An integral equation method in dynamic fracture mechanics

This paper contains a study of the problems of crack propagation under static stresses and under transient stress waves. Within the assumption of linear elastic fracture mechanics, an integral-equation method has been developed for the analysis of these problems. The method has been applied to: (a) the determination of the stress intensity factor at a given loading and a crack-tip velocity and (b) the determination of the crack-tip motion under a given transient loading.     

Computational model of boundary integral equation in solid mechanics

In the first part of the paper, the computational model of boundary integral equation in solid mechanics is presented while in the second part the model is used in the solution of two problems of solid mechanics.     

An integral equation method for stress concentration problems in cylindrical shells

A method is described for determining the stresses in perforated cylindrical shells. The method is applicable to cases where several holes of arbitrary form are present. The problem is formulated as a system of four coupled integral equations together with a number of compatibility relations in integral form. A numerical procedure for solving the equations is also described and some simple applications of the method including the case of one elliptical hole with arbitrary orientation relative to the generators are presented.

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Zusammenfassung Eine Methode zur Berechnung der Spannungen in zylindrischen Schalen mit Löchern willkürlicher Form wird beschrieben. Das Problem ist als ein System von vier gekuppelten Integralgleichungen mit einer Anzahl von Kompatibilitätsbedingungen auf Integralform formuliert. Ein numerisches Verfahren zur Lösung der Gleichungen ist beschrieben und auf einfache Rechnungsbeispiele angewandt, darunter der Fall, dass ein elliptisches Loch vorhanden ist, dessen Orientierung bezüglich der Erzeuger willkürlich ist.

     

An integral equation approach to self-similar plane-elastodynamic crack problems

The elastodynamic problem of an expanding crack under homogeneous polynomialform loading was reduced to the solution of a Cauchy singular integral equation. In this manner the solution of the original problem can be obtained by using well-known numerical treatments available for Cauchy SIEs. The procedure was accomplished by means of the Busemann-Chaplygin similarity technique and complex variable methods. The analysis has been restricted to the subsonic case.     

On the regularization method of the first kind of Fredholm integral equation with a complex kernel and its application

The regularized integrodifferential equation for the first kind of Fredholm integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-demensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given. Project supported by the National Natural Science Foundation, of China     

Eigenfunction expansion method to the solution of simultaneous equations and its application in mechanics

The solution of a homogeneous vector-matrix differential equation, when the operator is of linear second order, has been established by the method of eigenfunction approach. The uniqueness of the solution has been established for both the cases when the roots of the characteristic equations are distinct and when they are repeated. Finally, the theory has been applied on two problems of mechanics and the results are compared with the existing literature.     

A pseudo-time integral method for non-isothermal viscoelastic flows and its application to extrusion simulation

A pseudo-time integral scheme based on a finite streamline element method is developed to combine variable temperature with viscoelasticity. A specific KBKZ integral model for isothermal flow is transformed to its non-isothermal version by introducing a pseudo-time and applying the Morland-Lee hypothesis. The coupling between momentum and energy equations is through the time-temperature shifting factor by which the pseudo-time is defined. The observer time and the pseudo-time are simultaneously calculated when tracing the strain history for the stress calculation in a non-homogeneous temperature field. Using this scheme, a full non-isothermal numerical simulation of some IUPAC extrusion experiments is carried out. Results show that while the temperature distribution near the die exit plane is an important factor controlling extrudate swell, either self-heating inside the die tube or external cooling on the free surface dominantly determines the temperature distribution near the die exit when the wall temperature is kept constant, depending on whether the Péclet number is large or small. The hot layer effect predicted by the inelastic swell mechanism is confirmed and well illustrated by the computation. Calculations with reasonable thermal boundary conditions further convince us that the isothermal assumption in our earlier numerical simulation is a good approximation in this particular case.     

Applications of the integral equation method to delay differential equations

In this paper, we compare two approaches for determining the amplitude equations; namely, the integral equation method and the method of multiple scales. To describe and compare the methods, we consider three examples: the parametric resonance of a Van der Pol oscillator under state feedback control with a time delay, the primary resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay, and the primary resonance together with 1:1 internal resonance of a two degree-of-freedom model. Using the integral equation method and the method of multiple scales, the amplitude equations are obtained. The stability of the periodic solution is examined by using the Floquet theorem together with the Routh–Hurwitz criterion (without time delay) and the Nyquist criterion (with time delay). By comparison with the solution obtained by the numerical integration, we find that the accuracy of the integral equation method is much better.     

Retrospective time integral scheme and its applications to the advection equation

To put more information into a difference scheme of a differential equation for making an accurate prediction, a new kind of time integration scheme, known as the retrospective (RT) scheme, is proposed on the basis of the memorial dynamics. Stability criteria of the scheme for an advection equation in certain conditions are derived mathematically. The computations for the advection equation have been conducted with its RT scheme. It is shown that the accuracy of the scheme is much higher than that of the leapfrog (LF) difference scheme. The project supported by the National Key Program for Developing Basic Sciences (G1999043408 and G1998040901-1) and the National Natural Sciences Foundation of China (40175024 and 40035010)     

Experimental method of dynamic caustics and its application in fracture mechanics

The equation of static and dynamic caustics, and the formulae determining the position of crack tip and stress intensity factor are given. It is proven that for the case of low speed of crack propagation the static formula is applicable in calculation. A simple method to measure the static stress-optical constants is proposed. An Optical system which is suitable for the experiments of dynamic caustics was set-up and used to study the fracture in beam and rings with initial crack under impact loading. A series of dynamic caustics' photographs and curves showing the variations of corresponding crack lengths and dynamic stress intensity factors with time, are presented.     

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method (LBIEM),singularities in the local boundary integrals need to be treated specially. In the current paper,local integral equations are adopted for the nodes inside the domain trod moving least square approximation (MLSA) for the nodes on the global boundary,thus singularities will not occur in the new al- gorithm.At the same time,approximation errors of boundary integrals are reduced significantly.As applications and numerical tests,Laplace equation and Helmholtz equa- tion problems are considered and excellent numerical results are obtained.Furthermore, when solving the Hehnholtz problems,the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions.Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.     

An integral equation approach to finite deflection of elastic plates

The present paper concerns an approximate integral equation approach to finite deflection of elastic plates with aribitrary plan form. With the combined help of the Berger equation governing non-linear bending and a weighted residual technique for boundary-value problem, a boundary integral equation is formulated for immovable edge conditions. We here clarify the formulation and show that the calculation can be performed, with a slight modification, through a procedure similar to that conventionally employed in the linear bending analysis. Availability of the derived integral equation and the solution scheme is shown by way of simple numerical examples.     

An integral equation method for closely interacting surfactant-covered droplets in wall-confined Stokes flow

A highly accurate method for simulating surfactant-covered droplets in two-dimensional Stokes flow with solid boundaries is presented. The method handles both periodic channel flows of arbitrary shape and stationary solid constrictions. A boundary integral method together with a special quadrature scheme is applied to solve the Stokes equations to high accuracy, also for closely interacting droplets. The problem is considered in a periodic setting and Ewald decompositions for the Stokeslet and stresslet are derived. Computations are accelerated using the spectral Ewald method. The time evolution is handled with a fourth-order, adaptive, implicit-explicit time-stepping scheme. The numerical method is tested through several convergence studies and other challenging examples and is shown to handle drops in close proximity both to other drops and solid objects to high accuracy.     

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。     

动态设计变量优化方法及其在工程静力学通用程序设计中的应用

本文提出动态设计变量优化新方法并解决理论和工程问题，该方法首先构建目标函数框架，然后根据具体问题的输入条件，动态地进行设计变量分配和合理排序，形成实际问题的动态目标函数。基于动态设计变量优化方法，编制出一个能解决单刚体、刚体系的平面和空间问题、摩擦问题和桁架所有工程静力学平衡问题的通用程序，并通过实例分析验证。为解决更多工程领域问题提供有效新观点。     

M.E.SHVEZ ITERATIVE METHOD AND ITS APPLICATION IN MECHANICS

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A boundary integral equation method for Navier-Stokes equations—application to flow in annulus of eccentric cylinders

A novel Navier-Stokes solver based on the boundary integral equation method is presented. The solver can be used to obtain flow solutions in arbitrary 2D geometries with modest computational effort. The vorticity transport equation is modelled as a modified Helmholtz equation with the wave number dependent on the flow Reynolds number. The non-linear inertial terms partly manifest themselves as volume vorticity sources which are computed iteratively by tracking flow trajectories. The integral equation representations of the Helmholtz equation for vorticity and Poisson equation for streamfunction are solved directly for the unknown vorticity boundary conditions. Rapid computation of the flow and vorticity field in the volume at each iteration level is achieved by precomputing the influence coefficient matrices. The pressure field can be extracted from the converged streamfunction and vorticity fields. The solver is validated by considering flow in a converging channel (Hamel flow). The solver is then applied to flow in the annulus of eccentric cylinders. Results are presented for various Reynolds numbers and compared with the literature.     

Generalized Fourier transform and its application to the volume integral equation for elastic wave propagation in a half space

In the present study, a generalized Fourier transform for time harmonic elastic wave propagation in a half space is developed. The generalized Fourier transform is obtained from the spectral representation of the operator derived from the elastic wave equation. By means of the generalized Fourier transform, a volume integral equation method for the analysis of scattered elastic waves is presented. The proposed method is based on the Krylov subspace iteration technique. During the iterative process, the discrete generalized Fourier transform is used, where the derivation of a huge and dense matrix from the volume integral equation is not necessary.     

A coupling method of difference with high order accuracy and boundary integral equation for evolutionary equation and its error estimates

In the present paper,a new numerical method for solving initial-boundary valueproblems of evolutionary equations is proposed and studied,combining difference methodwith high accuracy with boundary integral equation method.The numerical approximateschemes for both problems on a bounded or unbounded domain in R~3 are proposed and theirprior error estimates are obtained.