FEM-1

Finite Element Method Theory and Application

Dr.. S. Kamran Afaq Dr Afaq HITEC University

(Gold Medalist )

• Ph.D. (Composite Material Structures) : University Paul Sabatier Sabatier,, France France • M.S. (Composite Material Structures) : University Paul Sabatier, France • B.E (Mechanical) : NED University University of Engi Engineer neering ing Techno Technology logy,, Pakistan Pakistan

FINITE ELEMENT METHOD Reference Books •

MATLAB Guide to Finite Elements By Peter Kattan

• Finite Element Method (Basic Concept and Application) By Chen Chennak nakesa esava va R. Ala Alaval vala a • Finite Element Method with Application in Engineering By Y. M. Desa Desaii • Fundamental of Finite Element Analysis By David V. Hutton • Numerical Methods for Engineers By

Steven C. Chapra

FINITE ELEMENT METHOD Reference Book MATLAB Guide to Finite Elements By Peter Kattan

FINITE ELEMENT METHOD

Introduction • Mathematical Modeling • Algorithm Design • Approximation and Errors • Matrix Algebra

FINITE ELEMENT METHOD Mathematical Modeling One of the most important things for engineers and scientists do is to model physical phenomena. Virtually every phenomena in nature, whether aerospace, biological, chemical, geological or mechanical can be described, with the aid of physics, or other fields in terms of algebraic, differential, and/or integral equations relating various quantities of interest . Mathematical Model Analytical description of a physical phenomena and processes are called ‘mathematical model’. A set of equations that expresses the essential features of a physical systems in terms of variables that describe the system.

FINITE ELEMENT METHOD Numerical Simulation The use of Numerical Method and a computer to evaluate the mathematical model of a process and estimate its characteristics is called a Numerical Simulation. Finite Element Method basically a Numerical Simulation of physical Phenomena.

Why Numerical Simulation?

Most practical problems involve complicated domains (both geometry and material), loads and nonlinearities that forbid the development of analytical solution. So, only alternative is to find out approximate solutions by Numerical Methods.

FINITE ELEMENT METHOD Mathematical Modeling Engineering Problem solving

• Requires understanding of engineering systems  By observation and experiment  Theoretical analysis and generalization

• Computers are great tools, however, without fundamental understanding of engineering problems, they will be useless.

FINITE ELEMENT METHOD Mathematical Modeling Engineering Problem solving • A mathematical model relationship of the form

is represented as a functional

 Independen t , parameters, forcing fu nctions   Dependent Variable  f   Variables 

• Dependent variable:

Characteristic that usually reflects the state of the system • Independent variables: Dimensions such as time and space along which the systems behavior is being determined • Parameters: reflect the system’s properties or composition • Forcing functions: external influences acting upon the system


Exercise Determine the mathematical model, i.e., governing equation of a free-falling body.

Model: Falling parachutist (Free falling body) Determine the terminal velocity (v) at any time ‘t’

Terminal Velocity (v)  f (t )

FINITE ELEMENT METHOD Mathematical Modeling Engineering Problem solving Newton’s 2nd law of Motion “the time rate of change of momentum of a body is equal to the resulting force acting on it .”

The model is formulated as;

F=ma F = net force acting on the body (N) m = mass of the object (kg) a = its acceleration (m/s2)


Dependent Variable

F=ma

Forcing function

a=F/m A parameter

FINITE ELEMENT METHOD Mathematical Modeling Engineering Problem solving Newton’s 2nd law of Motion

F=ma

a

F 

dv dt

m F



m

(A)


Forces

F  F d + F g F g



mg

F d

 

cv

Now,

(A)

dv dt



mg  cv m


dv dt

 g 

c m

v

(B)

• If the parachutist is initially at rest (v = 0 at t = 0), using calculus

Dependent Variable

(B)

Forcing Function

Independent Variable

v(t ) 

gm c

1  e

 ( c / m ) t



Parameters

FINITE ELEMENT METHOD Mathematical Modeling Engineering Problem solving Analytical Solution

m = 68.1 kg c = 12.5 kg/sec Gives,

v(t ) 

gm c

1  e

 ( c / m ) t

v(t )  53.39 1  e



0.18355t

FINITE ELEMENT METHOD Mathematical Modeling Engineering Problem solving Analytical Solution



v(t )  53.39 1  e

t

v(t)

0

0

2

16.4

45

4

27.77

35

6

35.64

v 25

0.18355t



50

40

30

20

8

41.1

10

44.87

12

47.49



53.39

15 10 5 0 0

5

10

t

15


Numerical Differentiation Approximated

y  f ( x)

Real

 y y ( xi +  x)  y ( xi )   x  x

dy dx

  x lim 0

y ( xi +  x)  y ( xi )

 x


Numerical Differentiation

Let f ( x)  ln x and f ' (1.8)  ? The exact value of Find an approximate value for

h

f (1.8)

f 1.8  0.5556

f 1.8

f (1.8 h) f (1.8) f (1.8 h) h

0.1 0.5877867 0.6418539

0.5406720

0.01 0.5877867 0.5933268

0.5540100

0.001 0.5877867 0.5883421

0.5554000

Assignment 2.5

Write a Matlab code for Numerical Differentiation

FINITE ELEMENT METHOD Mathematical Modeling Engineering Problem solving Numerical Solution

dv dt

 g 

c m

v

By the definition of Differentiation

  (t i +1 )   (t i )   dt t (t i +1  t i )

d 


Numerical Solution

v v(t i +1 )  v(t i )   dt t (t i +1  t i )

dv

v(t i +1 )  v(t i ) (t i +1  t i )

 g 

c m

v(t i )

c   v(t i +1 )  v(t i ) +  g  v(t i )(t i +1  t i ) m  


c   v(t i +1 )  v(t i ) +  g  v(t i ) (t i +1  t i ) m   At t = 0 => v = 0 (boundary condition)

t i+1= 2 sec

12.5   v(2)  0 + 9.8  (0) (2) 68.1   v  19.60


c   v(t i +1 )  v(t i ) +  g  v(t i ) (t i +1  t i ) m  

t

v(t)

0

0

60

2

19.6

50

4

32.00

40

6

39.85

8

44.82

v 30 20

Analytical

10

10

47.97 0

12

49.96



53.39

0

5

10

t

15

FINITE ELEMENT METHOD Assignment-FEM-1.1



Assignment-FEM-2.1 Write a code to evaluate area of a circle by a triangular element approximation. Display results with increasing ‘N’

FINITE ELEMENT METHOD Basic Concept : 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.

T

T Approximate Piecewise Linear Solution

Exact Analytical Solution

x

One-Dimensional Temperature Distribution

x

T


Discretization Concepts

E xact T emperature D istribution, T( x)

x

F inite E lement Discretization Linear I nterpolation Model (F our E lements)

T 1

T 2

T 2

T 3 T 3

Quadratic I nterpolation M odel (Two E lements) T 1 T 2

T 4 T 4

T 3 T 3

T 5

T

T 4

T 5

T

T 1

T 1 T 2

T 2 T 3

T 4

T 5

x Piecewise Linear Appr oximation Temperature Continuous but with

T 3

T 4

T 5

x Piecewise Quadratic A pproxi mation Temperature and Temperature Gradients

FINITE ELEMENT METHOD The Role of FEM in Numerical Simulations



FINITE ELEMENT METHOD Algorithm An Algorithm is the sequence of logical steps required to perform a specific task such as solving a problem.

• Each step must be deterministic; that is, nothing can be left to chance. • The process must always end after a finite number of steps. An Algorithm can not be open ended.


Flow Chart It is a visual or graphical representation of an algorithm. Start or end of program Flow of logic Process Input/Output

Decision


Example

Begin

Input

Process

Condition True

Output

End

False


Assignment-FEM-2.2 Write an algorithm to find out Terminal velocity of a parachutist by following relation, for a given step size of time.

 

v(t i +1 )  v(t i ) +  g 

c m

At t=0 => v=0 (boundary condition)

 

v(t i ) (t i +1  t i )



FINITE ELEMENT METHOD Approximations and Errors • For many engineering analytical solutions.

problems, we cannot obtain

• Numerical methods yield approximate results, results that are close to the exact analytical solution. • How confident we are in our approximate result?

The question is “how much error is present in our calculation and is it tolerable?”


• Accuracy

How close is a computed or measured value to the true value. • Precision

How close is a computed or measured value to previously computed or measured values.

FINITE ELEMENT METHOD Accuracy/Precesion


Error Definitions

• True Value = Approximation + Error

• E t = True value – Approximation (+/-)

True error


Error Definitions No Account of order of magnitude

• E t = True value – Approximation (+/-) Example True Value Approx. Value Et

Rivet

Bridge

10 cm

10,000 cm

9 cm

9,999 cm

1 cm

1 cm


Error Definitions

True fractional relative error 

True percent relative error,  t 

true error true value


100%


Error Definitions True percent relative error,  t 

Example True Value Approx. Value


Rivet

Bridge

10 cm

10,000 cm

9 cm

9,999 cm

True Error Et = 1 cm

True percent relative error  t 

 t 

100%

1 10

1 cm

100%  t  10%

 t

1 10000 

100%

0.01%


Error Definitions • For numerical methods, the true value will be known only when we deal with functions that can be solved analytically (simple systems). In real world applications, we usually not know the answer a priori. Then

 a  •

Iterative approach

 a



Approximate error Approximation

100%

(+ / -)

Current approximation - Previous approximation Current approximation

100%




Error Definitions Computations are repeated until stopping criterion is satisfied.

 a



 s

Pre-specified % tolerance based on the knowledge of your solution

• If the following criterion is met

 s  (0.5 10

(2- n)

)%

you can be sure that the result is correct to at least n significant figures.


Error Estimation Assignment 2.3

Mathematical functions can be represented by infinite series

e x  1 + x +

x 2 2!

+

x 3 3!

+ ............ +

• Find e0.5 (upto 3 significant digits)

x n n!


Error Estimation Assignment 2.4

Maclaurin Series expansion of Sin(x)

Sin ( x)  x 

x 3 3!

+

x 5 5!



x7 7!

+ ...............

• Find Sin(pi/2) (upto 4 significant figuers)




Matrix Algebra •

Matrix Definition

•

Order of a Matrix

•

Rectangular/Square/Row/Column Matrix

Operations: •

Addition/Subtraction

•

Scalar Multiplication

•

Multiplication

•

Transpose/Symmetric

•

Unit Matrix, Inverse Matrix

•

Orthogonal Matrix, Transformation Matrix

•

Simultaneous Linear Equations

FEM-1

Recommend Documents