The present paper discusses the boundary integral analysis technique and its characteristics as a stress analysis tool. In particular, the fundamental differences between this technique and the more familiar finite element method are discussed.
Initially, attention will focus on a discussion of the theoretical preliminaries necessary to the understanding of this modeling technique. This discussion is a broad overview of the manner in which the boundary integral equation is developed for the elastostatic problem. This discussion is followed by arguments that allow the conversion of the integral equation to a set of simultaneous algebraic equations that represent the physical problem under investigation.
Example problems document the effectiveness of this analysis technique and conclude this paper.
FROM A MATHEMATICAL POINT OF VIEW, the process of structural analysis may be viewed as the process of determining a solution to the governing differential equation which in turn satisfies all boundary conditions imposed on the system being examined. This is nothing more than a definition of the classic boundary value problem and will be used in this paper as the starting point in the development of the Boundary Integral Equation. In particular, the Boundary Integral method may be thought of as a series of mathematical manipulations that allows one to replace the boundary value problem with an equivalent statement known as the boundary integral equation. In turn, it is the boundary integral equation that is then approximated numerically.
If one can accept the fact that such a mathematical process is possible, several important observations can be made at this point. As the name implies, boundary integrals are integral to be carried out on the boundary or surface of the region being studied. Thus, the mathematical dimension of the problem has been reduced. For example, a two-dimensional problem becomes a series of line integral surrounding the region being examined. Three-dimensional problems will reduce to surface integral, again, that bound the region.
Although this may seem of only minor significance, it is at this very point that the boundary integral method achieves its greatest advantage over other numerical techniques. Since only integrals on the boundary of the region are to be determined, only the boundary of the problem needs to be described. In other words, to conduct an elastic stress analysis of a two-dimensional region will involve discretizing only the boundary of the region. No interior modeling as in the finite element method is necessary.
Discussion will now center on the manner in which the mathematical manipulation eluded to in the previous paragraph is carried out. For simplicity, a detailed discussion of the Laplace Equation will follow. Many ideas that are relevant to the boundary integral method are easily demonstrated with this problem. This will be followed by a discussion of the Navier problem.