On the boundary integral formulation of thermo-viscoelasticity theory

A formulation of the boundary integral equation method for generalized thermo-viscoelasticity is given. Fundamental solutions of the corresponding differential equations are obtained. An outline of the implementation of the boundary element method is discussed for the solution of the above boundary equations. Special emphasis is given to the representation of the primary fields, namely, temperature and displacement.     

基于中心流形理论的小水电并网系统Hopf分岔分析

针对小水电并网系统,用Matcont软件搜寻系统的Hopf分岔点绘制分岔图;利用中心流形理论将高维电力系统降到二维模型,并通过计算二维模型分岔稳定性指标的正负判定原系统Hopf分岔类型。结果表明,分岔稳定性指标大于零时电压失稳,小于零时电压稳定。用Matlab软件对讨论结果进行数值仿真,证明理论结果的正确性。     

Analysis of radiative heat-loss effects in thermal explosion of a gas using the integral manifold method

Semenov's classical model of thermal explosion in a combustible gas mixture is modified to include radiative (rather than conductive) heat-loss effects and gas-density changes. A geometrical asymptotic technique (the method of integral manifolds - MIM) is exploited to perform a qualitative analysis of the governing equations. The strength of this method lies in the compact, clear geometrical/analytical rendition and classification of all possible dynamical scenarios, in terms of the physico-chemical parameters of the system. It is found that there are two main dynamical regimes of the system: cooling regimes and fast explosive regimes. Peculiarities of these dynamical regimes are investigated and their dependence on physical system parameters is analyzed. A criterion for the occurrence of thermal explosion is disclosed. An estimate for the maximum mixture temperature is also derived analytically. It is found that, under certain operating conditions, the dynamics are such that the initial explosive stage of the process essentially behaves adiabatically before succumbing to the dominance of the radiative heat loss that brings the system down to the ambient temperature.     

Influence of laser parameters on the geometric characteristics of a beam transformed by an optical system

The influence of the parameters of a laser on the divergence and the light radii of a beam formed by a plane optical resonator and transformed by an optical system is considered. The studies are conducted for a broad range of Fresnel numbers characteristic of lasers. __________ Translated from Izmeritel’naya Tekhnika, No. 1, pp. 24–27, January, 2008.     

Transient pressure solution for a horizontal well in a petroleum reservoir by boundary integral methods

A novel application of the boundary integral method to horizontal well analysis in the field of petroleum engineering is presented. The transient pressure satisfies the heat equation, non‐local and non‐linear boundary conditions. The turbulent flow inside the well is modelled by considering a pressure gradient along the well. The heat potential is used and Chebyshev collocation along with a time discretization is employed. Some numerical results are presented to show the features of this new approach. Copyright © 2000 John Wiley & Sons, Ltd.     

On the solution of a minimum compliance topology optimisation problem by optimality criteria without a priori volume constraint specification

The paper presents a topology optimisation method based on optimality criteria for total potential energy maximisation with a volume constraint. The final volume of the optimal structural configuration has not to be specified a priori and is directly controlled by the stress, displacement or stiffness constraints defined at the design problem layout phase. The proposed method leads to the identification of well defined structures characterised by a small number of discrete elements with intermediate material properties within a limited number of iterations. The results obtained by solving several two dimensional benchmark problems are shown.     

Threshold-based method for elevating the system's constraint under theory of constraints

The principal tenet of theory of constraints (TOC) is that there is at least one constraint in each system that limits the ability of achieving higher levels of performance relative to its goal. Maximum utilisation of the constraint leads to maximum output of the system. However, activation of a non-constraint resource at 100% of its capacity does not increase output. Therefore, some resources are not fully utilised. In this paper, the authors use the left capacity of a non-constraint resource (NC) to elevate the system's constraint. It is assumed that the capacity-constrained resource (CCR) is a continuous time Markov process having a two-dimensional state space. The work in the NC is interruptible, allowing a worker in the NC to switch to CCR. The switch from NC to CCR would occur when the queue of waiting parts in the CCR becomes ‘too long’ and vice versa, when there are few parts in the CCR. Returning to the NC from the CCR may require some ‘re-orientation time’ on the part of the switched worker. The goal is to find the maximum output of CCR subject to the time-average number of workers in the NC must be greater than a pre-specified value.     

Investigation of anti-plane shear behavior of a Griffith crack in a piezoelectric material by using the non-local theory

In this paper, the behavior of a Griffith crack in a piezoelectric material under anti-plane shear loading is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with two pairs of dual integral equations. These equations are solved using Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present at the crack tip.     

Numerical resolution of the boundary integral equations for elastic scattering by a plane crack

The problem of wave scattering by a plane crack is solved, either in the case of acoustic waves or in the case of elastic waves incidence using the boundary integral equation method. A collocation method is often used to solve that equation, but here we will use a variational method, first writing the problem of Fourier variables, and then writing the associated integrals in the sesquilinear form with weak singularity kernels. This representation is used in the numerical approach, made with a finite element method in the surface of the crack. Numerical tests were made with circular and elliptical cracks, but this method can be extended to other shapes, with the same convergence profiles. Extensive results are given concerning the crack opening displacement, the scattering cross-section, the back-scattered amplitude and far-field patterns.     

Investigation of the scattering of harmonic elastic antiplane shear waves by a finite crack using the non-local theory

In this paper, the scattering of harmonic antiplane shear waves by a finite crack is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. Contrary to the classical elasticity solution, it is found that no stress singularity is presented at the crack tip. The non- local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations

Complex boundary integral equations (Fredholm‐type regular or Cauchy‐type singular or even Hadamard–Mangler‐type hypersingular) have been used for the numerical solution of general plane isotropic elasticity problems. The related Muskhelishvili and, particularly, Lauricella–Sherman equations are famous in the literature, but several more extensions of the Lauricella–Sherman equations have also been proposed. In this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries (for anyone of the above equations: regular or singular or hypersingular, but with a prerequisite their actual numerical solution) through the appropriate use of the even more classical elementary Cauchy theorem in complex analysis. This approach has been already used for the accurate numerical computation of analytic functions and their derivatives by Ioakimidis, Papadakis and Perdios (BIT 1991; 31 : 276–285), without applications to elasticity problems, but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non‐analytic complex density function (such as an edge dislocation/loading distribution density) is numerically available on the boundary. The present results are also directly applicable to inclusion problems, anisotropic elasticity, antiplane elasticity and classical two‐dimensional fluid dynamics, but, unfortunately, not to crack problems in fracture mechanics. Brief numerical results (for the complex potentials), showing the dramatic increase of the computational accuracy, are also displayed and few generalizations proposed. Copyright © 2000 John Wiley & Sons, Ltd.     

On the approximate solution of the third boundary-value problem of heat-conduction theory for a circle

An efficient analytical approximate representation of the solution of the third boundary-value problem of heatconduction theory for a circle is obtained. A uniform evaluation of the error of the approximate formula ensures the convergence of the numerical algorithm. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 2, pp. 403–408, March–April, 2009.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.     

Developing a new model for simultaneous scheduling of two grand projects based on game theory and solving the model with Benders decomposition

Grand infrastructure projects, such as dam, power plant, petroleum, and gas industry projects, have several contractors working on them in several independent sub-projects. The concern of reducing the duration of these projects is one of the important issues among various aspects; thus, our aim is to fulfill the requirements by using the game theory approach. In this study, a mixed-integer programming model consisting of game theory and project scheduling is developed to reduce the duration of projects with a minimum increase in costs. In this model, two contractors in successive periods are entered into a step-by-step competition by the employer during dynamic games, considering an exchange in their limited resources. The optimum solution of the game in each stage are selected as the strategy, and the resources during the game are considered to be renewable and limited. The strategy of each contractor can be described as follows: 1) share their resources with the other contractor and 2) not share the resources with the other contractor. This model can act dynamically in all circumstances during project implementation. If a player chooses a non-optimum strategy, then this strategy can immediately update itself at the succeeding time period. The proposed model is solved using the exact Benders decomposition method, which is coded in GAMS software. The results suggest the implementation of four step-by-step games between the contractors. Then, the results of our model are compared with those of the conventional models. The projects’ duration in our model is reduced by 22.2%. The nominal revenue of both contractors has also reached a significant value of 46078 units compared with the relative value of zero units in the original model. Moreover, we observed in both projects the decreases of 19.5%, 20.9%, and 19.7% in the total stagnation of resources of types 1, 2, and 3, respectively.     

Thermal analysis of carbon-nanotube composites using a rigid-line inclusion model by the boundary integral equation method

The boundary integral equation (BIE) method is applied for the thermal analysis of fiber-reinforced composites, particularly the carbon-nanotube (CNT) composites, based on a rigid-line inclusion model. The steady state heat conduction equation is solved using the BIE in a two-dimensional infinite domain containing line inclusions which are assumed to have a much higher thermal conductivity (like CNTs) than that of the host medium. Thus the temperature along the length of a line inclusion can be assumed constant. In this way, each inclusion can be regarded as a rigid line (the opposite of a crack) in the medium. It is shown that, like the crack case, the hypersingular (derivative) BIE can be applied to model these rigid lines. The boundary element method (BEM), accelerated with the fast multipole method, is used to solve the established hypersingular BIE. Numerical examples with up to 10,000 rigid lines (with 1,000,000 equations), are successfully solved by the BEM code on a laptop computer. Effective thermal conductivity of fiber-reinforced composites are evaluated using the computed temperature and heat flux fields. These numerical results are compared with the analytical solution for a single inclusion case and with the experimental one reported in the literature for carbon-nanotube composites for multiple inclusion cases. Good agreements are observed in both situations, which clearly demonstrates the potential of the developed approach in large-scale modeling of fiber-reinforced composites, particularly that of the emerging carbon-nanotube composites.     

On solvability of a boundary integral equation of the first kind for Dirichlet boundary value problems in plane elasticity

The solution of a Dirichlet boundary value problem of plane isotropic elasticity by the boundary integral equation (BIE) of the first kind obtained from the Somigliana identity is considered. The logarithmic function appearing in the integral kernel leads to the possibility of this operator being non-invertible, the solution of the BIE either being non-unique or not existing. Such a situation occurs if the size of the boundary coincides with the so-called critical (or degenerate) scale for a certain form of the fundamental solution used. Techniques for the evaluation of these critical scales and for the removal of the non-uniqueness appearing in the problems with critical scales solved by the BIE of the first kind are proposed and analysed, and some recommendations for BEM code programmers based on the analysis presented are given.     

Investigation of the scattering of harmonic elastic waves by two collinear symmetric cracks using non-local theory

In this paper, the scattering of harmonic waves by two collinear symmetric cracks is studied by use of non-local theory. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the dynamic problem to obtain the stress occurring at the crack tips. The Fourier transform is applied and a mixed boundary-value problem is formulated. The solutions are obtained by means of the Schmidt method. This method is more exact and more appropriate than Eringen's for solving this kind of problem. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the lattice parameter and the circular frequency of the incident wave.     

Boundary integral equations for the scattering of elastic waves by elastic inclusions with thin interface layers

Elastic waves are scattered by an elastic inclusion. The interface between the inclusion and the surrounding material is imperfect: the displacement and traction vectors on one side of the interface are assumed to be linearly related to both the displacement vector and the traction vector on the other side of the interface. The literature on such inclusion problems is reviewed, with special emphasis on the development of interface conditions modeling different types of interface layer. Inclusion problems are formulated mathematically, and uniqueness theorems are proved. Finally, various systems of boundary integral equations over the interface are derived.