"the finite element methods", in wiley encyclopedia of computer

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THE FINITE ELEMENT METHOD

INTRODUCTION

Finite element methods are now widely used to solvestructural, fluid, and multiphysics problems numerically(1). The methods are used extensively because engineersand scientists can mathematically model and numericallysolve very complex problems. The analyses in engineeringare performed to assess designs, and the analyses in thevarious scientific fields are carried out largely to obtaininsight into and ideally to predict natural phenomena. Theprediction of how a design will perform and whether andhow a natural phenomenon will occur is of much value:Designs can be made safer and more cost effective, whileinsight into and the prediction of nature can help, forexample, to prevent disasters. Thus, the use of the finiteelement method greatly enriches our lives.

As with many other important scientific developments,it is difficult to give an exact date of the ‘‘invention’’ of thefinite element method. Indeed, we could trace back thedevelopment of the method to the Greek philosophersand in modern times to physicists, mathematicians, andengineers (see the discussions in Refs. 2 and 3). However,the real impetus for the development of what is nowreferred to as the finite element method was provided bythe need to analyze complex structures in aeronauticalengineering and the availability of the electronic computer.Namely, when using the finite element method, large sys-tems of algebraic equations need to be assembled andsolved, and the computer provides the necessary meansto accomplish this task.

The papers of Argyris and Kelsey (4) and Turner et al. (5)were seminal contributions in the 1950s. The name ‘‘finiteelement method’’ was coined by R.W. Clough in 1960 (6).The potential of the finite element method for engineeringanalysis was clearly foreseen, and henceforth the researchon finite element methods started to accelerate in variousresearch centers in Europe and in the United States. Whilesignificant papers and books were written in the nextdecade (see for example the references in Refs. 2 and 3),the development and rapidly increasing use of some com-puter programs clearly contributed in a major way to theacceptance and advancement of the method. Indeed, withno computer programs ever developed, the finite elementmethod would have been a theoretical entity without muchattention given to it.

The three finite element programs that had a majorimpact were ASKA, NASTRAN, and SAP (7–9). TheNASTRAN and ASKA programs rapidly became majoranalysis tools in the aerospace and automotive industries.The SAP programs were largely used in civil and mechan-ical engineering industries and at universities. The avail-ability and wide use of the source codes of SAP IV andNONSAP with the description of the techniques usedtherein, see Bathe et al. (9–12), had a seminal effect on

the development of new algorithms, many finite elementprograms, and also finite element theory (3,13).

Today, finite element methods probably are used for theanalysis of every major engineering design and probably inevery branch of scientific studies. The method is now usedprimarily through the application of commercial finiteelement programs (see the section titled, ‘‘The Use ofthe Finite Element Method in Computer-Aided Engineer-ing’’). These programs are used on mainframes, worksta-tions and PCs and are employed to solve very complexproblems. While the finite element methods were usedoriginally for the analysis of solids and structures, theprocedures arenow employed also for the analysis of multi-physics problems, including fluid flows with fluid–struc-ture interactions (1–3).

The objective in the following sections is to brieflydescribe the finite element method and give some refer-ences that can be consulted for additional study. Thedescription and references, of course, are by no meansexhaustive. For the equations used, the notation of Ref. 3is employed.

THE FORMULATION OF THE FINITE ELEMENT METHOD

Figures 1 to 3 show typical finite element meshes (or finiteelement assemblages) modeling some solid, structural, andfluid systems. In each case the finite elements are used torepresent the volume of the system. The finite elements areconnected at the nodal points located at the corners andalong the sides and in the faces of the elements. However,nodal points can also be located within the volume of anelement. An important feature is that the finite elements donot overlap geometrically but together fill the completevolume of the solid or fluid.

Considering the analysis of a three-dimensional solid,like the wheel in Fig. 1, the geometry and the displacementsof each element (within each element including all elementsurfaces) are completely described by the geometric posi-tions and displacements of the element nodal points. Theelement nodal coordinates prior to any displacementsare known, of course, and the nodal displacements arethe unknowns to be calculated. The basic step is to inter-polate the element geometry and the element displace-ments by using the nodal values. Here the isoparametricelement interpolation, a most important development dueto Irons (14), is largely used. For an element with q nodalpoints we use

x ¼Xq

i¼1

hixi; y ¼Xq

i¼1

hiyi; z ¼Xq

i¼1

hizi

u ¼Xq

i¼1

hiui; v ¼Xq

i¼1

hivi; w ¼Xq

i¼1

hiwi ð1Þ

1

Wiley Encyclopedia of Computer Science and Engineering, edited by Benjamin Wah.Copyright # 2008 John Wiley & Sons, Inc.

where the hi are the given interpolation functions, (xi, yi, zi)are the known coordinates of nodal point i, and (ui, vi, wi) arethe unknown displacements of nodal point i. The hi arefunctions of the element natural coordinates (r, s, t), whichrelate uniquely to the global coordinates. In this assump-tion, the same interpolation is employed for the geometryand the displacements, and these equations are applicableto very general curved element configurations. However,in by far most analyses and certainly in nonlinear analy-ses, low-order linear, non-curved elements are preferred(4-node quadrilateral elements in the two-dimensional(2-D) analyses of solids, and 8-node brick elements in thethree-dimensional (3-D) analyses of solids), but quadraticelements (9-node elements in 2-D analyses and 27-nodeelements in3-D analyses) can bemoreeffective (3).Toobtaina more accurate solution, mostly the number of elements issimply increased; that is, the h-method is used. Alterna-tively, also the number and size of elements can be keptconstant and the order of the geometry and displacementinterpolations is increased; that is, the p-method is used. Ofcourse, these approaches can also be combined and then anh/p-method is employed (15).

While some of the first finite elements for structuralanalyses were largely formulated based on physicalinsight and intuition, the development and use of ageneral theory based on the principle of virtual work (orvirtual displacements) established a firm foundation anda general approach for the formulation of finite elements.

Today, the principle of virtual displacements is the basicequation used to formulate displacement-based finite ele-ments. Considering a general solid, this principle can bewritten as (3)

ZV

e TtdV ¼Z

S f

u SfT

fS f dSþZ

Vu TfBdV ð2Þ

where the unknown (real) stresses are t, the knownapplied body forces are fB, the known applied surfacetractions are fSf , the virtual displacements are u, andthe virtual strains that correspond to the virtual displace-ments are e . Of course, the stresses need to satisfy thegiven stress-strain laws, and the strains (the virtual andreal quantities) must satisfy the strain–displacementrelationships. Also, the real displacements must satisfythe given displacement boundary conditions and the vir-tual displacements must be zero at the locations of theprescribed real displacements. The integrations are per-formed over the volume, V, and traction-loaded surface,Sf, of the solid considered.

The principle of virtual displacements, with these con-ditions satisfied, is totally equivalent to the differentialformulation of the boundary value problem. The principleis referred to also as a variational formulation and a weakformulation (2,3).

In linear analysis, the displacements are assumed to beinfinitesimally small, the material laws are assumed tobe constant, and the volume and surfaces over which theintegrations are performed are known and constant, that is,unaffected by the displacements. Substituting from Equa-tion (1) into Equation (2) and invoking the principle ofvirtual displacements as many times as there are nodaldisplacement degrees of freedom, we obtain

KU ¼ R ð3Þ

where K is the stiffness matrix, U is the displacement vectorlisting all unknown nodal point displacements (the ui, vi, wi,for all nodes, i = 1, 2, . . ., as applicable), and R is a vector ofexternally applied forces at the nodal-point displacement

Figure 1. Finite element model of a wheel using three-dimensional brick elements, and a typical 8-node brick element(q=8).

Figure 2. Finite element model of a car body using predomi-nantly shell elements.

Figure 3. Finite element computational fluid dynamics (CFD)model of a manifold; FCBI elements, about 10 million equationssolved in less than 1 hour on a single-processor PC.

2 THE FINITE ELEMENT METHOD

degrees of freedom. The solution of Equation (3) yields thenodal displacements for Equation (1) and hence the strainsand stresses in all elements.

In dynamic analysis, the body forces include inertia anddamping effects, and these can be included directly in thesolution to obtain (3)

MUþCU:þKU ¼ R ð4Þ

where M is the mass matrix (usually a consistent massmatrix), C is a damping matrix (frequently the Rayleighdamping matrix is used), and U, U

:denote the nodal

accelerations and velocities, respectively. In structuralvibration analyses, these dynamic equilibrium equationsare solved by using mode superposition or implicit directtime integration (for example, by using the popular tra-pezoidal rule).

However, for wave propagation analyses, the dynamicequilibrium equations solved are generally (3)

MU ¼ R � F ð5Þ

where F represents the nodal forces that correspond to theelement stresses. Usually, a lumped (diagonal) massmatrix is used.

The above equations have been written for a three-dimensional solid, but also are applicable with appropri-ate modifications for the analysis of two-dimensionalsolids. Furthermore, the equations can be extended andthen also are applicable for the analysis of structuresmodeled by beams, plates, and shells. The extension isreached by introducing midsurfaces, displacements androtations, and the applicable kinematic and stressassumptions. For example, in the analysis of shells, theReissner–Mindlin kinematic assumption—that plane sec-tions originally plane and normal to the midsurface of theshell remain plane—is introduced, and the assumptionthat the stress normal to that midsurface is zero isimposed. Because the pure displacement-based finite ele-ments are not effective for thin shells (and plates andbeams), that is, they ‘lock’ in bending actions (see thesection below titled, ‘‘The Analysis of Shells’’), mixedand hybrid formulations are used to reach more effectiveelements. The construction of such elements in essence isbased on an extension of the principle of virtual displace-ments, namely the use of the Hu–Washizu variationalprinciple (2,3). Effective structural elements can thendirectly be employed in Equations (3–5) together withthe displacement-based solid elements (but of course Unow also includes nodal rotations).

Considering nonlinear analysis, the nonlinear effects canarise from nonlinear material behavior, like plasticity andcreep; from geometric nonlinear effects, that is, large dis-placements and large strains; and from contact betweenbodies established as a result of the displacements. In thesecases, the basic principle of virtual displacements and theextensions for structural elements still are directly applic-able, but the appropriate stress and strain measures need beused and the equilibrium need be considered in thedeformed geometry of the bodies. For such general non-

linear analyses, the total Lagrangian (TL) and the updatedLagrangian (UL) formulations provide a fundamental basisof solution (3,16) and are widely used. In these formulations,the nonlinear material behavior is introduced to establishthe stresses for ‘‘given’’ deformations, that is, not using arate formulation when the constitutive relations are rate-independent (see also Ref. 17). If required, contact condi-tions including frictional effects are imposed as constraintson the nodal displacements and forces over the contactingsurfaces. Because the large deformations and nonlinearmaterial conditions and the solution for the actual areasof contact introduce significant nonlinearities in the govern-ing equations, iteration for solution in general is required(see below).

Considering a dynamic solution based on implicit directtime integration, and assuming that contact conditions arenot present, the incremental nonlinear analysis is accom-plished by solving at discrete times Dt apart

M tþDtUþ C tþDtU:þtþDtF ¼tþDtR ð6Þ

where the equilibrium is considered at time t þ Dt, and thenodal forces tþDtF and tþDtR correspond to the internalelement stresses and the externally applied loads, respec-tively. In static analysis, simply the inertia and dampingeffects are neglected and the superscript t ¼ time denotesthe load level (but is also an actual physical variable when amaterial law is time-dependent).

Because the nodal forces highly depend on the deforma-tions and all the considered nonlinearities need be includedin calculating these forces, an iteration is necessary for thesolution of Equation (6). A Newton–Raphson iteration, orvariant thereof, is commonly performed, with the govern-ing equations

tþDtKði�1Þ

DUðiÞ ¼ tþDtR ði�1Þ � tþDtFði�1Þ

tþDtUðiÞ ¼ tþDtUði�1Þ þ DUðiÞð7Þ

where tþDtKði�1Þ is the effective tangent stiffness matrixthat corresponds to the linearization of the nodal forcevectors with respect to the incremental displacements

DUðiÞ. The hat in tþDtKði�1Þ

and tþDtRði�1Þ

is used to signifythat also mass and damping effects are included (3). Instatic analysis these effects of course are neglected. Theiteration is continued until appropriate convergence cri-teria (based on forces, displacements, or energy) are satis-fied. To reach fast convergence, and quadratic convergencewhen near the solution, it is important to use the linear-ization ‘‘consistent’’ with the stress integration proceduresand the force updating used to evaluate the right-hand sideof Equation (7) (3,17,18).

If contact conditions are to be included, then for exam-ple, for normal contact the following conditions need besatisfied (3):

tþDtl� 0; tþDtg� 0; tþDtl � tþDtg ¼ 0 ð8Þ

THE FINITE ELEMENT METHOD 3

where tþDtg represents the gap and tþDtl represents thenormal contact force. To reach stable and accurate solu-tions, appropriate interpolations of contact tractions needbe used (19). Of course, with the relations in Equation (8)imposed as conditions on Equation (6), penetration is pre-vented at the contacting surfaces. Frictional contact con-ditions are similarly solved for; see Ref. 3. These conditionsto enforce contact between bodies enter into the iterationfor equilibrium and compatibility; that is, Equation (7)needs to be amended to enforce the contact relations.

The use of Equation (7) (in general amended for contactconditions) for dynamic analysis of course implies that atransient solution with implicit time integration is pur-sued. If a short time duration or wave propagation analysisis to be performed (e.g., a crash simulation in the motor carindustry), then an explicit time integration frequently ismore effective. In this case, Equation (5), also amended toinclude contact conditions, is used with the nodal vectorscontinuously updated for all nonlinearities, which resultfrom large deformations, contact and nonlinear materialeffects in the step-by-step solution. The solution is attrac-tive because no iteration is performed like in Equation (7);however, the disadvantage is that the time step size in theforward integration needs to be very small (because thetime integration is only conditionally stable) (3).

Equations (7) and (5), as discussed above, are applicableto the nonlinear analysis of solids and structures; however,for structural elements, a mixed formulation need be used(see the section titled, ‘‘The Analysis of Shells’’). Also, inmaterial nonlinear analysis, frequently (almost) incom-pressible conditions are encountered, as when modelinga rubber-like material or elasto-plastic conditions. In thesecases, considering a solid, it is effective to use a mixedformulation in which the unknown displacements andpressure are interpolated; that is, the u/p formulation isemployed (see the section titled, ‘‘The Reliable and EffectiveAnalysis of Solids’’).

The same basic approach also can be followed toestablish finite element solutions of heat transfer pro-blems (considering solids or fluids) and of incompressiblefluid flows. In these cases, the temperature and thevelocities (and pressure) need to be interpolated; inessence the ‘‘principle of virtual temperatures’’ and the‘‘principle of virtual velocities’’ are employed. These prin-ciples in concept are similar to the principle of virtualdisplacements and hence of course also correspond toweak formulations. A major added difficulty arises inthe use of the Eulerian formulation of fluid flows becauseof the convective effects. A pure interpolation of velocitiesand temperature results in a unstable solution. Thisinstability can be circumvented, like in finite differencesolutions, by introducing some form of upwinding (seethe section titled, ‘‘Finite Element Discretizations forIncompressible Fluid Flows’’).

The Lagrangian formulations for the solids and struc-tures and the Eulerian formulations for fluid flows havebeen fully coupled by the use of arbitrary Lagangian–Eulerian formulations, referred to as ALE formulationsfor fluid–structure interactions. Such formulations arenow used to solve reliably very complex multiphysics

problems that involve displacements of solids and struc-tures, temperature distributions, fluid flows, electromag-netic effects, and so on (see for example Refs. 1–3, and 20).

EFFECTIVE FINITE ELEMENT PROCEDURES

The first requirement of any analysis is of course theselection of an appropriate mathematical model to repre-sent the physical problem (3). The next requirement is tosolve this model with reliable and effective numericalmethods. Because the use of reliable and effective finiteelement methods is very important, much research efforthas been expended to reach such procedures.

The Reliable and Effective Analysis of Solids

The analysis of solids is efficiently accomplished by usingthe pure displacement-based finite element methodreferred to above, unless incompressible or almost incom-pressible material conditions are considered, as in theanalysis of rubber-like materials or in elasto-plasticresponse. In incompressible analysis, the pure displace-ment-based discretizations ‘‘lock,’’ and a mixed interpola-tion based on displacements and pressure is much moreeffective. However, the appropriate combination of displa-cement and pressure interpolations must be used.

The phenomenon of locking is of fundamental impor-tance for the understanding of what we mean by a reliablefinite element formulation. Figure 4 shows schematicallythe error in some norm between the finite element solution,uh, and the exact solution, u, of the mathematical model tobe solved, as the mesh is refined. Of course, the exactsolution is in general not known, but we can think of aclose approximation to the exact solution, obtained by afinite element analysis using a very fine mesh. If for almostincompressible analysis, a pure displacement-based finiteelement formulation is used, then the convergence curvesin Figs. 4(a) and 4(b) are generally much below and abovethe respective curves obtained in the compressible analy-sis. Hence the error obtained for a given mesh depends onthe bulk modulus and significantly increases as incompres-sible conditions are approached. Because the displace-ments become small measured on what they should be,the phenomenon is referred to as ‘‘locking’’. Finite elementdiscretizations that lock, like schematically referred to inFig. 4, are not reliable because small changes in some datacan cause large changes in the solution error.

Hence a reliable finite element discretization displaysthe following convergence behavior in the appropriatenorm:

ku� uhk’chk ð9Þ

where c is a constant (that is, independent of the bulkmodulus), h is the generic element size, and k is the orderof interpolation used. To reach such finite element discre-tizations in almost, or fully, incompressible analysis, theuse of the displacement–pressure interpolation is effectivewhere the displacement and pressure interpolations usedshould satisfy two conditions (3,21–23). First, the ellipticity


condition must be satisfied

aðvh; vhÞ�akvhk2 8 vh 2Khð0Þ ð10Þ

where a (., .) is the (deviatoric strain energy) bilinear form ofthe problem, Khð0Þ represents all those finite elementinterpolations vh over the mesh which satisfy the discre-tized zero divergence condition, and a is a constant greaterthan zero. And second, the inf-sup condition should besatisfied for any mesh, and in particular as h! 0,

infqh 2Qh

supvh 2Vh

RVol qhdiv vh dVol

kvhkkqhk�b> 0 ð11Þ

where, for the mesh, Vh is the space of displacement inter-polations, Qh is the space of pressure interpolation, appro-priate norms are used, and b is a constant. Effective finiteelements that satisfy these two conditions are given, for

example, in Refs. 3 and 22. It is also possible to developfinite elements that bypass the inf-sup condition and do notlock, but these then depend on nonphysical numericalparameters (22,23).

The Analysis of Shells

The first developments of practical finite element methodswere directed toward the analysis of aeronautical struc-tures, that is, thin shell structures. Since these firstdevelopments, much further effort has been expendedto reach more general, reliable, and effective shell finiteelement schemes. Shells are difficult to analyze becausethey exhibit a variety of behaviors and can be very sensi-tive structures, depending on the curvatures, the thick-ness t, the span L, the boundary conditions, and theloading applied (23). A shell may carry the loading largelyby membrane stresses, largely by bending stresses, or bycombined membrane and bending actions. Shells that onlycarry loading by membrane stresses can be efficientlyanalyzed by using a displacement-based formulation(referred to above); however, such formulations ‘‘lock’’(like discussed above when considering incompressibleanalysis) when used in the analysis of shells subjectedto bending. To establish generally reliable and effectiveshell finite elements (that can be used for any shell ana-lysis problem), the major difficulty has been to overcomethe ‘‘locking’’ of discretizations, which can be severe whenthin shells are considered.

The principles summarized above for incompressibleanalysis apply, in essence, also in the analysis of plate andshell structures. The critical parameter is now the thick-ness to span ratio, t/L, and the relevant inf-sup expressionideally should be independent of this parameter. To obtaineffective elements, the mixed interpolation of displace-ments and strain components has been proposed and iswidely used in mixed-interpolated tensorial component(MITC) and related elements (3,23–25). The appropriateinterpolations for the displacements and strains havebeen chosen carefully for the shell elements and aretied at specific element points. The resulting elementsthen only have displacements and rotations as nodaldegrees of freedom, just as for the pure displacement-based elements. The effectiveness of the elements canbe tested numerically to see whether the consistency,ellipticity, and inf-sup conditions (23) are satisfied. How-ever, the inf-sup condition for plate and shell elements ismuch more complex to evaluate than for incompressibleanalysis (26), and the direct testing by solving appropri-ately chosen test problems is more straightforward(23,27,28). Some effective mixed-interpolated shell ele-ments are given in Refs. 3,23 and 25.

For plates and shells, instead of imposing the Reiss-ner–Mindlin kinematic assumption and the assumptionof ‘‘zero stress normal to the midsurface,’’ also three-dimensional solid elements without these assumptionscan be used (29–31). Of course, for these elements to beeffective, they also need to be formulated to avoid ‘‘lock-ing;’’ that is, they also need to satisfy the conditionsmentioned above and more (23).

Figure 4. Locking phenomenon (when using pure displacementinterpolations in almost incompressible analysis).


Finite Element Discretizations for Incompressible Fluid Flows

Considering all fluid flow problems, in engineering prac-tice most problems are solved by using finite volume andfinite difference methods. Various CFD computer pro-grams based on finite volume methods are in wide usefor high-speed compressible and incompressible fluidflows. However, much research effort has been expendedon the development of finite element methods, in parti-cular for incompressible flows. Considering such flows andan Eulerian formulation, stable finite element proceduresneed to use velocity and pressure interpolations like thoseused in the analysis of incompressible solids (but of coursevelocities are interpolated instead of displacements) andalso need to circumvent any instability that arises in thediscretization of the convective terms. This requirement isusually achieved by using some form of upwinding (see forexample Refs. 3, 32 and the references therein). However,another difficulty is that the traditional finite elementformulations do not satisfy ‘‘local’’ mass and momentumconservation. Because numerical solutions of incompres-sible fluid flows in engineering practice should satisfyconservation locally, the usual finite element methodshave been extended (see for example Ref. 33).

One simple approach that meets these requirements isgiven by the flow-condition-based interpolation (FCBI)formulation, in which finite element velocity andpressure interpolations are used to satisfy the inf-supcondition for incompressible analysis, flow-condition-dependent interpolations are used to reach stability inthe convective terms, and control volumes are employedfor integrations, like in finite volume methods (34). Hencestability is reached by the use of appropriate velocity andpressure interpolations, the conservation laws are satis-fied locally, and, also, the given interpolations can be usedto establish consistent Jacobian matrices for the Newton–Raphson type iterations to solve the governing algebraicequations (which correspond to the nodal conditions to besatisfied).

An important point is that the FCBI schemes of fluid flowsolution can be used directly to solve ‘‘fully coupled’’ fluidflows with structural interactions (20,35). The coupling ofarbitrary discretizations of structures and fluids in whichthe structures might undergo large deformations isachieved by satisfying the applicable fluid–structure inter-face force and displacement conditions (20). The completeset of interface relations is included in the governing nodalpoint equations to be solved. Depending on the problem andin particular the number of unknown nodal point variables,it may be most effective to solve the governing equations byusing partitioning (36). However, once the iterations haveconverged (to a reasonable tolerance), the solution of theproblem has been obtained irrespective of whether parti-tioning of the coefficient matrix has been used.

While the solution of fluid–structure interactions isencountered in many applications, the analysis of evenmore general and complex multiphysics problems includ-ing thermo-mechanical, electromagnetic, and chemicaleffects is also being pursued, and the same fundamentalprinciples apply (1).

Solution of Algebraic Equations

The finite element analysis of complex systems usuallyrequires the solution of a large number of algebraic equa-tions; to accomplish this solution effectively is an importantrequirement.

Consider that no parallel processing is used. In staticanalysis, ‘‘direct sparse solvers’’ based on Gauss elimina-tion are effective up to about one half of a million equationsfor three-dimensional solid models and up to about 3 millionequations for shell models. The essence of these solvers isthat first graph theory is used to identify an optimalsequence to eliminate variables and then the actual Gausselimination (that is, the factorization of the stiffness matrixand solution of the unknown nodal variables) is performed.

For larger systems, iterative solvers possibly combinedwith a direct sparse solver become effective, and here inparticular an algebraic multigrid solver is attractive. Mul-tigrid solvers can be very efficient in the solution of struc-tural equations but frequently are embedded in particularstructural idealizations only (like for the analysis of plates).An ‘‘algebraic’’ multigrid solver in principle can be used forany structural idealization because it operates directly onthe given coefficient matrix (37). Figure 5 shows a modelsolved by using an iterative scheme and gives a typicalsolution time.

Considering transient analysis, it is necessary to distin-guish between vibration analyses and wave propagationsolutions. For the linear analysis of vibration problems,mode superposition is commonly performed (3,38). In suchcases, frequencies and mode shapes of the finite elementmodels need be computed, and this is commonly achievedusing the Bathe subspace iteration method or the Lanczosmethod (3).

For the nonlinear analysis of vibration problems,usually a step-by-step direct time integration solution isperformed with an implicit integration technique, and fre-quently the trapezoidal rule is used (3). However, whenlarge deformation problems and relatively long time dura-tions are considered, the scheme in Ref. 39 can be much

Figure 5. Model of a rear axle; about a quarter of a millionelements, including contact solved in about 20 minutes on asingle-processor PC.


more effective. The solution of a finite element model of onehalf of a million equations solved with a few hundred timesteps would be considered a large problem.

Of course, the explicit solution procedures already men-tioned above are used for short duration transient and wavepropagation analyses (3,40).

In the simulations of fluid flows, the number of nodalunknowns usually is very large, and iterative methods needto be used for solution. Here algebraic multigrid solvers arevery effective, but important requirements are that boththe computation time and amount of memory used shouldincrease about linearly with the number of nodal unknownsto be solved for. Figure 3 gives a typical solution time for aNavier–Stokes fluid flow problem. It is seen that withrather moderate hardware capabilities large systems canbe solved.

Of course, these solution times are given merely toindicate some current state-of-the-art capabilities, andhave been obtained using ADINA (41). Naturally, thesolution times would be much smaller if parallel processingwere used (and then would depend on the number ofprocessors used, etc.) and surely will be much reducedover the years to come.

The given observations hold also of course for the solu-tion of multiphysics problems.

MESHING

A finite element analysis of any physical problem requiresthat a mesh of finite elements be generated. Because thegeneration of finite element meshes is a fundamental stepand can require significant human and computationaleffort, procedures have been developed and implementedthat automatize the mesh generation without human inter-vention as much as possible.

Some basic problems in automatic mesh generation are(1) that the given geometries can be complex with verysmall features (small holes, chamfers, etc.) embedded inotherwise rather large geometric domains, (2) that the useof certain element types is much preferable over otherelement types (for example, brick elements are more effec-tive than tetrahedral elements), (3) that graded meshesneed be used for effective solutions (that is, the mesh shouldbe finer in regions of stress concentrations and in boundarylayers in fluid flows or the analysis of shells), and (4) that ananisotropic mesh may be required. In addition, any valu-able mesh generation technique in a general purpose ana-lysis environment (like used in CAE solution packages, seethe section titled, ‘‘The Use of the Finite Element Method inComputer-Aided Engineering’’) must be able to mesh com-plex and general domains.

The accuracy of the finite element analysis results,measured on the exact solution of the mathematical model,highly depends on the use of an appropriate mesh, and thisholds true in particular when coarse meshes need be used toreduce the computer time employed for complex analyses.Hence, effective mesh generation procedures are mostimportant.

Various mesh generation techniques are in use (42).Generally, these techniques can be classified into mapped

meshing procedures, in which the user defines and con-trols the element spacings to obtain a relatively struc-tured mesh, and free-form meshing procedures, in whichthe user defines the minimum and maximum sizes ofelements in certain regions but mostly has little controlas to what mesh is generated, and the user obtains anunstructured mesh. Of course, in each case the user alsodefines for what elements the mesh is to be generated.Mapped meshing techniques in general can be used onlyfor rather regular structural and fluid domains andrequire some human effort to prepare the input butusually result in effective meshes, in the sense that theaccuracy of solution is high for the number of elementsused. The free-form meshing techniques in principle canmesh automatically any 3-D domain provided tetrahedralelements are used; however, a rather unstructuredmesh that contains many elements may be reached.The challenge in the development of free-form meshingprocedures has been to reach meshes that in general donot contain highly distorted elements (long, thin sliverelements must be avoided unless mesh anisotropy isneeded), that do not contain too many elements, andthat contain brick elements rather than tetrahedral ele-ments. Two fundamental approaches have been pursuedand refined, namely methods based on advancing frontmethodologies that generate elements from the boundaryinwards and methods based on Delaunay triangulariza-tions that directly mesh from coarse to fine over thecomplete domain. Although a large effort has alreadybeen expended on the development of effective meshgeneration schemes, improvements still are muchdesired, for example to reach more general and effectiveprocedures to mesh arbitrary three-dimensional geome-tries with brick elements.

Figure 1 shows a three-dimensional mapped mesh ofbrick elements, a largely structured mesh, for the analysisof a wheel, and Fig. 6 shows a three-dimensional mesh oftetrahedral elements, an unstructured mesh, for the ana-lysis of a helmet. It is important to be able to achieve thegrading in elements shown in Fig. 6 because the potentialarea of contact on the helmet requires a fine mesh.

Figure 7 shows another important meshing feature forfinite element analysis, namely the possibility to glue in a‘‘consistent manner’’ totally different meshes together (19).This feature provides flexibility in meshing different partsand allows multiscale analysis. The glueing of course isapplicable in all linear and nonlinear analyses.

Because the effective meshing of complex domains still isrequiring significant human and computational effort,some new discretization methods that do not require amesh in the traditional sense have been developed (43).These techniques are referred to as meshless or meshfreemethods but of course still require nodal points, with nodalpoint variables that are used to interpolate solid displace-ments, fluid velocities, temperatures, or any other conti-nuum variable. The major difference from traditional finiteelement methods is that in meshfree methods, the discretedomains, over which the interpolations take place, usuallyoverlap, whereas in the traditional finite element methods,the finite element discrete domains abut each other, andgeometric overlapping is not allowed. These meshfree


methods require appropriate nodal point spacing, and thenumerical integration is more expensive (43,44). However,the use of these procedures coupled with traditional finiteelements shows promise in regions where either traditionalfinite elements are difficult to generate or such elementsbecome highly distorted in geometric nonlinear analysis(43).

THE USE OF THE FINITE ELEMENT METHOD IN COMPUTER-AIDED ENGINEERING

The finite element method would not have become a suc-cessful tool to solve complex physical problems if themethod had not been implemented in computer programs.The success of the method is clearly due to the effectiveimplementation in computer programs and the possible

wide use of these programs in many industries andresearch environments. Hence, reliable and effective finiteelement methods were needed that could be trusted toperform well in general analysis conditions as mentionedabove. But also, the computer programs had to be easy touse. Indeed, the ease of use is very important for a finiteelement program to be used in engineering environments.

The ease of use of a finite element program is dependenton the availability of effective pre- and post-processingtools based on graphical user interfaces for input andoutput of data. The pre-processing embodies the use ofgeometries from computer-aided design (CAD) programsor the construction of geometries with CAD-like tools, andthe automatic generation of elements, nodal data, bound-ary conditions, material data, and applied loadings. Animportant ingredient is the display of the geometry andconstructed finite element model with the elements,nodes, and so on. The post-processing is used to list andgraphically display the computed results, such as thedisplacements, velocities, stresses, forces, and so on. Inthe post-processing phase, the computed results usuallyare looked at first to see whether the results make sense(because, for example, by an input error, a different thanintended finite element model may have been solved), andthen the results are studied in depth. In particular, theresults should give the answers to the questions thatprovided the stimulus for performing the finite elementanalysis in the first instance.

The use of finite element programs in computer-aidedengineering (CAE) frequently requires that geometries fromCAD programs need be employed. Hence, interfaces toimport these geometries into the finite element pre-proces-sing programs are important. However, frequently the com-puter-aided design geometry can not be used directly togenerate finite element meshes and needs to be changed(‘‘cleaned-up’’) to ensure well-connected domains, and bydeleting unnecessary details. The effective importing ofcomputer-aided design data is a most important step forthewideuseoffiniteelementanalysis inengineeringdesign.

As mentioned already, the purpose of a finite elementanalysis is to solve a mathematical model. Hence, in thepost-processing phase, the user of a finite element programideally would be able to ascertain that a sufficiently accu-rate solution has been obtained. This means that, ideally, ameasure of the error between the computed solution andthe ‘‘exact solution of the mathematical model’’ would beavailable. Because the exact solution is unknown, onlyestimates can be given. Some procedures for estimatingthe error are available (mostly in linear analysis) but needto be used with care (45,46), and in practice frequently theanalysis is performed simply with a very fine mesh toensure that a sufficiently accurate solution has beenobtained. Figure 8 shows an example of error estimation.Here, the scheme proposed by Sussman and Bathe (47) isused in a linear, nonlinear, and FSI solution using ADINA.

In the first decades of finite element analysis, manyfinite element programs were available and used. However,the ensuing strive for improved procedures in these pro-grams, in terms of generality, effectiveness, and ease of use,required a significant continuous development effort. Theresulting competition in the field to further develop and

Figure 6. Model of a bicycle helmet showing mesh gradation.

Figure 7. Glueing of dissimilar meshes.


Figure 8. Error estimation example: analysis of a cantilever with a hole. (a) Mesh of 9-node elements used for analysis of cantilever (withboundary conditions). (b) Cantilever structure — linear analysis, estimated error in region of interest shown. (c) Cantilever structure —linear analysis, exact error in region of interest shown. (d) Cantilever structure — nonlinear analysis, estimated error in region of interestshown. (e) Cantilever structure — nonlinear analysis, exact error in region of interest shown. (f) Cantilever structure — FSI analysis,estimated error in region of interest shown. (g) Cantilever structure — FSI analysis, exact error in region of interest shown.


continuously support a finite element program reduced thenumber of finite element systems now widely used to only afew: MSC NASTRAN, NX NASTRAN, ANSYS, ABAQUS,ADINA, and MARC are the primary codes used for staticand implicit dynamic analyses of structures, and LS-DYNA, RADIOSS, PAMCRASH are the primary codesused for explicit dynamic analyses. For multiphysics pro-blems, notably fluid–structure interactions, primarilyADINA and ANSYS are used.

CONCLUSIONS AND KEY CHALLENGES

Since the first publications on the practical use of the finiteelement method, the field of finite element analysis hasexploded in terms of research activities and applications.Numerous references are given on the World Wide Web.The method is now applied in practically all areas ofengineering and scientific analyses. Since some time, thefinite element method and related techniques frequentlyare simply referred to as ‘‘methods in computational fluidand solid mechanics.’’

With all these achievements in place and the abundantapplications of computational mechanics today, it is surely

of interest to ask, ‘‘What are the outstanding research anddevelopment tasks? What are the key challenges still inthe field?’’ These questions are addressed in the prefaces ofRef. 1, see the 2003 and 2005 volumes.

Although much has been accomplished, still, manyexciting research tasks exist in further developing finiteelement methods. These developments and the envisa-ged increased use of finite element methods not only willhave a continuous and very beneficial impact on tradi-tional engineering endeavors but also will lead to greatbenefits in other areas such as in the medical and healthsciences.

Specifically, the additional developments of finite ele-ment methods should not be directed only to the moreeffective analysis of single media but also must focus onthe solution of multiphysics problems that involve fluids,solids, their interactions, and chemical and electromag-netic effects from the molecular to the macroscopic scales,including uncertainties in the given data, and should alsobe directed to reach more effective algorithms for theoptimization of designs.

Based on these thoughts, we can identify at least eightkey challenges for research and development in finite ele-ment methods and numerical methods in general; these

Figure 8. (Continued )


challenges pertain to the more automatic solution of math-ematical models, to more effective and predictive numericalschemes for fluid flows, to mesh-free methods that arecoupled with traditional finite element methods, to finiteelement methods for multiphysics problems and multiscaleproblems, to the more direct modeling of uncertainties inanalysis data, to the analysis and optimization of completelifecycles of designed structures and fluid systems, andfinally, to providing effective education and educationaltools for engineers and scientists to use the given analysisprocedures in finite element programs correctly and to thefull capabilities (1).

Hence, although the finite element method already iswidely used, still, many exciting research and develop-ment efforts exist and will continue to exist for manyyears.

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KLAUS-JURGEN BATHE

Massachusetts Institute ofTechnology

Cambridge, Massachusetts


ABSTRACT

The objective in this article is to give an overview of finiteelement methods that currently are used extensively inacademia and industry. The method is described ingeneral terms, the basic formulation is presented, andsome issues regarding effective finite element proceduresare summarized. Various applications are given briefly toillustrate the current use of the method. Finally, thearticle concludes with key challenges for the additionaldevelopment of the method.

"the finite element methods", in wiley encyclopedia of computer

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