Chapter 16 – Numerical Finite and Boundary Element Methods

Analytical solutions to elasticity problems are normally accomplished for regions and loadings with relatively simple geometry. However, most real-world problems involve structures with complicated shape and loading. This has led to the development of many numerical solution methods for elastic stress analysis. Over the past several decades, two methods have emerged that provide necessary accuracy, general applicability, and ease of use, and this has led to their wide acceptance by the stress analysis community. The first of these techniques is known as the finite element method (FEM) and involves dividing the body under study into a number of pieces or subdomains called elements. Because element size, shape, and approximating scheme can be varied to suit the problem, the method can accurately simulate solutions to problems of complex geometry and loading, and thus it has become a primary tool for practical stress analysis. The second numerical scheme, called the boundary element method (BEM), is based on an integral statement of elasticity. This statement may be cast into a form with unknowns only over the boundary of the domain under study. The boundary integral equation can then be solved using finite element concepts, and thus the method can accurately solve a large variety of problems. This chapter provides an overview of each method, focusing on narrow applications for two-dimensional elasticity problems. The primary goal is to establish a basic level of understanding that will allow a quick look at applications and enable connections to be made between numerical solutions (simulations) and those developed analytically in the previous chapters.