lecture 01 - intro to fem

8/3/2019 Lecture 01 - Intro to FEM

1/56

1

.


2/56

2

Introduction to the Finite Element Method (FEM)

Steps in Using the FEM (an Example from SolidMechanics)

Examples

Commercial FEM Software Competing Technologies

Future Trends

Internet Resources References


3/56

3

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 briefhistory of its origins. The theoretical basis for the FEM isdiscussed, with emphasis on the basic methodologies,assumptions, and advantages (and limitations) of themethod. Next, the basic steps that must be performed inany FEM analysis are illustrated (using an example fromsolid mechanics), and FEM examples are provided forproblems from other engineering disciplines.

To aid the reader in selecting a FEM software package, abrief survey of currently available FEM software ispresented, together with a discussion of alternative analysistechniques that might be considered in lieu of the FEM.Finally, we examine future trends in the FEM.


4/56

4

Contents 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


5/56

5

Many problems in engineering and appliedscience are governed by differential or integralequations.

The solutions to these equations would providean exact, closed-form solution to the particular

problem being studied.

However, complexities in the geometry,properties and in the boundary conditions that

are seen in most real-world problems usuallymeans that an exact solution cannot be obtainedor obtained in a reasonable amount of time.

Finite Element Method Defined


6/56

6

Finite Element Method Defined (cont.)

Current product design cycle times imply thatengineers must obtain design solutions in a

short amount of time.

They are content to obtain approximate solutions

that can be readily obtained in a reasonable timeframe, and with reasonable effort. The FEM isone such approximate solution technique.

The FEM is a numerical procedure for obtainingapproximate solutions to many of the problemsencountered in engineering analysis.


7/56

7

Finite Element Method Defined (cont.) In the FEM, a complex region defining a continuum is

discretized into simple geometric shapes called elements.

The properties and the governing relationships are assumedover these elements and expressed mathematically interms of unknown values at specific points in the elementscalled nodes.

An assembly process is used to link the individual elementsto the given system. When the effects of loads andboundary conditions are considered, a set of linear ornonlinear algebraic equations is usually obtained.

Solution of these equations gives the approximate behaviorof the continuum or system.


8/56

8

Finite Element Method Defined (cont.) 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, finiteelementmethod.

The number of equations is usually rather large for most

real-world applications of the FEM, and requires thecomputational power of the digital computer. The FEM haslittle practical value if the digital computer were notavailable.

Advances in and ready availability of computers andsoftware has brought the FEM within reach of engineersworking in small industries, and even students.


9/56

9

Finite Element Method Defined (cont.) Two features of the finite element method are worth

noting.

The piecewise approximation of the physical field(continuum) on finite elements provides good precisioneven with simple approximating functions. Simply

increasing the number of elements can achieve increasingprecision.

The locality of the approximation leads to sparse equationsystems 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 systemscontaining a million primary unknowns.


10/56

10

Origins of the Finite Element Method 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 finite elementwas first coined by Clough in 1960.In the early 1960s, engineers used the method for

approximate solution of problems in stress analysis, fluidflow, heat transfer, and other areas.

The first book on the FEM by Zienkiewicz and Chung waspublished in 1967.

In the late 1960s and early 1970s, the FEM was applied toa wide variety of engineering problems.


11/56

11

Origins of the Finite Element Method

(cont.) The 1970s marked advances in mathematical treatments,

including the development of new elements, and

convergence studies.

Most commercial FEM software packages originated in the1970s (ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s

(FENRIS, LARSTRAN 80, SESAM 80.)

The FEM is one of the most important developments incomputational methods to occur in the 20th century. Injust a few decades, the method has evolved from one with

applications in structural engineering to a widely utilizedand richly varied computational approach for manyscientific and technological areas.


12/56

12

How can the FEM Help the Design

Engineer? The FEM offers many important advantages to the design

engineer:

Easily applied to complex, irregular-shaped objects composedof several different materials and having complex boundaryconditions.

Applicable to steady-state, time dependent and eigenvalueproblems.

Applicable to linear and nonlinear problems.

One method can solve a wide variety of problems, includingproblems in solid mechanics, fluid mechanics, chemicalreactions, electromagnetism, biomechanics, heat transfer andacoustics, to name a few.


13/56

13

How can the FEM Help the Design

Engineer? (cont.) General-purpose FEM software packages are available

at reasonable cost, and can be readily executed on

microcomputers, including workstations and PCs.

The FEM can be coupled to CAD programs to facilitatesolid modeling and mesh generation.

Many FEM software packages feature GUI interfaces,auto-meshers, and sophisticated postprocessors andgraphics to speed the analysis and make pre and post-processing more user-friendly.


14/56

14

How can the FEM Help the

Design Organization? Simulation using the FEM also offers important business

advantages to the design organization:

Reduced testing and redesign costs thereby shortening the productdevelopment time.

Identify issues in designs before tooling is committed.

Refine components before dependencies to other components prohibitchanges.

Optimize performance before prototyping.

Discover design problems before litigation.

Allow more time for designers to use engineering judgement, and lesstime turning the crank.


15/56

15Discretization error due to poor geometryrepresentation.

Discretization error effectively eliminated.

Sources of Error in the FEM The three main sources of error in a typical FEM solution are discretization

errors, formulation errors and numerical errors.

Discretization error results from transforming the physical system(continuum) into a finite element model, and can be related to modelingthe boundary shape, the boundary conditions, etc.


16/56

16

Formulation error results from the use of elements that don'tprecisely describe the behavior of the physical problem.

Elements which are used to model physical problems for whichthey are not suited are sometimes referred to as ill-conditioned ormathematically unsuitable elements.

For example a particular finite element might be formulated onthe assumption that displacements vary in a linear manner overthe domain. Such an element will produce no formulation error

when it is used to model a linearly varying physical problem(linear varying displacement field in this example), but wouldcreate a significant formulation error if it used to represent aquadratic or cubic varying displacement field.

Sources of Error in the FEM


17/56

17

Sources of Error in the FEM (cont.) Numerical error occurs as a result of numerical

calculation procedures, and includes truncationerrors and round off errors.

Numerical error is therefore a problem mainlyconcerning the FEM vendors and developers.

The user can also contribute to the numerical

accuracy, for example, by specifying a physicalquantity, say Youngs modulus, E, to aninadequate number of decimal places.


18/56

18

Advantages of the

Finite Element Method Can readily handle complex geometry:

The heart and power of the FEM.

Can handle complex analysis types: Vibration Transients Nonlinear

Heat transfer Fluids

Can handle complex loading: Node-based loading (point loads).

Element-based loading (pressure, thermal, inertialforces). Time or frequency dependent loading.

Can handle complex restraints: Indeterminate structures can be analyzed.


19/56

19

Advantages of the

Finite Element Method Can handle bodies comprised of nonhomogeneous

materials:

Every element in the model could be assigned a different setof material properties.

Can handle bodies comprised of nonisotropic materials: Orthotropic

Anisotropic Special material effects are handled:

Temperature dependent properties.

Plasticity

Creep

Swelling

Special geometric effects can be modeled: Large displacements.

Large rotations.

Contact (gap) condition.


20/56

20

Disadvantages of the

Finite Element Method A specific numerical result is obtained for a specific

problem. A general closed-form solution, which would

permit one to examine system response to changes invarious parameters, is not produced.

The FEM is applied to an approximation of themathematical model of a system (the source of so-called

inherited errors.)

Experience and judgment are needed in order to constructa good finite element model.

A powerful computer and reliable FEM software areessential.

Input and output data may be large and tedious to prepareand interpret.


21/56

21

Numerical problems: Computers only carry a finite number of significant

digits. Round off and error accumulation. Can help the situation by not attaching stiff (small)

elements to flexible (large) elements.

Susceptible to user-introduced modeling errors: Poor choice of element types. Distorted elements. Geometry not adequately modeled.

Certain effects not automatically included: Buckling Large deflections and rotations. Material nonlinearities . Other nonlinearities.

Disadvantages of the

Finite Element Method


22/56

22


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

Examples



Future Trends

Internet Resources

References


23/56

23

Six Steps in the Finite Element Method Step 1 - Discretization:

The problem domain is discretized into a

collection of simple shapes, or elements.

Step 2 - Develop Element Equations:Developed using the physics of the problem, and

typically Galerkins Method or variationalprinciples.

Step 3 - Assembly:

The element equations for each element in theFEM mesh are assembled into a set of globalequations that model the properties of the entiresystem.


24/56

24

Six Steps in the Finite Element Method Step 4 - Application of Boundary Conditions:

Solution cannot be obtained unless boundary

conditions are applied. They reflect the knownvalues for certain primary unknowns. Imposingthe boundary conditions modifies the globalequations.

Step 5 - Solve for Primary Unknowns:The modified global equations are solved for theprimary unknowns at the nodes.

Step 6 - Calculate Derived Variables:Calculated using the nodal values of the primaryvariables.


25/56

25

Analysis of solids

Static Dynamics

Behavior of Solids

Linear Nonlinear

Material

Fracture

GeometricLarge Displacement

Instability

PlasticityViscoplasticity

GeometricClassification of solids

Skeletal Systems1D Elements

Plates and Shells2D Elements

Solid Blocks3D Elements

TrussesCablesPipes

Plane StressPlane StrainAxisymmetricPlate BendingShells with flat elementsShells with curved elements

Brick ElementsTetrahedral ElementsGeneral Elements

Elementary Advanced

Stress Stiffening

Classification of Solid-Mechanics

Problems


26/56

26

StartProblemDefinition

Pre-processor

Reads or generatesnodes and elements(ex: ANSYS)

Reads or generatesmaterial property data.

Reads or generatesboundary conditions(loads andconstraints.)

Processor

Generateselement shapefunctions

Calculates master

element equations Calculates

transformationmatrices

Maps elementequations into

global system Assembles

element equations Introduces

boundaryconditions

Performs solutionprocedures

Post-processor

Prints or plotscontours of stresscomponents.

Prints or plotscontours ofdisplacements.

Evaluates andprints errorbounds.

Analysis anddesign decisions

Stop

Step 1, Step 4

Step 6

Steps 2, 3, 5

Process Flow in a Typical FEM Analysis


27/56

27

Geometry

from CADprogram

meshgenerator

surface model

12

34 5 11

121314

meshed model

Step 1: Discretization

Mesh Generation


28/56

28

Displacements DOF constraints usually specified at model boundaries to define rigid

supports.

Forces and Moments Concentrated loads on nodes usually specified on the model exterior.

Pressures Surface loads usually specified on the model exterior.

Temperatures Input at nodes to study the effect of thermal expansion or

contraction.

Inertia Loads Loads that affect the entire structure (ex: acceleration, rotation).

Step 4: Boundary Conditions for a Solid

Mechanics Problem


29/56

29

150

175

200

225

250

275

300

200

225

250

275

300

150175

Temp mapper

Nodes fromFE Modeler

ThermalSoln Files

Step 4: Applying Boundary Conditions

(Thermal Loads)


30/56

30



Examples



Future Trends

Internet Resources

References


31/56

31

Heat transfer analysis Temperature

Heat fluxes Thermal gradients

Heat flow from convectionfaces

Fluid analysis Pressures

Gas temperatures

Convection coefficients

Velocities

Static analysis: Deflection

Stresses Strains

Forces

Energies

Dynamic analysis: Frequencies

Deflection (mode shape)

Stresses

Strains

Forces

Energies

Information Available from Various

Types of FEM Analysis


32/56

32

Example FEM Application Areas Automotive industry

Static analyses

Modal analyses Transient dynamics Heat transfer Mechanisms Fracture mechanics

Metal forming Crashworthiness

Architectural Soil mechanics

Rock mechanics Hydraulics Fracture mechanics Hydroelasticity

Aerospace industry Static analyses Modal analyses Aerodynamics Transient dynamics Heat transfer Fracture mechanics Creep and plasticity analyses

Composite materials Aeroelasticity Metal forming Crashworthiness


33/56

33

The FEM has been applied to a richly diversearray of scientific and technological problems.

The next few slides present some examples ofthe FEM applied to a variety of real-world design

and analysis problems.

Variety of FEM Solutions is

Wide and Growing Wider


34/56

34

This example shows an intravenous pump modeled

using hexahedral elements.


35/56

35

Car tires require sophisticated analysis because of their complex geometry, large

deformations, nonlinear material behavior, and varying contact conditions. Brick

elements are used to represent the tread and steel bead, while shell elements areused in the wall area. Membrane elements are used to represent the tire cords.


36/56

36

This forging example is a simulation of a bulk forming

process with multiple stages. This axisymmetric analysis

begins with a cylinder of metal meshed very simply.


37/56

37

A 3-D finite element model of an instrumented canine cervical spine.

The model consisted of four vertebrae (C3-C6), a titanium alloy plate,

and two screws attached to the back of two vertebrae (C4-C5).


38/56

38

Finite element analysis works on the premise that a complex

structure like the helicopter shown here can be simulated on

a computer screen so that the helicopter's physicalproperties can be studied to determine how well the design

will perform under real-world conditions. The computer

models permit the design team to examine a wide range of

options and to detect design flaws long before the prototypestage.


39/56

39

This guitar features two strips of

graphite running the length of the

neck. This FEM model was used to

study how much the neck movedwhen string forces were applied and

moisture content

changed.

Using the FEM calculations,

designers could try different

reinforcement scenarios to increase

neck stability.


40/56

40

The boats hull consists of a thick core material sandwiched between two thinner

layers of plys oriented in different directions. The initial analysis work focused

on maximizing the hull's overall stiffness by examining different core-materialdensities and varying the ply thickness and orientations.

Dynamic analysis of a tuning fork to find it's first eight modes of vibration


41/56

41

Dynamic analysis of a tuning fork, to find it s first eight modes of vibration.

1

2

3

4

5

6

7

8


42/56

42



Examples



Future Trends

Internet Resources

References

Commercially Available


43/56

43

Here we present a survey of some of the better-knownintegratedFEM software packages. These integrated

systems allow users to perform all facets of FEM analysis,including modeling, meshing, solution and post-processing.

The Internet provides a vast new resource for individualsinterested in the FEM. See the Reference section of this

paper for interesting FEM links to start your Internetresearch.

In addition to the integrated FEM packages listed below,

many vendors offer dedicated software for solid modeling,mesh generation, FE equation generation and solution, andpost-processing.

Commercially Available

FEM Software Suites

Commercially Available FEM Software


44/56

44

Software Package Introduced Comments

ABAQUS 1978 General purpose, with special emphasis on advanced linear and nonlinear structures and heat transfer applications.

ADINA 7.0 1975 Optimized for structural and heat transfer applications. Limited element library. Extensive material model library.

ALGOR 1984 First FEM package available for PC use.

ANSYS/LS-DYNA N/A For solving highly nonlinear structural dynamics problems (impact, large deformation, nonlinear materials, etc.)

ANSYS/MECHANICAL 1970 Probably the best-known and most widely-used FEM software. Complete structures/thermal/acoustics modleing.

ANSYS/Multiphysics N/A Coupled-field, multidisciplinary FEM program.

ELFEN N/A Includes linear and nonlinear buckling, modal analysis, transient heat transfer analysis, impact and fragmentation.

GENESIS N/A Fully integrated finite element analysis and numerical optimization software for structural analysis.

LUSAS N/A Includes automatic meshing, advanced non-linear analysis, and composites analysis.

MARC 6.2 1970 3D automated contact analysis capabilities suited for studying tough manufacturing problems, (metal forming/ etc.)

MSC/FEA 1971 MSC participated in the 1965 development of NASA's public-domain FEM code, NASTRAN.

MSC/NASTRAN for Windows N/A Handles stress, vibration, dynamic, nonlinear, heat transfer, and fluid flow analyses of mechanical components.

NISA/DISPLAY 1973 A family of general purpose FEM programs for PCs and workstations. Modular design.PAM 1973 FEM software optimized to study restraint systems (PAM-SAFE), impacts (PAM-SHOCK) and metal forming.

SAMCEF 1965 One of the oldest FEM codes available. A powerful FEM package for structural and heat transfer analysis.

STARDYNE 1967 The world's first commercially available Finite Element Analysis software.

STARS N/A Integrated, general-purpose, finite element software. Developed by NASA.

Commercially Available FEM Software

Suites (cont.) (partial list)


45/56

45



Examples



Future Trends

Internet Resources

References

Technologies That Compete With the


46/56

46

Technologies That Compete With the

FEM Other numerical solution methods:

Finite differences Approximates the derivatives in the differential equation using

difference equations. Useful for solving heat transfer and fluid mechanics problems. Works well for two-dimensional regions with boundaries parallel

to the coordinate axes. Cumbersome when regions have curved boundaries.

Weighted residual methods (not confined to a smallsubdomain): Collocation Subdomain Least squares* Galerkins method*

Variational Methods* (not confined to a small subdomain)

* Denotes a method that has been used to formulate finiteelement solutions.

Technologies that Compete With the


47/56

47

Technologies that Compete With the

FEM (cont.) Prototype Testing

Reliable. Well-understood. Trusted by regulatory agencies (FAA, DOT, etc.) Results are essential for calibration of simulation software. Results are essential to verify modeled results from

simulation. Non destructive testing (NDT) is lowering costs of testing in

general. Expensive, compared to simulation. Time consuming. Development programs that rely too much on testing are

increasingly less competitive in todays market. Faster product development schedules are pressuring the

quality of development test efforts. Data integrity is more difficult to maintain, compared to

simulation.


48/56

48



Examples



Future Trends

Internet Resources

References

Future Trends in the


49/56

49

The FEM in particular, and simulation in general, arebecoming integrated with the entire product developmentprocess (rather than just another task in the product

development process): FEM cannot become the bottleneck.

A broader range of people are using the FEM:

Not just hard-core analysts.

Increased data sharing between analysis data sources(CAD, testing, FEM software, ERM software.)

FEM software is becoming easier to use: Improved GUIs, automeshers. Increased use of sophisticated shellscripts and wizards.


FEM and Simulation



50/56

50


FEM and Simulation (cont.) Enhanced multiphysics capabilities are coming:

Coupling between numerous physical phenomena. Ex: Fluid-structural interaction is the most common example.

Ex: Semiconductor circuits, EMI and thermal buildup vary with currentdensities.

Improved life predictors, improved service estimations.

Increasing use of non-deterministic analysis and design methods: Statistical modeling of material properties, tolerances, and anticipatedloads.

Sensitivity analyses.

Faster and more powerful computer hardware. Massively parallel

processing.

Decreasing reliance on testing.

FEM and simulation software available via Internet subscription.


51/56

51



Examples



Future Trends

Internet Resources

References


52/56

52

The internet offers virtually unlimited resources to personsinterested in the FEM.

The following links are a small sample of FEM sites on theInternet which the author has found useful. Thousandsmore (at least!) are readily available.

Most commercial FEM developers have extensive presenceon the Internet, with web pages that include companyhistories, descriptions of software products, and exampleFEM solutions.

Other good FEM resources on the web originate withacademia, government, and discussion and user groups.

Selected FEM Resources on the Internet


53/56

53

http://www.engineeringzones.com -A website created to educate people in the latestengineering technologies, manufacturing techniques and

software tools. Exellent FEM links, including links to allcommercial providers of FEM software.

http://www.comco.com/feaworld/feaworld.html -Extensive FEM links, categorized by analysis type

(mechanical, fluids, electromagnetic, etc.)

http://femur.wpi.edu Extensive collection of elementary and advanced materialrelating to the FEM.

http://www.engr.usask.ca/%7Emacphed/finite/fe_resources/fe_resources.html -Lists many public domain and shareware programs.



54/56

54

http://sog1.me.qub.ac.uk/dermot/ferg/ferg.html#Finite -Home page of the the Finite Element Research Group at

The Queen's University of Belfast. Excellent set of FEMlinks.

http://www.tenlinks.com/cae/ -Hundreds of links to useful and interesting CAE cited,

including FEM, CAE, free software, and career information.

http://www.geocities.com/SiliconValley/5978/fea.html -Extensive FEM links.

http://www.nafems.org/ -National Agency for Finite Element Methods and Standards(NAFEMS).



55/56

55



Examples



Future Trends

Internet Resources

References


56/56

56

Cashman, J., 2000. Future of Engineering Simulation,ANSYSSolutions, Vol. 2, No. 1, pp. 3-4.

Chandrupatla, T. R. and Ashok D. Belegundu, 1997. Introduction

to Finite Elements in Engineering, Prentice Hall, Upper SaddleRiver, New Jersey. Kardestuncer, H., 1987. Finite Element Handbook, McGraw-Hill,

New York. Krouse, J., 2000. Physical Testing Gets a Bum Rap,ANSYS

Solutions, Vol. 2, No. 2, p. 2. Lentz, J., 1994. Finite Element Analysis Cross Training,

unpublished lecture notes, Honeywell Engines and Systems,Phoenix, Az.

Nikishkov, G.V., 1998. Introduction to the Finite Element Method,unpublished lecture notes, University of Arizona, Tucson, Az.

Rajan, S.D., 1998. Finite Elements for Engineers, unpublishedlecture notes, Arizona State University, Tempe, Az.

Segerlind, L. J., 1984.Applied Finite Element Analysis, John Wileyand Sons, New York.

References

lecture 01 - intro to fem

Documents