FEM Lecture Notes_2001

8/6/2019 FEM Lecture Notes_2001

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INTRODUCTIONTO THE FINITE ELEMENT METHOD

AMEC 3508 Mechanics of Solids 2B

AMEC 3706 Aircraft Structures 1B

Mr. M. [email protected]

School ofAerospace andMechanicalEngineering


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Australian Defence Force Academy

School of Aerospace and Mechanical Engineering

INTRODUCTION TO THE FINITE ELEMENT METHOD

AMEC 3508 & AMEC 3706by Murat TAHTALI. 1

Chapter I: INTRODUCTION

1.1 Need for Approximate Analysis

It is not possible to obtain closed form analytical solutions for most of the actual complex

engineering problems. Therefore, we try to approximate the physical nature of the problem

such that an acceptable solution, ie-acceptable accuracy, can be obtained in reasonable time

and at reasonable cost.

As engineers we seek for convergence of the approximate solution: we must be sure that if we

increase the degree of approximation we will obtain better results.

We are asked to measure the circumference of a circle with unit diameter (exact result:

=3.14). However, the only equipment supplied is a ruler for measuring straight lines. Whatwe will do is to approximate the circle by polygons:

1

n = 4 n = 6 n = 8

Figure I-1 Approximating a circle by polygons

Convergence Characteristics

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

4.1

4 5 6 7 8 9 10 11 12 13 14 15

# of segments (n)MeasuredCircumferen

ce

Internal polygonExternal polygon

AVR

Figure I-2 Convergence of circumference


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Hence: Our method of approximation converges if we increase the degree of approximation

(here: n). Therefore it is of practical use.

Note that knowing the bounding nature of two different approximations, their average can

give an even better approximation. Here we know that the circumference of the internalpolygon will be always below the actual value and the circumference of the external polygon

will be always above the actual value, ie bounding the actual value. In this case the average

will give a much better approximation.

1.2 Numerical Approximation: Discretisation

The basic idea of any numerical approximation is DISCRETISATION, ie, the reduction of

the infinitely many unknowns to a finite number of them.

Given the continuous functiony=x4

, find the derivativedy/dx in the interval [1,4].

The closed form analytical solution, obtained by the application of differential calculus, is:

34xdx

dy= (exact)

Assume that the function is given only at two discrete points:

( )

( ) 4,

1,

4

4

===

===

bbbfy

aaafy

b

a

0 116

81

256y = 85x - 169

-150

-100

-50

0

50

100

150

200

250

300

0 1 2 3 4

x

y

assumed

exact variation

Figure I-3 Function f(x) and linear approximation

The variation off() within the interval must be assumed: += xy

Using two discrete points and the assumed variation, the approximate result is


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dx

dy

Where is found by solving the algebraic system of equations:

+=

+=

by

ay

b

a

85 =

=

ab

yy ab

0 4

32

108

256

0

50

100

150

200

250

300

0 1 2 3 4

x

dy/dxapproximate

numerical result

exact y=4 x^3

error e(x)

emax =171

Figure I-4 Derivative dy/dx and constant approximation

Now, if the error is acceptable (this is an engineering judgement), there is no problem: the

analytical problem of findingdy/dx is converted into a numerical problem of finding . Butif the error is too much, we have two alternatives to modify the approximation:

a) Increase the number of discrete points keeping assumed linear form of the function

between those discrete points. Or,

b) Increase the order of the approximation, here form linear to quadratic.

Taking on option b) the variation off() within the interval can be assumed:

++= xxy 2

Using three discrete points and the assumed variation, the approximate result is

+ xdx

dy

2


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Where and is found by solving the algebraic system of equations:

++=

++=

++=

ccy

bby

aay

c

b

a

2

2

2

=

=90

35

-50

0

50

100

150

200

250

300

0 1 2 3 4

x

y

approximate

quadratic

exact y= x^4

Figure I-5 Function f(x) and quadratic approximation

0 432

108

256

-150

-100

-50

0

50100

150

200

250

300

0 1 2 3 4

x

dy/dx

approximate

linear

exact y=4 x^3

emax =66

Figure I-6 Derivative dy/dx and linear approximation


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Chapter II: SCOPE OF THE FEM, DIRECT APPROACH

In the previous section, we said that the FEM offers a way to solve a complex continuum

problem by subdividing it into simpler ones that can be solved individually without much

effort. By continuum, we mean a body of matter or simply a region of space in which aparticular phenomenon is occurring. This can be a piece of metal subjected to a temperature

difference, a region of space subjected to a magnetic field, or a fluid subjected to a pressure

difference. In any case, we are after the distribution of the field variable resulting from the

imposed boundary conditions. The simpler is the continuum the easier is the solution.

One of the basic and intuitive discretisation we can consider in mechanics is to represent an

elastic structure simply by its stiffness and its mass. This is called lumping the distributed

material properties into simple distinct elements. For a simple static problem, we can even

consider the stiffness alone and represent it by a linear massless spring. This approximation

may not always represent the actual problem due to shape irregularities, however, it may be

suitable to represent a smaller part of the problem. Thus the complete solution of the problemcan be obtained as an assemblage of solutions. In the case of a system made up uniquely of

interconnected springs, the solution would be a series of displacement values at the

interconnection points that we will call nodes. Each spring in the system may have a different

stiffness constant but the governing equation for each spring has the same form all over the

system. Each governing equation can be represented as a matrix equation having a stiffness

matrix multiplied by a displacement vector equal to a force vector. Then the individual

matrices can be combined together using the fact that the displacement at a shared node is the

same for the springs sharing it. The result would be the representation of the governing

equations for the whole system in matrix form.

2.1 Common Procedure of FE-Approach in Solid Mechanics

Regardless of the geometry, material, boundary conditions and type of the problem, the finite

element method follows a general, well-defined step-by-step procedure:

Step 1: IDEALISATION

The continuum is divided into a finite number of ideal elements bearing the following

simplification w.r.t. the actual elements:

a) Ideal geometry usually curved boundaries are replaced by straight ones.

b) Ideal element response e.g. real displacement field is approximated.

c) Others e.g. boundary conditions and/or material properties are simplified.

Remarks:

1) This step is completely an engineering judgement.

2) This step is not necessary for discrete systems.

Step 2: DISCRETISATION

Reduce the number of infinite unknowns (degrees of freedom DOF) to a finite number:


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u(x,y)

v(x,y)

1 2

3

4 u(4)

v(4)NODES

ELEMENT

Interpolation (shape)

functions

Figure II-1 Displacement Discretisation

Interpolation functions usually polynomials of low orders (linear, quadratic,)

Similarly, the stress state is expressed by fictitious nodal forces:

Traction t

1 2

3

4

fy(4)

fx(4)

Figure II-2 Force Discretisation

Hence, at a general point within the element:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

u

vuvuvuvuyxv

yxu44332211

,,,,,,,ofFunction),(

),(


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( )

( )

( )

( )OOOOOOO `OOOOOOO UQ

f

yxyxyxyx

xy

yy

xx

ffffffff

yx

yx

yx)4()4()3()3()2()2()1()1( ,,,,,,,ofFunction

,

,

,

Remark:We are not completely free in selecting interpolation functions (see chapter IV).

Step 3: DETERMINATION OF ELEMENT PROPERTIES (STIFFNESSES)

Once the finite element model is established - ie the continuum is idealised by elements, and

discretised by nodal point unknowns and interpolation functions-, we can determine the

relationship between unknown displacements and known forces at the nodes(= element

response) as:

{ } [ ]{ }ukf = Eq. II-1

Where, [k] is the element stiffness matrix. There are three possible approaches to derive [k]:

a) Direct Approach:

intuitive, restricted to very simple elements (chapter II)

b) Variational Approach:

general and powerful for any type of problems possessing variational statements

(chapter III)c) Weighted Residual Methods:

applicable to any type of differential problems (chapter VII)

Step 4: ASSEMBLY OF ELEMENT STIFFNESSES

TO DISCRETISE THE WHOLE CONTINUUM

Assembly is performed according to the within the

scope of the direct approach:

Condition (1): Compatibility


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1 2

3

4

[1]v(3)

v

system

node

u

[2]v(1)

[2]u(1)

1

2

34

Element [2]Element [1]

X

Y

0,0

Two adjacent elements

Figure II-3 Compatibility of Nodal Displacements

Hence:

==

==>< ===2

1

1

1

1]1[1

1

1][1ukukffF

i

i

>< ===3

2

2

2

3]1[1

1

3][3ukukffF

i

i

Or, in matrix notation:

+

=

>

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FEM Lecture Notes_2001

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