Boundary element solution of the transonic integro-differential equation

The transonic integro-differential equation for two-dimensional flows is solved by boundary element methods. In addition to constant and quadrilateral elements we develop hybrid elements based on constant elements in the streamwise direction and variable elements in the transverse direction. Computation is carried out for parabolic-arc and NACA0012 airfoils and the results, which converge fast, compare favourably with finite-difference solutions. The hybrid elements are to be preferred because they yield results which are more accurate than constant elements without the computational complexity associated with quadrilateral elements. Moreover, they can be applied with a small number of nodes by using only one strip of rectangular elements.     

Boundary integral equation solution of the single sided linear induction pump problem

This paper utilizes the Boundary Integral Equation Method to analyse the single sided linear induction pump. The advantage offered by the method is that transverse edge effects are not neglected. The equations necessary for the electromagnetic field and force analysis of the pump are developed and the numerical solution of the equations is described. Particular attention is given to means of calculating the electromagnetic fields within the molten metal secondary. Typical field and force distributions are presented.     

Boundary element analysis of ductile crack growth using the traction-free fundamental solution

A new boundary element formulation in two-dimensional rate-independent plasticity is given. This new formulation uses a so-called traction-free fundamental solution so that the resulting boundary integral equation converges in the normal sense, and more important, a formal differentiation of the boundary integral equation leads to a valid integral representation for the in-plane stress component on the boundary. No finite difference approximation is needed to construct the stress recovery routine. The new boundary element method is then used to solve the problem of quasi-static ductile crack growth. Numerical simulations based on a set of experimental data have been carried out to evaluate a new path-independent integral,T* _M . TheT* _M ,-integral is a modified version of Atluri'sT*-integral. This modified version has an advantage of having a less singular domain integral near the crack flank so that it is numericaly preferable toT*.     

An efficient time-truncated boundary element formulation applied to the solution of the two-dimensional scalar wave equation

An efficient time-truncation algorithm applied to the boundary element solution of the two-dimensional scalar wave equation is proposed. Rational interpolation functions are here employed to compute time-domain influence matrices, in appropriate instants of time, improving previous time-truncation strategies presented in the literature. Two numerical examples are considered at the end of the paper, illustrating that the present scheme is fairly suitable for both bounded and unbounded models.     

Boundary element analysis of two-dimensional elastoplastic contact problems

A general boundary element formulation for contact problems, capable of dealing with local elastoplastic effects and friction, is presented. Both conforming and non-conforming problems may be analysed. The contact problem is solved by means of a direct constraint technique, in which compatibility and equilibrium conditions are directly enforced in the general system of equations. The contact areas are modelled with linear interpolation functions, and quadratic interpolation functions are used everywhere else. Elastoplasticity is solved by a BEM initial strain approach The Von Mises yield criterion with its associated flow rule is adopted. Both perfectly plastic and work hardening materials are studied in the proposed formulation.

An incremental loading technique is proposed, which allows accurate development of the loading history of the problem. The non-linear nature of these problems demands the use of an iterative procedure, to determine the correct frictional conditions at every node of the contact area and the value of the plastic strains at selected points where local yielding may have occurred. Several numerical examples are presented to demonstrate the efficiency of the proposed formulation.     

Boundary element analysis of two-dimensional elastoplastic contact problems

In this paper a boundary element method for two-dimensional elastoplastic stress analysis of frictional contact problems is presented. The bodies in contact are treated as separate regions. The contact conditions are used to join the different system of equations for different regions of contacted bodies and, hence, an overall system of equation is obtained. An incremental and iterative procedure can be used to find the contact load, or the contact extent and the proper contact conditions. To include the plastic deformation in the analysis, the initial strain algorithm is employed. Elastic-perfectly plastic or work-hardening material behaviour can be assumed. For the numerical analysis, an isoparametric three-noded line elements are used to represent the boundary and eight-noded quadrilateral or six-noded triangular elements are used for the interior of the domain. The displacement rates and traction rates are assumed to vary quadratically and the shape functions for the interior strain rates are also of quadratic type. As an example, the behaviour of an elastic and elastoplastic body with a smooth, circular inclusion under the increasing load is presented.     

Boundary element analysis of reinforced concrete structural elements

A simple and practical technique for the discrete representation of reinforcement in two-dimensional boundary element analysis of reinforced concrete structural elements is presented. The bond developed over the surface of contact between the reinforcing steel and concrete is represented using fictitious one-dimensional spring elements. Potentials of the model developed are demonstrated using a number of numerical examples. The results are seen to be in good agreement with the results obtained using standard finite element software.     

Boundary integral equation methods for two-dimensional models of crack-field interactions

An introduction to the application of surface integral equation methods to the calculation of eddy current-flaw interactions is presented. Two two-dimensional problems are presented which are solved by the boundary integral equation method. Application of collocation methods reduces the problems to systems of linear algebraic equations. The first problem is that of a closed surface crack in a flat slab with an AC magnetic field parallel to the plane of the crack. The second is that of av-groove crack in the AC field of a pair of parallel wires placed parallel to the vertex of the crack. In both cases, maps of the current densities at the surface are displayed, as well as the impedance changes due to the cracks.     

Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation

The dispersive properties of finite element semidiscretizations of the two-dimensional wave equation are examined. Both bilinear quadrilateral elements and linear triangular elements are considered with diagonal and nondiagonal mass matrices in uniform meshes. It is shown that mass diagonalization and underintegration of the stiffness matrix of the quadrilateral element markedly increases dispersive errors. The dispersive properties of triangular meshes depends on the mesh layout; certain layouts introduce optical modes which amplify numerically induced oscillations and dispersive errors. Compared to the five-point Laplacian finite difference operator, rectangular finite element semidiscretizations with consistent mass matrices provide superior fidelity regardless of the wave direction.     

Accurate approximation to the double sine-Gordon equation

In general, this paper deals with nonlinear double sine-Gordon equation with even potential energy which has arisen in many physical phenomena. The nonlinear dispersion problems without a small perturbation parameter are difficult to be solved analytically. Hence, the main concern is focused on solving the traveling wave of the double sine-Gordon equation. As commonly known, the perturbation method is for solving problems with small parameters, and the analytical representation thus derived has, in most cases, a small range of validity. For some nonlinear problems, although an exact analytical solution can be achieved, they often appear in terms of sophisticated implicit functions, and are not convenient for application. Although a variety of transformation methods has been developed for solving the nonlinear dispersion problems, such transformed equations still include nonlinear terms. To overcome these difficulties, a new approach for Newton-harmonic balance (NHB) method with Fourier–Bessel series is presented here. It is applied to solve the higher-order analytical approximations for dispersion relation in double sine-Gordon equation. The Fourier–Bessel series with the NHB method presents excellent improvement from lower-order to higher-order analytical approximations involving the nonlinear terms in the double sine-Gordon equation. Not restricted by the existence of a small perturbation parameter, the method is suitable for small as well as large amplitudes of wavetrains. Excellent agreement with exact solutions is presented in some practical examples.     

Boundary value techniques for finite element solutions of the diffusion-convection equation

In this paper, the formulation of six-point and nine-point finite element equations for the solution of the diffusion-convection equation is presented. The six-point equation requires the solution of a tridiagonal system of equations and the nine-point centred equation is treated as a solution of a boundary value problem which leads to a large linear system of equations. Some numerical experiments are presented and the comparison with existing methods is included.     

Boundary element analysis of 2D thin walled structures with high-order geometry elements using transformation

Thin structures have been widely designed and utilized in many industries. However, the analysis of the mechanical behavior of such structures represents a very challenging and attractive task to scientists and engineers because of their special geometrical shapes. The major difficulty in applying the boundary element method (BEM) to thin structures is the coinstantaneous existence of the singular and nearly singular integrals in conventional boundary integral equation (BIE). In this paper, a non-linear transformation over curved surface elements is introduced and applied to the indirect regularized boundary element method for 2-D thin structural problems. The developed transformation can remove or damp out the nearly singular properties of the integral kernels, based on the idea of diminishing the difference of the orders of magnitude or the scale of change of operational factors. For the test problems studied, very promising results are obtained when the thickness to length ratio is in the orders of 1E?01 to 1E?06, which is sufficient for modeling most thin structures in industrial applications.     

A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation

In this paper a Galerkin least-squares (GLS) finite element method, in which residuals in least-squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one-dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two-dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher-order quadratic interpolations is also presented.     

Approximate solution of the linear heat equation in heterogeneous media

The asymptotic Vishik-Lyusternik method is used to solve the linear heat equations in heterogeneous media.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 5, pp. 912–915, November, 1981.     

Finite element solution for electromagnetic scattering from two-dimensional bodies

The problem of scattering from bodies in free space is formulated using a differential equation approach. The finite element mesh extends outward into the far field region of the scattering body, where the outer boundary condition is evaluated using the asymptotic expression for the scattered field. Numerical results for two scattering bodies are presented and discussed. Non-physical, standing waves appear in the results due to the inadequacy of the outer boundary condition in fulfilling the radiation condition for the scattered field. The differential equation approach does not appear to be competitive with integral equation approaches for thin bodies, but seems promising for handling scattering from thick inhomogeneous bodies into which the field penetrates.     

Boundary integral equation solution of viscous flows with free surfaces

Summary solutions of the biharmonic equation governing steady two-dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.An iterative modification of the classical BBIE is presented which is able to solve a large class of (nonlinear) viscous free surface flows for a wide range of surface tensions. The method requires a knowledge of the asymptotic behaviour of the free surface profile in the limiting case of infinite surface tension but this can usually be obtained from a perturbation analysis. Unlike space discretisation techniques such as finite difference or finite element, the BBIE evaluates only boundary information on each iteration. Once the solution is evaluated on the boundary the solution at interior points can easily be obtained.     

Boundary integral equation solution of moving boundary phase change problems

Boundary integral equation methods are presented for the solution of some two-dimensional phase change problems. Convection may enter through boundary conditions, but cannot be considered within phase boundaries. A general formulation based on space-time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation. The latter is pursued and applied in detail. An elementary, noniterative system is constructed, featuring linear interpolation over elements on a polygonal boundary. Nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution. The system performs well against basic analytical solutions, using these values in the interphase jump condition, with the simplest formulation of the surface normal at boundary vertices. Because the discretized surface changes automatically to fit the scale of the problem, the method appears to offer many of the advantages of moving mesh finite element methods. However, it only requires the manipulation of a surface mesh and solution for surface variables. In some applications, coarse meshes and very large time steps may be used, relative to those which would be required by fixed grid domain methods. Computations are also compared to original lab data, describing two-dimensional soil freezing with a time-dependent boundary condition. Agreement between simulated and measured histories is good.     

Standing-wave solutions of the Enneper (sine-Gordon) equation

The present paper investigates the real solutions of the partial differential equations (∂²ω/∂u²) − (∂²ω/∂ν²) = sin ω (Enneper or sine-Gordon equation) that are of the FORM = F(|₁(u) · |₂(ν)). In the application of the Enneper equations to crystal physics such solutions may represent standing waves. A complete classification of these solutions and its degenerate cases is given.     

Boundary element solution of thermal creep flow in microfluidic devices

Flow in rarefied gases can be caused by a tangential temperature gradient along the contour boundaries (tangential heat flux), without the presence of any other external driven force, inducing a fluid motion from colder to hotter regions. This phenomenon is known as thermal creep and has gained importance in recent years in connection with micro-scale gas flow systems. Prediction of the flow field in micro-systems can be obtained by using continuum based models under appropriate boundary conditions accounting for the slip velocity due to tangential shear rate and heat flux. In this work a boundary integral equation formulation for Stokes slip flow, based on the normal and tangential projection of the Green's integral representational formulae for the velocity field is presented. The tangential heat flux is evaluated in terms of the tangential gradient of the temperature integral representational formulae presenting singularities of the Cauchy type, which are removed by the use of an auxiliary potential field. These formulations are used to evaluate the performance of different microfluidic devices.     

Boundary element analysis of an integral equation for three-dmensional crack problems

In another paper, the authors proposed an integral equation for arbitrary shaped three-dimensional cracks. In the present paper, a discretization of this equation using a tensor formalism is formulated. This approach has the advantage of providing the displacement discontinuity vector in the local basis which varies as a function of the point of the crack surface. This also facilitates the computation of the stress intensity factors along the crack edge. Numerical examples reported for a circular crack and a semi-elliptical surface crack in a cylindrical bar show that one can obtain good results, using few Gaussian points and no singular elements.