Three‐dimensional steady thermal stress analysis by triple‐reciprocity boundary element method

The steady thermal stress problems without heat generation can be solved easily by the boundary element method. However, for the case with arbitrary heat generation, the domain integral is necessary. In this paper, it is shown that the problems of three‐dimensional steady thermal stress with heat generation can be approximately solved without the domain integral by the triple‐reciprocity boundary element method. In this method, an arbitrary distribution of heat generation is interpolated by boundary integral equations. In order to solve the problem, the values of heat generation at internal points and on the boundary are used. Copyright © 2005 John Wiley & Sons, Ltd.     

Two-dimensional steady heat conduction in functionally gradient materials by triple-reciprocity boundary element method

Homogeneous heat conduction can be easily solved by means of the boundary element method. However, domain integrals are generally necessary to solve the heat conduction problem in the functionally gradient materials. This paper shows that the two-dimensional heat conduction problem in the functionally gradient materials can be solved approximately without a domain integral by the triple-reciprocity boundary element method. In this method, the distribution of domain effects is interpolated by integral equations. A new computer program is developed and applied to several problems.     

Two‐dimensional unsteady heat conduction analysis with heat generation by triple‐reciprocity BEM

If the initial temperature is assumed to be constant, a domain integral is not needed to solve unsteady heat conduction problems without heat generation using the boundary element method (BEM).However, with heat generation or a non‐uniform initial temperature distribution, the domain integral is necessary. This paper demonstrates that two‐dimensional problems of unsteady heat conduction with heat generation and a non‐uniform initial temperature distribution can be solved approximately without the domain integral by the triple‐reciprocity boundary element method. In this method, heat generation and the initial temperature distribution are interpolated using the boundary integral equation. Copyright © 2001 John Wiley & Sons, Ltd.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Three-dimensional thermo-elastoplastic analysis by triple-reciprocity boundary element method

In general, internal cells are required to solve thermo-elastoplastic problems using a conventional boundary element method (BEM). However, in this case, the merit of the BEM, which is ease of data preparation, is lost. The triple-reciprocity BEM can be used to solve two-dimensional thermo-elastoplasticity problems with a small plastic deformation without using internal cells. In this study, it is shown that three-dimensional thermo-elastoplastic problems with heat generation can be solved by the triple-reciprocity BEM without using internal cells. Initial strain and stress formulations are adopted and the initial strain or stress distribution is interpolated using boundary integral equations. A new computer program is developed and applied to solve several problems.     

Three-dimensional unsteady thermal stress analysis by triple-reciprocity boundary element method

The conventional boundary element method (BEM) requires a domain integral in unsteady thermal stress analysis with heat generation and/or an initial temperature distribution. In this paper, it is shown that the three-dimensional unsteady thermal stress problem can be solved effectively using the triple-reciprocity boundary element method without internal cells. In this method, the distributions of heat generation and initial temperature are interpolated using integral equations and higher order time-dependent fundamental solutions. A new computer program was developed and applied for solving several test problems.     

A meshless solution for two dimensional density-driven groundwater flow

Density-driven groundwater flow is a complicated nonlinear problem in groundwater hydraulics. The local boundary integral method is a promising meshless scheme that is used for solving several difficult problems in different areas. This method applies the boundary integral equations to the local domain around every node. The nodes can be randomly distributed in the domain and on the global boundary. Therefore, this method is characterised as meshless. The unknown potentials and concentrations in all of the nodes are approximated by interpolation to obtain a system of linear equations. Solving this system of equations leads to the numerical solution for the main problem. In this paper, a combination of the radial basis function interpolation and the local boundary element method is used to solve groundwater flow problem combined with the transport of pollution, which also influences the density of groundwater.     

An interface integral equation method for solving general multi‐medium mechanics problems

In this paper, based on the general stress–strain relationship, displacement and stress boundary‐domain integral equations are established for single medium with varying material properties. From the established integral equations, single interface integral equations are derived for solving general multi‐medium mechanics problems by making use of the variation feature of the material properties. The displacement and stress interface integral equations derived in this paper can be applied to solve non‐homogeneous, anisotropic, and non‐linear multi‐medium problems in a unified way. By imposing some assumptions on the derived integral equations, detailed expressions for some specific mechanics problems are deduced, and a few numerical examples are given to demonstrate the correctness and robustness of the derived displacement and stress interface integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical solutions of singular integral equations having Cauchy-type singular kernel by means of expansion method

This paper is concerned with numerical solutions of singular integral equations with Cauchy-type singular kernel. It is well-known that this type of singular integral equations appears in the analysis of crack problems using the continuously distributed dislocation method. In addition, it also appears in the analysis of notch problems using the body force method. In the present analysis, the unknown function of densities of dislocations and body forces are approximated by the product of the fundamental density functions and polynomials. The accuracy of stress intensity factors and stress concentration factors obtained by the present method is verified through the comparison with the exact solution and the reliable numerical solution obtained by other researchers. The present method is found to give good convergency of the numerical results for notch problem as well as internal and edge crack problems.     

Stress analysis of arbitrarily distributed elliptical inclusions under longitudinal shear loading

This paper deals with an interaction problem of arbitrarily distributed elliptical inclusions under longitudinal shear loading. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are the densities of body forces distributed in the longitudinal directions of infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical inclusions, four kinds of fundamental density functions are introduced in a similar way of previous papers treating plane stress problems. Then the body force densities are approximated by a linear combination of those fundamental density functions and polynomials. In the analysis, elastic constants of matrix and inclusion are varied systematically; then the magnitude and position of the maximum stress are shown in tables and the stress distributions along the boundary are shown in figures. For any fixed shape, size and elastic constant of inclusions, the relationships between number of inclusions and maximum stress are investigated for several arrangements.     

Body-force linear elastic stress intensity factor calculation using fractal two level finite element method

Fractal two level finite element method (F2LFEM) for the analysis of linear fracture problems subjected to body force loading is presented. The main objective here is to show that by employing the F2LFEM a highly accurate and an efficient linear analysis of fracture bodies subjected to internal loading can be obtained as it is hard to find any analytical and exact values of stress intensity factor (SIF) for any kind of geometry subjected to internal loading. Also in this paper, a fast method to transform the body force to the reduced force vector is presented and has been effectively employed. The problems solved here include both the single mode or mixed mode cracks subjected to internal body-force or external loading. In comparison with other numerical algorithms, it seems that with a small amount of computational time and computer storage, highly accurate results can be obtained.     

A green's function method for large deflection analysis of plates

Summary An integral equation method is presented for large deflection analysis of thin elastic plates, whose behaviour is governed by the von Kármán equations. The method uses the Green function of the biharmonic equation to establish integral representations of the deflection and stress function for the linear part of the governing operator while the nonlinearities are treated as loading forces. Six nonlinear domain integral equations are formulated which are solved to yield the curvature tensors of the deflection and stress function surfaces and thereby the deflections and stress resultants. The nonlinear integral equations are solved numerically by developing an effective technique based on Gaussian quadrature over domains of arbitrary shape. For domains of simple geometry ready-to-use Green functions are employed whereas for regions of arbitrary shape the Green function is established numerically using BEM. The efficiency of the method is demonstrated and attested by analyzing a circular clamped plate with movable edge.With 4 Figures     

Meshless thermo‐elastoplastic analysis by triple‐reciprocity boundary element method

In general, internal cells are required to solve thermo‐elastoplasticity problems by a conventional boundary element method (BEM). However, in this case, the merit of BEM, which is the easy preparation of data, is lost. A conventional multiple‐reciprocity boundary element method (MRBEM) cannot be used to solve elastoplasticity problems, because the distribution of initial strain or stress cannot be determined analytically. In this study, it is shown that without the use of internal cells, two‐dimensional thermo‐elastoplasticity problems can be solved by a triple‐reciprocity BEM using a thin plate spline. Initial strain and stress formulations are adopted and the initial strain or stress distribution is interpolated using boundary integral equations. A new computer program was developed and applied to solve several problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A time domain boundary element method for modeling the quasi-static viscoelastic behavior of asphalt pavements

A time domain boundary element method (BEM) is presented to model the quasi-static linear viscoelastic behavior of asphalt pavements. In the viscoelastic analysis, the fundamental solution is derived in terms of elemental displacement discontinuities (DDs) and a boundary integral equation is formulated in the time domain. The unknown DDs are assumed to vary quadratically in the spatial domain and to vary linearly in the time domain. The equation is then solved incrementally through the whole time history using an explicit time-marching approach. All the spatial and temporal integrations can be performed analytically, which guarantees the accuracy of the method and the stability of the numerical procedure. Several viscoelastic models such as Boltzmann, Burgers, and power-law models are considered to characterize the time-dependent behavior of linear viscoelastic materials. The numerical method is applied to study the load-induced stress redistribution and its effects on the cracking performance of asphalt pavements. Some benchmark problems are solved to verify the accuracy and efficiency of the approach. Numerical experiments are also carried out to demonstrate application of the method in pavement engineering.     

A comprehensive study on the 2D boundary element method for anisotropic thermoelectroelastic solids with cracks and thin inhomogeneities

This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.     

A boundary element method without internal cells for solving viscous flow problems

In this paper, a new boundary element method without internal cells is presented for solving viscous flow problems, based on the radial integration method (RIM) which can transform any domain integrals into boundary integrals. Due to the presence of body forces, pressure term and the non-linearity of the convective terms in Navier–Stokes equations, some domain integrals appear in the derived velocity and pressure boundary-domain integral equations. The body forces induced domain integrals are directly transformed into equivalent boundary integrals using RIM. For other domain integrals including unknown quantities (velocity product and pressure), the transformation to the boundary is accomplished by approximating the unknown quantities with the compactly supported fourth-order spline radial basis functions combined with polynomials in global coordinates. Two numerical examples are given to demonstrate the validity and effectiveness of the proposed method.     

Three-dimensional unsteady heat conduction analysis by triple-reciprocity boundary element method

The conventional boundary element method (BEM) requires a domain integral in heat conduction analysis with heat generation or an initial temperature distribution. In this paper it is shown that the three-dimensional heat conduction problem can be solved effectively using the triple-reciprocity BEM without internal cells. In this method, the distributions of heat generation and initial temperature are interpolated using integral equations and time-dependent fundamental solutions are used. A new computer program was developed and applied to solving several problems.     

Lower‐bound limit analysis by using the EFG method and non‐linear programming

Intended to avoid the complicated computations of elasto‐plastic incremental analysis, limit analysis is an appealing direct method for determining the load‐carrying capacity of structures. On the basis of the static limit analysis theorem, a solution procedure for lower‐bound limit analysis is presented firstly, making use of the element‐free Galerkin (EFG) method rather than traditional numerical methods such as the finite element method and boundary element method. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration. The reduced‐basis technique is adopted to solve the mathematical programming iteratively in a sequence of reduced self‐equilibrium stress subspaces with very low dimensions. The self‐equilibrium stress field is expressed by a linear combination of several self‐equilibrium stress basis vectors with parameters to be determined. These self‐equilibrium stress basis vectors are generated by performing an equilibrium iteration procedure during elasto‐plastic incremental analysis. The Complex method is used to solve these non‐linear programming sub‐problems and determine the maximal load amplifier. Numerical examples show that it is feasible and effective to solve the problems of limit analysis by using the EFG method and non‐linear programming. Copyright © 2007 John Wiley & Sons, Ltd.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part II: Numerical analysis

In this paper, elliptical cracks and rectangular cracks embedded in a three-dimensional infinite transversely isotropic piezoelectric solid are analyzed under combined mechanical tension and electric fields. The hypersingular integral equation method is used to solve the mentioned problems. The unknown function in the hypersingular integral equations is approximated with a product of the fundamental density function and polynomials. The hypersingular integrals can be numerically evaluated by using a method of Taylor series expansion. Therefore, the hypersingular integral equations for the crack problems can be solved immediately. Finally, numerical examples of the stress and electric displacement intensity factors as well as the energy release rates for these crack configurations are presented. The numerical results demonstrate the present approach to be very efficient.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.