Matlab for Engineering

The following sections are included:

• Preface
• Contents

Matlab¹ derives its power from its extensive capability of manipulating matrices. Therefore, this topic is discussed in the first section of this chapter. The name “Matlab” cames from Mat(rix) lab(oratory). Matlab programs are written in files with an .m extension. There are two kinds of m-files: script files and function files. The latter are much more interesting than the former. Script files are illustrated in Sec. 1.2 while function files are discussed in Sec. 1.3. See Moler (2011), which could be useful for the topics in Sec. 1.1.

The chapter presents the Finite Difference Method (FDM). This method dates back to Euler¹ who introduced it in Institutiones calculi Differentialis (1755). The modern researches on the FDM started after the paper by Courant, et al. (1928), where the method was used to obtain approximated solutions to Partial Differential Equations (PDEs). In this field, the method was improved mainly after the Second World War when powerful computers were available. The books by Collatz (1966); Forsythe and Wasov (1960), and Richtmyer and Morton (1967) had a great role in stimulating research on the FDM. Other books by Cooper (1998); Kharab and Guenther (2002) considered Matlab applications too. Today, the FDM is considered a consolidated tool that is able to provide reliable solutions of PDEs and is used by scientists and technicians in many scientific areas, e.g., D’Acunto (2004); de Vahl Davis (1986). In this chapter, FDM will be applied to the heat equation by introducing these noteworthy methods: Explicit Euler Method Implicit Euler Method and Crank–Nicolson Method…

The convection-diffusion physical process is very important since it is involved in many engineering problems. The convection-diffusion equation, named the advection-diffusion equation in some contexts, was derived in Chap. 2. This equation models two physical processes: the diffusive process, governed by the diffusion equation, and the convection (or advection) process, governed by the convection (or advection) equation Crank (1979). The Finite Difference Method for the convection-diffusion equation is presented in the first section of this chapter Lapidus and Pinter (1982); Mitchell and Griffiths (1995); Necati Ozisik (1994)…

The Finite Element Method (FEM) is introduced in this chapter Hutton (2004). Scientific research on the FEM began after the Second World War and was partially supported by the American aerospace industry. The name appeared for the first time in the paper by Clough (1960). The Engineering community immediately understood the power of the new method. The structural engineers made a great contribution to its development. Expressions such as stiffness matrix, load vector, and others initially introduced for applications in Structural Mechanics, are used today for FEM applications in different scientific fields Fenner (2005). Matlab applications are presented in this book by Kwon and Bang (2000)…

The Finite Element Method (FEM) in two spatial dimensions is presented. Since the method is applied to elliptic partial differential equations, the section below is devoted to this topic, where the uniqueness of the solutions of the main boundary value problems is discussed. In addition, Green’s identities are introduced, as they are related to the weak formulation…

The equation governing the lateral vibrations of a beam is derived in the framework of the Euler–Bernoull theory. The weak formulation is presented and the Finite Element Method (FEM) is introduced Fenner (2005); Rao (2005). The application of the method is illustrated with problems in Statics. The Matlab codes consider beams subjected to several constraints and loads. A section is devoted to beams subjected to concentrated forces. The weak form for the related equation is introduced and Matlab applications are illustrated in this new situation.

The following sections are included:

• Bibliography
• Index