The Finite Element Method

Basic Concepts and Applications with MATLAB, MAPLE, and COMSOL, Third Edition

Darrell W. Pepper, Juan C. Heinrich

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781315395081
• Edition: 3
• Topic: Business
• Subtopic: Business generale
• Index: Business

This self-explanatory guide introduces the basic fundamentals of the Finite Element Method in a clear manner using comprehensive examples. Beginning with the concept of one-dimensional heat transfer, the first chapters include one-dimensional problems that can be solved by inspection. The book progresses through more detailed two-dimensional elements to three-dimensional elements, including discussions on various applications, and ending with introductory chapters on the boundary element and meshless methods, where more input data must be provided to solve problems. Emphasis is placed on the development of the discrete set of algebraic equations. The example problems and exercises in each chapter explain the procedure for defining and organizing the required initial and boundary condition data for a specific problem, and computer code listings in MATLAB and MAPLE are included for setting up the examples within the text, including COMSOL files.

Widely used as an introductory Finite Element Method text since 1992 and used in past ASME short courses and AIAA home study courses, this text is intended for undergraduate and graduate students taking Finite Element Methodology courses, engineers working in the industry that need to become familiar with the FEM, and engineers working in the field of heat transfer. It can also be used for distance education courses that can be conducted on the web. Highlights of the new edition include: - Inclusion of MATLAB, MAPLE code listings, along with several COMSOL files, for the example problems within the text. Power point presentations per chapter and a solution manual are also available from the web.

- Additional introductory chapters on the boundary element method and the meshless method.
- Revised and updated content.
-Simple and easy to follow guidelines for understanding and applying the Finite Element Method.

CHAPTER 1

Introduction

1.1 BACKGROUND

The finite element method is a numerical technique that gives approximate solutions to differential equations that model problems arising in physics and engineering. As with the more commonly used finite difference schemes, the finite element method reduces problems defined in geometrical space (or domain), to finding a solution in a finite number of points by subdividing the domain into smaller regions (a mesh).

In finite difference methods in the past, the mesh consisted of rows and columns of orthogonal lines (in computational space—a requirement now relaxed through the use of coordinate transformations and unstructured mesh generators); in finite elements, each subregion or “element” is unique and need not be orthogonal to the others. For example, triangles or quadrilaterals can be used in two dimensions, and tetrahedra or hexahedra in three dimensions. Over each finite element, the unknown variables (e.g., temperature, velocity, etc.) are approximated using known functions; these functions are usually polynomials that can be linear or higher-order expansions based on the geometrical locations of a few points (nodes) used to define the finite element shape. In contrast to finite difference procedures (conventional finite differences, as opposed to the finite volume method, which is integrated), the governing equations in the finite element method are integrated over each finite element, and the contributions summed (“assembled”) over the entire problem domain. As a consequence of this procedure, a set of finite linear equations is obtained in terms of the values of the unknown parameters at the elements nodes. Solutions of these equations are achieved using linear algebra techniques.

1.2 SHORT HISTORY

The history of the finite element method is particularly interesting, especially since the method as such has only been in existence since the mid-1950s. The early work on numerical solution of boundary-value problems can be traced to the use of finite difference schemes; Southwell (1946) discusses the use of such methods in his book published in the mid-1940s. The beginnings of the finite element method actually stem from these early numerical methods and the frustration associated with attempting to use finite difference methods on more difficult, geometrically irregular problems (Roache, 1972, 1998).

Beginning in the mid-1950s, efforts to solve continuum problems in elasticity using small, discrete “elements” to describe the overall behavior of simple elastic bars began to appear. Argyris (1954) and Turner et al. (1956) were the first to publish use of such techniques for the aircraft industry. Actual coining of the term “finite element” appeared in a paper by Clough (1960).

The early use of finite elements lay in the application of such techniques for structural-related problems. However, others soon recognized the versatility of the method and its underlying rich mathematical basis for application in nonstructural areas. Zienkiewicz and Cheung (1965) were among the first to apply the finite element method to field problems (e.g., heat conduction, irrotational fluid flow, etc.) involving solution of Laplace and Poisson equations. Much of the early work on nonlinear problems can be found in Oden (1972). Early efforts to model heat transfer problems with complex boundaries are discussed in Huebner (1975); a comprehensive 3-D finite element model for heat conduction is described by Heuser (1972). Early application of the finite element technique to viscous fluid flow is given in Baker (1971).

Since these early works, rapid growth in usage of the method has continued since the mid-1970s. Numerous articles and texts have been published, and new applications appear routinely in the literature. Excellent reviews and descriptions of the method can be found in some of the earlier texts by Finlayson (1972), Desai (1979), Becker et al. (1981), Baker (1983), Fletcher (1984), Reddy (1984), Segerlind (1984), Bickford (1990), Zienkiewicz and Taylor (1989), and Reddy (2006). A vigorous mathematical discussion is given in the text by Johnson (1987), and programming the finite element method is described by Smith (1982). A short monograph on development of the finite element method is given by Owen and Hinton (1980).

The underlying mathematical basis of the finite element method first lies with the classical Rayleigh–Ritz and variational calculus procedures introduced by Rayleigh (1877) and Ritz (1909). These theories provided the reasons why the finite element method worked well for the class of problems in which variational statements could be obtained (e.g., linear diffusion type problems). However, as interest expanded in applying the finite element method to more types of problems, the use of classical theory to describe such problems became limited and could not be applied (this is particularly evident in fluid-related problems).

Extension of the mathematical basis to nonlinear and nonstructural problems was achieved through the method of weighted residuals, originally conceived by Galerkin (1915) in the early twentieth century. The method of weighted residuals was found to provide the ideal theoretical basis for a much wider range of problems as opposed to the Rayleigh–Ritz method. Basically, the method requires the governing differential equation to be multiplied by a set of predetermined weights and the resulting product integrated over space; this integral is then required to vanish. Technically, Galerkin’s method is a subset of the general weighted residuals procedure, since various types of weights can be utilized; in the case of Galerkin’s method, the weights are chosen to be the same as the functions used to define the unknown variables.

Galerkin and Rayleigh–Ritz approximations yield identical results whenever a proper variational statement exists and the same basis functions are used. By using constant weights instead of functions, the weighted residual method yields the finite volume technique. A more rigorous description of the method of weighted residuals can be found in Finlayson (1972). More detailed information regarding the method is discussed in the earlier works by Portela and Charafi (2002), Chandrupatla and Belegundu (2002), Liu and Quek (2003), Hollig (2003), Bohn and Garboczi (2003), Hutton (2004), Solin et al. (2004), Reddy (2006), Becker (2004), Ern and Guermond (2004), Thompson (2005), Gosz (2006), Kattan (2007), Moaveni (2008), and more recently in Dow (2012), and Bathe (2014).

Most practitioners of the finite element method now employ Galerkin’s method to establish the approximations to the governing equations. The underlying theme in this book likewise follows Galerkin’s method. The simplicity and richness of the method pays for itself as the user progresses into more complicated and demanding types of problems. Once this fundamental concept is grasped, application of the finite element method unfolds quickly.

1.3 ORIENTATION

This book is designed to serve as a simple introductory text and self-explanatory guide to the finite element method. Beginning with the concept of one-dimensional heat transfer (which is relatively easy to follow), the book progresses through two-dimensional elements to three-dimensional elements, with a discussion on various applications, including fluid flow. We conclude with a brief introduction to boundary elements, a logical extension to finite elements, and then finish with a brief chapter on meshless (meshfree) methods—a mesh is no longer required.

Particular emphasis is placed on the development of the one-dimensional element. All the principles and formulation of the finite element method can be found in the class of one-dimensional elements; extrapolation to two and three dimensions is mostly straightforward.

Each chapter contains a set of example problems and some exercises that can be verified manually. In most cases, the exact solution is obtained from either inspection or an analytical equation. By concentrating on example problems, the manner and procedure for defining and organizing the requisite initial and boundary condition data for a specific problem becomes apparent. In the first few examples, the solutions are apparent; as the succeeding problems become progressively more involved, more input data must be provided.

For those problems requiring more extensive calculations (which is quickly discovered when dealing with matrices), a set of computer codes is available on the web (www.ncacm.unlv.edu). These source codes are written in MATLAB^®, including FORTRAN modules, with occasional MAPLE and MATHCAD listings (Kattan, 2003; MAPLE 18, 2014; MATHCAD 15, 2014; MATLAB and SIMULINK, 2015). All the codes run on PCs. The purpose of these codes is to illustrate simple finite element programming and to provide the reader with a set of programs that will assist in solving the examples and most of the exercises. The computer codes are fairly generic and have been written with the intention of instruction and ease of use. The reader may modify and optimize them as desired. Additional codes are also available from the web (see http://www.femcodes.nscee.edu). One is written in C/C++ and the other is written in JAVA. Both permit 2-D heat transfer calculations to be run in real time under WINDOWS and on the WEB. These two codes include simple pre- and postprocessing of meshes and results.

A set of files from COMSOL, a multiphysics finite element code developed by COMSOL, COMSOL 5.2 (2015) is also included here. COMSOL is a commercial finite element package, originally written to run with MATLAB, which is easy to use yet handles a wide variety of problems. The software can be used to solve 1-, 2-, and 3-D problems in structural analysis, heat transfer, fluid flow, and electrodynamics, and employs a rather sophisticated, but easy to use mesh generator. The software also permits the user to employ meshes ranging from coarse to very fine density (but one must be careful to make sure the mesh is sufficient to yield convergence and accuracy).

A discussion of the method of weighted residuals is given in Chapter 2. This chapter provides the underlying mathematical basis of the Galerkin procedure that is basic to the finite element method. Chapter 3 serves as the actual beginning of the finite element method, utilizing the one-dimensional element—in fact, the entire framework of the method is presented in this chapter. Reinforcement of the basic concepts is achieved in Chapters 4, 5 and 6 as the reader progresses through the class of two-dimensional elements. In Chapter 7, simple three-dimensional elements are discussed, utilizing a single element heat conduction problem with various boundary conditions, including radiation. Chapter 8 describes applications to solid mechanics and the role of multiple degrees of freedom (e.g., displacement in x and y) with example problems in two dimensions. Chapter 9 discusses applications to convective transport, using examples from potential flow and species dispersion. In Chapter 10, the reader is introduced to viscous fluid flow and the nonlinear equations of fluid motion for both incompressible and compressible flows. COMSOL is particularly effective at solving fluid flow problems. The more advanced book by Heinrich and Pepper (1999) discusses fluid flow in greater detail, and includes both 2-D penalty and primitive equation approaches (the more practical and widely used method employed today) for solving incompressible fluid flow. Chapter 11 introduces the concept of the boundary element method (BEM), which essentially adds one more step to the method of weighted residuals and employing the Green–Gauss theorem to reduce a problem by one dimension (a 3-D problem becomes a 2-D problem; a 2-D problem becomes a 1-D problem). The meshless method is described in Chapter 12, and extends beyond conventional schemes that require meshes to discretize and solve problems to a technique where only node points are required—with no connectivity, or meshes. This method uses radial basis functions, similar to basis functions in finite elements (Pepper et al., 2014).

The finite element method has essentially become the de facto standard for numerical approximation of the partial differential equations that define structural engineering, and is now widely accepted for a multitude of other engineering and scientific problems. Most of the commercial computer codes today are finite element based—even the finite volume computational fluid dynamics cod...