In this chapter we discuss the mathematical foundation for obtaining the finite element equations for a general engineering problem or a physical system. Before studying the material in this chapter, it is essential to review the concepts of matrix algebra and indicial notation discussed in Appendix A. The chapter will provide a detailed discussion of how to formulate finite element equations using variational principles and weighted residual methods. The development will start with simple one-dimensional problems and will then proceed to full three-dimensional cases. The focus will be on deriving the FE equations in linear elasticity and heat transfer applications. The equations developed and presented here will be the basis for our discussion in the following chapters: the linear elasticity and heat transfer chapters.

In this chapter, we provide a simple introduction to the finite element method (FEM) and how it is related to other solution methods for engineering and physics problems. Throughout this chapter and the next, we avoid rigorous mathematical developments and equations and use simple examples that are easily understood by all students. We identify five basic steps or stages for any type of finite element analysis. These are: modeling and discretization; formulation and element equations; assembly; boundary conditions and solution; and finally postprocessing. In commercial FE programs these steps are lumped into three stages: modeling; solution; and postprocessing. One of the five steps, namely the assembly process, is quite simple and straightforward. A student may actually write a simple and general program in just one page that will do the assembly process. On the other hand, most of the research done and the textbooks written in the finite element area involve one or more of the other basic steps or stages. Throughout the introduction of the basic steps of the FE method, we introduce definitions of conceptual terminology that are common in the FE field, e.g., elements, nodes, boundary conditions, degrees of freedom (DOFs). To enable the students to start the modeling step we highlight the common elements used in most commercial programs and their geometry, nodes and DOFs.

This chapter, then, provides a brief account of the history of the development of the FE method. This is presented in two parts: the history of the development of the formulation and algorithms of the method; and the history of the development of computer hardware and software related to the application of the method. The final section of the chapter presents some typical applications of the FEM, mostly from the work of the author. These are meant to give students an overview of the capabilities and limitations of the method in various fields.

Acid deposition, better known as acid rain,1 goes far beyond the deposition of acid pollutants. It is atmospheric deposition, a phenomenon in which airborne chemicals and particulates – whether acids, metals, organic chemicals, microbes, or pollens – deposit from air onto land and water. You became familiar with fine particulates, PM2.5, in Chapter 5 and know that many chemicals can be associated with particulate matter.

We are reminded in Section 4.1 that a hazardous chemical is a risk only if there is exposure to it. Section 4.2 summarizes the ongoing studies of the US Centers for Disease Control and Prevention (CDC) on human exposure to more than 300 chemicals. Section 4.3 introduces epidemiology, a discipline that studies possible relationships between human exposure to a particular chemical and its possible adverse health effects. We see the many difficulties posed in doing a good epidemiological study, but that they are important. Section 4.4 introduces chemical risk assessment, which is done to calculate a dose that is safe for human exposure; the four steps of an assessment are explained. Box 4.1 describes the use of factors of 10. An example of how risk assessment can lead to controversy is explained in Box 4.2. Section 4.5 explains that the purpose of risk management is reducing exposure to the chemical in question. Regulatory methods are often used, but other approaches are also noted. Section 4.6 describes why reducing risk to children has special importance. It also looks at chemical risks in less-developed countries versus developed countries. Section 4.7 discusses how chemical risk assessment continues to be done in animals; however, that may rapidly change with Tox21, a major effort that is radically changing chemical risk assessment. This uses automated means of testing chemical toxicity and does so rapidly and without animals. Section 4.8 concludes the chapter.