An Overview of the FINITE ELEMENT METHOD

(FEM)

By

VIJAY G. S.

Senior Lecturer, Dept. of Mechanical Engg.,

NMAM Institute of Technology,

Nitte

IntroductionFor studying physical phenomena, engineers and scientists are involved with two major tasks:

Mathematical formulation of the physical problem –behaviour/governing equations

Numerical analysis of the mathematical model – numerical method & computer

Problems in engineering may be:

1. Boundary Value Problems

2. Initial Value Problems

3. Boundary and Initial Value

Problems

4. Eigen Value Problems

1. Boundary Value Problems – values of dependent variables (and its derivatives) are known on the continuum (boundary) of the problem.

Example:

01

0 ,)0( gdx

duadu

x

10

xforfdx

dua

dx

dis the governing equation

are the boundary conditions

2. Initial Value Problems – values of the dependent variables (and its derivatives) are known at an initial instant (i.e., at t=0). These are time dependent problems.

Example:

is the governing equation02

2

0 ttforfaudt

ud

00

0 ,)0( vdt

duuu

t

are the boundary conditions

3. Boundary and Initial Value Problems – values of the dependent variables (and its derivatives) are known on the boundary at specific time instants.Example:

00

10),(

tt

xfortxf

t

u

x

ua

x

)()0,(),(),(),0( 001

0 xuxutgx

uatdtu

x

is the governing equation

are the boundary conditions

4. Eigen Value Problems – the problem of determining value of the constant λ such that:

10

xforudx

dua

dx

d

0,0)0(1

xdx

duau

is called the eigen value problem for the above differential equation.

In all the above problems of engineering, we may require to find the value of the dependent variable at any specified point in the continuum.

For this, the above governing differential equations must be solved to get the value of the dependent variable.

But, in actual practice, engineering problems involve complicated geometries (continuums), loadings and varying material properties.

Due to this, it may be impossible to specify the boundary conditions, consider material properties and solve the governing differential equation.

In such a situation, we go for numerical methods such as “Finite Element Method (FEM)” to get approximate but acceptable solutions.

The Finite Element Method is a numerical method for solving problems of engineering and mathematical physics where their behaviour/governing equations are expressed by integral or differential equations.

The FEM formulation of the problem results in a set of simultaneous algebraic equations for solution, instead of requiring the solution of the governing differential equation

This yields approximate values of the variables at discrete points in the continuum

Irregular Geometry

Area = Length Breadth

Regular Geometry

Area = (area of sub divisions)

Basic Concept of FEM

General Steps of the Finite Element Method

1. Select Element type and discretize the continuum

2. Select a Displacement Function3. Define the Strain/Displacement and

Stress/Strain Relationships4. Derive the Element Stiffness Matrix

and Element Equations5. Assemble the Element Equations to

obtain Global Equations6. Apply Boundary Conditions and

modify the Global Equations7. Solve for unknown variables8. Solve for Element Strains and

Stresses9. Interpret the results

Some terminologies used in FEMDiscretization (Meshing): The process of dividing the model of the problem continuum into a finite number of regular subdivisions

Elements: Each subdivision is called an “Element”

Nodes: The grid (connection) points at which the elements meet each other are called “nodes”

Degrees of Freedom (DOF): The total number of variables (displacements) that are associated with each node

Boundary Conditions:Known values of the variable at the continuum boundary

Displacement Function:It is an assumed polynomial expression which closely represents the anticipated variation of the unknown variable over the element domain.

Discretization

Form the element

equations

{F}e=[k]e{q}e1. Direct stiffness method

2. Energy method

3. Weighted residual method

Assemble the element

equations to form global equations

{F}g=[K]g{Q}g

Apply Boundary Conditions

Solve the set of global simultaneous

equations

{Q}g=Inv([K]g){F}g

Obtain the stress and strain in each

element

PQ

• One Dimensional Problems: Variable along only one coordinate axis is required to

represent the problem behaviour Line Elements are used to model 1D problems

• Two Dimensional Problems:

Variable components along two coordinate axes are required to represent the problem behaviour Area Elements are used to model 2D problems

• Three Dimensional Problems:

Variable components along three coordinate axes are

required to represent the problem behaviour Volume Elements are used to model 2D problems

Problem Dimensions

2D Elements (Area Elements) 3D Elements (Volume Elements)

1D Element (Line Element)

BASIC

ELEMENT

TYPES

IN

FEM

Triangular Element

Quadrilateral Element

Brick Element

Tetrahedron Element

Convergence: The monotonic approach of the FEM Solution to the Actual Solution

Coarse MeshNo. of Elements=n1

Element Size = h1

Medium MeshNo. of Elements=n2

Element Size = h2

Fine MeshNo. of Elements=n3

Element Size = h3

n1 < n2 <n3

h1 > h2 > h3

No. of Elements

Sol

uti

on E

rror

FEM Solution

Basic form of the Element Equations:

ee

Q

QK

F

F

2

1

2

1

11

11

eee QKF

where, {F}e is called the ELEMENT FORCE VECTOR

[K]e is called the ELEMENT STIFFNESS MATRIX

{Q}e is called the ELEMENT DISPLACEMENT VECTOR

Element equation form for a Structural Problem

eee

Q

Q

L

AE

F

F

2

1

2

1

11

11

1 m 0.6 m

20 kN 10 kN

1 2 31 2

eee QKF

Element equation form for a Conduction Heat Transfer Problem

e

e

ee

eo

T

T

L

kA

Q

QLAq

2

1

2

1

11

11

1

1

2

T1T2 T3 T4

1 2 3Inside

temperature T0

Outside temperature

T5

5 cm 3 cm 1 cm

k1k2 k3

1 2 3 4

eee QKF

Element equation form for a Hydraulic Network Problem

128

,11

11

2

14

2

1

cwhereP

P

L

dc

Q

Qe

e

e

e

Q1

2 3

4P = 0

R1R2

R4

R3

R5

4

32

1

1

2

4

3

5

eee QKF

Element equation form for a DC Electric Network ProblemR3

R1

R2 R4

R5

E1E2

2

1

34

e

e

e

V

V

RI

I

2

1

2

1

11

111

eee QKF

20 kN 10 kNA1, E1, L1

A2, E2, L2

A3, E3, L3

1 2 3

1 2 3 4

Assembly of Element Equations to form Global Equations

Example: Consider the structural problem

1

1

111 L

EAK

21

2

222 L

EAK

2 32 3 43

3

333 L

EAK

12

11

11

11

12

11

Q

Q

KK

KK

F

F1 2

1

2

23

22

22

22

23

22

Q

Q

KK

KK

F

F2 3

2

3

34

33

33

33

34

33

Q

Q

KK

KK

F

F3 4

3

4

ggg QKF

Assembled Force Vector Assembled Stiffness Matrix

Assembled Displacement

Vector

34

33

23

22

12

11

33

3322

2211

11

34

33

23

22

12

11

00

0

0

00

Q

QQ

QQ

Q

KK

KKKK

KKKK

KK

F

FF

FF

F2

2

1

1 3

3

4

4

Basic form of the Global Equations:

nnnnn

n

n

n Q

Q

Q

KKK

KKK

KKK

F

F

F

..

..

....

..........

..........

....

....

..

..2

1

21

22221

11211

2

1

where, {F}g is called the GLOBAL FORCE VECTOR

[K]g is called the GLOBAL STIFFNESS MATRIX

the elements of [K]g (K11, K12,….Knn) are called INFLUENCE

COEFFICIENTS

{Q}g is called the DISPLACEMENT VECTOR

ggg QKF

Application of FEMStructural: Stress analysis of truss and frame, stress

concentration problems Buckling problems Vibration analysis

Non Structural: Heat Transfer Fluid flow, including seepage through porous media Distribution of electric or magnetic potential Acoustics

Others: Biomedical engineering problems – analysis of

human spine, skull, hip joints, jaw/gum tooth implants, heart and eyes.

Advantages of FEM Model irregularly shaped bodies quite easily. Handle general load conditions without

difficulty. Model bodies composed of different materials

because their element equations are evaluated individually.

Handle unlimited numbers and kinds of Boundary Conditions

Vary the size of the elements to make it possible to use small elements where ever necessary.

Alter the FEM Model relatively easily & cheaply Include dynamic effects Handle nonlinear behaviour existing with large

deformations and non linear materials

PRE PROCESSOR

Model the continuum (Coordinate data, constants & material properties)Select Element Type & Discretize(Mesh) the continuumStore all the input data

PROCESSOR

Compute element coefficient matrices and column vectorsAssemble element equationsImpose Boundary ConditionsSolve equations

POST PROCESSOR

Compute solution at points other than at the nodesLists the resultsPlots the resultsProvides simulation

Module 1

Module 2

Module 3

General Structure of Commercially available FEM software

Use of a computer in FEM:

To provide a User Interactive Graphics Environment while

modeling

the problem continuum.

To create a data base of the input data.

To evaluate the element equations and store them in matrix

form.

To assemble the element equations (by superposition) and

form the

global equations.

To solve the very large set of simultaneous algebraic

equations by

tools like ‘Gauss Elimination Method’.

To evaluate the unknown element parameters.

To create a data base of the results.

To graphically display the results

To provide the simulation.

20

20

60

80

R = 10

R = 50

R = 30