Basic Concepts of the Finite Element Method：有限元法的基

Hutton: Fundamentals of

1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of

1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the

Finite Element Method

Text? The McGraw?Hill

Companies, 2004

1.2How Does the Finite Element Method Work?3

If the values of the ?eld variable are computed only at nodes, how are values obtained at other points within a ?nite element? The answer contains the crux of the ?nite element method: The values of the ?eld variable computed at the nodes are used to approximate the values at nonnodal points (that is, in the element interior) by interpolation of the nodal values. For the three-node triangle exam-ple, the nodes are all exterior and, at any other point within the element, the ?eld variable is described by the approximate relation

?(x,y)=N1(x,y)?1+N2(x,y)?2+N3(x,y)?3(1.1) where?1,?2, and ?3are the values of the ?eld variable at the nodes, and N1,N2, and N3are the interpolation functions,also known as shape functions or blend-ing functions. In the ?nite element approach, the nodal values of the ?eld vari-able are treated as unknown constants that are to be determined. The interpola-tion functions are most often polynomial forms of the independent variables, derived to satisfy certain required conditions at the nodes. These conditions are discussed in detail in subsequent chapters. The major point to be made here is that the interpolation functions are predetermined, known functions of the inde-pendent variables; and these functions describe the variation of the ?eld variable within the ?nite element.

The triangular element described by Equation 1.1 is said to have 3 degrees of freedom,as three nodal values of the ?eld variable are required to describe the?eld variable everywhere in the element. This would be the case if the ?eld variable represents a scalar ?eld, such as temperature in a heat transfer problem (Chapter 7). If the domain of Figure 1.1 represents a thin, solid body subjected to plane stress (Chapter 9), the ?eld variable becomes the displacement vector and the values of two components must be computed at each node. In the latter case, the three-node triangular element has 6 degrees of freedom. In general, the num-ber of degrees of freedom associated with a ?nite element is equal to the product of the number of nodes and the number of values of the ?eld variable (and pos-sibly its derivatives) that must be computed at each node.

How does this element-based approach work over the entire domain of in-terest? As depicted in Figure 1.1c, every element is connected at its exterior nodes to other elements. The ?nite element equations are formulated such that, at the nodal connections, the value of the ?eld variable at any connection is the same for each element connected to the node. Thus, continuity of the ?eld vari-able at the nodes is ensured. In fact, ?nite element formulations are such that continuity of the ?eld variable across interelement boundaries is also ensured. This feature avoids the physically unacceptable possibility of gaps or voids oc-curring in the domain. In structural problems, such gaps would represent physi-cal separation of the material. In heat transfer, a “gap” would manifest itself in the form of different temperatures at the same physical point.

Although continuity of the ?eld variable from element to element is inherent to the ?nite element formulation, interelement continuity of gradients (i.e., de-rivatives) of the ?eld variable does not generally exist. This is a critical observa-tion. In most cases, such derivatives are of more interest than are ?eld variable values. For example, in structural problems, the ?eld variable is displacement but

Hutton: Fundamentals of 1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of

1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of

1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of

1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of 1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of

1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the

Finite Element Method

Text? The McGraw?Hill

Companies, 2004

10C H A P T E R1Basic Concepts of the Finite Element Method

Probably the most descriptive way to contrast the two methods is to note that the ?nite difference method models the differential equation(s) of the problem

and uses numerical integration to obtain the solution at discrete points. The ?nite

element method models the entire domain of the problem and uses known phys-

ical principles to develop algebraic equations describing the approximate solu-

tions. Thus, the ?nite difference method models differential equations while the

?nite element method can be said to more closely model the physical problem at

hand. As will be observed in the remainder of this text, there are cases in which

a combination of ?nite element and ?nite difference methods is very useful and

ef?cient in obtaining solutions to engineering problems, particularly where dy-

namic (transient) effects are important.

1.3 A GENERAL PROCEDURE FOR FINITE

ELEMENT ANALYSIS

Certain steps in formulating a ?nite element analysis of a physical problem are

common to all such analyses, whether structural, heat transfer, ?uid ?ow, or

some other problem. These steps are embodied in commercial ?nite element

software packages (some are mentioned in the following paragraphs) and are

implicitly incorporated in this text, although we do not necessarily refer to the

steps explicitly in the following chapters. The steps are described as follows.

1.3.1Preprocessing

The preprocessing step is, quite generally, described as de?ning the model and

includes

De?ne the geometric domain of the problem.

De?ne the element type(s) to be used (Chapter 6).

De?ne the material properties of the elements.

De?ne the geometric properties of the elements (length, area, and the like).

De?ne the element connectivities (mesh the model).

De?ne the physical constraints (boundary conditions).

De?ne the loadings.

The preprocessing (model de?nition) step is critical. In no case is there a better

example of the computer-related axiom “garbage in, garbage out.” A perfectly

computed ?nite element solution is of absolutely no value if it corresponds to the

wrong problem.

1.3.2Solution

During the solution phase, ?nite element software assembles the governing alge-

braic equations in matrix form and computes the unknown values of the primary

?eld variable(s). The computed values are then used by back substitution to

Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the

Finite Element Method

Text? The McGraw?Hill

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1.4Brief History of the Finite Element Method11

compute additional, derived variables, such as reaction forces, element stresses, and heat ?ow.

As it is not uncommon for a ?nite element model to be represented by tens of thousands of equations, special solution techniques are used to reduce data storage requirements and computation time. For static, linear problems, a wave front solver,based on Gauss elimination (Appendix C), is commonly used. While a complete discussion of the various algorithms is beyond the scope of this text, the interested reader will ?nd a thorough discussion in the Bathe book [1].

1.3.3Postprocessing

Analysis and evaluation of the solution results is referred to as postprocessing. Postprocessor software contains sophisticated routines used for sorting, printing, and plotting selected results from a ?nite element solution. Examples of opera-tions that can be accomplished include

Sort element stresses in order of magnitude.

Check equilibrium.

Calculate factors of safety.

Plot deformed structural shape.

Animate dynamic model behavior.

Produce color-coded temperature plots.

While solution data can be manipulated many ways in postprocessing, the most important objective is to apply sound engineering judgment in determining whether the solution results are physically reasonable.

1.4BRIEF HISTORY OF THE FINITE

ELEMENT METHOD

The mathematical roots of the ?nite element method dates back at least a half century. Approximate methods for solving differential equations using trial solu-tions are even older in origin. Lord Rayleigh [2] and Ritz [3] used trial functions (in our context, interpolation functions) to approximate solutions of differential equations. Galerkin [4] used the same concept for solution. The drawback in the earlier approaches, compared to the modern ?nite element method, is that the trial functions must apply over the entire domain of the problem of concern. While the Galerkin method provides a very strong basis for the ?nite element method (Chapter 5), not until the 1940s, when Courant [5] introduced the con-cept of piecewise-continuous functions in a subdomain, did the ?nite element method have its real start.

In the late 1940s, aircraft engineers were dealing with the invention of the jet engine and the needs for more sophisticated analysis of airframe structures to withstand larger loads associated with higher speeds. These engineers, without the bene?t of modern computers, developed matrix methods of force analysis,

Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the

Finite Element Method

Text? The McGraw?Hill

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12C H A P T E R1Basic Concepts of the Finite Element Method

collectively known as the ?exibility method,in which the unknowns are the

forces and the knowns are displacements. The ?nite element method, in its most

often-used form, corresponds to the displacement method,in which the un-

knowns are system displacements in response to applied force systems. In this

text, we adhere exclusively to the displacement method. As will be seen as we

proceed, the term displacement is quite general in the ?nite element method and

can represent physical displacement, temperature, or ?uid velocity, for example.

The term ?nite element was ?rst used by Clough [6] in 1960 in the context of

plane stress analysis and has been in common usage since that time.

During the decades of the 1960s and 1970s, the ?nite element method was extended to applications in plate bending, shell bending, pressure vessels, and

general three-dimensional problems in elastic structural analysis [7–11] as well

as to ?uid ?ow and heat transfer [12, 13]. Further extension of the method to

large de?ections and dynamic analysis also occurred during this time period

[14,15]. An excellent history of the ?nite element method and detailed bibliog-

raphy is given by Noor [16].

The ?nite element method is computationally intensive, owing to the required operations on very large matrices. In the early years, applications were performed

using mainframe computers, which, at the time, were considered to be very pow-

erful, high-speed tools for use in engineering analysis. During the 1960s, the ?nite

element software code NASTRAN [17] was developed in conjunction with the

space exploration program of the United States. NASTRAN was the ?rst major

?nite element software code. It was, and still is, capable of hundreds of thousands

of degrees of freedom (nodal ?eld variable computations). In the years since the

development of NASTRAN, many commercial software packages have been in-

troduced for ?nite element analysis. Among these are ANSYS [18], ALGOR [19],

and COSMOS/M [20]. In today’s computational environment, most of these

packages can be used on desktop computers and engineering workstations to

obtain solutions to large problems in static and dynamic structural analysis, heat

transfer, ?uid ?ow, electromagnetics, and seismic response. In this text, we do not

utilize or champion a particular code. Rather, we develop the fundamentals for

understanding of ?nite element analysis to enable the reader to use such software

packages with an educated understanding.

1.5EXAMPLES OF FINITE ELEMENT

ANALYSIS

We now present, brie?y, a few examples of the types of problems that can be

analyzed via the ?nite element method. Figure 1.7 depicts a rectangular region

with a central hole. The area has been “meshed” with a ?nite element grid of two-

dimensional elements assumed to have a constant thickness in the z direction.

Note that the mesh of elements is irregular: The element shapes (triangles and

quadrilaterals) and sizes vary. In particular, note that around the geometric dis-

continuity of the hole, the elements are of smaller size. This represents not only

Hutton: Fundamentals of 1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of

1. Basic Concepts of the Text? The McGraw?Hill

Hutton: Fundamentals of 1. Basic Concepts of the Text? The McGraw?Hill

(b)

(a) A ?nite element model of a prosthetic hand for weightlifting. (b)Completed

prototype of a prosthetic hand, attached to a bar.

(Courtesy of Payam Sadat. All rights reserved.)

OBJECTIVES OF THE TEXT

Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the

Finite Element Method

Text? The McGraw?Hill

Companies, 2004

References17

As the reader progresses through the text, he or she will discern that we cover a signi?cant amount of ?nite element theory in addition to application examples. Given the availability of many powerful and sophisticated ?nite element software packages, why study the theory? The ?nite element method is a tool, and like any other tool, using it without proper instruction can be quite dangerous. My premise is that the proper instruction in this context includes understanding the basic theory underlying formulation of ?nite element models of physical problems. As stated previously, critical analysis of the results of a ?nite element model computation is essential, since those results may eventually become the basis for design. Knowledge of the theory is necessary for both proper modeling and evaluation of computational results. REFERENCES

1.Bathe, K-J. Finite Element Procedures. Englewood Cliffs, NJ: Prentice-Hall,

1996.

2.Lord Rayleigh. “On the Theory of Resonance.” Transactions of the Royal Society

(London)A161 (1870).

3.Ritz, W. “Uber eine neue Methode zur Losung gewissen Variations-Probleme der

mathematischen Physik.” J. Reine Angew. Math.135 (1909).

4.Galerkin, B. G. “Series Solution of Some Problems of Elastic Equilibrium of Rods

and Plates” [in Russian]. Vestn. Inzh. Tekh. 19 (1915).

5.Courant, R. “Variational Methods for the Solution of Problems of Equilibrium and

Vibrations.”Bulletin of the American Mathematical Society49 (1943).

6.Clough, R. W. “The Finite Element Method in Plane Stress Analysis.”

Proceedings, American Society of Civil Engineers, Second Conference on

Electronic Computation, Pittsburgh, 1960.

7.Melosh, R. J. “A Stiffness Method for the Analysis of Thin Plates in Bending.”

Journal of Aerospace Sciences 28, no. 1 (1961).

8.Grafton, P. E., and D. R. Strome. “Analysis of Axisymmetric Shells by the Direct

Stiffness Method.” Journal of the American Institute of Aeronautics and

Astronautics1, no. 10 (1963).

9.Gallagher, R. H. “Analysis of Plate and Shell Structures.” Proceedings,

Symposium on the Application of Finite Element Methods in Civil Engineering,

Vanderbilt University, Nashville, 1969.

10.Wilson, E. L. “Structural Analysis of Axisymmetric Solids.” Journal of the

American Institute of Aeronautics and Astronautics3, (1965).

11.Melosh, R. J. “Structural Analysis of Solids.” Journal of the Structural Division,

Proceedings of the American Society of Civil Engineers, August 1963.

12.Martin, H. C. “Finite Element Analysis of Fluid Flows.” Proceedings of the

Second Conference on Matrix Methods in Structural Mechanics,Wright-Patterson Air Force Base, Kilborn, Ohio, October 1968.

13.Wilson, E. L., and R. E. Nickell. “Application of the Finite Element Method to

Heat Conduction Analysis.” Nuclear Engineering Design4 (1966).

Hutton: Fundamentals of Finite Element Analysis 1. Basic Concepts of the

Finite Element Method

Text? The McGraw?Hill

Companies, 2004

18C H A P T E R1Basic Concepts of the Finite Element Method

14.Turner, M. J., E. H. Dill, H. C. Martin, and R. J. Melosh. “Large De?ections of

Structures Subjected to Heating and External Loads.” Journal of Aeronautical

Sciences27 (1960).

15.Archer, J. S. “Consistent Mass Matrix Formulations for Structural Analysis Using

Finite Element Techniques.” Journal of the American Institute of Aeronautics and

Astronautics3, no. 10 (1965).

16.Noor, A. K. “Bibliography of Books and Monographs on Finite Element

Technology.” Applied Mechanics Reviews44, no. 6 (1991).

17.MSC/NASTRAN. Lowell, MA: MacNeal-Schwindler Corp.

18.ANSYS. Houston, PA: Swanson Analysis Systems Inc.

19.ALGOR. Pittsburgh: Algor Interactive Systems.

20.COSMOS/M. Los Angeles: Structural Research and Analysis Corp.

21.Hutton, D. V. “Modal Analysis of a Deployable Truss Using the Finite Element

Method.”Journal of Spacecraft and Rockets21, no. 5 (1984).

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