Hutton - Fundamentals of Finite Element Analysis, страница 3

Thus, continuity of the field variable at the nodes is ensured. In fact, finite element formulations are such thatcontinuity of the field variable across interelement boundaries is also ensured.This feature avoids the physically unacceptable possibility of gaps or voids occurring in the domain. In structural problems, such gaps would represent physical separation of the material. In heat transfer, a “gap” would manifest itself inthe form of different temperatures at the same physical point.Although continuity of the field variable from element to element is inherentto the finite element formulation, interelement continuity of gradients (i.e., derivatives) of the field variable does not generally exist.

This is a critical observation. In most cases, such derivatives are of more interest than are field variablevalues. For example, in structural problems, the field variable is displacement but3Hutton: Fundamentals ofFinite Element Analysis41. Basic Concepts of theFinite Element MethodCHAPTER 1Text© The McGraw−HillCompanies, 2004Basic Concepts of the Finite Element Methodthe true interest is more often in strain and stress. As strain is defined in terms offirst derivatives of displacement components, strain is not continuous across element boundaries.

However, the magnitudes of discontinuities of derivatives canbe used to assess solution accuracy and convergence as the number of elementsis increased, as is illustrated by the following example.1.2.1 Comparison of Finite Element and Exact SolutionsThe process of representing a physical domain with finite elements is referred toas meshing, and the resulting set of elements is known as the finite element mesh.As most of the commonly used element geometries have straight sides, it is generally impossible to include the entire physical domain in the element mesh if thedomain includes curved boundaries. Such a situation is shown in Figure 1.2a,where a curved-boundary domain is meshed (quite coarsely) using square elements.

A refined mesh for the same domain is shown in Figure 1.2b, usingsmaller, more numerous elements of the same type. Note that the refined meshincludes significantly more of the physical domain in the finite element representation and the curved boundaries are more closely approximated. (Triangularelements could approximate the boundaries even better.)If the interpolation functions satisfy certain mathematical requirements(Chapter 6), a finite element solution for a particular problem converges to theexact solution of the problem.

That is, as the number of elements is increased andthe physical dimensions of the elements are decreased, the finite element solutionchanges incrementally. The incremental changes decrease with the mesh refinement process and approach the exact solution asymptotically.

To illustrateconvergence, we consider a relatively simple problem that has a known solution.Figure 1.3a depicts a tapered, solid cylinder fixed at one end and subjected toa tensile load at the other end. Assuming the displacement at the point of loadapplication to be of interest, a first approximation is obtained by consideringthe cylinder to be uniform, having a cross-sectional area equal to the average area(a)(b)Figure 1.2(a) Arbitrary curved-boundary domain modeled using square elements. Stippledareas are not included in the model.

A total of 41 elements is shown. (b) Refinedfinite element mesh showing reduction of the area not included in the model. Atotal of 192 elements is shown.Hutton: Fundamentals ofFinite Element Analysis1. Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 20041.2 How Does the Finite Element Method Work?of the cylinder (Figure 1.3b). The uniform bar is a link or bar finite element(Chapter 2), so our first approximation is a one-element, finite element model.The solution is obtained using the strength of materials theory. Next, we modelthe tapered cylinder as two uniform bars in series, as in Figure 1.3c. In the twoelement model, each element is of length equal to half the total length of thecylinder and has a cross-sectional area equal to the average area of the corresponding half-length of the cylinder. The mesh refinement is continued using afour-element model, as in Figure 1.3d, and so on.

For this simple problem, thedisplacement of the end of the cylinder for each of the finite element models is asshown in Figure 1.4a, where the dashed line represents the known solution. Convergence of the finite element solutions to the exact solution is clearly indicated.roxrLAAo AL2rLF(a)(b)Element 1Element 2(c)(d)Figure 1.3(a) Tapered circular cylinder subjected to tensile loading:r(x) r0 (x/L)(r0 rL). (b) Tapered cylinder as a single axial(bar) element using an average area. Actual tapered cylinderis shown as dashed lines.

(c) Tapered cylinder modeled astwo, equal-length, finite elements. The area of each elementis average over the respective tapered cylinder length.(d) Tapered circular cylinder modeled as four, equal-lengthfinite elements. The areas are average over the respectivelength of cylinder (element length L兾4).5Hutton: Fundamentals ofFinite Element Analysis61. Basic Concepts of theFinite Element MethodCHAPTER 1Text© The McGraw−HillCompanies, 2004Basic Concepts of the Finite Element MethodExactFour elements␦( Lx )␦(x L)Exact123Number of elements(a)40.250.50.751.0xL(b)Figure 1.4(a) Displacement at x L for tapered cylinder in tension of Figure 1.3. (b) Comparison of the exact solutionand the four-element solution for a tapered cylinder in tension.On the other hand, if we plot displacement as a function of position along thelength of the cylinder, we can observe convergence as well as the approximatenature of the finite element solutions.

Figure 1.4b depicts the exact strength ofmaterials solution and the displacement solution for the four-element models.We note that the displacement variation in each element is a linear approximationto the true nonlinear solution. The linear variation is directly attributable to thefact that the interpolation functions for a bar element are linear.

Second, we notethat, as the mesh is refined, the displacement solution converges to the nonlinearsolution at every point in the solution domain.The previous paragraph discussed convergence of the displacement of thetapered cylinder. As will be seen in Chapter 2, displacement is the primary fieldvariable in structural problems. In most structural problems, however, we areinterested primarily in stresses induced by specified loadings. The stresses mustbe computed via the appropriate stress-strain relations, and the strain components are derived from the displacement field solution. Hence, strains andstresses are referred to as derived variables.

For example, if we plot the elementstresses for the tapered cylinder example just cited for the exact solution as wellas the finite element solutions for two- and four-element models as depicted inFigure 1.5, we observe that the stresses are constant in each element and represent a discontinuous solution of the problem in terms of stresses and strains.

Wealso note that, as the number of elements increases, the jump discontinuities instress decrease in magnitude. This phenomenon is characteristic of the finite element method. The formulation of the finite element method for a given problemis such that the primary field variable is continuous from element to element butHutton: Fundamentals ofFinite Element Analysis1. Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 20041.2 How Does the Finite Element Method Work?4.54.0ExactTwo elementsFour elements3.5␴␴03.02.52.01.51.00.250.5xL0.751.0Figure 1.5Comparison of the computed axial stress value in atapered cylinder: 0 F兾A0.the derived variables are not necessarily continuous.

In the limiting process ofmesh refinement, the derived variables become closer and closer to continuity.Our example shows how the finite element solution converges to a knownexact solution (the exactness of the solution in this case is that of strength ofmaterials theory). If we know the exact solution, we would not be applying thefinite element method! So how do we assess the accuracy of a finite element solution for a problem with an unknown solution? The answer to this question is notsimple. If we did not have the dashed line in Figure 1.3 representing the exactsolution, we could still discern convergence to a solution. Convergence of anumerical method (such as the finite element method) is by no means assurancethat the convergence is to the correct solution.

A person using the finite elementanalysis technique must examine the solution analytically in terms of (1) numerical convergence, (2) reasonableness (does the result make sense?), (3) whether thephysical laws of the problem are satisfied (is the structure in equilibrium? Does theheat output balance with the heat input?), and (4) whether the discontinuities invalue of derived variables across element boundaries are reasonable. Manysuch questions must be posed and examined prior to accepting the results of a finiteelement analysis as representative of a correct solution useful for design purposes.1.2.2 Comparison of Finite Element and FiniteDifference MethodsThe finite difference method is another numerical technique frequently used toobtain approximate solutions of problems governed by differential equations.Details of the technique are discussed in Chapter 7 in the context of transient heat7Hutton: Fundamentals ofFinite Element Analysis81.

Basic Concepts of theFinite Element MethodCHAPTER 1Text© The McGraw−HillCompanies, 2004Basic Concepts of the Finite Element Methodtransfer. The method is also illustrated in Chapter 10 for transient dynamic analysis of structures. Here, we present the basic concepts of the finite differencemethod for purposes of comparison.The finite difference method is based on the definition of the derivative of afunction f (x ) that isd f (x )f (x + x ) − f (x )= limx→0dxx(1.2)where x is the independent variable.

In the finite difference method, as impliedby its name, derivatives are calculated via Equation 1.2 using small, but finite,values of x to obtaind f (x )f (x + x ) − f (x )≈dxx(1.3)A differential equation such asdf+x =0dx0≤x ≤1(1.4)f (x + x ) − f (x )+x =0x(1.5)is expressed asin the finite difference method. Equation 1.5 can be rewritten asf (x + x ) = f (x ) − x (x )(1.6)where we note that the equality must be taken as “approximately equals.” Fromdifferential equation theory, we know that the solution of a first-order differentialequation contains one constant of integration.