Introduction to Finite Element Methods.pdf

8/14/2019 Introduction to Finite Element Methods.pdf

1/34

Need for Computational Methods Solutions Using Either Strength of Materials or Theory ofElasticity Are Normally Accomplished for Regions and

Loadings With Relatively Simple Geometry

Many Applicaitons Involve Cases with Complex Shape,Boundary Conditions and Material Behavior

Therefore a Gap Exists Between What Is Needed in

Applications and What Can Be Solved by Analytical Closed-

form Methods

This Has Lead to the Development of SeveralNumerical/Computational Schemes Including: Finite

Difference, Finite Element and Boundary Element Methods

Introduction to Finite Element Methods

M.THIRUMALAIMUTHUKUMARAN AP/MECH Dr.NGPIT


2/34


3/34

Advantages of Finite Element Analysis

- Models Bodies of Complex Shape

- Can Handle General Loading/Boundary Conditions

- Models Bodies Composed of Composite and Multiphase Materials

- Model is Easily Refined for Improved Accuracy by Varying

Element Size and Type (Approximation Scheme)

- Time Dependent and Dynamic Effects Can Be Included

- Can Handle a Variety Nonlinear Effects Including Material

Behavior, Large Deformations, Boundary Conditions, Etc.


4/34

Basic Concept of the Finite Element Method

Any continuous solution field such as stress, displacement,

temperature, pressure, etc. can be approximated by a

discrete model composed of a set of piecewise continuous

functions defined over a finite number of subdomains.

Exact Analytical Solution

x

T

Approximate Piecewise

Linear Solution

x

T

One-Dimensional Temperature Distribution



5/34

Two-Dimensional Discretization

-1-0.5

00.5

11.5

22.5

3

1

1.5

2

2.5

3

3.54

-3

-2

-1

0

1

2

xy

u(x,y)

Approximate Piecewise

Linear Representation


6/34

Discretization Concepts

x

T

Exact T emperatu re Dis tributio n, T( x)

Fin ite E lem en t D isc retization

Li near In terpo lat ion Mo del (Four Elements)

Quadratic Interpolation M odel (Two Elements)

T1

T2 T2T3 T3

T4 T4T5

T1

T2

T3T4 T5

Piece wi se Linear Ap proxim ation

T

x

T1

T2T3 T3

T4 T5

T

T1

T2

T3T4 T5

Piece wi se Qua dra tic Approxim ati on

x

Temperature Continuous but with

Discon tinu ous T emp erature G radients

Temperature and Temp erature Gradients

Continuous


7/34

Common Types of Elements

One-Dimensional Elements

Line

Rods, Beams, Trusses, Frames

Two-Dimensional ElementsTriangular, Quadrilateral

Plates, Shells, 2-D Continua

Three-Dimensional Elements

Tetrahedral, Rectangular Prism (Brick)

3-D Continua


8/34

Discretization Examples

One-Dimensional

Frame Elements

Two-Dimensional

Triangular Elements

Three-Dimensional

Brick Elements


9/34

Basic Steps in the Finite Element Method

Time Independent Problems- Domain Discretization

- Select Element Type (Shape and Approximation)

- Derive Element Equations (Variational and Energy Methods)

- Assemble Element Equations to Form Global System

[K]{U} = {F}

[K] = Stiffness or Property Matrix

{U} = Nodal Displacement Vector

{F} = Nodal Force Vector

- Incorporate Boundary and Initial Conditions

- Solve Assembled System of Equations for Unknown Nodal

Displacements and Secondary Unknowns of Stress and Strain Values


10/34

Common Sources of Error in FEA

Domain Approximation

Element Interpolation/Approximation

Numerical Integration Errors

(Including Spatial and Time Integration)

Computer Errors (Round-Off, Etc., )


11/34

Measures of Accuracy in FEA

Accuracy

Error = |(Exact Solution)-(FEM Solution)|

Convergence

Limit of Error as:

Number of Elements (h-convergence)

or

Approximation Order (p-convergence)

IncreasesIdeally, Error 0 as Number of Elements or

Approximation Order


12/34

Two-Dimensional Discretization Refinement

(Discretization with 228 Elements)

(Discretization with 912 Elements)

(Triangular Element)

(Node)



13/34

One Dimensional Examples

Static Case

1 2

u1 u2

Bar Element

Uniaxial Deformation of Bars

Using Strength of Materials Theory

Beam Element

Deflection of Elastic Beams

Using Euler-Bernouli Theory

1 2

w1 w2

q2

q1

dx

duau

qcuaudx

d

,

:ionSpecificatCondtionsBoundary

0)(

:EquationalDifferenti

)(,,,


)()(

:EquationalDifferenti

2

2

2

2

2

2

2

2

dx

wdb

dx

d

dx

wdb

dx

dww

xf

dx

wdb

dx

d


14/34

Two Dimensional Examples

u1

u2

1

2

3 u3

v1

v2

v3

1

2

3

f1

f2

f3

Triangular Element

Scalar-Valued, Two-DimensionalField Problems

Triangular Element

Vector/Tensor-Valued, Two-Dimensional Field Problems

yx ny

nxdn

d

yxfyx

,


),(

:EquationialDifferentExample

2

2

2

2

yxy

yxx

y

x

n

y

vC

x

uCn

x

v

y

uCT

nx

v

y

uCn

y

vC

x

uCT

Fy

v

x

u

y

Ev

Fy

v

x

u

x

Eu

221266

661211

2

2

ConditonsBoundary

0)1(2

0)1(2

entsDisplacemofTermsinEquationsFieldElasticity


15/34

Development of Finite Element Equation The Finite Element Equation Must Incorporate the Appropriate Physicsof the Problem

For Problems in Structural Solid Mechanics, the Appropriate Physics

Comes from Either Strength of Materials or Theory of Elasticity

FEM Equations are Commonly Developed UsingDirect, Variational-

Virtual Work or Weighted ResidualMethods

Variational-Virtual Work Method

Based on the concept of virtual displacements, leads to relations between internal and

external virtual work and to minimization of system potential energy for equilibrium

Weighted Residual Method

Starting with the governing differential equation, special mathematical operations

develop the weak form that can be incorporated into a FEM equation. This

method is particularly suited for problems that have no variational statement. For

stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that

found by variational methods.

Direct Method

Based on physical reasoning and limited to simple cases, this method is

worth studying because it enhances physical understanding of the process


16/34

Simple Element Equation Example

Direct Stiffness Derivation

1 2k

u1 u2

F1 F2

}{}]{[

rmMatrix Foinor

2NodeatmEquilibriu

1NodeatmEquilibriu

2

1

2

1

212

211

FuK

F

F

u

u

kk

kk

kukuF

kukuF

Stiffness Matrix Nodal Force Vector


17/34

Common Approximation Schemes

One-Dimensional Examples

Linear Quadratic Cubic

Polynomial Approximation

Most often polynomials are used to construct approximation

functions for each element. Depending on the order of

approximation, different numbers of element parameters are

needed to construct the appropriate function.

Special Approximation

For some cases (e.g. infinite elements, crack or other singular

elements) the approximation function is chosen to have special

properties as determined from theoretical considerations


18/34

One-Dimensional Bar Element

udVfuPuPedV jjii

}]{[:LawStrain-Stress

}]{[}{][

)(:Strain

}{][)(:ionApproximat

dB

dBdN

dN

EEe

dx

dux

dx

d

dx

due

uxu

k

kk

k

kk

L

TT

j

iTL

TTfdxA

P

PdxEA

00][}{}{}{][][}{ NdddBBd

L

TL

TfdxAdxEA

00][}{}{][][ NPdBB

VectorentDisplacemNodal}{

VectorLoading][}{

MatrixStiffness][][][

0

0

j

i

LT

j

i

LT

u

u

fdxAP

P

dxEAK

d

NF

BB

}{}]{[ FdK



19/34

One-Dimensional Bar Element

A = Cross-sectional Area

E = Elastic Modulusf(x) = Distributed Loading

dVuFdSuTdVe iV

iS

i

n

iijV

ijt

Virtual Strain Energy = Virtual Work Done by Surface and Body Forces

For One-Dimensional Case

udVfuPuPedV jjii

W

(i) (j)

Axial Deformation of an Elastic Bar

Typical Bar Element

dx

duAEP ii

dx

duAEP

j

j iu ju

L

x

(Two Degrees of Freedom)


20/34

Linear Approximation Scheme


MatrixFunctionionApproximat][

}]{[1

)()(

1

2

1

2

1

21

2211

2112

1

212

11

21

d

N

dN

ntDisplacemelasticeApproximat

u

u

L

x

L

x

u

u

u

uxux

uL

xu

L

xx

L

uuuu

Laau

au

xaau

x (local coordinate system)(1) (2)

iu ju

L

x(1) (2)

u(x)

x(1) (2)

y 1(x) y 2(x)

1

y k(x) Lagrange Interpolation Functions


21/34

Element EquationLinear Approximation Scheme, Constant Properties


1

1

2][}{

11

1111

1

1

][][][][][

2

1

2

1

02

1

02

1

00

u

u

LAf

P

Pdx

L

xL

x

AfP

PfdxAP

P

L

AEL

LL

L

LAEdxAEdxEAK

oL

o

L T

LT

LT

d

NF

BBBB

1

1

211

11}{}]{[

2

1

2

1 LAf

P

P

u

u

L

AE oFdK


22/34

Quadratic Approximation Scheme

}]{[

)()()(

42

3

2

1

321

332211

2

3213

2

3212

11

2321

dN

ntDisplacemeElasticeApproximat

u

u

u

u

uxuxuxu

LaLaau

LaLaau

au

xaxaau

x(1) (3)

1u 3u

(2)

2u

L

u(x)

x(1) (3)(2)

x(1) (3)(2)

1

y 1(x) y 3(x)

y 2(x)

3

2

1

3

2

1

781

8168

187

3F

F

F

u

u

u

L

AE

quationElement


23/34

Lagrange Interpolation FunctionsUsing Natural or Normalized Coordinates

11

(1) (2))1(

2

1

)1(2

1

2

1

)1(2

1

)1)(1(

)1(21

3

2

1

)1)(3

1)(

3

1(

16

9

)3

1)(1)(1(

16

27

)3

1)(1)(1(

16

27

)

3

1)(

3

1)(1(

16

9

4

3

2

1

(1) (2) (3)

(1) (2) (3) (4)

ji

jiji

,0,1)(


24/34

Simple ExampleP

A1,E1,L1 A2,E2,L2

(1) (3)(2)

1 2

0000

011

011

1ElementEquationGlobal

)1(

2

)1(1

3

2

1

1

11 P

P

U

U

U

L

EA

)2(

2

)2(

1

3

2

1

2

22

0

110

110

000

2ElementEquationGlobal

P

P

U

U

U

L

EA

3

2

1

)2(

2

)2(

1

)1(

2

)1(1

3

2

1

2

22

2

22

2

22

2

22

1

11

1

11

1

11

1

11

0

0

EquationSystemGlobalAssembled

P

P

P

P

PP

P

U

U

U

L

EA

L

EA

L

EA

L

EA

L

EA

L

EAL

EA

L

EA

0

LoadingedDistributZeroTake

f


25/34

Simple Example Continued

P

A1,E1,L1 A2,E2,L2

(1) (3)(2)

1 2

0

0ConditionsBoundary

)2(

1

)1(

2

)2(

2

1

PP

PP

U

P

P

U

U

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

LEA

LEA

0

0

0

0

EquationSystemGlobal Reduced

)1(

1

3

2

2

22

2

22

2

22

2

22

1

11

1

11

1

11

1

11

PU

U

L

EA

L

EA

LEA

LEA

LEA

0

3

2

2

22

2

22

2

22

2

22

1

11

LEA ,,Properties

mFor Unifor

PU

U

L

AE 0

11

12

3

2

PPAE

PLU

AE

PLU )1(132 ,

2,Solving


26/34

One-Dimensional Beam ElementDeflection of an Elastic Beam

2

2423

1

1211

2

2

2

4

2

2

2

3

12

2

21

2

2

1

,,,

,

,

dx

dwuwu

dx

dwuwu

dx

wdEIQ

dx

wdEI

dx

dQ

dx

wd

EIQdx

wd

EIdx

d

Q

I = Section Moment of Inertia

E = Elastic Modulus

f(x) = Distributed Loading

W

(1) (2)

Typical Beam Element

1w

L

2w1 2

1M2M

1V 2V

x

Virtual Strain Energy = Virtual Work Done by Surface and Body Forces

wdVfwQuQuQuQedV 44332211

L

TL

dVfwQuQuQuQdxEI0

443322110

][}{][][ NdBB T

(Four Degrees of Freedom)


27/34

Beam Approximation FunctionsTo approximate deflection and slope at each

node requires approximation of the form3

4

2

321)( xcxcxccxw

Evaluating deflection and slope at each node

allows the determination ofci thus leading to

FunctionsionApproximatCubicHermitethearewhere

,)()()()()( 44332211

i

uxuxuxuxxw


28/34

Beam Element Equation

LT

L

dVfwQuQuQuQdxEI0443322110

][}{][][ NdBB T

4

3

2

1

}{

u

u

u

u

d ][][

][ 4321

dx

d

dx

d

dx

d

dx

d

dx

d

NB

22

22

30

233

3636

323

3636

2][][][

LLLL

LL

LLLL

LL

L

EIdxEI

L

BBK T

L

LfL

Q

Q

Q

Q

u

u

u

u

LLLL

LL

LLLL

LL

L

EI

6

6

12

233

3636

323

3636

2

4

3

2

1

4

3

2

1

22

22

3

L

LfLdxfdxf

LLT

6

6

12][

0

4

3

2

1

0N


29/34

FEA Beam Problem

f

a b

UniformEI

0

0

0

0

6

6

12

000000

000000

00/2/3/1/3

00/3/6/3/6

00/1/3/2/3

00/3/6/3/6

2)1(

4

)1(

3

)1(

2

)1(

1

6

5

4

3

2

1

22

2323

22

2323

Q

Q

Q

Q

a

a

fa

U

U

U

U

U

U

aaaa

aaaa

aaaa

aaaa

EI

1Element

)2(

4

)2(

3

)2(

2

)2(

1

6

5

4

3

2

1

22

2323

22

2323

0

0

/2/3/1/300

/3/6/3/600

/1/3/2/300

/3/6/3/600

000000

000000

2

Q

Q

Q

Q

U

U

U

U

U

U

bbbb

bbbb

bbbb

bbbbEI

2Element

(1) (3)(2)

1 2


30/34

FEA Beam Problem

)2(

4

)2(

3

)2(

2

)1(

4

)2(

1

)1(

3

)1(

2

)1(

1

6

5

4

3

2

1

23

2

232233

2

2323

0

0

6

6

12

/2

/3/6

/1/3/2/2

/3/6/3/3/6/6

00/1/3/2

00/3/6/3/6

2

Q

Q

QQ

QQ

Q

Q

a

a

fa

U

U

U

U

U

U

a

aa

aaba

aababa

aaa

aaaa

EI

SystemAssembledGlobal

0,0,0 )2(4)2(

3

)1(

12

)1(

11 QQUwU

ConditionsBoundary

0,0 )2(2)1(

4

)2(

1

)1(

3 QQQQ

ConditionsMatching

0

0

0

0

0

0

6

12

/2

/3/6

/1/3/2/2

/3/6/3/3/6/6

2

4

3

2

1

23

2

332233

afa

U

U

U

U

a

aa

aaba

aababa

EI

SystemReduced

Solve System for Primary Unknowns U1,U2,U3,U4

Nodal Forces Q1and Q2Can Then Be Determined

(1) (3)(2)

1 2


31/34

Special Features of Beam FEA

Analytical Solution Gives

Cubic Deflection Curve

Analytical Solution Gives

Quartic Deflection Curve

FEA Using Hermit Cubic InterpolationWill Yield Results That Match Exactly

With Cubic Analytical Solutions


32/34

Truss ElementGeneralization of Bar Element With Arbitrary Orientation

x

y

k=AE/L

cos,sin cs


33/34

Frame ElementGeneralization of Bar and Beam Element with Arbitrary Orientation

W

(1) (2)

1w

L

2w1 2

1M2M

1V 2V

2P1P1u 2u

4

3

2

2

1

1

2

2

2

1

1

1

22

2323

22

2323

460

260

6120

6120

0000

260

460

6120

6120

0000

QQ

P

Q

Q

P

w

u

w

u

L

EI

L

EI

L

EI

L

EIL

EI

L

EI

L

EI

L

EI L

AE

L

AEL

EI

L

EI

L

EI

L

EIL

EI

L

EI

L

EI

L

EIL

AE

L

AE

Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation


34/34

Some Standard FEA References

Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, 1982, 1995.

Beer, G. and Watson, J.O., Introduction to Finite and Boundary Element Methods for Engineers,John Wiley, 1993

Bickford, W.B., A First Course in the Finite Element Method, Irwin, 1990.

Burnett, D.S., Finite Element Analysis, Addison-Wesley, 1987.

Chandrupatla, T.R. and Belegundu, A.D., Introduction to Finite Elements in Engineering, Prentice-Hall, 2002.

Cook, R.D., Malkus, D.S. and Plesha, M.E., Concepts and Applications of Finite Element Analysis, 3rdEd., John Wiley,

1989.

Desai, C.S., Elementary Finite Element Method, Prentice-Hall, 1979.

Fung, Y.C. and Tong, P., Classical and Computational Solid Mechanics, World Scientific, 2001.

Grandin, H., Fundamentals of the Finite Element Method, Macmillan, 1986.

Huebner, K.H., Thorton, E.A. and Byrom, T.G., The Finite Element Method for Engineers, 3rdEd., John Wiley, 1994.

Knight, C.E., The Finite Element Method in Mechanical Design, PWS-KENT, 1993.

Logan, D.L., A First Course in the Finite Element Method, 2ndEd., PWS Engineering, 1992.

Moaveni, S., Finite Element Analysis Theory and Application with ANSYS,2ndEd., Pearson Education, 2003.

Pepper, D.W. and Heinrich, J.C., The Finite Element Method: Basic Concepts and Applications, Hemisphere, 1992.

Pao, Y.C., A First Course in Finite Element Analysis, Allyn and Bacon, 1986.

Rao, S.S., Finite Element Method in Engineering, 3rdEd., Butterworth-Heinemann, 1998.

Reddy, J.N., An Introduction to the Finite Element Method, McGraw-Hill, 1993.Ross, C.T.F., Finite Element Methods in Engineering Science,Prentice-Hall, 1993.

Stasa, F.L., Applied Finite Element Analysis for Engineers, Holt, Rinehart and Winston, 1985.

Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, Fourth Edition, McGraw-Hill, 1977, 1989.

Documents

Introduction to Finite Element Methods.pdf