Dual Reciprocity Boundary Element Method

The boundary element method (BEM) is now a well-established numerical technique which provides an efficient alternative to the prevailing finite difference and finite element methods for the solution of a wide range of engineering problems. The main advantage of the BEM is its unique ability to provide a complete problem solution in terms of boundary values only, with substantial savings in computer time and data preparation effort. An initial restriction of the BEM was that the fundamental solution to the original partial differential equation was required in order to obtain an equivalent boundary in­ tegral equation. Another was that non-homogeneous terms accounting for effects such as distributed loads were included in the formulation by means of domain integrals, thus making the technique lose the attraction of its "boundary-only" character. Many different approaches have been developed to overcome these problems. It is our opinion that the most successful so far is the dual reciprocity method (DRM), which is the subject matter of this book. The basic idea behind this approach is to employ a fundamental solution corresponding to a simpler equation and to treat the remaining terms, as well as other non-homogeneous terms in the original equation, through a procedure which involves a series expansion using global approximating functions and the application of reciprocity principles.

1 Introduction.- 2 The Boundary Element Method for Equations ?2u = 0 and ?2u = b.- 2.1 Introduction.- 2.2 The Case of the Laplace Equation.- 2.2.1 Fundamental Relationships.- 2.2.2 Boundary Integral Equations.- 2.2.3 The Boundary Element Method for Laplace’s Equation.- 2.2.4 Evaluation of Integrals.- 2.2.5 Linear Elements.- 2.2.6 Treatment of Corners.- 2.2.7 Quadratic and Higher-Order Elements.- 2.3 Formulation for the Poisson Equation.- 2.3.1 Basic Relationships.- 2.3.2 Cell Integration Approach.- 2.3.3 The Monte Carlo Method.- 2.3.4 The Use of Particular Solutions.- 2.3.5 The Galerkin Vector Approach.- 2.3.6 The Multiple Reciprocity Method.- 2.4 Computer Program 1.- 2.4.1 MAINP1.- 2.4.2 Subroutine INPUT1.- 2.4.3 Subroutine ASSEM2.- 2.4.4 Subroutine NECMOD.- 2.4.5 Subroutine SOLVER.- 2.4.6 Subroutine INTERM.- 2.4.7 Subroutine OUTPUT.- 2.4.8 Results of a Test Problem.- 2.5 References.- 3 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y).- 3.1 Equation Development.- 3.1.1 Preliminary Considerations.- 3.1.2 Mathematical Development of the DRM for the Poisson Equation.- 3.2 Different f Expansions.- 3.2.1 Case f = r.- 3.2.2 Case f = 1+ r.- 3.2.3 Case f = 1 at One Node and f = r at Remaining Nodes.- 3.3 Computer Implementation.- 3.3.1 Schematized Matrix Equations.- 3.3.2 Sign of the Components of r and its Derivatives.- 3.4 Computer Program 2.- 3.4.1 MAINP2.- 3.4.2 Subroutine INPUT2.- 3.4.3 Subroutine ALFAF2.- 3.4.4 Subroutine RHSVEC.- 3.4.5 Comparison of Results for a Torsion Problem using Different Approximating Functions.- 3.4.6 Data and Output for Program 2.- 3.5 Results for Different Functions b = b(x,y).- 3.5.1 The Case ?2u = ?x.- 3.5.2 The Case ?2u = ?x2.- 3.5.3 The Case ?2u = a2 ? x2.- 3.5.4 Results using Quadratic Elements.- 3.6 Problems with Different Domain Integrals on Different Regions.- 3.6.1 The Subregion Technique.- 3.6.2 Integration over Internal Region.- 3.7 References.- 4 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u).- 4.1 Introduction.- 4.2 The Convective Case.- 4.2.1 Results for the Case ?2u = ??u/?x.- 4.2.2 Results for the Case ?2u = ?(?u/?x+ ?u/?y).- 4.2.3 Internal Derivatives of the Problem Variables.- 4.3 The Helmholtz Equation.- 4.3.1 DRM Formulations.- 4.3.2 DRM Results for Vibrating Beam.- 4.3.3 Results for Non-Inversion DRM.- 4.4 Non-Linear Cases.- 4.4.1 Burger’s Equation.- 4.4.2 Spontaneous Ignition: The Steady-State Case.- 4.4.3 Non-Linear Material Problems.- 4.5 Computer Program 3.- 4.5.1 MAINP3.- 4.5.2 Subroutine ALFAF3.- 4.5.3 Subroutine RHSMAT.- 4.5.4 Subroutine DERIVXY.- 4.5.5 Results of Test Problems.- 4.6 Three-Dimensional Analysis.- 4.6.1 Equations of the Type ?2u = b(x, y, z).- 4.6.2 Equations of the Type ?2u = b(x, y, z, u).- 4.7 References.- 5 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u, t).- 5.1 Introduction.- 5.2 The Diffusion Equation.- 5.3 Computer Program 4.- 5.3.1 MAINP4.- 5.3.2 Subroutine ASSEMB.- 5.3.3 Subroutine VECTIN.- 5.3.4 Subroutine BOUNDC.- 5.3.5 Results of a Test Problem.- 5.3.6 Data Input.- 5.3.7 Computer Output.- 5.3.8 Further Applications.- 5.3.9 Other Time-Stepping Schemes.- 5.4 Special f Expansions.- 5.4.1 Axisymmetric Diffusion.- 5.4.2 Infinite Regions.- 5.5 The Wave Equation.- 5.5.1 Infinite and Semi-Infinite Regions.- 5.6 The Transient Convection-Diffusion Equation.- 5.7 Non-Linear Problems.- 5.7.1 Non-Linear Materials.- 5.7.2 Non-Linear Boundary Conditions.- 5.7.3 Spontaneous Ignition: Transient Case.- 5.8 References.- 6 Other Fundamental Solutions.- 6.1 Introduction.- 6.2 Two-Dimensional Elasticity.- 6.2.1 Static Analysis.- 6.2.2 Treatment of Body Forces.- 6.2.3 Dynamic Analysis.- 6.3 Plate Bending.- 6.4 Three-Dimensional Elasticity.- 6.4.1 Computational Formulation.- 6.4.2 Gravitational Load.- 6.4.3 Centrifugal Load.- 6.4.4 Thermal Load.- 6.5 Transient Convection-Diffusion.- 6.6 References.- 7 Conclusions.- Appendix 1.- Appendix 2.- The Authors.