Finite Element Method (FEM) is a numerical method for solving differential equations that describe many engineering problems. One of the reasons for FEM's popularity is that the method results in computer programs versatile in nature that can solve many practical problems with a small amount of training. Obviously, there is a danger in using computer programs without proper understanding of the theory behind them, and that is one of the reasons to have a thorough understanding of the theory behind FEM.

Many universities teach FEM to students at the junior/senior level. One of the biggest challenges to the instructor is finding a textbook appropriate to the level of students. In the past, FEM was taught only to graduate students who would carry out research in that field. Accordingly, many textbooks focus on theoretical development and numerical implementation of the method. However, the goal of an undergraduate FEM course is to introduce the basic concepts so that the students can use the method efficiently and interpret the results properly. Furthermore, the theoretical aspects of FEM must be presented without too much mathematical niceties. Practical applications through several design projects can help students to understand the method clearly.

This book is suitable for junior/senior level undergraduate students and beginning graduate students in engineering mechanics, mechanical, civil, aerospace, biomedical and industrial engineering as well as researchers and design engineers in the above fields.

The textbook is organized into ten chapters. The Appendix at the end summarizes most mathematical preliminaries that are repeatedly used in the text. The Appendix is by no means a comprehensive mathematical treatment of the subject. Rather, it provides a common notation and the minimum amount of mathematical knowledge that will be required in using the book effectively. This includes basics of matrix algebra, minimization of quadratic functions, and techniques for solving linear equations that are commonly used in commercial finite element programs.

The book begins with the introduction of finite element concepts via the direct stiffness method using spring elements. The concepts of nodes, elements, internal forces, equilibrium, assembly, and applying boundary conditions are presented in detail. The spring element is then extended to the uniaxial bar element without introducing interpolation. The concept of local (elemental) and global coordinates and their transformations and element connectivity tables are introduced via two– and three–dimensional truss elements. Four design projects are provided at the end of the chapter, so that students can apply the method to real life problems. The direct method in Chapter 1 provides a clear physical insight into FEM and is preferred in the beginning stages of learning the principles. However, it is limited in its application in that it can be used to solve one–dimensional problems only.

The direct stiffness method becomes impractical for more realistic problems especially multi‐dimensional problems. In Chapter 2, we introduce more general approaches, such as, the Weighted Residual Methods and, in particular, the Galerkin Method. Similarity to energy methods in solid and structural mechanics problems is discussed. We include a simple 1–D variational formulation in Chapter 2 using boundary value problems. The concept of polynomial approximation and domain discretization is introduced. The formal procedure of finite element analysis is also presented in this chapter. Chapter 2 is written in such way that it can be left out in elementary level courses.

The 1–D formulation is further extended to beams and plane frames in Chapter 3. At this point, the direct method is not useful because the stiffness matrix generated from the direct method cannot provide a clear physical interpretation. Accordingly, we use the principle of minimum potential energy to derive the matrix equation at the element level. The 1–D beam element is extended to 2–D frame element by using coordinate transformation. A 2–D bicycle frame design project is included at the end of this chapter. Buckling of beams and plane frames is included in the revised second edition. First, the concepts of linear buckling of beam is introduced using the Rayleigh-Ritz method. Then the corresponding energy terms are derived in the finite element context.

The finite element formulation is extended to the steady–state heat transfer problem in Chapter 4. Both direct and Galerkin's methods along with convective boundary conditions are included. Two-dimensional heat transfer problems are discussed in the second edition. Practical issues in modeling 2D heat transfer problems are also discussed.

Before proceeding to solid elements in Chapter 6, a review of solid mechanics is provided in Chapter 5. The concepts of stress and strain are presented followed by constitutive relations and equilibrium equations. We limit our interest to linear, isotropic materials in order to make the concepts simple and clear. However, advanced concepts such as transformation of stress and strain, and the eigen value problem for calculating the principal values, are also included. Since, in practice, FEM is used mostly for designing a structure or a mechanical system, failure/yield criteria are also introduced in this chapter.

In Chapter 6, we introduce 2–D solid elements. The governing variational equation is developed using the principle of minimum potential energy. The finite element concepts are explained in detail using only triangular and rectangular elements. Numerical performance of each element is discussed through examples. A new addition to the second edition is the axisymmetric element as it is essentially a plane problem.

The concept of isoparametric mapping is introduce in a separate chapter (Chapter 7) as most practical problems require irregular elements such as linear or higher order quadrilateral elements. Three-dimensional solid elements are introduced in this chapter. Numerical integration and FE modeling practices for isoparametric elements are also included.

Dynamic problems is another addition to the second edition. The concept of free vibration, calculation of natural frequencies and mode shapes, various time integration methods and mode superposition method, are all explained using 1-D structural elements such as uniaxial bars and beams.

In Chapter 9, we discuss traditional finite element analysis procedures, including preliminary analysis, pre-processing, solving matrix equations, and post-processing. Emphasis is on selection of element types, approximating the part geometry, different types of meshing, convergence, and taking advantage of symmetry. A design project involving 2–D analysis is provided at the end of the chapter.

As one of the important goals of FEM is to use the tool for engineering design, the last chapter (Chapter 10) is dedicated to the topic of structural design using FEM. The basic concept of design parameterization and the standard design problem formulation are presented. This chapter is self contained and can be skipped depending on the schedule and content of the course.

Each chapter contains a comprehensive set of homework problems, some of which require commercial FEA programs. A total of nine design projects are provided in the book.

We are thankful to several instructors across the country who used the first edition and provided feedback. We are grateful for their valuable suggestions especially regarding example and exercise problems.