The BEASY code is based on the Boundary Element Method (BEM) of Analysis which offers important advantages over the classical finite element approach. In BEM only the boundary needs to be discretized thus reducing by one dimension the data required to run a problem. Direct boundary element solution of the type used in BEASY gives displacements and stresses (or temperature and fluxes, etc.) with the same degree of accuracy, which makes the technique well suited to problems — for example involving stress concentration — where reliable and highly accurate results are required. Boundary elements are also well adapted to problems with infinite domains.

Over the past decade the finite element method (FEM) has become established as a valuable tool in the solution of a wide variety of problems in engineering. The FEM may be seen as a method of solving boundary value problems where the phenomenon in the domain being studied obeys known differential equations. In the FEM the domain is discretized into a number of elements in each of which the solution of the governing equation is approximated by some function which satisfies the boundary conditions. A set of equations is then set up which when solved forces the solution at various points in the domain, known as nodal points, to the best approximation allowed by the approximating functions and the boundary conditions.

BEASY is a Boundary Element Analysis System for the solution of problems in engineering. The method's chief advantage over techniques such as Finite Difference and Finite Element lies in the necessity of modelling the problem boundary only. BEASY is ideally suited for the solution of Laplace's equation in potential theory and for the solution of problems in linear elastostatics but may also be used to solve the diffusion and wave equations and problems in elastoplasticity.

Traditional formulations of the Boundary Element Method (BEM) applied to elastostatics [1] [2] are extremely convenient when the loading on the body under analysis is limited to surface loading, since it is necessary to discretize only the boundary of the body and not the whole domain as must be done when using a technique such as the Finite Element Method (FEM). When body forces are present these have usually been handled by evaluating a domain integral [3]. Unfortunately this requires the domain of the problem to be divided into integration cells since for any practical problem the domain integral must be evaluated numerically. This greatly increases the amount of data preparation required and causes the BEM to lose much of its advantage over domain type methods. However, Cruse [4] and Cruse, Snow and Wilson [5] have shown that for certain types of commonly encountered body forces the domain integral may be transformed to a boundary integral or boundary integrals which may be evaluated at the same time as the boundary integrals involving the surface displacements and tractions. Ref. [5] is concerned exclusively with axisymmetric geometry; however, the authors’ use of the Galerkin vector to achieve the required transformation from domain to boundary integrals provides the key to the present paper which relates to two and three dimensional geometry. The three dimensional case was derived in Ref. [4] without resort to the Galerkin vector. However, the Galerkin vector approach is used below both to demonstrate the power of the technique and to present the results of [4] in a slightly more general form. The two dimensional formulation is also presented.

1. Weighted Residual Formulation of Approximate Methods.- 1.1. Introduction.- 1.2. Basic Definition.- 1.3. Approximate Solutions.- 1.4. Method of Weighted Residuals.- 1.5. Weak Formulations.- 1.6. The Inverse Problem.- 1.7. Conclusions.- References.- 2. Boundary Element Methods.- 2.1. Fundamentals of Functional Analysis.- 2.2. Generalized Green's Formula.- 2.3. Variational Formulation.- 2.4. Weighted Residual Scheme.- 2.5. Boundary Element Formulation of Poisson's Equation.- 2.6. Fundamental Solutions.- 2.7. Boundary Discretisation and Systems Equations.- 2.8. Computation of Integrals - 2D case.- 3. Boundary Integral Equations.- 3.1. Simple-layer Formulations.- 3.2. Double-layer Formulations.- 3.3. Direct Formulations.- 3.4. Indirect Vector Formulations.- 3.5. Direct Formulations.- References.- 4. Scalar and Vector Potential Theory.- 4.1. The Simple-layer Potential.- 4.2. The Double-layer Potential.- 4.3. Green's Formula.- 4.4. Identification of Scalar and Vector Symbolism.- 4.5. Somigliana's Identity.- 4.6. Rigid-body Displacement Field.- References.- 5. Potential Problems in Two Dimensions.- 5.1. Introduction.- 5.2. Flow past an Obstacle.- 5.3. Discretisation.- 5.4. Green's Boundary Formula.- 5.5. Applications.- 5.6. Boundary Singularities.- 5.7. Composite Domains.- 5.8. Conclusion.- References.- 6. Three-dimensional Axisymmetrical Potential Problems.- 6.1. Introduction.- 6.2. The Newtonian Potential.- 6.3. Discretisation.- 6.4. General Domain.- 6.5. Axisymmetric Problems.- 6.6. Conclusion.- References.- 7. Heat Transfer Applications.- 7.1. Introduction.- 7.2. Integral Equations associated with Steady Heat Conduction Problems.- 7.3. Numerical Solution of the Integral Equations.- 7.4. Poisson's Equation.- 7.5. Non-homogeneous Bodies Method of Subregions.- 7.6. Anisotropic Bodies.- References.- 8. Numerical Integration and other Computational Techniques.- 8.1. Introduction.- 8.2. Isoparametric Elements.- 8.3. Numerical Integration.- References.- 9. Starting to work with Boundary Elements.- 9.1. Introduction.- 9.2. The Boundary Element Method.- 9.3. Advantages and Disadvantages of the BEM compared to FEM.- 9.4. Introduction to BEASY.- 9.5. Examples.- 9.6. Conclusions.- References.- 10. Experiences in Boundary Element Applications.- 10.1. Introduction.- 10.2. Pre- and Post Processing.- 10.3. C.A.D. Coupling.- 10.4. Installation on Different Computers.- 10.5. Recommendations for BEM use.- 11. Electrostatics Problems.- 11.1. Introduction.- 11.2. Theoretical Basis.- 11.3. Boundary Elements.- 11.4. Applications.- 11.5. Conclusions.- References.- 12. A Boundary Element Solution of the Wave Equation.- 12.1. Introduction.- 12.2. Theoretical Development.- 12.3. Boundary Conditions.- 12.4. Numerical Implementation.- 12.5. Velocities and Pressures.- 12.6. Identification of Areas in Shadow.- 12.7. Test Example.- 12.8. Conclusions.- References.- 13. Elasticity Problems.- 13.1. Introduction.- 13.2. Governing Equations.- 13.3. Boundary Integral Formulation.- 13.4. Two Dimensional Elasticity Problems.- 13.5. Three Dimensional Elasticity Problems.- 13.6. Axisymmetric Elasticity Problems.- References.- 14. Elasticity Problems with Body Forces.- 14.1. Introduction.- 14.2. Transformation to Boundary Integrals.- 14.3. 2D Body Forces.- 14.4. 3D Body Forces.- 14.5. Axisymmetric Body Forces.- References.- 15. Time Dependent Problems.- 15.1. Introduction.- 15.2. Time Dependent Diffusion.- 15.3. The Scalar Wave Equation.- 15.4. Transient Elastodynamics.- 15.5. Mass Matrix Representation.- 15.6. Conclusions.- References.- 16. Time Dependent Potential Problems.- 16.1. Introduction.- 16.2. Integral Formulation of Heat Conduction Problems.- 16.3. Numerical Solution of the Integral Equations.- 16.4. Conclusions.- References.- 17. Plate Bending Problems.- 17.1. Preliminaries.- 17.2. Reciprocal Work Relation.- 17.3. Boundary Integral Representations.- 17.4. Concluding Remarks.- References.- 18. A Choice of Fundamental Solutions.- 18.1. Introductory Remarks.- 18.2. A simple example: 2D Heat Conduction.- 18.3. A more significant example: Plane Elastostatics.- 18.4. Concluding Remarks.- References.- 19. Formulation for Cracks in Plate Bending.- 19.1. Fundamental Solutions for Cracks.- 19.2. Augmented Boundary Integral Equations.- 19.3. Concluding Remarks.- References.- 20. Fracture Mechanics Stress Analysis, I..- 20.1. Introduction.- 20.2. Stress Intensity Factors.- 20.3. Integral Equation Methods for Crack Tip Stress Analysis.- References.- 21. Fracture Mechanics Stress Analysis, II.- 21.1. Introduction.- 21.2. Invariant Integral based on the Energy Momentum Tensor.- 21.3 Invariant Integrals deduced from Betti's Reciprocal Theorem.- 21.4. Some Numerical Results for a Nocht Problem.- 21.5. A Problem of Debond Stress Analysis.- References.- 22. BEM in Geomechanics.- 22.1. Introduction.- 22.2. Notation and some Basic Ideas.- 22.3. BEM applied to the Interaction between Structures and the Supporting Ground.- 22.4. Inhomogeneity, Zoning and Layering.- 22.5. Elastoplasticity.- 22.6. Concluding Remarks.- References.- 23. An Asymptotic Error Analysis and Underlying Mathematical Principles for Boundary Element Methods.- 23.1. Projection Methods and Garding's Inequality.- 23.2. Examples of Strongly Elliptic Boundary Integral Equations.- 23.3. Asymptotic Convergence of Galerkin type Boundary Element Methods.- 23.4. Asymptotic Convergence of Collocation Methods.- References.