Über die Autorin bzw. den Autor

KENNETH H. HUEBNER, PhD, is retired from Ford Motor Company, where he was manager of the Computer-Aided Engineering Research Staff. He received his PhD from Purdue University in 1969.

DONALD L. DEWHIRST, PhD, has many years of experience in the aerospace and automotive industries. He is recently retired from Ford Motor Company. He received his PhD from the Department of Theoretical and Applied Mechanics at the University of Illinois.

DOUGLAS E. SMITH, PhD, is Assistant Professor, Division of Engineering at the Colorado School of Mines, Golden, Colorado. He has been involved with finite element analysis (FEA) for over fifteen years while working at IBM Corporation and Ford Motor Company. He received his PhD from the University of Illinois, Urbana-Champaign.

TED G. BYROM, PhD, is an independent consulting engineer and a former technology consultant with Oryx Energy Co., Inc. He received his PhD in aerospace engineering from Texas A&M University.

Von der hinteren Coverseite

A useful balance of theory, applications, and real-world examples
The Finite Element Method for Engineers, Fourth Edition presents a clear, easy-to-understand explanation of finite element fundamentals and enables readers to use the method in research and in solving practical, real-life problems. It develops the basic finite element method mathematical formulation, beginning with physical considerations, proceeding to the well-established variation approach, and placing a strong emphasis on the versatile method of weighted residuals, which has shown itself to be important in nonstructural applications.
The authors demonstrate the tremendous power of the finite element method to solve problems that classical methods cannot handle, including elasticity problems, general field problems, heat transfer problems, and fluid mechanics problems. They supply practical information on boundary conditions and mesh generation, and they offer a fresh perspective on finite element analysis with an overview of the current state of finite element optimal design.
Supplemented with numerous real-world problems and examples taken directly from the authors' experience in industry and research, The Finite Element Method for Engineers, Fourth Edition gives readers the real insight needed to apply the method to challenging problems and to reason out solutions that cannot be found in any textbook.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

The Finite Element Method for Engineers

John Wiley & Sons

Chapter One

1.1 What Is the Finite Element Method?

1.2 How the Finite Element Method Works

1.3 A Brief History of the Method

1.4 Range of Applications

1.5 Commercial Finite Element Software

1.6 The Future of the Finite Element Method References

1.1 WHAT IS THE FINITE ELEMENT METHOD?

The finite element method is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. Although originally developed to study stresses in complex airframe structures, it has since been extended and applied to the broad field of continuum mechanics. Because of its diversity and flexibility as an analysis tool, it is receiving much attention in engineering schools and in industry.

Although this brief comment on the finite element method answers the question posed by the section heading, it does not give us the operational definition we need to apply the method to a particular problem. Such an operational definition-along with a description of the fundamentals of the method-requires considerably more than one paragraph to develop. Hence Part I of this book is devoted to basic concepts and fundamental theory. Before discussing more aspects of the finite element method, we should first consider some of the circumstances leading to its inception, and we should briefly contrast it with other numerical schemes.

In more and more engineering situations today, we find that it is necessary to obtain approximate numerical solutions to problems rather than exact closed-form solutions. For example, we may want to find the load capacity of a plate that has several stiffeners and odd-shaped holes, the concentration of pollutants during nonuniform atmospheric conditions, or the rate of fluid flow through a passage of arbitrary shape. Without too much effort, we can write down the governing equations and boundary conditions for these problems, but we see immediately that no simple analytical solution can be found. The difficulty in these three examples lies in the fact that either the geometry or some other feature of the problem is irregular or "arbitrary." Analytical solutions to problems of this type seldom exist; yet these are the kinds of problems that engineers are called upon to solve.

The resourcefulness of the analyst usually comes to the rescue and provides several alternatives to overcome this dilemma. One possibility is to make simplifying assumptions-to ignore the difficulties and reduce the problem to one that can be handled. Sometimes this procedure works; but, more often than not, it leads to serious inaccuracies or wrong answers. Now that computers are widely available, a more viable alternative is to retain the complexities of the problem and find an approximate numerical solution.

Several approximate numerical analysis methods have evolved over the years; a commonly used method is the finite difference [1] scheme. The familiar finite difference model of a problem gives a pointwise approximation to the governing equations. This model (formed by writing difference equations for an array of grid points) is improved as more points are used. With finite difference techniques we can treat some fairly difficult problems; but, for example, when we encounter irregular geometries or an unusual specification of boundary conditions, we find that finite difference techniques become hard to use.

Unlike the finite difference method, which envisions the solution region as an array of grid points, the finite element method envisions the solution region as built up of many small, interconnected subregions or elements. A finite element model of a problem gives a piecewise approximation to the governing equations. The basic premise of the finite element method is that a solution region can be analytically modeled or approximated by replacing it with an assemblage of discrete elements. Since these elements can be put together in a variety of ways, they can be used to represent exceedingly complex shapes.

As an example of how a finite difference model and a finite element model might be used to represent a complex geometrical shape, consider the turbine blade cross section in Figure 1.1. For this device we may want to find the distribution of displacements and stresses for a given force loading or the distribution of temperature for a given thermal loading. The interior coolant passage of the blade, along with its exterior shape, gives it a nonsimple geometry.

A uniform finite difference mesh would reasonably cover the blade (the solution region), but the boundaries must be approximated by a series of horizontal and vertical lines (or "stair steps"). On the other hand, the finite element model (using the simplest two-dimensional element-the triangle) gives a better approximation to the region. Also, a better approximation to the boundary shape results because the curved boundary is represented by straight lines of any inclination. This example is not intended to suggest that finite element models are decidedly better than finite difference models for all problems. The only purpose of the example is to demonstrate that the finite element method is particularly well suited for problems with complex geometries.

Still another numerical analysis method is the boundary element method (boundary integral equation method) [2-4]. This method uses Green's theorem to reduce the dimensionality of the problem; a volume problem is reduced to a surface problem, a surface problem is reduced to a line problem. The turbine blade cross section example of Figure 1.1 would have no interior mesh, but rather a mesh of connected points along the exterior boundary and a mesh of connected points along the interior boundary. This method is computationally less efficient than finite elements and is not widely used in industry. It is popular for acoustic problems and is sometimes used as a hybrid method in conjunction with finite elements.

1.2 HOW THE FINITE ELEMENT METHOD WORKS

We have been alluding to the essence of the finite element method, but now we shall discuss it in greater detail. In a continuum problem of any dimension the field variable (whether it is pressure, temperature, displacement, stress, or some other quantity) possesses infinitely many values because it is a function of each generic point in the body or solution region. Consequently, the problem is one with an infinite number of unknowns. The finite element discretization procedures reduce the problem to one of a finite number of unknowns by dividing the solution region into elements and by expressing the unknown field variable in terms of assumed approximating functions within each element. The approximating functions (sometimes called interpolation functions) are defined in terms of the values of the field variables at specified points called nodes or nodal points. Nodes usually lie on the element boundaries where adjacent elements are connected. In addition to boundary nodes, an element may also have a few interior nodes. The nodal values of the field variable and the interpolation functions for the elements completely define the behavior of the field variable within the elements.

For the finite element representation of a problem the nodal values of the field variable become the unknowns. Once these unknowns are found, the interpolation functions define the field variable throughout the assemblage of elements.

Clearly, the nature of the solution and the degree of approximation depend not only on the size and number of the elements used but also on the interpolation functions selected. As one would expect, we cannot choose functions arbitrarily, because certain compatibility conditions should be satisfied. Often functions are chosen so that the field variable or its derivatives are continuous across adjoining element boundaries. The essential guidelines for choosing interpolation functions are discussed in Chapters 3 and 5. These are applied to the formulation of different kinds of elements.

Thus far we have briefly discussed the concept of modeling an arbitrarily shaped solution region with an assemblage of discrete elements, and we have pointed out that interpolation functions must be defined for each element. We have not yet mentioned, however, an important feature of the finite element method that sets it apart from other numerical methods. This feature is the ability to formulate solutions for individual elements before putting them together to represent the entire problem. This means, for example, that if we are treating a problem in stress analysis, we find the force-displacement or stiffness characteristics of each individual element and then assemble the elements to find the stiffness of the whole structure. In essence, a complex problem reduces to considering a series of greatly simplified problems.

Another advantage of the finite element method is the variety of ways in which one can formulate the properties of individual elements. There are basically three different approaches. The first approach to obtaining element properties is called the direct approach because its origin is traceable to the direct stiffness method of structural analysis. Although the direct approach can be used only for relatively simple problems, it is presented in Chapter 2 because it is the easiest to understand when meeting the finite element method for the first time. The direct approach suggests the need for matrix algebra (Appendix A) in dealing with the finite element equations.

Element properties obtained by the direct approach can also be determined by the variational approach. The variational approach relies on the calculus of variations (Appendix B) and involves extremizing a functional. For problems in solid mechanics the functional turns out to be the potential energy, the complementary energy, or some variant of these, such as the Reissner variational principle. Knowledge of the variational approach (Chapter 3) is necessary to work beyond the introductory level and to extend the finite element method to a wide variety of engineering problems. Whereas the direct approach can be used to formulate element properties for only the simplest element shapes, the variational approach can be employed for both simple and sophisticated element shapes.

A third and even more versatile approach to deriving element properties has its basis in mathematics and is known as the weighted residuals approach (Chapter 4). The weighted residuals approach begins with the governing equations of the problem and proceeds without relying on a variational statement. This approach is advantageous because it thereby becomes possible to extend the finite element method to problems where no functional is available. The method of weighted residuals is widely used to derive element properties for nonstructural applications such as heat transfer and fluid mechanics.

Regardless of the approach used to find the element properties, the solution of a continuum problem by the finite element method always follows an orderly step-by-step process. To summarize in general terms how the finite element method works we will succinctly list these steps now; they will be developed in detail later.

1. Discretize the Continuum. The first step is to divide the continuum or solution region into elements. In the example of Figure 1.1 the turbine blade has been divided into triangular elements that might be used to find the temperature distribution or stress distribution in the blade. A variety of element shapes (such as those cataloged in Chapter 5) may be used, and different element shapes may be employed in the same solution region. Indeed, when analyzing an elastic structure that has different types of components such as plates and beams, it is not only desirable but also necessary to use different elements in the same solution. Although the number and the type of elements in a given problem are matters of engineering judgment, the analyst can rely on the experience of others for guidelines. The discussion of applications in Chapters 6-9 reveals many of these useful guidelines.

2. Select Interpolation Functions. The next step is to assign nodes to each element and then choose the interpolation function to represent the variation of the field variable over the element. The field variable may be a scalar, a vector, or a higher-order tensor. Often, polynomials are selected as interpolation functions for the field variable because they are easy to integrate and differentiate. The degree of the polynomial chosen depends on the number of nodes assigned to the element, the nature and number of unknowns at each node, and certain continuity requirements imposed at the nodes and along the element boundaries. The magnitude of the field variable as well as the magnitude of its derivatives may be the unknowns at the nodes.

3. Find the Element Properties. Once the finite element model has been established (that is, once the elements and their interpolation functions have been selected), we are ready to determine the matrix equations expressing the properties of the individual elements. For this task we may use one of the three approaches just mentioned: the direct approach, the variational approach, or the weighted residuals approach.

4. Assemble the Element Properties to Obtain the System Equations. To find the properties of the overall system modeled by the network of elements we must "assemble" all the element properties. In other words, we combine the matrix equations expressing the behavior of the elements and form the matrix equations expressing the behavior of the entire system. The matrix equations for the system have the same form as the equations for an individual element except that they contain many more terms because they include all nodes.

The basis for the assembly procedure stems from the fact that at a node, where elements are interconnected, the value of the field variable is the same for each element sharing that node. A unique feature of the finite element method is that the system equations are generated by assembly of the individual element equations. In contrast, in the finite difference method the system equations are generated by writing nodal equations. In Chapter 2 we demonstrate how the assembly process leads to the system equations.

5. Impose the Boundary Conditions. Before the system equations are ready for solution they must be modified to account for the boundary conditions of the problem. At this stage we impose known nodal values of the dependent variables or nodal loads. In Chapter 2 we will see examples of how nodal boundary conditions are introduced.

6. Solve the System Equations. The assembly process gives a set of simultaneous equations that we solve to obtain the unknown nodal values of the problem. If the problem describes steady or equilibrium behavior, then we must solve a set of linear or nonlinear algebraic equations. In Chapter 10 we briefly discuss standard solution techniques for solving these equations. If the problem is unsteady, the nodal unknowns are a function of time, and we must solve a set of linear or nonlinear ordinary differential equations. We describe techniques for solving time-dependent equations in Part II of the book in Chapters 6-9.

7. Make Additional Computations If Desired. Many times we use the solution of the system equations to calculate other important parameters. For example, in a structural problem the nodal unknowns are displacement components. From these displacements we calculate element strains and stresses. Similarly, in a heat-conduction problem the nodal unknowns are temperatures, and from these we calculate element heat fluxes.

(Continues...)

Excerpted from The Finite Element Method for Engineersby Kenneth H. Huebner Donald L. Dewhirst Douglas E. Smith Ted G. Byrom Copyright © 2001 by Kenneth H. Huebner. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.