INTRODUCTION TO FINITE ELEMENT METHOD

PROF. DR. AHMED AMER

Introduction to Finite Element Method

Mathematic Model

Finite Element Method

Historical Background

Analytical Process of FEM

Applications of FEM

Computer Programs for FEM

Mathematical Model. 1

Physical Problems

Mathematical Model

Solution

Identify control variablesAssumptions (empirical law)

(1) Modeling

(2) Types of solution

Exact Eq. Approx. Eq.

Exact Sol. ◎ ◎Approx. Sol. ◎ ◎

Eq.Sol.

(3) Methods of Solution

(3) Method of Solution

A. Classical methodsThey offer a high degree of insight, but the problems are difficult or impossible to solve for anything but simple geometries and loadings.

B. Numerical methods(I) Energy: Minimize an expression for the potential energy of the

structure over the whole domain.(II) Boundary element: Approximates functions satisfying the

governing differential equations not the boundary conditions.

(III) Finite difference: Replaces governing differential equations and boundary conditions with algebraic finite difference equations.

(IV) Finite element: Approximates the behavior of an irregular, continuous structure under general loadings and constraints with an assembly of discrete elements.

3. Historical Background

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. Analytical Processes of Finite Element Method(1) Structural stress analysis problem

A. Conditions that solution must satisfya. Equilibriumb. Compatibilityc. Constitutive lawd. Boundary conditions

Above conditions are used to generate a system of equations representing system behavior.

B. Approacha. Force (flexibility) method: internal forces as unknowns.b. Displacement (stiffness) method: nodal disp. As unknowns.

For computational purpose, the displacement method is more desirable because its formulation is simple. A vast majority of general purpose FE software's have incorporated the displacement method for solving structural problems.

(2) Analysis procedures of linear static structural analysis

A. Build up geometric model

a. 1D problem

line

b. 2D problem

surface

c. 3D problem

solid

B. Construct the finite element model

a. Discretize and select the element types

(a) element type

1D line element

2D element

3D brick element

(b) total number of element (mesh)

1D:

2D:

3D:

b. Select a shape function

1D line element: u=ax+b

c. Define the compatibility and constitutive law

d. Form the element stiffness matrix and equations

(a) Direct equilibrium method

(b) Work or energy method

(c) Method of weight Residuals

e. Form the system equation

Assemble the element equations to obtain global system equation and introduce boundary conditions

C. Solve the system equations

a. elimination method

Gauss’s method (Nastran)

b. iteration method

Gauss Seidel’s method

Displacement field strain field stress field

D. Interpret the results (postprocessing)

a. deformation plot b. stress contour

5. Applications of Finite Element Method

Structural Problem Non-structural Problem

Stress Analysis

- truss & frame analysis

- stress concentrated

problem

Buckling problem

Vibration Analysis

Impact Problem

Heat Transfer

Fluid Mechanics

Electric or Magnetic

Potential

6. Computer Programs for Finite Element Method

ANSYS ◎ D ◎ ◎ D ◎ ◎ ◎

NASTRAN ◎ D ◎ ◎ D ◎ ◎

ABAQUS ◎ ◎ ◎ ◎ ◎

MARC ◎ ◎ ◎ ◎ ◎

LS-DYNA3D ◎

MSC/DYNA ◎

ADAMS/

DADS

◎

COSMOS ◎ D ◎ ◎ D ◎ ◎

MOLDFLOW ◎

C-FLOW ◎

PHOENICS ◎ ◎

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 sub-domains.

Exact Analytical Solution

x

T

Approximate Piecewise Linear Solution

x

T

Dimensional Temperature Distribution-One

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

Exact Temperature Distribution, T(x)

Finite Element Discretization

Linear Interpolation Model (Four Elements)

Quadratic Interpolation Model (Two Elements)

T1

T2 T2T3 T3

T4 T4

T5

T1

T2

T3

T4 T5

Piecewise Linear Approximation

T

x

T1

T2

T3 T3

T4 T5

T

T1

T2

T3T4 T5

Piecewise Quadratic Approximation

x

Temperature Continuous but with Discontinuous Temperature Gradients

Temperature and Temperature GradientsContinuous

Common Types of Elements

Dimensional Elements-OneLine

Rods, Beams, Trusses, Frames

Dimensional Elements-TwoTriangular, Quadrilateral

Plates, Shells, 2-D Continua

Dimensional Elements-ThreeTetrahedral, Rectangular Prism (Brick)

3-D Continua

Discretization Examples

One-Dimensional Frame Elements

Two-Dimensional Triangular Elements

Three-Dimensional Brick Elements

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)

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

Virtual Work Method-Variational

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 variationalmethods.

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 ExampleDirect Stiffness Derivation

1 2k

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 SchemesOne-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

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

TT fdxAP

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 AreaE = Elastic Modulus

f(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

(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

Vectorent DisplacemNodal}{

Matrix FunctionionApproximat][

}]{[1

)()(

1

2

1

2

1

21

2211

2112

1

212

11

21

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)

1(x) 2(x)

1

k(x) – Lagrange Interpolation Functions

Element EquationLinear Approximation Scheme, Constant Properties

Vectorent DisplacemNodal}{

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

u

u

u

u

uxuxuxu

LaLaau

La

Laau

au

xaxaau

x(1) (3)

1u3u

(2)

2u

L

u(x)

x(1) (3)(2)

x(1) (3)(2)

1

1(x) 3(x)2(x)

3

2

1

3

2

1

781

8168

187

3F

F

F

u

u

u

L

AE

EquationElement

Lagrange Interpolation FunctionsUsing Natural or Normalized Coordinates

11

(1) (2))1(

2

1

)1(2

1

2

1

)1(2

1

)1)(1(

)1(2

1

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)(

Simple ExampleP

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 ElementDeflection 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

I = Section Moment of InertiaE = Elastic Modulus

f(x) = Distributed Loading

(1) (2)

Typical Beam Element

1w

L

2w1 2

1M 2M

1V2V

x

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

wdVfwQuQuQuQedV 44332211

L

TL

dVfwQuQuQuQdxEI0

443322110

][}{][][ NdBBT

(Four Degrees of Freedom)

Beam Approximation FunctionsTo 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

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

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

L

LfLdxfdxf

LLT

6

6

12][

0

4

3

2

1

0N

FEA Beam Problemf

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 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 GivesCubic Deflection Curve

Analytical Solution GivesQuartic Deflection Curve

FEA Using Hermit Cubic Interpolation Will Yield Results That Match Exactly

With Cubic Analytical Solutions

Truss ElementGeneralization of Bar Element With Arbitrary Orientation

x

y

k=AE/L

cos,sin cs

Frame ElementGeneralization of Bar and Beam Element with Arbitrary Orientation

(1) (2)

1w

L

2w

1 2

1M 2M

1V2V

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

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, 1993Bickford, 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.