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This text grew out of a set of notes, originally created for a graduate class on Finite Element (FE) analysis of structures that I teach in the Civil and Environmental Engineering department of Virginia Tech.
Fundamentals of Finite Element Analysis attempts to provide an understanding or the salient features of linear FE analysis, beginning with the mathematics describing an actual physical problem and continuing with the introduction of the FE approximation and the calculation of pertinent arrays and vectors.
The mathematical description of various physical problems in Fundamentals of Finite Element Analysis (primarily elasticity and heat conduction) employs the “strong form–weak form” paradigm. Given a physical problem, the strong form (governing differential equations and boundary conditions) is established.
Then, a weak form is obtained, into which the finite element approximation is subsequently introduced. I believe the strong form–weak form paradigm is a more efficient way of teaching the method than, for example, variational principles, because it is fairly straightforward and also more powerful from a teaching point of view.
For example, the derivation of the weak form for solid and structural mechanics provides a formal proof of the principle of virtual work. Fundamentals of Finite Element Analysis contains a total of 19 chapters and 4 appendices. Chapter 1 constitutes an introduction, with an explanation of the necessity of numerical simulation, the essence of FE approximations and a brief presentation of the early history of FE analysis.
Chapters 2 and 3 present the conceptual steps required for setting up and solving the FE equations of a one-dimensional physical problem.
Specifically, Chapter 2 describes the process by which we can establish the governing differential equations and boundary conditions, which we collectively call the strong form of a problem, and how to obtain the corresponding weak form, which is an alternative mathematical statement of the governing physics, fully equivalent to the strong form.
The weak form turns out to be more convenient for the subsequent introduction of the FE approximation. The latter is described in Chapter 3, wherein we see how the discretization of a domain into a finite element mesh and the stipulation of a piecewise approximation allows us to transform the weak form to a system of linear equations.
Computational procedures accompanying finite element analysis, such as Gaussian quadrature, are also introduced in Chapter 3. Chapter 4 establishes some necessary mathematical preliminaries for multidimensional problems, and Chapters 5 and 6 are focused on the strong form, weak form, and FE solution for scalar-field problems.
Emphasis is laid on heat conduction, but other scalar field problems are discussed, such as flow in porous media, chemical diffusion, and inviscid, incompressible, irrotational fluid flow. The similarity of the mathematical structure of these problems with that of heat conduction and the expressions providing the various arrays and vectors for their FE analysis is established.
Chapter 7 introduces fundamental concepts for multidimensional linear elasticity. Any reader who wants to focus on multidimensional solid and structural mechanics should read this chapter.
Chapters 8 and 9 present the FE solution of two-dimensional and three-dimensional problems in elasticity (more specifically, elastostatics, wherein we do not have inertial effects).
Chapter 10 is devoted to some important practical aspects of finite element problems, such as the treatment of constraints, field-dependent natural boundaries (e.g., spring supports), and how to take advantage of symmetry in analysis.
Chapters 11 and 12 are primarily focused on the application of the method to solid mechanics. Chapter 11 discusses advanced topics such as the use of incompatible modes to alleviate the effect of parasitic shear stiffness, and volumetric locking and its remedy through use of uniform- or selective-reduced integration.
The side-effect of spurious zero-energy (hourglass) modes for uniform-reduced integration is identified, and a procedure to establish hourglass control, by means of artificial stiffness, is described.
Chapter 12 is an introduction to multifield (mixed) finite element formulations, wherein the finite element approximation includes more than one field.