Hutton - Fundamentals of Finite Element Analysis, страница 4

The constant of integration mustbe determined such that one given condition (a boundary condition or initial condition) is satisfied. In the current example, we assume that the specified conditionis x (0) = A = constant. If we choose an integration step x to be a small, constant value (the integration step is not required to be constant), then we can writex i+1 = x i + xi = 0, N(1.7)where N is the total number of steps required to cover the domain. Equation 1.6is thenf i+1 = f i − x i (x )f0 = Ai = 0, N(1.8)Equation 1.8 is known as a recurrence relation and provides an approximation tothe value of the unknown function f (x) at a number of discrete points in the domain of the problem.To illustrate, Figure 1.6a shows the exact solution f (x ) = 1 − x 2 /2 and afinite difference solution obtained with x = 0.1.

The finite difference solution isshown at the discrete points of function evaluation only. The manner of variationHutton: Fundamentals ofFinite Element Analysis1. Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 20041.2 How Does the Finite Element Method Work?10.8f (x)0.60.40.2000.20.60.40.81xFigure 1.6Comparison of the exact and finite differencesolutions of Equation 1.4 with f0 A 1.of the function between the calculated points is not known in the finite differencemethod. One can, of course, linearly interpolate the values to produce an approximation to the curve of the exact solution but the manner of interpolation isnot an a priori determination in the finite difference method.To contrast the finite difference method with the finite element method,we note that, in the finite element method, the variation of the field variable inthe physical domain is an integral part of the procedure.

That is, based on theselected interpolation functions, the variation of the field variable throughout afinite element is specified as an integral part of the problem formulation. In thefinite difference method, this is not the case: The field variable is computed atspecified points only. The major ramification of this contrast is that derivatives(to a certain level) can be computed in the finite element approach, whereas thefinite difference method provides data only on the variable itself. In a structuralproblem, for example, both methods provide displacement solutions, but thefinite element solution can be used to directly compute strain components (firstderivatives).

To obtain strain data in the finite difference method requires additional considerations not inherent to the mathematical model.There are also certain similarities between the two methods. The integrationpoints in the finite difference method are analogous to the nodes in a finiteelement model. The variable of interest is explicitly evaluated at such points.Also, as the integration step (step size) in the finite difference method is reduced,the solution is expected to converge to the exact solution.

This is similar to theexpected convergence of a finite element solution as the mesh of elements isrefined. In both cases, the refinement represents reduction of the mathematicalmodel from finite to infinitesimal. And in both cases, differential equations arereduced to algebraic equations.9Hutton: Fundamentals ofFinite Element Analysis101. Basic Concepts of theFinite Element MethodCHAPTER 1Text© The McGraw−HillCompanies, 2004Basic Concepts of the Finite Element MethodProbably the most descriptive way to contrast the two methods is to note thatthe finite difference method models the differential equation(s) of the problemand uses numerical integration to obtain the solution at discrete points.

The finiteelement method models the entire domain of the problem and uses known physical principles to develop algebraic equations describing the approximate solutions. Thus, the finite difference method models differential equations while thefinite element method can be said to more closely model the physical problem athand. As will be observed in the remainder of this text, there are cases in whicha combination of finite element and finite difference methods is very useful andefficient in obtaining solutions to engineering problems, particularly where dynamic (transient) effects are important.1.3 A GENERAL PROCEDURE FOR FINITEELEMENT ANALYSISCertain steps in formulating a finite element analysis of a physical problem arecommon to all such analyses, whether structural, heat transfer, fluid flow, orsome other problem.

These steps are embodied in commercial finite elementsoftware packages (some are mentioned in the following paragraphs) and areimplicitly incorporated in this text, although we do not necessarily refer to thesteps explicitly in the following chapters. The steps are described as follows.1.3.1 PreprocessingThe preprocessing step is, quite generally, described as defining the model andincludesDefine the geometric domain of the problem.Define the element type(s) to be used (Chapter 6).Define the material properties of the elements.Define the geometric properties of the elements (length, area, and the like).Define the element connectivities (mesh the model).Define the physical constraints (boundary conditions).Define the loadings.The preprocessing (model definition) step is critical.

In no case is there a betterexample of the computer-related axiom “garbage in, garbage out.” A perfectlycomputed finite element solution is of absolutely no value if it corresponds to thewrong problem.1.3.2 SolutionDuring the solution phase, finite element software assembles the governing algebraic equations in matrix form and computes the unknown values of the primaryfield variable(s). The computed values are then used by back substitution toHutton: Fundamentals ofFinite Element Analysis1. Basic Concepts of theFinite Element MethodText1.4 Brief History of the Finite Element Methodcompute additional, derived variables, such as reaction forces, element stresses,and heat flow.As it is not uncommon for a finite element model to be represented by tensof thousands of equations, special solution techniques are used to reduce datastorage requirements and computation time.

For static, linear problems, a wavefront solver, based on Gauss elimination (Appendix C), is commonly used. Whilea complete discussion of the various algorithms is beyond the scope of this text,the interested reader will find a thorough discussion in the Bathe book [1].1.3.3 PostprocessingAnalysis 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 finite element solution. Examples of operations that can be accomplished includeSort 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 mostimportant objective is to apply sound engineering judgment in determiningwhether the solution results are physically reasonable.1.4 BRIEF HISTORY OF THE FINITEELEMENT METHODThe mathematical roots of the finite element method dates back at least a halfcentury.

Approximate methods for solving differential equations using trial solutions are even older in origin. Lord Rayleigh [2] and Ritz [3] used trial functions(in our context, interpolation functions) to approximate solutions of differentialequations. Galerkin [4] used the same concept for solution.

The drawback in theearlier approaches, compared to the modern finite element method, is that thetrial functions must apply over the entire domain of the problem of concern.While the Galerkin method provides a very strong basis for the finite elementmethod (Chapter 5), not until the 1940s, when Courant [5] introduced the concept of piecewise-continuous functions in a subdomain, did the finite elementmethod have its real start.In the late 1940s, aircraft engineers were dealing with the invention of the jetengine and the needs for more sophisticated analysis of airframe structures towithstand larger loads associated with higher speeds. These engineers, withoutthe benefit of modern computers, developed matrix methods of force analysis,© The McGraw−HillCompanies, 200411Hutton: Fundamentals ofFinite Element Analysis121.

Basic Concepts of theFinite Element MethodCHAPTER 1Text© The McGraw−HillCompanies, 2004Basic Concepts of the Finite Element Methodcollectively known as the flexibility method, in which the unknowns are theforces and the knowns are displacements. The finite element method, in its mostoften-used form, corresponds to the displacement method, in which the unknowns are system displacements in response to applied force systems. In thistext, we adhere exclusively to the displacement method.