the class of displacement–temperature variation fields whose dissipation energy has a temporal behavior lower than an appropriate growing exponential.     

Solution of a multidimensional impact deformation problem for an elastic half-space with curved boundary on the basis of a modified ray method

A generalization of the method for constructing approximate solutions of boundary value problems of impact deformation dynamics in the form of ray expansions for two-dimensional plane deformation problems is presented. For each shock wave, the solution near its front is determined on the basis of ray coordinates consistent with this wave. The nonlinear divergence of curvilinear rays is taken into account. A mechanism of transformation from one ray coordinate system to another, which is crucially important in the ray method, is described. The developed technique is illustrated by solving the impact deformation problem for a half-space with boundary of nonzero curvature.     

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.     

The analysis of crack problems with non-local elasticity

IntroductionTheclassicalconhnuummechanicshasbeenusedtosolvemanyproblemsinmacrofracturemechanics,butencountersdifficulheswhentheeffectofITilcrocharacteristicdimensionshouldbetakenintoaccount.Thestressfieldverynearthecracktipisstillnotclear.Somephenomenaofshortcrackscannotbeexplained["']andsomemechanismoffracturehasnotbeensolvedyet.Thenon-localelashcitytheoryseemsattractivetotheseproblems.Thetheoryofnon-localelasticity,establishedanddevelopedbyEringenetal[3),connectstheclassicalcontinuummechan…     

The boundary expanding-contracting method

The boundary element method (KEM) which a used to solve the elastic problems has more advantages than other numerical methods. Especially, it can resolve rapidly varying internal stress and strain fields more accurately. However, it is of en fails in the region near the boundary because of the singularity of the solutions. Though we can increase the boundary meshes more and more, the solutions of stress on the boundary can’t be given directly; which has obstructed the applications of the HEM to some extent.In this paper we proposed the boudary expanding-contracting principle and the boundary expanding-contracting method (BECM) based on the principle. With this method, not only the solutions in the region near or on the boundary can be obtained directly, but the iterative processes can also he used conveniently to improve the accuracy of the solutions.     

Periodic and “slightly” periodic boundary value problems in elastostatics on bodies bounded in all but one direction

General properties of solutions to elastostatic boundary value problems in which some or all of the functions involved are periodic are studied with particular attention given to problems on bodies unbounded in one direction only. It is shown that, even though the displacement corresponding to a periodic strain may, in a very nontrivial sense, be nonperiodic, it does satisfy a semiperiodicity condition. In addition, a theorem of work and energy is derived for periodic strain states on bodies unbounded in only one direction. This formulation of the theorem of work and energy includes extra terms arising from the possible semiperiodicity of the displacement but only explicitly involves one component of the mean stress. This leads to a discussion of the uniqueness of periodic strain solutions to various boundary value problems. Conditions insuring uniqueness are obtained with the necessity of these conditions demonstrated by counter-examples. The degree to which uniqueness can fail is also studied and is shown to be limited.The next portion of the paper discusses the question of whether periodic boundary value problems must have, in some sense, periodic solutions. This leads naturally to the question of the uniqueness of solutions to boundary value problems which, in themselves, are not necessarily periodic but whose corresponding null boundary value problem is periodic. Positive results to both questions are obtained for several fairly broad classes of problems. Counter-examples are then cited to show the necessity of many of the assumptions used in deriving these results.     

A class of singular perturbation solutions to semilinear equations of fourth order

A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.     

改进的奇异边界法模拟三维位势问题

奇异边界法是与基本解法相对应的一种边界型无网格数值离散方法. 该方法提出了源点强度因子的概念, 克服了传统基本解方法中最复杂最头疼的虚拟边界问题.基于边界元法中处理奇异积分的数值处理技术, 导出了源点强度因子的解析表达式, 提出了改进的无网格奇异边界法, 并进一步将该方法应用于三维位势问题. 该方法消除了传统方法中样本点的选取, 在不增加计算量的前提下, 极大地提高了奇异边界法的计算精度与稳定性.      

一类线性Liénard方程周期边值问题解的存在唯一性的构造性证明

利用初值问题方法给出了一类线性Liénard方程周期边值问题解的存在唯一性的构造性证明,利用数值延拓方法,并给出了计算实例和一种大范围求解这类方程周期解的方法。     

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEMS FOR SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS ON INFINITE INTERVAL(Ⅱ)

In this paper the existence of solutions of the singularly perturbed boundary valueproblems on infinite interval for the second order nonlinear equation containing a smallparameterε>0 :is examined,whereα_i,βare constants,and i=0,1 .Moreover,asymptoticestimates of the solutions for the above problems are given.     

Boundary Value Problems of Plane Elasticity Involving Orientations of Displacements and Tractions

This paper presents unconventional formulations of boundary problems of plane elasticity formulated in terms of orientations of tractions and displacements on a closed contour separating internal and external domains as the boundary conditions. These are combined with the conditions of continuity of tractions or displacements across the boundary. Therefore the magnitudes of neither tractions nor displacements are assumed on the contour. Four boundary value problems for both external and internal domains are investigated by analyzing the solvability of the corresponding singular integral equations. It is shown that all considered problems can have non-unique solutions expressed as linear combinations of particular solutions and, hence, contain free arbitrary parameters, the number of which is finite and determined by the contour orientations of tractions and/or displacements     

BLOW－UP AND DIE－OUT OF SOLUTIONS OF NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS OF GENERALIZED NERVE CONDUCTION TYPE

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Invariant submodels of the Westervelt model with dissipation

We study three-dimensional Westervelt model of nonlinear hydroacoustics with dissipation. We received all its invariant submodels. With the help of invariant solutions, we explored some wave processes, specifying their physical meaning. The boundary value problems describing these processes are reduced to the nonlinear integro-differential equations. We established the existence and uniqueness of the solutions of these boundary value problems under some additional conditions. Also we considered the invariant solutions of rank 2 and 3. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters.     

On the approximate solution of elliptic,self adjoint boundary value problems

Summary Integral representations for the solutions to linear elliptic self adjoint boundary value problems are derived in terms of two functions which are generalisations of the single and double layer potentials used in the theory of harmonic functions. The generalised potentials are constructed in terms of a fundamental solution which is an approximation to the exact kernel of the boundary value problem in question. The representations so obtained are shown to provide a basis from which strict approximations to the solutions of boundary value problems can be developed. In particular the structure of the integral equation representing the given boundary value problem is precisely determined.     

求解非线性动力系统周期解的改进打靶法

针对有周期解的动力系统边值问题可以转化为初值问题这一特点，改进了周期解的打靶 法数值求解. 在计算边界条件代数方程关于待定初值参数导数的过程中利用前一次 Runge-Kutta方法计算得到的节点函数值并通过再次利用Runge-Kutta方法获得了该导数值. 用此方法求解了Duffing方程及非线性转子---轴承系统的周期解,用Floquet理论判断了 周期解的稳定性,与普通打靶法作了比较,验证了方法的有效性.     

悬臂半无限大板弯曲的边界积分公式

根据双调和方程边值问题的边界积分公式,求得了一般载荷作用下悬臂半无限大板的弯曲解,在此基础上求解了不同边界条件下几种半无限大板的弯曲问题.其解式收敛速度快、计算精度高,计算过程相对简单,并与相应有限元解进行了对比分析.     

On the boundary value problems for ordinary differential equations with turning points

In this paper,we consider the boundary value problems of the formsy″-f(x,ε)y′ g(x,ε)=0 (-a≤x≤b,0≤ε《1 )y(-a)=a,y(b)=βwhere f(x,0)has several and multiple zeros on the interval[-a,b].The conditions forexhibiting boundary and interior layers are given,and the corresponding asymptoticexpansions of solutions are constructed.