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CHAPTER 1 1-1 NUMERICAL METHODS For the last two or three decades, scientists and engineers have used numerical methods as an important tool in many different areas. This significant fact has its inexorable historical trend and it is the inevitable outcome of the recent developments in science, technology and industry. Analytical methods have been developed for a long period and have produced a great amount of successful results, but they failed to solve most practical engineering problems with complicated boundary conditions or irregular geometry. It is also very difficult to solve non-linear or time-dependent problems using analytical approaches, even if they are very simple. On the other hand, research on analytical methods has provided a solid foundation for different types of numerical methods. Because of the rapid developments of science and technology it is now necessary to solve complicated problems using more efficient and accurate approaches than before. Not only problems with complicated boundary conditions or irregular configurations require solutions but also non-linear or time-dependent problems must be solved. Computer hardware and software have developed at an unexpected high speed. During the last thirty years, ithaz become possible for scientists and engineers to use numerical methods with computers easily. This has 2 stimulated scientists and engineers to improve some classical numerical methods (such as finite difference method) and to establish new numerical methods (such as the finite element method and boundary element method). For all these reasons, numerical methods have rapidly developed in the areas of mechanics and engineering.
List of contents
1 General Introduction.- 1-1 Numerical Methods.- 1-2 Domain Methods.- 1-3 Boundary Element Method.- 1-4 The Main Procedures and Features of BEM.- 1-5 The Subject of this Work.- 1-6 Contents of the Present Work.- 1-7 The Cartesian Tensor Notation.- 2 Potential Problems.- 2-1 Introduction.- 2-2 The Boundary Integral Formulation for Potential Problems.- 2-3 The Boundary Element Method for Potential Problems.- 2-4 Motivation and General Ideas.- 2-5 Fourier Analysis.- 2-6 Basic Formulations for Transforming the Domain Integrals into the Boundary for 2-D Problems.- 2-7 Numerical Approaches.- 2-8 Numerical Accuracy of the Transformation Formula.- 2-9 Some Further Discussions.- 2-10 Examples.- 2-11 The Transformation Formula for 3-D Poisson's Equation.- 2-12 Applications in Time-dependent Problems.- 2-13 Application in Non-linear Problems.- 3 Linear Elastostatics.- 3-1 Introduction.- 3-2 Basic Relationships of Elasticity.- 3-3 Fundamental Solution for Elastostatics.- 3-4 Somigliana Identity.- 3-5 The Boundary Integral Equations of Elastostatics.- 3-6 The Boundary Element Method in Elasticity.- 3-7 Basic Formulations for Transforming 2-D Elasticity Domain Integrals to the Boundary.- 3-8 Numerical Implementation.- 3-9 Results of Numerical Experiments.- 4 Applications in Elasticity and Elasto-Plasticity.- 4-1 Introduction.- 4-2 An Example of Gravitational Loading.- 4-3 An Example with a More General Type of Distributed Loading.- 4-4 Relationship between Plastic Stresses and Plastic Strains.- 4-5 The Governing Equations for Elasto-Plasticity.- 4-6 Numerical Analysis using Finite Fourier Series.- 4-7 Application to Elasto-plastic Problems.- 5 Programming.- 5-1 Potential Problems.- 5-2 Elasticity Problems.- 5-3 Elasto-Plasticity Problems.- 6 General Discussion and Conclusions.-References.
