New numerical method for volterra integral equation of the second kind in piezoelastic dynamic problems

The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.     

Solution of integral equations for plane elastostatical problems with discontinuously prescribed boundary values

Engineers are often confronted with boundary value problems of plane elastostatics where the boundary tractions or displacements or their derivatives have a jump. The discontinuities represent by no means impediments to treating the problems economically with the aid of integral equations. However, it is necessary to know the structure of the solutions before starting numerical calculations.In this paper singular and regular integral equations of the second and of the first kind are investigated by methods which are based mainly on mechanical ideas. The essential terms of the solutions are determined for boundary values with a jump or a jumping derivative. The solutions contain both discontinuous or discontinuously differentiable terms and also logarithmically diverging terms.Particular attention is paid to the most frequently applied integral equation of the indirect method for the plane problem with prescribed tractions. The solutions of this equation for elastic slices loaded by concentrated forces and moments are deduced as special cases of the general results.(For an extensive survey of this paper: see Chapter 1.)

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Zusammenfassung Ingenieure werden oft mit Randwertproblemen der ebenen Elastostatik konfrontiert, bei denen die Randspannungen oder Randverschiebungen oder deren Ableitungen Sprünge aufweisen. Die Unstetigkeiten stellen keine grundsätzlichen Hindernisse dar, die Probleme auf ökonomische Weise mit Hilfe von Integralgleichungen zu behandeln. Jedoch ist es dazu unabdingbar, die Struktur der Lösungen zu kennen, bevor man mit den numerischen Rechnungen anfängt.In diesem Aufsatz werden singuläre und reguläre Integralgleichungen zweiter und erster Art mit Methoden untersucht, die hauptsächlich auf mechanischen Gesichtspunkten basieren. Die wesentlichen Terme der Lösungen werden für Randwerte mit einem Sprung oder einer unstetigen Ableitung bestimmt. Die Lösungen enthalten sowohl unstetige als auch unstetig differenzierbare Summanden und darüber hinaus auch logarithmisch divergierende Terme.Besondere Aufmerksamkeit wird der am häufigsten benutzten Integralgleichung der indirekten Methode für das ebene Problem mit vorgeschriebenen Spannungen geschenkt. Die Lösungen dieser Integralgleichung werden für elastische, durch Einzelkräfte und Einzelmomente belastete Scheiben als Spezialfälle der allgemeinen Resultate hergeleitet.(In Kapitel 1 befindet sich ein ausführlicher Überblick über den Aufsatz.)

     

General stress analysis method by means of integral equations and boundary elements

Summary This paper concerns an integral equation method for the numerical solution of the linear problems of elasticity of the homogeneous continuum. The mathematical form adopted is sufficiently general to refer to both plane and three dimensional analysis as well as to analysis of plates and specialized axisymmetric cases.Having indirectly deduced the integral equations, the resolving procedure is based on variational principles and uses a convenient discretization of the boundary through the use of finite elements.Particular attention is given to the definition of the asymptotic conditions of the elementary displacement field to theoretically ensure a priori the conditioning of the resolving algebraic equations without any restriction as regards the nature of the boundary and the conditions thereby imposed.Efficiency, praticality and flexibility of use are characteristics particular of this method and have already been widely tested by the author in varying applications. Further information on the above can be found in the bibliography cited.

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Sommario Nella presente memoria si delinea un metodo di calcolo fondato sulle equazioni integrali per la soluzione numerica dei problemi lineari di elasticità del continuo omogeneo. La veste matematica adottata é sufficientemente generale perché possa riferirsi all'analisi sia dello stato triplo sia dello stato piano di deformazione o di tensione, come pure all'analisi delle lastre e dei casi specializzati assialsimmetrici.Dedotte per via indiretta le equazioni integrali, il procedimento risolutivo è fondato su principi variazionali e si avvale di una conveniente discretizzazione del contorno mediante elementi finiti.Particolare attenzione è rivolta a definire le condizioni asintotiche dei campi elementari di spostamento al fine di assicurare teoricamente a priori il condizionamento delle equazioni algebriche risolventi, senza alcuna restrizione per quanto riguarda la natura del contorno e le condizioni ivi imposte.L'efficienza, la praticità e la flessibilità d'impiego sono caratteristiche peculiari del metodo, per altro già ampiamente collaudato dall'autore in diverse applicazioni di cui possono trovarsi notizie nella bibliografia citata.

     

热弹性平面问题的规则化边界积分方程

对于热弹性平面问题,过去广泛集中在直接变量边界元法研究,本文研究间接变量规则化边界元法,建立了间接变量规则化边界积分方程。和直接边界元法相比,间接法具有降低密度函数的连续性要求、位移梯度方程中的热载荷体积分具有较弱奇异性等优点。数值实施中,用精确单元描述边界几何,不连续插值函数逼近边界量。算例表明,本文方法效率高,所得数值结果与精确解相当吻合。     

Some remarks about the formulation of three-dimensional thermoelastoplastic problems by integral equations

The extended form of the identity of Somigliana, giving the displacement rate in terms of the traction and the displacement rate on the boundary, involves plastic and thermal strain rate. Because of strong singularity of some kernel, die gradient of the displacement rate may be strictly evaluated by an integro-differential expression. The paper shows how to derive the integral form of the gradient rate which corrects some erroneous expressions reported in the literature.     

Finite elements for exterior problems using integral equations

We present some integral methods for exterior problems for the Laplace equation. Then we give finite element approximations for these equations and some errors estimates. Finally, we indicate how these integral equations can be coupled with a usual finite element method on a bounded domain to solve an exterior non-linear problem which is linear far away.     

从动能定理到第二类拉格朗日方程

第二类拉格朗日方程是处理质系(尤其是多自由度、非自由质系)动力学问题的重要理论基础,被列为理论力学多学时教学大纲的基本要求. 第二类拉格朗日方程的导出过程涉及较多的数学变换,如何揭示这些抽象数学变换背后的物理意义成为教学的一个难点.借鉴学生所熟知的动能定理,介绍一种在物理意义指导下逐步进行数学变换的讲授方法.     

Solution of filtration problems in cracked and porous media by integral methods

The integral-equation methods first proposed for solving filtration problems by Barenblatt [1] were subsequently used to good effect when studying cracked and porous media [2, 3]. However, the realization of the integral-equation method in the latter two cases [2, 3] contradicted the idea of interpenetrating continua which was used as a basis for the model of filtration in cracked and porous media [4]. This contradiction was first noted by Nikolaevskii et al. [5], who indicated the necessity of introducing two zones of liquid-pres sure variation corresponding to the propagation of perturbation in porous blocks and cracks, respectively [6]. The formal use of the method of integral equations in [2, 3] had the effect that the size of the perturbation zone differed from zero at the initial instant of time. In this paper we shall demonstrate that the introduction of two such zones not only eliminates this shortcoming but also easily generalizes the results to the case of gas filtration.     

On the stability of integral equations of the first kind with logarithmic kernels

This paper discusses the stability of the Galerkin method for a class of boundary integral equations of the first kind. These integral equations arise in acoustics, elasticity, and hydrodynamics, and the kernels of the principal parts of the corresponding integral operators all have logarithmic singularities. It is shown that an optimal choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence of the approximate solutions. The results here are consistent with those obtained by the Tikhonov regularization procedure.     

NOVEL REGULARIZED BOUNDARY INTEGRAL EQUATIONS FOR POTENTIAL PLANE PROBLEMS

The universal practices have been centralizing on the research of regulariza-tion to the direct boundary integal equations (DBIEs). The character is elimination of singularities by using the simple solutions. However, up to now the research of regular ization to the first kind integral equations for plane potential problems has never been found in previous literatures. The presentation is mainly devoted to the research on the regularization of the singular boundary integral equations with indirect unknowns. A novel view and idea is presented herein, in which the regularized boundary integral equations with indirect unknowns without including the Cauchy principal value (CPV) and Hadamard-finite-part (HFP) integrals are established for the plane potential problems. With some numerical results, it is shown that the better accuracy and higher efficiency, especially on the boundary, can be achieved by the present system.     

Some inverse problems in rheology leading to integral equations

Two inverse problems of the integral type, which are of the general importance to rheology, are discussed. The first of them is the possibility of calculating the molecular weight distribution (MWD) from the flow curve and the second one is the interrelation between creep and relaxation functions. It was shown that the first problem is incorrect and any solution is unstable in respect to minor experimental errors. It means that the general solution of this problem is impossible in principle and only estimations of the width of unimodal MWD can be received from the curvature of the flow curve. The possibility of the correct calculation of the creep function exists in case the relaxation curve being approximated by the sum of exponential members. But the approximation of the relaxation curve within the experimental accuracy is the necessary, but not sufficient condition for the correct solution of this integral inverse problem, because not every mode of approximation leads to the satisfactory prediction of creep function.Delivered as the Courtaulds Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Boundary integral equations in quasisteady problems of capillary fluid mechanics

Summary In this paper the motion of a viscous multiphase liquid occupying all the space under the action of surface tension is considered. Applying the quasisteady approximation, i.e. when mass forces including inertia terms are neglected, the flow is completely defined by the location of an interface for which the abstract Cauchy problem with nonlocal normal velocity operator is formulated. In order to obtain this operator a standard auxiliary problem for the Stokes system is to be solved which can be reduced to the Fredholm boundary integral equations by means of the application of the hydrodynamic potentials of simple layers. The stability problem of a spherical drop drift is investigated as an illustration of this method.

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Sommario Si studia il moto di un liquido viscoso a più fasi che occupa l'intero spazio ed è soggetto all'azione della tensione superficiale. Applicando l'approssimazione quasistazionaria (ossia trascurando forze di massa e termini di inerzia) il flusso è determinato dalla posizione di un'interfaccia per la quale viene formulato un problema di Cauchy astratto con operatore di velocità normale non locale. Per la definizione di questo operatore viene risolto un problema ausiliario standard per il sistema di Stokes, che può essere ridotto ad un'equazione integrale di contorno del tipo Fredholm, facendo uso della rappresentazione in semplice strato del potenziale idrodinamico. Come illustrazione del metodo viene trattato il problema della stabilità di una goccia sferica.

     

Lane-Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere

We study a modified version of the Lane-Emden equation of the second kind modelling a thermal explosion in an infinite cylinder and a sphere. We first show that the solution to the relevant boundary value problem is bounded and that the solutions are monotone decreasing. The upper bound, the value of the solution at zero, can be approximated analytically in terms of the physical parameters. We obtain solutions to the boundary value problem, using both the Taylor series （which work well for weak nonlinearity） and the b-expansion method （valid for strong nonlinearity）. From here, we are able to deduce the qualitative behavior of the solution profiles with a change in any one of the physical parameters.     

Boundary value problems for second order delay differential equations

In this paper,using a fixed point principle and existence principle given in[1],westudy the boundary value problems for second order differential equations.Some newexistence results are obtained.     

Boundary integral equations for inverse problems in the elasticity theory

The paper is concerned with the application of the boundary integral equation method for the holomorphic vector to nonclassical problems in the static theory of elasticity. Problems are considered for the case when on part of a body conditions are overvalued, i.e. the vectors of displacements and loads are prescribed, and on the other part of the body conditions are unknown (so-called (u, p) problem). It is shown that the method proposed is efficient and can be applied to the problems of computer defectoscopy in stationary potential fields.     

Singular integral equations in Hilbert space applied to crack problems

Crack problems are solved by application of a system of singular integral equations in Hilbert space. The method consists of replacing the singular integral equation by a system of linear algebraic equations for the values of the unknown function at specially chosen points within the range of integration. Obtained is a solution for a nonsymmetric cross-shaped crack in an infinite and isotropic solid subjected to a constant pressure, the symmetric configuration being a special case.     

Solution of a dual integral equation for crack and indentation problems

Using the Sonine integrals, some dual integral equations with the Bessel kernels were reduced to a single integral equation. Then the closed form solutions of these dual integral equations were obtained. Based on the method, the anti-plane shear of an elastic layer was solved exactly.