Enayat Mahajerin

A comparison of the boundary element and superposition methods

Abstract For some time now, the Boundary Integral Equation (BIE) Method or as it is alternately called the Boundary Element Method (BEM) has been hailed as the technique best suited to problems in elasticity and related fields, both for accuracy and efficiency. The authors demonstrate by example that this is not the case. A much simpler and more versatile technique, the Superposition Method (SUP), is introduced and is shown to outperform BEM in both areas. Followed by a discussion of the merits and drawbacks of each method, compact computer programs for both BEM and SUP and numerical results for nine different example problems are presented to support the authors claim.

Boundary Element Study of a Loaded Hole in an Orthotropic Plate

The purpose of this paper is to develop an efficient boundary element method (BEM) for use in the analysis of composite structures, principally loaded holes in mechanically fastened composites. The idea is to modify the BEM by an analytical scheme to evaluate boundary stresses and deformations accurately. The validity of the method is verified by computing the stress concentration factor in a double lap joint for different hole sizes and various boundary conditions.

The fundamental collocation method applied to the nonlinear poisson equation in two dimensions

Abstract The fundamental collocation method is adapted to the nonlinear Poisson equation in two dimensions with mixed boundary conditions of the Dirichlet and Neumann type. The technique is an iterative collocation procedure which requires a representation of the boundary of a finite region by N points and of the interior by M points. The order of the problem as determined by the dimensions of the collocation matrices is N × N for each iteration. The method also employs an adjustable parameter S which can be used to check for stability. The accuracy and efficiency are shown to be quite good on three example problems, two of which are for heat-transfer and non-Newtonian laminar flow. Suggestions for improving the method are made.

An extension of the superposition method for plane anisotropic elastic bodies

Abstract The determination of the elastic field in anisotropic bodies subjected to mixed and nonhomogeneous boundary conditions has not been developed to the extent of its isotropic counterpart. This is mainly due to the complexity of the anisotropic fundamental solution. In this paper, the superposition method (SUP) is extended to allow for anisotropy. In particular, the method is developed for the plane orthotropic case. Using some examples, it is shown that the method is both efficient and very accurate. A general FORTRAN computer program based on the formulation presented here is included.

A numerical method for laterally loaded thin plates

Abstract A numerical method for dealing with laterally loaded thin plates is presented and compared to the similar but more basic numerical technique used in [1, 2]. This method considers the plate to be embedded in the infinite plane and uses point load sources external to the plate boundary to satisfy boundary conditions as in collocation. The exact solution for a constant lateral load acting over an arbitrary polygon in the infinite plane is first derived and then used to couple the effects of lateral loading to the collocation method described above. The results show that less than half the number of collocation points used in [1, 2] is needed for the same accuracy and that the computer execution time is reduced several hundred times. This method is not limited to particular plate shapes, boundary condition types or load distributions. Five examples are used to illustrate the method.

Rotational fluid flow using a least squares collocation technique

Abstract A potential theory approach for incompressible viscous flow which leads to the biharmonic equation is first developed. A numerical least squares collocation technique using fundamental singular solutions of the biharmonic equation is then applied to a rotational flow problem with moving boundaries that produce discontinuous boundary conditions associated with the biharmonic. It is shown that the least squares technique smoothes out local disturbances in boundary data of the type which are likely to present difficulties to the more commonly used boundary element method. A compact computer program for the method and the results for the problem of a rectangular channel with one moving boundary are included along with an experimental verification of the results using the thin plate bending analogy.

An analytical contour integration method for handling body forces in elasticity

Abstract In the boundary element and superposition methods for two-dimensional elasticity problems, the presence of body forces requires the integration of the basic point load solution against the body force field over a specified area. For polynomial body forces, x m y n , these area integrals may be transformed into contour integrals which have closed form analytic representations when the area is a polygon. This paper shows that commonly used numerical integration techniques developed for area integrals are unsuitable for handling body forces using the above named methods. Not only are these techniques inaccurate but expensive in terms of execution time in comparison to the analytical algorithm developed here. A computer program and four example problems are included.

Analysis of torsion of orthotropic bars using personal computers

Abstract A simple procedure for analysis of torsion of orthotropic bars is presented. It considers an equivalent isotropic region for which the warping function satisfies Laplaces equation. The method requires minimum computer time and memory and is suited to personal computers. To demonstrate the accuracy as well as the simplicity of the method, a straightforward algorithm and an example problem are included.

A unified computer program for the solution of ordinary differential equations

Abstract This paper describes a compact program for the solution of ordinary differential equations. The program generalizes all one-step methods including Eulers Heuns, Ralstons, Runge-Kutta, Gills and other approaches. The idea is to consider any individual method as a particular case of a more general ‘weighted-Average-Slope’ method. A list of the program in BASIC along with an example problem to illustrate its versatility are included.

Exact evaluation of ∫dsr2 over a circular element

Abstract A simple analytical scheme to evaluate the essential integral in the Boundary Element Method (BEM) is derived. It is shown that better accuracy is achieved whilethe computer time is considerably reduced. Application to noncircular elements is also discussed and an example problem is included.