mce 565 wave motion & vibration in continuous...

MCE 565

Wave Motion & Vibration in Continuous Media

Spring 2005

Professor M. H. Sadd

Introduction to Finite Element Methods

Need for Computational Methods

• Solutions Using Either Strength of Materials or Theory of

Elasticity 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 Several

Numerical/Computational Schemes Including: Finite

Difference, Finite Element and Boundary Element Methods

Introduction to Finite Element Analysis

The finite element method is a computational scheme to solve field problems in

engineering and science. The technique has very wide application, and has been used on

problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations,

electrical and magnetic fields, etc. The fundamental concept involves dividing the body

under study into a finite number of pieces (subdomains) called elements (see Figure).

Particular assumptions are then made on the variation of the unknown dependent

variable(s) across each element using so-called interpolation or approximation functions.

This approximated variation is quantified in terms of solution values at special element

locations called nodes. Through this discretization process, the method sets up an

algebraic system of equations for unknown nodal values which approximate the

continuous solution. Because element size, shape and approximating scheme can be

varied to suit the problem, the method can accurately simulate solutions to problems of

complex geometry and loading and thus this technique has become a very useful and

practical tool.

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.

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

Two-Dimensional Discretization

-1-0.5

00.5

11.5

22.5

3

1

1.5

2

2.5

3

3.5

4

-3

-2

-1

0

1

2

xy

u(x,y)

Approximate Piecewise

Linear Representation

Discretization Concepts

x

T

E xact T emperatu re D is tribution, T(x)

Fin ite Elemen t Discretization

Linear Interpo lation Model (Four E lements)

Q uadra tic Interpo la tion M odel

(Tw o E lements)

T1

T 2 T 2T 3 T3

T 4 T4

T5

T1

T 2

T3

T4 T5

Piecewise Linear Approx im ation

T

x

T1

T 2T 3 T 3

T 4 T5

T

T 1

T 2

T3T4 T 5

Piecewise Quadra tic Approximation

x

Tem perature Conti nuous bu t w ith D iscontinuous Temperature G radients

Temperature and Temperature Gradien tsCont inuous

Common Types of Elements

One-Dimensional Elements

Line

Rods, Beams, Trusses, Frames

Two-Dimensional Elements

Triangular, Quadrilateral

Plates, Shells, 2-D Continua

Three-Dimensional Elements

Tetrahedral, Rectangular Prism (Brick)

3-D Continua

Discretization Examples

One-Dimensional

Frame Elements

Two-Dimensional

Triangular Elements

Three-Dimensional

Brick Elements

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

Common Sources of Error in FEA

• Domain Approximation

• Element Interpolation/Approximation

• Numerical Integration Errors

(Including Spatial and Time Integration)

• Computer Errors (Round-Off, Etc., )

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)

Increases

Ideally, Error →→→→ 0 as Number of Elements or

Approximation Order →→→→ ∞∞∞∞

Two-Dimensional Discretization Refinement

(Discretization with 228 Elements)

(Discretization with 912 Elements)

(Triangular Element)

(Node)

••••

••••

••••

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

θθθθ2

θθθθ1

dx

duau

qcuaudx

d

,

:ionSpecificat CondtionsBoundary

0)(

: EquationalDifferenti

=−+−

)(,,,


)()(

: EquationalDifferenti

2

2

2

2

2

2

2

2

dx

wdb

dx

d

dx

wdb

dx

dww

xfdx

wdb

dx

d=−

Two Dimensional Examples

u1

u2

1

2

3 u3

v1

v2

v3

1

2

3

φφφφ 1111

φφφφ 2222

φφφφ 3333

Triangular Element

Scalar-Valued, Two-Dimensional

Field Problems

Triangular Element

Vector/Tensor-Valued, Two-

Dimensional Field Problems

yxn

yn

xdn

d

yxfyx

∂φ∂

+∂

φ∂=

φφ

=∂

φ∂+

∂

φ∂

,


),(

: Equationial DifferentExample

2

2

2

2

yxy

yxx

y

x

ny

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

∂∂

+∂∂

+

∂∂

+∂∂

=

∂∂

+∂∂

+

∂∂

+∂∂

=

=+

∂∂

+∂∂

∂∂

ν−+∇µ

=+

∂∂

+∂∂

∂∂

ν−+∇µ

221266

661211

2

2

Conditons Boundary

0)1(2

0)1(2

ents Displacemof Terms in Equations FieldElasticity

Development of Finite Element Equation • The Finite Element Equation Must Incorporate the Appropriate Physics

of 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 Using Direct, Variational-

Virtual Work or Weighted Residual Methods

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

Simple Element Equation Example

Direct Stiffness Derivation

1 2 k

u1 u2

F1 F2

][

rm Matrix Foinor

2 Nodeat mEquilibriu

1 Nodeat mEquilibriu

2

1

2

1

212

211

FuK

F

F

u

u

kk

kk

kukuF

kukuF

=

=

−

−

+−=⇒

−=⇒

Stiffness Matrix Nodal Force Vector

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

One-Dimensional Bar Element

⇒δ++=δσ ∫∫ ΩΩudVfuPuPedV

jjii

][ : LawStrain-Stress

][][

)( :Strain

][)( :ionApproximat

dB

dBdN

dN

EEe

dx

dux

dx

d

dx

due

uxu

k

kk

k

kk

==σ

==ψ==

=ψ=

∑

∑

⇒+

= ∫∫L

TT

j

iTL

TTfdxA

P

PdxEA

00][][][ NδdδddBBδd

∫∫ +=L

TL

T fdxAdxEA00

][][][ NPdBB

Vectorent DisplacemNodal

Vector Loading][

MatrixStiffness][][][

0

0

=

=

=+

=

==

∫

∫

j

i

LT

j

i

LT

u

u

fdxAP

P

dxEAK

d

NF

BB

][ FdK =

One-Dimensional Bar Element

A = Cross-sectional Area

E = Elastic Modulus

f(x) = Distributed Loading

dVuFdSuTdVe iV

iS

i

n

iijV

ijt

δ+δ=δσ ∫∫∫

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

For One-Dimensional Case

∫∫ ΩΩδ++=δσ udVfuPuPedV jjii

Ω

(i) (j)

Axial Deformation of an Elastic Bar

Typical Bar Element

dx

duAEP i

i−=

dx

duAEP

j

j−=

iu

ju

L

x

(Two Degrees of Freedom)

Linear Approximation Scheme

[ ]


Matrix FunctionionApproximat][

][1

)()(

1

2

1

2

1

21

2211

2112

1

212

11

21

=

=

=

−=

ψψ=

ψ+ψ=

+

−=

−+=⇒

+=

=⇒+=

d

N

dN

ntDisplacemeElasticeApproximat

u

u

L

x

L

x

u

uu

uxux

uL

xu

L

xx

L

uuuu

Laau

auxaau

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

iu

ju

L

x (1) (2)

u(x)

x (1) (2)

ψ1(x) ψ2(x)

1

ψk(x) – Lagrange Interpolation Functions

Element Equation Linear 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

PfdxA

P

P

L

AEL

LL

L

LAEdxAEdxEAK

oL

o

LT

LT

LT

d

NF

BBBB

+

=

−

−⇒=

1

1

211

11][

2

1

2

1 LAf

P

P

u

u

L

AE oFdK

Quadratic Approximation Scheme

[ ] ][

)()()(

42

3

2

1

321

332211

2

3213

2

3212

11

2

321

dN

ntDisplacemeElasticeApproximat

=

ψψψ=

ψ+ψ+ψ=

++=

++=

=

⇒++=

u

u

u

u

uxuxuxu

LaLaau

La

Laau

au

xaxaau

x (1) (3)

1u

3u

(2)

2u

L

u(x)

x (1) (3) (2)

x (1) (3) (2)

1

ψ1(x) ψ3(x)

ψ2(x)

=

−

−−

−

3

2

1

3

2

1

781

8168

187

3F

F

F

u

u

u

L

AE

EquationElement

Lagrange Interpolation Functions Using Natural or Normalized Coordinates

11 ≤ξ≤−

ξ

(1) (2) )1(

2

1

)1(2

1

2

1

ξ+=ψ

ξ−=ψ

)1(2

1

)1)(1(

)1(2

1

3

2

1

ξ+ξ=ψ

ξ+ξ−=ψ

ξ−ξ−=ψ

)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

ξ+ξ−ξ+−=ψ

ξ+ξ+ξ−=ψ

ξ−ξ+ξ−=ψ

ξ−ξ+ξ−−=ψ

ξ

(1) (2) (3)

ξ

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

≠

==ξψ

ji

jiji

,0

,1)(

Simple Example P

A1,E1,L1 A2,E2,L2

(1) (3) (2)

1 2

=

−

−

0000

011

011

1 Element EquationGlobal

)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

2 Element EquationGlobal

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

EquationSystem Global Assembled

P

P

P

P

PP

P

U

U

U

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

L

EA

0

Loadinged DistributZero Take

=f

Simple Example Continued

P

A1,E1,L1 A2,E2,L2

(1) (3) (2)

1 2

0

0

ConditionsBoundary

)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

L

EA

L

EA

0

0

0

0

EquationSystem Global 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

L

EA

L

EA

L

EA

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

One-Dimensional Beam Element Deflection of an Elastic Beam

2

2423

1

1211

2

2

2

4

2

2

2

3

1

2

2

2

1

2

2

1

,,,

,

,

dx

dwuwu

dx

dwuwu

dx

wdEIQ

dx

wdEI

dx

dQ

dx

wdEIQ

dx

wdEI

dx

dQ

−=θ==−=θ==

−=

−=

=

=

I = Section Moment of Inertia

E = Elastic Modulus

f(x) = Distributed Loading

Ω (1) (2)

Typical Beam Element

1w

L

2w

1θ

2θ

1M 2

M

1V

2V

x

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

⇒δ++++=δσ ∫∫ ΩΩwdVfwQuQuQuQedV

44332211

∫∫ ++++=L

TL

dVfwQuQuQuQdxEI0

443322110

][][][ NdBBT

(Four Degrees of Freedom)

Beam Approximation Functions To approximate deflection and slope at each

node requires approximation of the form

3

4

2

321)( xcxcxccxw +++=

Evaluating deflection and slope at each node

allows the determination of ci thus leading to

FunctionsionApproximat Cubic Hermite theare where

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

i

uxuxuxuxxw

φ

φ+φ+φ+φ=

Beam Element Equation

∫∫ ++++=L

TL

dVfwQuQuQuQdxEI0

443322110

][][][ NdBBT

=

4

3

2

1

u

u

u

u

d ][][

][ 4321

dx

d

dx

d

dx

d

dx

d

dx

d φφφφ==

NB

−

−

−

−−−

== ∫22

22

30

233

3636

323

3636

2][][][

LLLL

LL

LLLL

LL

L

EIdxEI

L

BBKT

−+

=

−

−

−

−−−

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

−=

φ

φ

φ

φ

= ∫∫L

LfLdxfdxf

LLT

6

6

12][

0

4

3

2

1

0N

FEA Beam Problem

f

a b

Uniform EI

+

−

−=

−

−

−

−−−

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

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

System AssembledGlobal

0,0,0 )2(

4

)2(

3

)1(

12

)1(

11 ===θ=== QQUwU

Conditions Boundary

0,0 )2(

2

)1(

4

)2(

1

)1(

3 =+=+ QQQQ

Conditions Matching

+

−=

⋅⋅⋅

⋅⋅

+⋅

−−−+

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

System Reduced

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

Nodal Forces Q1 and Q2 Can Then Be Determined

(1) (3) (2)

1 2

Special Features of Beam FEA

Analytical Solution Gives

Cubic Deflection Curve

Analytical Solution Gives

Quartic Deflection Curve

FEA Using Hermit Cubic Interpolation

Will Yield Results That Match Exactly

With Cubic Analytical Solutions

Truss Element Generalization of Bar Element With Arbitrary Orientation

x

y

k=AE/L

θ=θ= cos,sin cs

Frame Element Generalization of Bar and Beam Element with Arbitrary Orientation

Ω (1) (2)

1w

L

2w

1θ 2

θ

1M 2

M

1V

2V

2P1

P1u

2u

=

θ

θ

−

−−−

−

−

−

−

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

Q

Q

P

Q

Q

P

w

u

w

u

L

EI

L

EI

L

EI

L

EIL

EI

L

EI

L

EI

L

EIL

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

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, 3rd Ed., 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, 3rd Ed., 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, 2nd Ed., PWS Engineering, 1992.

Moaveni, S., Finite Element Analysis – Theory and Application with ANSYS, 2nd Ed., 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, 3rd Ed., 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.

mce 565 wave motion & vibration in continuous...

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