Boundary Element Techniques

Engineers and physical scientists have in recent years become very conversant with numerical techniques of analysis. These techniques are based on the approximate solution of an equation or set of equations describing a physical problem. The first widely known approximate method was finite differences which approximates the governing equations of the problem using local expansions for the variables, generally truncated Taylor series. The technique can be interpreted as a special case of the more general weighted residual methods as shown in Section 1.8.

The boundary element method is now firmly established as an important alternative technique to the prevailing numerical methods of analysis in continuum mechanics. One of the most important applications is for the solution of a range of problems such as temperature diffusion, some types of fluid flow motion, flow in porous media, electrostatics, and many others which can be written as a function of a potential and whose governing equation is the classical Laplace or Poisson equation. All the cases are potential problems and they can generally be efficiently and economically analyzed using boundary elements.

It was assumed in Chapter 2 that u and q were constant over each element. In general, however, u and q can have a linear or higher-order variation. In addition, their functional behavior need not be of the same order and, for instance, it may be more consistent to take q of one order less than u since q is given by the derivative of the potential. In practice it is simpler to take both functions u and q of the same order as otherwise the computational procedures become more involved.

In this chapter, we study boundary integral solutions to the diffusion equation with boundary conditions of the following types:

This chapter is partly devoted to introducing some basic concepts of the theory of elasticity needed for developing boundary element models. The chapter starts by reviewing the small strain theory of elasticity in accordance with standard texts on the subject [1–5].

Although applications of integral equations in elasticity were already known in the 1960s, it was only during the last decade that the first papers on nonlinear material problems appeared. The first publication on this subject was due to Swedlow and Cruse [1] in 1971. The article was concerned with the generalization of the strain hardening elastoplastic constitutive equations, previously presented by the first author, to compressible and anisotropic plastic flow, and presented an extended form of Somigliana’s identity including plastic strain rates. In addition, the starting boundary integral equation for the direct boundary element formulation was first introduced, for three-dimensional problems, but examples were not shown nor was the integral expression for internal stresses given. The authors, however, pointed out the existence of a domain integral which accounts for the plastic strains contribution to the formulation.

In this chapter the boundary element equations presented in Chapter 6 are employed to solve problems concerned with the inviscid or classical theory of plasticity. An application of the initial strain equations for incompressible plastic strains is first introduced in conjunction with the von Mises yield criterion and Mendelson’s successive elastic solutions method [4]. This simple solution technique, also called “elastic predictor-radial corrector method” by Schreyer et al. [10], has proved to be very efficient and stable with reference to the load increment size. The initial stress equations, on the other hand, are more general and are here implemented to handle four different yield criteria (Tresca, Mises, Mohr-Coulomb, and Drucker-Prager) with two different iterative routines. The first is a pure incremental technique comparable to what was used by Zienkiewicz et al. [11] for finite elements. The second deals with accumulated values of the initial stresses in a similar fashion to the initial strain implementation.

In the present chapter an application of the boundary element equations to viscoplasticity is presented. The procedure can be used for creep problems as well. The Perzyna [1–3] approach has been adopted since it is appropriate for computer applications and — as indicated in Chapter 6 — can be used to simulate pure elastoplastic solutions. The time-dependent solution is obtained by a simple Euler one-step procedure and some guidelines for the selection of the time step length are discussed. In addition, problems involving no-tension materials are also presented and illustrated by examples.

This chapter deals with the boundary integral theory of plate bending and some of its applications using boundary elements. The usual assumptions of thin plate bending theory are reviewed and applied in a weighted residual manner, following the concepts presented by Washizu [1] and other authors [2]. Plates with transverse shear deformation present a much simpler formulation than the one for thin plates. This occurs because when shear deformation is included in the formulation, the displacements and rotations are independent of each other, while for thin plates they are not.

The phenomenon of wave propagation is frequently encountered in a variety of engineering disciplines. For instance, in the design of antennas, it is important to know the interaction with electromagnetic waves. In earthquake analysis, knowledge of the elastodynamic wave propagation is essential. Problems of radiation and scattering of water waves are common features in fluid mechanics where they appear in the design of harbors, breakwaters, off-shore structures, etc.

Some applications of the boundary element method in fluid mechanics have already been discussed in Chapters 2–4 and 9. However, applications of the method in this field are by no means restricted to the cases treated in those chapters. In fact, a wide variety of fluid mechanics problems, some of which involving rather complex features such as nonlinearities, moving boundaries, etc., have been successfully dealt with using boundary elements.

Many engineering problems present a certain amount of coupling or interaction between different parts or systems. For instance, systems representing structure, fluid, and soil can be coupled within the same problem, each part represented by a physical region over which a particular numerical solution can be applied. Fluids such as water, air, or lubricants may be interacting against structural elements such as buildings, dams, offshore structures, mechanical components, pressure vessels, etc. Surface structures interact with the soil through their foundations and the behavior of buried structures is strongly coupled with the surrounding rock or soil strata. In many cases it is possible to assume that for all practical purposes, the effect of one system upon the other does not occur concurrently. Typical examples of this uncoupled behavior are wind forces on stiff buildings and hydrodynamic forces on massive off-shore gravity platforms. For these cases the forces on the structure can be computed assuming that the structure is rigid and neglecting the interaction with the surrounding fluid. Boundary elements are recommended to solve these problems due to their ability to model domains extending to infinity. Problems such as wave diffraction, harbor resonance, fluid flow, etc., have been frequently solved using boundary elements. For these cases the boundary element technique offers a very simple data input by comparison with methods such as finite elements of finite differences.

We will now describe a simple Fortran computer program for the solution of two-dimensional elastostatic problems using linear boundary elements, i.e., elements with linear variations of displacements and tractions.