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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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Australian Defence Force Academy
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INTRODUCTION TO THE FINITE ELEMENT METHOD
AMEC 3508 & AMEC 3706by Murat TAHTALI. 2
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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Australian Defence Force Academy
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INTRODUCTION TO THE FINITE ELEMENT METHOD
AMEC 3508 & AMEC 3706by Murat TAHTALI. 3
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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Australian Defence Force Academy
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INTRODUCTION TO THE FINITE ELEMENT METHOD
AMEC 3508 & AMEC 3706by Murat TAHTALI. 4
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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Australian Defence Force Academy
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INTRODUCTION TO THE FINITE ELEMENT METHOD
AMEC 3508 & AMEC 3706by Murat TAHTALI. 5
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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INTRODUCTION TO THE FINITE ELEMENT METHOD
AMEC 3508 & AMEC 3706by Murat TAHTALI. 6
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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Australian Defence Force Academy
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INTRODUCTION TO THE FINITE ELEMENT METHOD
AMEC 3508 & AMEC 3706by Murat TAHTALI. 7
( )
( )
( )
( )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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Australian Defence Force Academy
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INTRODUCTION TO THE FINITE ELEMENT METHOD
AMEC 3508 & AMEC 3706by Murat TAHTALI. 8
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:
+
=
>