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Finite Element Primer for Engineers

Mike Barton & S. D. Rajan

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Finite Element Primer for Engineers

Introduction to the Finite Element Method (FEM)

Steps in Using the FEM: An Example from Solid Mechanics

Examples

Commercial FEM Software

Competing Technologies

Future Trends

Internet Resources

References

Contents

©, 2000, Barton & Rajan



Foreword

This document was submitted as a term paper for the graduate engineering course

CEE598 Finite Elements for Engineers, offered at Arizona State University.

The objective of this article is to provide engineers with a brief introduction to the

finite element method (FEM). The article includes an overview of the FEM,

including a brief history of its origins. The theoretical basis for the FEM is discussed,

with emphasis on the basic methodologies, assumptions, and advantages (and

limitations) of the method. Next, the basic steps that must be performed in any FEManalysis are illustrated (using an example from solid mechanics), and FEM examples

are provided for problems from other engineering disciplines.

To aid the reader in selecting a FEM software package, a brief survey of currently

available FEM software is presented, together with a discussion of alternative analysis

techniques that might be considered in lieu of the FEM. Finally, we examine futuretrends in the FEM.

References are provided for those desiring further information on the FEM (including

selected Internet references.)





Contents

Introduction to the Finite Element Method (FEM)

Steps in Using the FEM (an Example from Solid Mechanics)Steps in Using the FEM (an Example from Solid Mechanics)Steps in Using the FEM (an Example from Solid Mechanics)

Examples Examples Examples

Commercial FEM Software Commercial FEM Software Commercial FEM Software

Competing Technologies Competing Technologies Competing Technologies

Future Trends Future Trends Future Trends

Internet Resources Internet Resources Internet Resources

References References References




Finite El ement M ethod D efined

Many problems in engineering and applied science aregoverned by differential or integral equations.

The solutions to these equations would provide an exact,

closed-form solution to the particular problem being studied.

However, complexities in the geometry, properties and inthe boundary conditions that are seen in most real-world

problems usually means that an exact solution cannot beobtained or obtained in a reasonable amount of time.




Finite El ement M ethod D efined (co nt.)

Current product design cycle times imply that engineersmust obtain design solutions in a µshort¶ amount of time.

They are content to obtain approximate solutions that can be readily obtained in a reasonable time frame, and with

reasonable effort. The FEM is one such approximatesolution technique.

The FEM is a numerical procedure for obtainingapproximate solutions to many of the problems encountered

in engineering analysis.





In the FEM, a complex region defining a continuum isdiscretized into simple geometric shapes called el ement s.

The properties and the governing relationships are assumedover these elements and expressed mathematically in terms of unknown values at specific points in the elements called node s.

An assembly process is used to link the individual elements tothe given system. When the effects of loads and boundaryconditions are considered, a set of linear or nonlinear algebraicequations is usually obtained.

Solution of these equations gives the appr oximate behavior of the continuum or system.





The continuum has an infinite number of degrees-of-freedom(DOF), while the discretized model has a finite number of DOF.

This is the origin of the name, f inite el ement method.

The number of equations is usually rather large for most real-

world applications of the FEM, and requires the computational

power of the digital computer. The FEM has l itt l e practical

value i f the digit al com puter wer e not availabl e.

Advances in and ready availability of computers and software

has brought the FEM within reach of engineers working in

small industries, and even students.





Two features of the finite element method are worth noting.

The piecewise approximation of the physical field

(continuum) on finite elements provides good precision even

with simple approximating functions. Simply increasing the

number of elements can achieve increasing precision.

The locality of the approximation leads to sparse equation

systems for a discretized problem. This helps to ease the

solution of problems having very large numbers of nodalunknowns. It is not uncommon today to solve systems

containing a million primary unknowns.




Or igins o f the Finite El ement M ethod

It is difficult to document the exact origin of the FEM, because the basic concepts have evolved over a period of 150 or more years.

The term f inite el ement was first coined by Clough in 1960. In the

early 1960s, engineers used the method for approximate solution of

problems in stress analysis, fluid flow, heat transfer, and other areas.

The first book on the FEM by Zienkiewicz and Chung was

published in 1967.

In the late 1960s and early 1970s, the FEM was applied to a widevariety of engineering problems.




Or igins o f the Finite El ement M ethod (co nt.)

The 1970s marked advances in mathematical treatments, includingthe development of new elements, and convergence studies.

Most commercial FEM software packages originated in the 1970s

(ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s

(FENRIS, LARSTRAN µ80, SESAM µ80.)

The FEM is one of the most important developments in

computational methods to occur in the 20th century. In just a few

decades, the method has evolved from one with applications in

structural engineering to a widely utilized and richly variedcomputational approach for many scientific and technological areas.




How ca n the FEM Help the D esign Engineer?

The FEM offers many important advantages to the de sign engineer :

Easily applied to complex, irregular-shaped objects composed

of several different materials and having complex boundary

conditions.

Applicable to steady-state, time dependent and eigenvalue

problems.

Applicable to linear and nonlinear problems.

One method can solve a wide variety of problems, including

problems in solid mechanics, fluid mechanics, chemical

reactions, electromagnetics, biomechanics, heat transfer and

acoustics, to name a few.




How ca n the FEM Help the D esign Engineer? (co nt.)

General-purpose FEM software packages are available atreasonable cost, and can be readily executed on

microcomputers, including workstations and PCs.

The FEM can be coupled to CAD programs to facilitate solid

modeling and mesh generation.

Many FEM software packages feature GUI interfaces, auto-

meshers, and sophisticated postprocessors and graphics to speed

the analysis and make pre and post-processing more user-

friendly.




How ca n the FEM Help the D esign Or ganization?

Simulation using the FEM also offers important business advantages to

the de sign or g anization:

Reduced testing and redesign costs thereby shortening the product

development time.

Identify issues in designs before tooling is committed.

Refine components before dependencies to other components

prohibit changes.

Optimize performance before prototyping.

Discover design problems before litigation.

Allow more time for designers to use engineering judgement, and

less time ³turning the crank.´




Theor etical B asis: For mulating Element Equations

Several approaches can be used to transform the physicalformulation of a problem to its finite element discrete analogue.

If the physical formulation of the problem is described as a

differential equation, then the most popular solution method is

the Method o f Weighted Re sid uals.

If the physical problem can be formulated as the minimization

of a functional, then the Var iational For mulation is usually

used.




Theor etical B asis: MWR

One family of methods used to numer ically solve differ ential equations ar e called the

methods of weighted r esiduals (MWR).

In the MWR, an app r oximate solution is substituted into the differ ential equation. S ince the app r oximate solution does not identically satisfy the equation, a r esidual, or err or ter m, r esults.

Consider a differ ential equationDy¶¶(x ) + Q = 0 (1)

Suppose that y = h(x ) is an app r oximate solution to (1). Substitution then g ives Dh¶¶(x ) + Q= R, wher e R is a nonzer o r esidual. The MWR then r equir es that

´Wi(x )R(x ) = 0 (2)

wher e Wi(x ) ar e the weighting f unctions. The number of weighting f unctions equals the number of unknown coefficients in the app r oximate solution.




Theor etical B asis: Galer kin¶s Method

Ther e ar e sever al choices for the weighting f unctions, Wi.

In the Galer kin¶s method, the weighting f unctions ar e the same f unctions that wer e used in the app r oximating equation.

The Galer kin¶s method yields the same

r esults as the va

r iational method when applied to differ ential equations that ar e self-adj oint.

The MWR is ther efor e an integr al solution method.

Many r eader s may find it unusual to see a numer ical solution that is based on an

integr al for mulation.




Theor etical B asis: Var iational Method

The var iational method involves the integr al of a f unction that pr oduces a

number . Each new f unction pr oduces a new number .

The f unction that pr oduces the lowest number has the additional pr oper ty of satisfying a specific differ ential equation.

Consider the integr alT !´?D/2 y¶¶(x) - Qy]dx = 0. (1)

The numer ical value of Tcan be calculated given a specific equation y =f (x). Var iational calculus shows that the par ticular equation y = g(x) which

yields the lowest numer ical value for Tis the solution to the differ ential equation

Dy¶¶(x) + Q = 0. (2)




Theor etical B asis: Var iational Method (cont.)

In solid mechanics, the so-called Rayeigh-Ritz technique uses the Theor em of Minimum P otential Ener gy (with the potential ener gy being the f unctional, T) to develop the element equations.

The tr ial solution that gives the minimum value of Tis the appr oximate

solution.

In other specialty ar eas, a var iational pr inciple can usually be found.




S our ces of Err or in the FEM

The thr ee main sour ces of err or in a typical FEM solution ar e discr etization err or s,

for mulation err or s and numer ical err or s.

Discr etization err or r esults f r om tr ansfor ming the physical system (continuum) into a finite element model, and can be r elated to modeling the boundar y shape, the boundar y conditions, etc.

Discretization error due to poor geometry

representation.

Discretization error effectively eliminated.




S our ces of Err or in the FEM (cont.)

For mulation err or r esults f r om the use of elements that don't pr ecisely descr ibe the behavior

of the physical pr oblem.

Elements which ar e used to model physical pr oblems for which they ar e not suited ar e sometimes r eferr ed to as ill-conditioned or mathematically unsuitable elements.

For example a par ticular finite element might be for mulated on the assumption that displacements var y in a linear manner over the domain. Such an element will pr oduce no for mulation err or when it is used to model a linear ly var ying physical pr oblem (linear var ying

displacement field in this example), but would cr eate a significant for mulation err or if it used to r epr esent a quadr atic or cubic var ying displacement field.




S our ces of Err or in the FEM (cont.)

Numer ical err or occur s as a r esult of numer ical calculation pr ocedur es, and includes tr uncation err or s and r ound off err or s.

Numer ical err or is ther efor e a pr oblem mainly concer ning the FEM vendor s and developer s.

The user can also contr ibute to the numer ical accur acy, for example, by specifying a physical quantity, say oung¶s modulus, E, to an inadequate number of decimal places.




Advantages of the Finite Element Method

Can r eadily handle complex geometr y:

» The hear t and power of the FEM.

Can handle complex analysis types:

» Vibr ation

» Tr ansients

» Nonlinear » Heat tr ansfer

» Fluids

Can handle complex loading:

» Node-based loading (point loads).

» Element-based loading (pr essur e, ther mal, iner tial for ces).

» Time or f r equency dependent loading.

Can handle complex r estr aints:

» Indeter minate str uctur es can be analyzed.




Advantages of the Finite Element Method (cont.)

Can handle bodies compr ised of nonhomogeneous mate

r ials:

» Ever y element in the model could be assigned a differ ent set of mater ial pr oper ties.

Can handle bodies compr ised of nonisotr opic mater ials:

» Or thotr opic

» Anisotr opic

Special mater ial effects ar e handled:

» Temper atur e dependent pr oper ties.

» P lasticity

» Cr eep

» Swelling Special geometr ic effects can be modeled:

» Lar ge displacements.

» Lar ge r otations.

» Contact (gap) condition.




Disadvantages of the Finite Element Method

A specific numer ical

r esult is obtained fo

r a specific p

r oblem. A gene

r al closed-for m solution, which would per mit one to examine system r esponse to

changes in var ious par ameter s, is not pr oduced.

The FEM is applied to an appr oximation of the mathematical model of a system (the sour ce of so-called inher ited err or s.)

Exper ience and judgment ar e needed in or der to constr uct a good finite element model.

A power f ul computer and r eliable FEM softwar e ar e essential.

Input and output data may be lar ge and tedious to pr epar e and inter pr et.




Disadvantages of the Finite Element Method (cont.)

Numer ical p

r oblems:

» Computer s only carr y a finite number of significant digits.

» Round off and err or accumulation.

» Can help the situation by not attaching stiff (small) elements to flexible (lar ge) elements.

Susceptible to user -intr oduced modeling err or s:» P oor choice of element types.

» Distor ted elements.

» Geometr y not adequately modeled.

Cer tain effects not automatically included:

» Buckling

» Lar ge deflections and r otations.

» Mater ial nonlinear ities .

» Other nonlinear ities.

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FEMPrimer-Part1 [EDocFind.com]