Stratified newtonian flow calculations by the boundary element method

A boundary integral equation formulation is used to model the stratified flow of two Newtonian viscous liquids, in which the geometrical detail of the interface between the layers is not known in advance. The technique is tested by comparing predictions with the results of previous finite element solutions, and found to perform well. Finally, the method is used to examine the two-layer jet theory of Tanner (1980) as a means of simplifying the modelling of complex extrudate swell problems.     

Solution of magnetohydrodynamic flow problems using the boundary element method

A boundary element solution is implemented for magnetohydrodynamic (MHD) flow problem in ducts with several geometrical cross-section with insulating walls when a uniform magnetic field is imposed perpendicular to the flow direction. The coupled velocity and induced magnetic field equations are first transformed into uncoupled inhomogeneous convection–diffusion type equations. After introducing particular solutions, only the homogeneous equations are solved by using boundary element method (BEM). The fundamental solutions of the uncoupled equations themselves (convection–diffusion type) involve the Hartmann number (M) through exponential and modified Bessel functions. Thus, it is possible to obtain results for large values of M (M≤300) using only the simplest constant boundary elements. It is found that as the Hartmann number increases, boundary layer formation starts near the walls and there is a flattening tendency for both the velocity and the induced magnetic field. Also, velocity becomes uniform at the center of the duct. These are the well-known behaviours of MHD flow. The velocity and the induced magnetic field contours are graphically visualized for several values of M and for different geometries of the duct cross-section.     

Simulation of 3D flow in porous media by boundary element method

In the present paper problem of natural convection in a cubic porous cavity is studied numerically, using an algorithm based on a combination of single domain and subdomain boundary element method (BEM). The modified Navier–Stokes equations (Brinkman-extended Darcy formulation with inertial term included) were adopted to model fluid flow in porous media, coupled with the energy equation using the Boussinesq approximation. The governing equations are transformed by the velocity–vorticity variables formulation which separates the computation scheme into kinematic and kinetic parts. The kinematics equation, vorticity transport equation and energy equation are solved by the subdomain BEM, while the boundary vorticity values, needed as a boundary conditions for the vorticity transport equation, are calculated by single domain BEM solution of the kinematics equation. Computations are performed for steady state cases, for a range of Darcy numbers from 10^?6 to 10^?1, and porous Rayleigh numbers ranging from 50 to 1000. The heat flux through the cavity and the flow fields are analyzed for different cases of governing parameters and compared to the results in some published studies.     

A boundary element method for steady incompressible thermoviscous flow

A boundary element formulation is presented for moderate Reynolds number, steady, incompressible, thermoviscous flows. The governing integral equations are written exclusively in terms of velocities and temperatures, thus eliminating the need for the computation of any gradients. Furthermore, with the introduction of reference velocities and temperatures, volume modelling can often be confined to only a small portion of the problem domain, typically near obstacles or walls. The numerical implementation includes higher order elements, adaptive integration and multiregion capability. Both the integral formulation and implementation are discussed in detail. Several examples illustrate the high level of accuracy that is obtainable with the current method.     

Probabilistic fracture mechanics by the boundary element method

In this paper, a new boundary element formulation is presented for probabilistic analysis of crack problems in context of linear elastic fracture mechanics. The method involves an implicit differentiation method for calculating fracture response derivatives with respect to random parameters and applies first-order reliability method to predicting the reliability of cracked structures. Direct differentiation is used to obtain the derivatives of fracture parameters with respect to crack size which is required for probabilistic analysis. Numerical examples are presented to illustrate the proposed method. The accuracy of the proposed method is demonstrated by comparing with either analytical solution or other established method.     

Building floor analysis by the boundary element method

In this work, a boundary element formulation to analyse slabs reinforced by beams, combined or not to define a grid sub-system, is proposed. Kirchhoff s hypothesis is assumed for the plate elements. The beams elements are not required to be displayed over the plate surface, therefore eccentricity effects are taken into account. The formulation is derived by assuming a zoned body where beam elements are introduced by degenerating plate sub-regions. After finding properly a single reciprocity for the whole body, the required integral representations are derived. The integral representations derived for this complex structural element take into account the bending and stretching effects of booth structural elements working together. The standard equilibrium and compatibility conditions along interface are naturally imposed. Moreover, the amount of degrees of freedom required along the interfaces is substantially reduced, leading therefore to small and more accurate algebraic system of linear equations. Several examples are then shown to illustrate the accuracy of the formulation, comparing the obtained results with analytical and other numerical solutions.The authors wish to thank FAPESP (São Paulo State Foundation for Scientific Research) for the financial support and Prof. Selma Hissae Shimura da Nobrega by the collaboration to run the ADINA code.     

Plate buckling loads by the boundary element method

A boundary element solution of the plate buckling problem is extended and refined to accommodate any combination of loading and support conditions. It is complemented by a similar boundary element formulation yielding the prebuckling membrane state of stress for non-uniform distributions of edge loads. Both continuous and discontinuous models of various orders of approximation for boundary as well as domain unknowns have been employed and their effect on the accuracy of the solution algorithm assessed. The computer code incorporates automatic mesh generation schemes for both the boundary and the domain through which a wide range of plate geometries can be discretized and analysed. Further, gains in both efficiency and accuracy are achieved by taking advantage of any support and loading symmetries and reducing the formulation to a standard eigenvalue problem. A large number of plate examples are solved to optimize the choice of various modelling parameters and to establish the reliability of the proposed analysis.     

Probabilistic boundary element method

In this paper, attention is focused on obtaining probabilistic density function (PDF) random variables in complex structures by the use of a Boundary Element Method (BEM). The method of incorporating random variables into convention BEM and the post-process treatment of the PDF of random variables are presented in this paper. As a numerical example, a plane crack problem is analysed, in which the length of the crack and the applied loads are taken as random variables.     

Use of finite elements in calculations of the flow of non-Newtonian media

The finite-element method (FEM) is applied to the solution of rheodynamic problems. As an example, the flow of non-Newtonian fluids in a channel is examined.The paper was translated from German into Russian and edited by Professor Z. P. Shul'man.Translated from Inzhenerno-Fizicheskii Zhurnal, Vo. 41, No. 5, pp. 827–836, November, 1981.     

Stiffened plate bending analysis by the boundary element method

 In this work, the plate bending formulation of the boundary element method (BEM) based on the Kirchhoff's hypothesis, is extended to the analysis of stiffened elements usually present in building floor structures. Particular integral representations are derived to take directly into account the interactions between the beams forming grid and surface elements. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composite structure as a single body. Two possible procedures are shown for dealing with plate domain stiffened by beams. In the first, the beam element is considered as a stiffer region requiring therefore the discretization of two internal lines with two unknowns per node. In the second scheme, the number of degrees of freedom along the interface is reduced by two by assuming that the cross-section motion is defined by three independent components only. Received 6 November 2000     

Optimal loading of solids by the boundary element method

Optimal boundary loading of solids in elastostatics is analyzed within the framework of optimal control problems. First, the necessary conditions of optimality are found by means of calculus of variations. The resulting sets of vector PDE's are then discretized in space by using the boundary element method, while an integral function (the performance index of the control problem) is minimized by the conjugate gradient method of optimization. Numerical results are provided for a thin plate of annular sector.     

Multidimensional interpolation and numerical integration by boundary element method

This paper presents a multidimensional interpolation method and a numerical integration for a bounded region using boundary integral equations and a polyharmonic function. In the method using B-spline, points must be assigned in a gridiron layout for two-dimensional cases. In the method presented in this paper, using the polyharmonic function, arbitrary points can be assigned instead of using a gridiron layout, making interpolation an easy process. This method requires the use of boundary geometry of the region and arbitrary internal points. Values at an arbitrary point and the integral value are calculated after solving the discretized boundary integral equations. Numerical integration is performed using this interpolation. In order to investigate the efficiency of this method, several examples are given.     

Axisymmetric non-Newtonian drops treated with a boundary integral method

A boundary integral method for the simulation of the time-dependent deformation of axisymmetric Newtonian or non-Newtonian drops suspended in a Newtonian fluid subjected to an axisymmetric flow field is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. By transforming the integral representation for the velocity to cylindrical coordinates we can reduce the dimension of the computational problem. The integral equation for the velocity remains of the same form as in Cartesian coordinates, and the Green's functions are transformed explicitly to cylindrical coordinates. Besides a numerical validation of the method we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for the Newtonian drop are consistent with results from the literature. The deformation process of the non-Newtonian drop for small capillary numbers appears to be governed by two relaxation times.     

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.     

On the efficiency and accuracy of the boundary element method and the finite element method

The efficiency and computational accuracy of the boundary element and finite element methods are compared in this paper. This comparison is carried out by employing different degrees of mesh refinement to solve a specific illustrative problem by the two methods.     

An adaptive boundary element method

This paper explores the concept of moving singularities in the boundary element analysis. The singularities are placed on an auxiliary boundary which is located outside the domain of the problem and are allowed to move as part of the solution process. This results in a highly adaptive but non-linear method. Examples involving the two- and three-dimensional Laplace's equations are solved. Excellent agreement with exact solutions is obtained using a minimal number of singularities. Also, the trajectories of the singularity motion are plotted. The behaviour seen here is that, as the solution approaches convergence, the singularities exhibit a general trend of moving away from the domain of the problem.     

An analysis of solidification problems by the boundary element method

The problem of interest in this paper is the calculation of the motion of the solid–liquid interface and the time-dependent temperature field during solidification of a pure metal. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time-dependent Green's functions and convolution integrals. The BEM approach requires discretization of only the surface of the solidifying body. Thus, the numerical method closely follows the physics of the problems and is intuitively very appealing. The formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems. Emphasis is given to two-dimensional problems. Comparison with existing semi-analytical solutions and other numerical solutions from the literature reveals that the method is fast, accurate and without major time step limitations.